src/HOL/Analysis/Linear_Algebra.thy
 author immler Thu May 03 16:17:44 2018 +0200 (12 months ago) changeset 68074 8d50467f7555 parent 68073 fad29d2a17a5 child 68224 1f7308050349 permissions -rw-r--r--
fixed HOL-Analysis
1 (*  Title:      HOL/Analysis/Linear_Algebra.thy
2     Author:     Amine Chaieb, University of Cambridge
3 *)
5 section \<open>Elementary linear algebra on Euclidean spaces\<close>
7 theory Linear_Algebra
8 imports
9   Euclidean_Space
10   "HOL-Library.Infinite_Set"
11 begin
13 lemma linear_simps:
14   assumes "bounded_linear f"
15   shows
16     "f (a + b) = f a + f b"
17     "f (a - b) = f a - f b"
18     "f 0 = 0"
19     "f (- a) = - f a"
20     "f (s *\<^sub>R v) = s *\<^sub>R (f v)"
21 proof -
22   interpret f: bounded_linear f by fact
23   show "f (a + b) = f a + f b" by (rule f.add)
24   show "f (a - b) = f a - f b" by (rule f.diff)
25   show "f 0 = 0" by (rule f.zero)
26   show "f (- a) = - f a" by (rule f.neg)
27   show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scale)
28 qed
30 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x \<in> (UNIV::'a::finite set)}"
31   using finite finite_image_set by blast
34 subsection%unimportant \<open>More interesting properties of the norm.\<close>
36 notation inner (infix "\<bullet>" 70)
38 text\<open>Equality of vectors in terms of @{term "(\<bullet>)"} products.\<close>
40 lemma linear_componentwise:
41   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
42   assumes lf: "linear f"
43   shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
44 proof -
45   interpret linear f by fact
46   have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
48   then show ?thesis
49     by (simp add: euclidean_representation sum[symmetric] scale[symmetric])
50 qed
52 lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
53   (is "?lhs \<longleftrightarrow> ?rhs")
54 proof
55   assume ?lhs
56   then show ?rhs by simp
57 next
58   assume ?rhs
59   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
60     by simp
61   then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
62     by (simp add: inner_diff inner_commute)
63   then have "(x - y) \<bullet> (x - y) = 0"
64     by (simp add: field_simps inner_diff inner_commute)
65   then show "x = y" by simp
66 qed
68 lemma norm_triangle_half_r:
69   "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
70   using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
72 lemma norm_triangle_half_l:
73   assumes "norm (x - y) < e / 2"
74     and "norm (x' - y) < e / 2"
75   shows "norm (x - x') < e"
76   using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
77   unfolding dist_norm[symmetric] .
79 lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
80   by (rule norm_triangle_ineq [THEN order_trans])
82 lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
83   by (rule norm_triangle_ineq [THEN le_less_trans])
85 lemma abs_triangle_half_r:
86   fixes y :: "'a::linordered_field"
87   shows "abs (y - x1) < e / 2 \<Longrightarrow> abs (y - x2) < e / 2 \<Longrightarrow> abs (x1 - x2) < e"
88   by linarith
90 lemma abs_triangle_half_l:
91   fixes y :: "'a::linordered_field"
92   assumes "abs (x - y) < e / 2"
93     and "abs (x' - y) < e / 2"
94   shows "abs (x - x') < e"
95   using assms by linarith
97 lemma sum_clauses:
98   shows "sum f {} = 0"
99     and "finite S \<Longrightarrow> sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
100   by (auto simp add: insert_absorb)
102 lemma sum_norm_bound:
103   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
104   assumes K: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> K"
105   shows "norm (sum f S) \<le> of_nat (card S)*K"
106   using sum_norm_le[OF K] sum_constant[symmetric]
107   by simp
109 lemma sum_group:
110   assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
111   shows "sum (\<lambda>y. sum g {x. x \<in> S \<and> f x = y}) T = sum g S"
112   unfolding sum_image_gen[OF fS, of g f]
113   by (auto intro: sum.neutral sum.mono_neutral_right[OF fT fST])
115 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
116 proof
117   assume "\<forall>x. x \<bullet> y = x \<bullet> z"
118   then have "\<forall>x. x \<bullet> (y - z) = 0"
120   then have "(y - z) \<bullet> (y - z) = 0" ..
121   then show "y = z" by simp
122 qed simp
124 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
125 proof
126   assume "\<forall>z. x \<bullet> z = y \<bullet> z"
127   then have "\<forall>z. (x - y) \<bullet> z = 0"
129   then have "(x - y) \<bullet> (x - y) = 0" ..
