src/HOL/Analysis/Linear_Algebra.thy
 author immler Thu May 03 15:07:14 2018 +0200 (12 months ago) changeset 68073 fad29d2a17a5 parent 68072 493b818e8e10 parent 68069 36209dfb981e child 68074 8d50467f7555 permissions -rw-r--r--
merged; resolved conflicts manually (esp. lemmas that have been moved from Linear_Algebra and Cartesian_Euclidean_Space)
```     1 (*  Title:      HOL/Analysis/Linear_Algebra.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3 *)
```
```     4
```
```     5 section \<open>Elementary linear algebra on Euclidean spaces\<close>
```
```     6
```
```     7 theory Linear_Algebra
```
```     8 imports
```
```     9   Euclidean_Space
```
```    10   "HOL-Library.Infinite_Set"
```
```    11 begin
```
```    12
```
```    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
```
```    29
```
```    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
```
```    32
```
```    33
```
```    34 subsection%unimportant \<open>More interesting properties of the norm.\<close>
```
```    35
```
```    36 notation inner (infix "\<bullet>" 70)
```
```    37
```
```    38 text\<open>Equality of vectors in terms of @{term "(\<bullet>)"} products.\<close>
```
```    39
```
```    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"
```
```    47     by (simp add: inner_sum_left)
```
```    48   then show ?thesis
```
```    49     by (simp add: euclidean_representation sum[symmetric] scale[symmetric])
```
```    50 qed
```
```    51
```
```    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
```
```    67
```
```    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
```
```    71
```
```    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] .
```
```    78
```
```    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])
```
```    81
```
```    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])
```
```    84
```
```    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
```
```    89
```
```    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
```
```    96
```
```    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)
```
```   101
```
```   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
```
```   108
```
```   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])
```
```   114
```
```   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"
```
```   119     by (simp add: inner_diff)
```
```   120   then have "(y - z) \<bullet> (y - z) = 0" ..
```
```   121   then show "y = z" by simp
```
```   122 qed simp
```
```   123
```
```   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"
```
```   128     by (simp add: inner_diff)
```
```   129   then have "(x - y) \<bullet> (x - y) = 0" ..
```
```   130   then show "x = y" by simp
```
```   131 qed simp
```
```   132
```
```   133
```
```   134 subsection \<open>Orthogonality.\<close>
```
```   135
```
```   136 definition%important (in real_inner) "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
```
```   137
```
```   138 context real_inner
```
```   139 begin
```
```   140
```
```   141 lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0"
```
```   142   by (simp add: orthogonal_def)
```
```   143
```
```   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
```
```   156
```
```   157 end
```
```   158
```
```   159 lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
```
```   160   by (simp add: orthogonal_def inner_commute)
```
```   161
```
```   162 lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x"
```
```   163   by (rule ext) (simp add: orthogonal_def)
```
```   164
```
```   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)
```
```   169
```
```   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)
```
```   173
```
```   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)
```
```   177
```
```   178 lemma norm_add_Pythagorean:
```
```   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
```
```   185     by (simp add: power2_norm_eq_inner)
```
```   186 qed
```
```   187
```
```   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
```
```   199     by (simp add: pairwise_insert norm_add_Pythagorean)
```
```   200 qed
```
```   201
```
```   202
```
```   203 subsection \<open>Bilinear functions.\<close>
```
```   204
```
```   205 definition%important "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
```
```   206
```
```   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)
```
```   209
```
```   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)
```
```   212
```
```   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)
```
```   215
```
```   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)
```
```   218
```
```   219 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
```
```   220   by (drule bilinear_lmul [of _ "- 1"]) simp
```
```   221
```
```   222 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
```
```   223   by (drule bilinear_rmul [of _ _ "- 1"]) simp
```
```   224
```
```   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
```
```   227
```
```   228 lemma bilinear_lzero:
```
```   229   assumes "bilinear h"
```
```   230   shows "h 0 x = 0"
```
```   231   using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
```
```   232
```
```   233 lemma bilinear_rzero:
```
```   234   assumes "bilinear h"
```
```   235   shows "h x 0 = 0"
```
```   236   using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
```
```   237
```
```   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)
```
```   240
```
```   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)
```
```   243
```
```   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"
```
```   251     by (simp add: l.sum)
```
```   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
```
```   257
```
```   258
```
```   259 subsection \<open>Adjoints.\<close>
```
```   260
```
```   261 definition%important "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
```
```   262
```
```   263 lemma adjoint_unique:
```
```   264   assumes "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   265   shows "adjoint f = g"
```
```   266   unfolding adjoint_def
```
```   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"
```
```   276     by (simp add: inner_diff_right)
```
```   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
```
```   283
```
```   284 text \<open>TODO: The following lemmas about adjoints should hold for any
```
```   285   Hilbert space (i.e. complete inner product space).