130   then show "x = y" by simp
131 qed simp
134 subsection \<open>Orthogonality.\<close>
136 definition%important (in real_inner) "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
138 context real_inner
139 begin
141 lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0"
144 lemma orthogonal_clauses:
145   "orthogonal a 0"
146   "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
147   "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
148   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
149   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
150   "orthogonal 0 a"
151   "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
152   "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
153   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
154   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
155   unfolding orthogonal_def inner_add inner_diff by auto
157 end
159 lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
160   by (simp add: orthogonal_def inner_commute)
162 lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x"
163   by (rule ext) (simp add: orthogonal_def)
165 lemma pairwise_ortho_scaleR:
166     "pairwise (\<lambda>i j. orthogonal (f i) (g j)) B
167     \<Longrightarrow> pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B"
168   by (auto simp: pairwise_def orthogonal_clauses)
170 lemma orthogonal_rvsum:
171     "\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (sum f s)"
172   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
174 lemma orthogonal_lvsum:
175     "\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (sum f s) y"
176   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
179   assumes "orthogonal a b"
180     shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2"
181 proof -
182   from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)"
183     by (simp add: algebra_simps orthogonal_def inner_commute)
184   then show ?thesis
186 qed
188 lemma norm_sum_Pythagorean:
189   assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I"
190     shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
191 using assms
192 proof (induction I rule: finite_induct)
193   case empty then show ?case by simp
194 next
195   case (insert x I)
196   then have "orthogonal (f x) (sum f I)"
197     by (metis pairwise_insert orthogonal_rvsum)
198   with insert show ?case
200 qed
203 subsection \<open>Bilinear functions.\<close>
205 definition%important "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
207 lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
208   by (simp add: bilinear_def linear_iff)
210 lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
211   by (simp add: bilinear_def linear_iff)
213 lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
214   by (simp add: bilinear_def linear_iff)
216 lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
217   by (simp add: bilinear_def linear_iff)
219 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
220   by (drule bilinear_lmul [of _ "- 1"]) simp
222 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
223   by (drule bilinear_rmul [of _ _ "- 1"]) simp
225 lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
226   using add_left_imp_eq[of x y 0] by auto
228 lemma bilinear_lzero:
229   assumes "bilinear h"
230   shows "h 0 x = 0"
233 lemma bilinear_rzero:
234   assumes "bilinear h"
235   shows "h x 0 = 0"
238 lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
239   using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
241 lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
242   using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
244 lemma bilinear_sum:
245   assumes "bilinear h"
246   shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
247 proof -
248   interpret l: linear "\<lambda>x. h x y" for y using assms by (simp add: bilinear_def)
249   interpret r: linear "\<lambda>y. h x y" for x using assms by (simp add: bilinear_def)
250   have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S"
252   also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S"
253     by (rule sum.cong) (simp_all add: r.sum)
254   finally show ?thesis
255     unfolding sum.cartesian_product .
256 qed
261 definition%important "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
264   assumes "\<forall>x y. inner (f x) y = inner x (g y)"
265   shows "adjoint f = g"
267 proof (rule some_equality)
268   show "\<forall>x y. inner (f x) y = inner x (g y)"
269     by (rule assms)
270 next
271   fix h
272   assume "\<forall>x y. inner (f x) y = inner x (h y)"
273   then have "\<forall>x y. inner x (g y) = inner x (h y)"
274     using assms by simp
275   then have "\<forall>x y. inner x (g y - h y) = 0"
277   then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
278     by simp
279   then have "\<forall>y. h y = g y"
280     by simp
281   then show "h = g" by (simp add: ext)
282 qed
284 text \<open>TODO: The following lemmas about adjoints should hold for any
285   Hilbert space (i.e. complete inner product space).
287 \<close>
290   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
291   assumes lf: "linear f"
292   shows "x \<bullet> adjoint f y = f x \<bullet> y"
293 proof -
294   interpret linear f by fact
295   have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
296   proof (intro allI exI)
297     fix y :: "'m" and x
298     let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
299     have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
301     also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
302       by (simp add: sum scale)
303     finally show "f x \<bullet> y = x \<bullet> ?w"
304       by (simp add: inner_sum_left inner_sum_right mult.commute)
305   qed
306   then show ?thesis
308     by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
309 qed
312   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
313   assumes lf: "linear f"
314   shows "x \<bullet> adjoint f y = f x \<bullet> y"
315     and "adjoint f y \<bullet> x = y \<bullet> f x"
319   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
320   assumes lf: "linear f"
322   by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
326   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
327   assumes lf: "linear f"
332 subsection%unimportant \<open>Interlude: Some properties of real sets\<close>
334 lemma seq_mono_lemma:
335   assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
336     and "\<forall>n \<ge> m. e n \<le> e m"
337   shows "\<forall>n \<ge> m. d n < e m"
338   using assms by force
340 lemma infinite_enumerate:
341   assumes fS: "infinite S"
342   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)"
343   unfolding strict_mono_def
344   using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
346 lemma approachable_lt_le: "(\<exists>(d::real) > 0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
347   apply auto
348   apply (rule_tac x="d/2" in exI)
349   apply auto
350   done
352 lemma approachable_lt_le2:  \<comment> \<open>like the above, but pushes aside an extra formula\<close>
353     "(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
354   apply auto
355   apply (rule_tac x="d/2" in exI, auto)
356   done
358 lemma triangle_lemma:
359   fixes x y z :: real
360   assumes x: "0 \<le> x"
361     and y: "0 \<le> y"
362     and z: "0 \<le> z"
363     and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
364   shows "x \<le> y + z"
365 proof -
366   have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
367     using z y by simp
368   with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
369     by (simp add: power2_eq_square field_simps)
370   from y z have yz: "y + z \<ge> 0"
371     by arith
372   from power2_le_imp_le[OF th yz] show ?thesis .
373 qed
377 subsection \<open>Archimedean properties and useful consequences\<close>
379 text\<open>Bernoulli's inequality\<close>
380 proposition%important Bernoulli_inequality:
381   fixes x :: real
382   assumes "-1 \<le> x"
383     shows "1 + n * x \<le> (1 + x) ^ n"
384 proof%unimportant (induct n)
385   case 0
386   then show ?case by simp
387 next
388   case (Suc n)
389   have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
391   also have "... = (1 + x) * (1 + n*x)"
392     by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
393   also have "... \<le> (1 + x) ^ Suc n"
394     using Suc.hyps assms mult_left_mono by fastforce
395   finally show ?case .
396 qed
398 corollary Bernoulli_inequality_even:
399   fixes x :: real
400   assumes "even n"
401     shows "1 + n * x \<le> (1 + x) ^ n"
402 proof (cases "-1 \<le> x \<or> n=0")
403   case True
404   then show ?thesis
405     by (auto simp: Bernoulli_inequality)
406 next
407   case False
408   then have "real n \<ge> 1"
409     by simp
410   with False have "n * x \<le> -1"
411     by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
412   then have "1 + n * x \<le> 0"
413     by auto
414   also have "... \<le> (1 + x) ^ n"
415     using assms
416     using zero_le_even_power by blast
417   finally show ?thesis .
418 qed
420 corollary real_arch_pow:
421   fixes x :: real
422   assumes x: "1 < x"
423   shows "\<exists>n. y < x^n"
424 proof -
425   from x have x0: "x - 1 > 0"
426     by arith
427   from reals_Archimedean3[OF x0, rule_format, of y]
428   obtain n :: nat where n: "y < real n * (x - 1)" by metis
429   from x0 have x00: "x- 1 \<ge> -1" by arith
430   from Bernoulli_inequality[OF x00, of n] n
431   have "y < x^n" by auto
432   then show ?thesis by metis
433 qed
435 corollary real_arch_pow_inv:
436   fixes x y :: real
437   assumes y: "y > 0"
438     and x1: "x < 1"
439   shows "\<exists>n. x^n < y"
440 proof (cases "x > 0")
441   case True
442   with x1 have ix: "1 < 1/x" by (simp add: field_simps)
443   from real_arch_pow[OF ix, of "1/y"]
444   obtain n where n: "1/y < (1/x)^n" by blast
445   then show ?thesis using y \<open>x > 0\<close>
446     by (auto simp add: field_simps)
447 next
448   case False
449   with y x1 show ?thesis
450     by (metis less_le_trans not_less power_one_right)
451 qed
453 lemma forall_pos_mono:
454   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
455     (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
456   by (metis real_arch_inverse)
458 lemma forall_pos_mono_1:
459   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
460     (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
461   apply (rule forall_pos_mono)
462   apply auto
463   apply (metis Suc_pred of_nat_Suc)
464   done
467 subsection%unimportant \<open>Euclidean Spaces as Typeclass\<close>
469 lemma independent_Basis: "independent Basis"
470   by (rule independent_Basis)
472 lemma span_Basis [simp]: "span Basis = UNIV"
473   by (rule span_Basis)
475 lemma in_span_Basis: "x \<in> span Basis"
476   unfolding span_Basis ..
479 subsection%unimportant \<open>Linearity and Bilinearity continued\<close>
481 lemma linear_bounded:
482   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
483   assumes lf: "linear f"
484   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
485 proof
486   interpret linear f by fact
487   let ?B = "\<Sum>b\<in>Basis. norm (f b)"
488   show "\<forall>x. norm (f x) \<le> ?B * norm x"
489   proof
490     fix x :: 'a
491     let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
492     have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
493       unfolding euclidean_representation ..
494     also have "\<dots> = norm (sum ?g Basis)"
495       by (simp add: sum scale)
496     finally have th0: "norm (f x) = norm (sum ?g Basis)" .
497     have th: "norm (?g i) \<le> norm (f i) * norm x" if "i \<in> Basis" for i
498     proof -
499       from Basis_le_norm[OF that, of x]
500       show "norm (?g i) \<le> norm (f i) * norm x"
501         unfolding norm_scaleR  by (metis mult.commute mult_left_mono norm_ge_zero)
502     qed
503     from sum_norm_le[of _ ?g, OF th]
504     show "norm (f x) \<le> ?B * norm x"
505       unfolding th0 sum_distrib_right by metis
506   qed
507 qed
509 lemma linear_conv_bounded_linear:
510   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
511   shows "linear f \<longleftrightarrow> bounded_linear f"
512 proof
513   assume "linear f"
514   then interpret f: linear f .
515   show "bounded_linear f"
516   proof
517     have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
518       using \<open>linear f\<close> by (rule linear_bounded)
519     then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
521   qed
522 next
523   assume "bounded_linear f"
524   then interpret f: bounded_linear f .
525   show "linear f" ..
526 qed
528 lemmas linear_linear = linear_conv_bounded_linear[symmetric]
530 lemma linear_bounded_pos:
531   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
532   assumes lf: "linear f"
533  obtains B where "B > 0" "\<And>x. norm (f x) \<le> B * norm x"
534 proof -
535   have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
536     using lf unfolding linear_conv_bounded_linear
537     by (rule bounded_linear.pos_bounded)
538   with that show ?thesis
539     by (auto simp: mult.commute)
540 qed
542 lemma linear_invertible_bounded_below_pos:
543   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
544   assumes "linear f" "linear g" "g \<circ> f = id"
545   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
546 proof -
547   obtain B where "B > 0" and B: "\<And>x. norm (g x) \<le> B * norm x"
548     using linear_bounded_pos [OF \<open>linear g\<close>] by blast
549   show thesis
550   proof
551     show "0 < 1/B"
552       by (simp add: \<open>B > 0\<close>)
553     show "1/B * norm x \<le> norm (f x)" for x
554     proof -
555       have "1/B * norm x = 1/B * norm (g (f x))"
556         using assms by (simp add: pointfree_idE)
557       also have "\<dots> \<le> norm (f x)"
558         using B [of "f x"] by (simp add: \<open>B > 0\<close> mult.commute pos_divide_le_eq)
559       finally show ?thesis .
560     qed
561   qed
562 qed
564 lemma linear_inj_bounded_below_pos:
565   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
566   assumes "linear f" "inj f"
567   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
568   using linear_injective_left_inverse [OF assms]
569     linear_invertible_bounded_below_pos assms by blast
571 lemma bounded_linearI':
572   fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
573   assumes "\<And>x y. f (x + y) = f x + f y"
574     and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
575   shows "bounded_linear f"
576   using assms linearI linear_conv_bounded_linear by blast
578 lemma bilinear_bounded:
579   fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
580   assumes bh: "bilinear h"
581   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
582 proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
583   fix x :: 'm
584   fix y :: 'n
585   have "norm (h x y) = norm (h (sum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (sum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
587   also have "\<dots> = norm (sum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
588     unfolding bilinear_sum[OF bh] ..
589   finally have th: "norm (h x y) = \<dots>" .
590   have "\<And>i j. \<lbrakk>i \<in> Basis; j \<in> Basis\<rbrakk>
591            \<Longrightarrow> \<bar>x \<bullet> i\<bar> * (\<bar>y \<bullet> j\<bar> * norm (h i j)) \<le> norm x * (norm y * norm (h i j))"
592     by (auto simp add: zero_le_mult_iff Basis_le_norm mult_mono)
593   then show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
594     unfolding sum_distrib_right th sum.cartesian_product
595     by (clarsimp simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
596       field_simps simp del: scaleR_scaleR intro!: sum_norm_le)
597 qed
599 lemma bilinear_conv_bounded_bilinear:
600   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
601   shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
602 proof
603   assume "bilinear h"
604   show "bounded_bilinear h"
605   proof
606     fix x y z
607     show "h (x + y) z = h x z + h y z"
608       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
609   next
610     fix x y z
611     show "h x (y + z) = h x y + h x z"
612       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
613   next
614     show "h (scaleR r x) y = scaleR r (h x y)" "h x (scaleR r y) = scaleR r (h x y)" for r x y
615       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
616       by simp_all
617   next
618     have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
619       using \<open>bilinear h\<close> by (rule bilinear_bounded)
620     then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
622   qed
623 next
624   assume "bounded_bilinear h"
625   then interpret h: bounded_bilinear h .
626   show "bilinear h"
627     unfolding bilinear_def linear_conv_bounded_linear
628     using h.bounded_linear_left h.bounded_linear_right by simp
629 qed
631 lemma bilinear_bounded_pos:
632   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
633   assumes bh: "bilinear h"
634   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
635 proof -
636   have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
637     using bh [unfolded bilinear_conv_bounded_bilinear]
638     by (rule bounded_bilinear.pos_bounded)
639   then show ?thesis
640     by (simp only: ac_simps)
641 qed
643 lemma bounded_linear_imp_has_derivative: "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
644   by (auto simp add: has_derivative_def linear_diff linear_linear linear_def
645       dest: bounded_linear.linear)
647 lemma linear_imp_has_derivative:
648   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
649   shows "linear f \<Longrightarrow> (f has_derivative f) net"
650   by (simp add: bounded_linear_imp_has_derivative linear_conv_bounded_linear)
652 lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net"
653   using bounded_linear_imp_has_derivative differentiable_def by blast
655 lemma linear_imp_differentiable:
656   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
657   shows "linear f \<Longrightarrow> f differentiable net"
658   by (metis linear_imp_has_derivative differentiable_def)
661 subsection%unimportant \<open>We continue.\<close>
663 lemma independent_bound:
664   fixes S :: "'a::euclidean_space set"
665   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
666   by (metis dim_subset_UNIV finiteI_independent dim_span_eq_card_independent)
668 lemmas independent_imp_finite = finiteI_independent
670 corollary
671   fixes S :: "'a::euclidean_space set"
672   assumes "independent S"
673   shows independent_card_le:"card S \<le> DIM('a)"
674   using assms independent_bound by auto
676 lemma dependent_biggerset:
677   fixes S :: "'a::euclidean_space set"
678   shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
679   by (metis independent_bound not_less)
681 text \<open>Picking an orthogonal replacement for a spanning set.\<close>
683 lemma vector_sub_project_orthogonal:
684   fixes b x :: "'a::euclidean_space"
685   shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
686   unfolding inner_simps by auto
688 lemma pairwise_orthogonal_insert:
689   assumes "pairwise orthogonal S"
690     and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
691   shows "pairwise orthogonal (insert x S)"
692   using assms unfolding pairwise_def
693   by (auto simp add: orthogonal_commute)
695 lemma basis_orthogonal:
696   fixes B :: "'a::real_inner set"
697   assumes fB: "finite B"
698   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
699   (is " \<exists>C. ?P B C")
700   using fB
701 proof (induct rule: finite_induct)
702   case empty
703   then show ?case
704     apply (rule exI[where x="{}"])
705     apply (auto simp add: pairwise_def)
706     done
707 next
708   case (insert a B)
709   note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
710   from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
711   obtain C where C: "finite C" "card C \<le> card B"
712     "span C = span B" "pairwise orthogonal C" by blast
713   let ?a = "a - sum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
714   let ?C = "insert ?a C"
715   from C(1) have fC: "finite ?C"
716     by simp
717   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
719   {
720     fix x k
721     have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
723     have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C"
724       apply (simp only: scaleR_right_diff_distrib th0)
726       apply (rule span_scale)
727       apply (rule span_sum)
728       apply (rule span_scale)
729       apply (rule span_base)
730       apply assumption
731       done
732   }
733   then have SC: "span ?C = span (insert a B)"
734     unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
735   {
736     fix y
737     assume yC: "y \<in> C"
738     then have Cy: "C = insert y (C - {y})"
739       by blast
740     have fth: "finite (C - {y})"
741       using C by simp
742     have "orthogonal ?a y"
743       unfolding orthogonal_def
744       unfolding inner_diff inner_sum_left right_minus_eq
745       unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
746       apply (clarsimp simp add: inner_commute[of y a])
747       apply (rule sum.neutral)
748       apply clarsimp
749       apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
750       using \<open>y \<in> C\<close> by auto
751   }
752   with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
753     by (rule pairwise_orthogonal_insert)
754   from fC cC SC CPO have "?P (insert a B) ?C"
755     by blast
756   then show ?case by blast
757 qed
759 lemma orthogonal_basis_exists:
760   fixes V :: "('a::euclidean_space) set"
761   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and>
762   (card B = dim V) \<and> pairwise orthogonal B"
763 proof -
764   from basis_exists[of V] obtain B where
765     B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
766     by force
767   from B have fB: "finite B" "card B = dim V"
768     using independent_bound by auto
769   from basis_orthogonal[OF fB(1)] obtain C where
770     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
771     by blast
772   from C B have CSV: "C \<subseteq> span V"
773     by (metis span_superset span_mono subset_trans)
774   from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
776   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
777   have iC: "independent C"
779   from C fB have "card C \<le> dim V"
780     by simp
781   moreover have "dim V \<le> card C"
782     using span_card_ge_dim[OF CSV SVC C(1)]
783     by simp
784   ultimately have CdV: "card C = dim V"
785     using C(1) by simp
786   from C B CSV CdV iC show ?thesis
787     by auto
788 qed
790 text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
792 lemma span_not_univ_orthogonal:
793   fixes S :: "'a::euclidean_space set"
794   assumes sU: "span S \<noteq> UNIV"
795   shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
796 proof -
797   from sU obtain a where a: "a \<notin> span S"
798     by blast
799   from orthogonal_basis_exists obtain B where
800     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B"
801     "card B = dim S" "pairwise orthogonal B"
802     by blast
803   from B have fB: "finite B" "card B = dim S"
804     using independent_bound by auto
805   from span_mono[OF B(2)] span_mono[OF B(3)]
806   have sSB: "span S = span B"
808   let ?a = "a - sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
809   have "sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
810     unfolding sSB
811     apply (rule span_sum)
812     apply (rule span_scale)
813     apply (rule span_base)
814     apply assumption
815     done
816   with a have a0:"?a  \<noteq> 0"
817     by auto
818   have "?a \<bullet> x = 0" if "x\<in>span B" for x
819   proof (rule span_induct [OF that])
820     show "subspace {x. ?a \<bullet> x = 0}"
822   next
823     {
824       fix x
825       assume x: "x \<in> B"
826       from x have B': "B = insert x (B - {x})"
827         by blast
828       have fth: "finite (B - {x})"
829         using fB by simp
830       have "?a \<bullet> x = 0"
831         apply (subst B')
832         using fB fth
833         unfolding sum_clauses(2)[OF fth]
834         apply simp unfolding inner_simps
836         apply (rule sum.neutral, rule ballI)
837         apply (simp only: inner_commute)
838         apply (auto simp add: x field_simps
839           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
840         done
841     }
842     then show "?a \<bullet> x = 0" if "x \<in> B" for x
843       using that by blast
844     qed
845   with a0 show ?thesis
846     unfolding sSB by (auto intro: exI[where x="?a"])
847 qed
849 lemma span_not_univ_subset_hyperplane:
850   fixes S :: "'a::euclidean_space set"
851   assumes SU: "span S \<noteq> UNIV"
852   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
853   using span_not_univ_orthogonal[OF SU] by auto
855 lemma lowdim_subset_hyperplane:
856   fixes S :: "'a::euclidean_space set"
857   assumes d: "dim S < DIM('a)"
858   shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
859 proof -
860   {
861     assume "span S = UNIV"
862     then have "dim (span S) = dim (UNIV :: ('a) set)"
863       by simp
864     then have "dim S = DIM('a)"
865       by (metis Euclidean_Space.dim_UNIV dim_span)
866     with d have False by arith
867   }
868   then have th: "span S \<noteq> UNIV"
869     by blast
870   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
871 qed
873 lemma linear_eq_stdbasis:
874   fixes f :: "'a::euclidean_space \<Rightarrow> _"
875   assumes lf: "linear f"
876     and lg: "linear g"
877     and fg: "\<And>b. b \<in> Basis \<Longrightarrow> f b = g b"
878   shows "f = g"
879   using linear_eq_on_span[OF lf lg, of Basis] fg
880   by auto
883 text \<open>Similar results for bilinear functions.\<close>
885 lemma bilinear_eq:
886   assumes bf: "bilinear f"
887     and bg: "bilinear g"
888     and SB: "S \<subseteq> span B"
889     and TC: "T \<subseteq> span C"
890     and "x\<in>S" "y\<in>T"
891     and fg: "\<And>x y. \<lbrakk>x \<in> B; y\<in> C\<rbrakk> \<Longrightarrow> f x y = g x y"
892   shows "f x y = g x y"
893 proof -
894   let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
895   from bf bg have sp: "subspace ?P"
896     unfolding bilinear_def linear_iff subspace_def bf bg
897     by (auto simp add: span_zero bilinear_lzero[OF bf] bilinear_lzero[OF bg]
900   have sfg: "\<And>x. x \<in> B \<Longrightarrow> subspace {a. f x a = g x a}"
901     apply (auto simp add: subspace_def)
902     using bf bg unfolding bilinear_def linear_iff
903       apply (auto simp add: span_zero bilinear_rzero[OF bf] bilinear_rzero[OF bg]
906     done
907   have "\<forall>y\<in> span C. f x y = g x y" if "x \<in> span B" for x
908     apply (rule span_induct [OF that sp])
909     using fg sfg span_induct by blast
910   then show ?thesis
911     using SB TC assms by auto
912 qed
914 lemma bilinear_eq_stdbasis:
915   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
916   assumes bf: "bilinear f"
917     and bg: "bilinear g"
918     and fg: "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> f i j = g i j"
919   shows "f = g"
920   using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis]] fg by blast
922 subsection \<open>Infinity norm\<close>
924 definition%important "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
926 lemma infnorm_set_image:
927   fixes x :: "'a::euclidean_space"
928   shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
929   by blast
931 lemma infnorm_Max:
932   fixes x :: "'a::euclidean_space"
933   shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)"
934   by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)
936 lemma infnorm_set_lemma:
937   fixes x :: "'a::euclidean_space"
938   shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
939     and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}"
940   unfolding infnorm_set_image
941   by auto
943 lemma infnorm_pos_le:
944   fixes x :: "'a::euclidean_space"
945   shows "0 \<le> infnorm x"
946   by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
948 lemma infnorm_triangle:
949   fixes x :: "'a::euclidean_space"
950   shows "infnorm (x + y) \<le> infnorm x + infnorm y"
951 proof -
952   have *: "\<And>a b c d :: real. \<bar>a\<bar> \<le> c \<Longrightarrow> \<bar>b\<bar> \<le> d \<Longrightarrow> \<bar>a + b\<bar> \<le> c + d"
953     by simp
954   show ?thesis
955     by (auto simp: infnorm_Max inner_add_left intro!: *)
956 qed
958 lemma infnorm_eq_0:
959   fixes x :: "'a::euclidean_space"
960   shows "infnorm x = 0 \<longleftrightarrow> x = 0"
961 proof -
962   have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
963     unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
964   then show ?thesis
965     using infnorm_pos_le[of x] by simp
966 qed
968 lemma infnorm_0: "infnorm 0 = 0"
971 lemma infnorm_neg: "infnorm (- x) = infnorm x"
972   unfolding infnorm_def by simp
974 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
975   by (metis infnorm_neg minus_diff_eq)
977 lemma absdiff_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
978 proof -
979   have *: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
980     by arith
981   show ?thesis
982   proof (rule *)
983     from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
984     show "infnorm x \<le> infnorm (x - y) + infnorm y" "infnorm y \<le> infnorm (x - y) + infnorm x"
985       by (simp_all add: field_simps infnorm_neg)
986   qed
987 qed
989 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
990   using infnorm_pos_le[of x] by arith
992 lemma Basis_le_infnorm:
993   fixes x :: "'a::euclidean_space"
994   shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
997 lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
998   unfolding infnorm_Max
999 proof (safe intro!: Max_eqI)
1000   let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
1001   { fix b :: 'a
1002     assume "b \<in> Basis"
1003     then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
1004       by (simp add: abs_mult mult_left_mono)
1005   next
1006     from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
1007       by (auto simp del: Max_in)
1008     then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis"
1009       by (intro image_eqI[where x=b]) (auto simp: abs_mult)
1010   }
1011 qed simp
1013 lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
1014   unfolding infnorm_mul ..
1016 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
1017   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
1019 text \<open>Prove that it differs only up to a bound from Euclidean norm.\<close>
1021 lemma infnorm_le_norm: "infnorm x \<le> norm x"
1022   by (simp add: Basis_le_norm infnorm_Max)
1024 lemma norm_le_infnorm:
1025   fixes x :: "'a::euclidean_space"
1026   shows "norm x \<le> sqrt DIM('a) * infnorm x"
1027   unfolding norm_eq_sqrt_inner id_def
1028 proof (rule real_le_lsqrt[OF inner_ge_zero])
1029   show "sqrt DIM('a) * infnorm x \<ge> 0"
1030     by (simp add: zero_le_mult_iff infnorm_pos_le)
1031   have "x \<bullet> x \<le> (\<Sum>b\<in>Basis. x \<bullet> b * (x \<bullet> b))"
1032     by (metis euclidean_inner order_refl)
1033   also have "... \<le> DIM('a) * \<bar>infnorm x\<bar>\<^sup>2"
1034     by (rule sum_bounded_above) (metis Basis_le_infnorm abs_le_square_iff power2_eq_square real_abs_infnorm)
1035   also have "... \<le> (sqrt DIM('a) * infnorm x)\<^sup>2"
1037   finally show "x \<bullet> x \<le> (sqrt DIM('a) * infnorm x)\<^sup>2" .
1038 qed
1040 lemma tendsto_infnorm [tendsto_intros]:
1041   assumes "(f \<longlongrightarrow> a) F"
1042   shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F"
1043 proof (rule tendsto_compose [OF LIM_I assms])
1044   fix r :: real
1045   assume "r > 0"
1046   then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
1047     by (metis real_norm_def le_less_trans absdiff_infnorm infnorm_le_norm)
1048 qed
1050 text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close>
1052 lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
1053   (is "?lhs \<longleftrightarrow> ?rhs")
1054 proof (cases "x=0")
1055   case True
1056   then show ?thesis
1057     by auto
1058 next
1059   case False
1060   from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
1061   have "?rhs \<longleftrightarrow>
1062       (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
1063         norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
1064     using False unfolding inner_simps
1065     by (auto simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
1066   also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)"
1067     using False  by (simp add: field_simps inner_commute)
1068   also have "\<dots> \<longleftrightarrow> ?lhs"
1069     using False by auto
1070   finally show ?thesis by metis
1071 qed
1073 lemma norm_cauchy_schwarz_abs_eq:
1074   "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow>
1075     norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
1076   (is "?lhs \<longleftrightarrow> ?rhs")
1077 proof -
1078   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a"
1079     by arith
1080   have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
1081     by simp
1082   also have "\<dots> \<longleftrightarrow> (x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
1083     unfolding norm_cauchy_schwarz_eq[symmetric]
1084     unfolding norm_minus_cancel norm_scaleR ..
1085   also have "\<dots> \<longleftrightarrow> ?lhs"
1086     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
1087     by auto
1088   finally show ?thesis ..
1089 qed
1091 lemma norm_triangle_eq:
1092   fixes x y :: "'a::real_inner"
1093   shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
1094 proof (cases "x = 0 \<or> y = 0")
1095   case True
1096   then show ?thesis
1097     by force
1098 next
1099   case False
1100   then have n: "norm x > 0" "norm y > 0"
1101     by auto
1102   have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
1103     by simp
1104   also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
1105     unfolding norm_cauchy_schwarz_eq[symmetric]
1106     unfolding power2_norm_eq_inner inner_simps
1107     by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
1108   finally show ?thesis .
1109 qed
1112 subsection \<open>Collinearity\<close>
1114 definition%important collinear :: "'a::real_vector set \<Rightarrow> bool"
1115   where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
1117 lemma collinear_alt:
1118      "collinear S \<longleftrightarrow> (\<exists>u v. \<forall>x \<in> S. \<exists>c. x = u + c *\<^sub>R v)" (is "?lhs = ?rhs")
1119 proof
1120   assume ?lhs
1121   then show ?rhs
1123 next
1124   assume ?rhs
1125   then obtain u v where *: "\<And>x. x \<in> S \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
1126     by (auto simp: )
1127   have "\<exists>c. x - y = c *\<^sub>R v" if "x \<in> S" "y \<in> S" for x y
1128         by (metis *[OF \<open>x \<in> S\<close>] *[OF \<open>y \<in> S\<close>] scaleR_left.diff add_diff_cancel_left)
1129   then show ?lhs
1130     using collinear_def by blast
1131 qed
1133 lemma collinear:
1134   fixes S :: "'a::{perfect_space,real_vector} set"
1135   shows "collinear S \<longleftrightarrow> (\<exists>u. u \<noteq> 0 \<and> (\<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u))"
1136 proof -
1137   have "\<exists>v. v \<noteq> 0 \<and> (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v)"
1138     if "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R u" "u=0" for u
1139   proof -
1140     have "\<forall>x\<in>S. \<forall>y\<in>S. x = y"
1141       using that by auto
1142     moreover
1143     obtain v::'a where "v \<noteq> 0"
1144       using UNIV_not_singleton [of 0] by auto
1145     ultimately have "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v"
1146       by auto
1147     then show ?thesis
1148       using \<open>v \<noteq> 0\<close> by blast
1149   qed
1150   then show ?thesis
1151     apply (clarsimp simp: collinear_def)
1152     by (metis scaleR_zero_right vector_fraction_eq_iff)
1153 qed
1155 lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S"
1156   by (meson collinear_def subsetCE)
1158 lemma collinear_empty [iff]: "collinear {}"
1161 lemma collinear_sing [iff]: "collinear {x}"
1164 lemma collinear_2 [iff]: "collinear {x, y}"
1166   apply (rule exI[where x="x - y"])
1167   by (metis minus_diff_eq scaleR_left.minus scaleR_one)
1169 lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
1170   (is "?lhs \<longleftrightarrow> ?rhs")
1171 proof (cases "x = 0 \<or> y = 0")
1172   case True
1173   then show ?thesis
1174     by (auto simp: insert_commute)
1175 next
1176   case False
1177   show ?thesis
1178   proof
1179     assume h: "?lhs"
1180     then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u"
1181       unfolding collinear_def by blast
1182     from u[rule_format, of x 0] u[rule_format, of y 0]
1183     obtain cx and cy where
1184       cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
1185       by auto
1186     from cx cy False have cx0: "cx \<noteq> 0" and cy0: "cy \<noteq> 0" by auto
1187     let ?d = "cy / cx"
1188     from cx cy cx0 have "y = ?d *\<^sub>R x"
1189       by simp
1190     then show ?rhs using False by blast
1191   next
1192     assume h: "?rhs"
1193     then obtain c where c: "y = c *\<^sub>R x"
1194       using False by blast
1195     show ?lhs
1196       unfolding collinear_def c
1197       apply (rule exI[where x=x])
1198       apply auto
1199           apply (rule exI[where x="- 1"], simp)
1200          apply (rule exI[where x= "-c"], simp)
1201         apply (rule exI[where x=1], simp)
1202        apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
1203       apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
1204       done
1205   qed
1206 qed
1208 lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
1209 proof (cases "x=0")
1210   case True
1211   then show ?thesis
1212     by (auto simp: insert_commute)
1213 next
1214   case False
1215   then have nnz: "norm x \<noteq> 0"
1216     by auto
1217   show ?thesis
1218   proof
1219     assume "\<bar>x \<bullet> y\<bar> = norm x * norm y"
1220     then show "collinear {0, x, y}"
1221       unfolding norm_cauchy_schwarz_abs_eq collinear_lemma
1222       by (meson eq_vector_fraction_iff nnz)
1223   next
1224     assume "collinear {0, x, y}"
1225     with False show "\<bar>x \<bullet> y\<bar> = norm x * norm y"
1226       unfolding norm_cauchy_schwarz_abs_eq collinear_lemma  by (auto simp: abs_if)
1227   qed
1228 qed
1230 end