```
```   286   (see \<^url>\<open>http://en.wikipedia.org/wiki/Hermitian_adjoint\<close>)
```
```   287 \<close>
```
```   288
```
```   289 lemma adjoint_works:
```
```   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"
```
```   300       by (simp add: euclidean_representation)
```
```   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
```
```   307     unfolding adjoint_def choice_iff
```
```   308     by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
```
```   309 qed
```
```   310
```
```   311 lemma adjoint_clauses:
```
```   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"
```
```   316   by (simp_all add: adjoint_works[OF lf] inner_commute)
```
```   317
```
```   318 lemma adjoint_linear:
```
```   319   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   320   assumes lf: "linear f"
```
```   321   shows "linear (adjoint f)"
```
```   322   by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
```
```   323     adjoint_clauses[OF lf] inner_distrib)
```
```   324
```
```   325 lemma adjoint_adjoint:
```
```   326   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   327   assumes lf: "linear f"
```
```   328   shows "adjoint (adjoint f) = f"
```
```   329   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
```
```   330
```
```   331
```
```   332 subsection%unimportant \<open>Interlude: Some properties of real sets\<close>
```
```   333
```
```   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
```
```   339
```
```   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
```
```   345
```
```   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
```
```   351
```
```   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
```
```   357
```
```   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
```
```   374
```
```   375
```
```   376
```
```   377 subsection \<open>Archimedean properties and useful consequences\<close>
```
```   378
```
```   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"
```
```   390     by (simp add: algebra_simps)
```
```   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
```
```   397
```
```   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
```
```   419
```
```   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
```
```   434
```
```   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
```
```   452
```
```   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)
```
```   457
```
```   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
```
```   465
```
```   466
```
```   467 subsection%unimportant \<open>Euclidean Spaces as Typeclass\<close>
```
```   468
```
```   469 lemma independent_Basis: "independent Basis"
```
```   470   by (rule independent_Basis)
```
```   471
```
```   472 lemma span_Basis [simp]: "span Basis = UNIV"
```
```   473   by (rule span_Basis)
```
```   474
```
```   475 lemma in_span_Basis: "x \<in> span Basis"
```
```   476   unfolding span_Basis ..
```
```   477
```
```   478
```
```   479 subsection%unimportant \<open>Linearity and Bilinearity continued\<close>
```
```   480
```
```   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
```
```   508
```
```   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"
```
```   520       by (simp add: mult.commute)
```
```   521   qed
```
```   522 next
```
```   523   assume "bounded_linear f"
```
```   524   then interpret f: bounded_linear f .
```
```   525   show "linear f" ..
```
```   526 qed
```
```   527
```
```   528 lemmas linear_linear = linear_conv_bounded_linear[symmetric]
```
```   529
```
```   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
```
```   541
```
```   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
```
```   563
```
```   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
```
```   570
```
```   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
```
```   577
```
```   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))"
```
```   586     by (simp add: euclidean_representation)
```
```   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
```
```   598
```
```   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"
```
```   621       by (simp add: ac_simps)
```
```   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
```
```   630
```
```   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
```
```   642
```
```   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)
```
```   646
```
```   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)
```
```   651
```
```   652 lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net"
```
```   653   using bounded_linear_imp_has_derivative differentiable_def by blast
```
```   654
```
```   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)
```
```   659
```
```   660
```
```   661 subsection%unimportant \<open>We continue.\<close>
```
```   662
```
```   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)
```
```   667
```
```   668 lemmas independent_imp_finite = finiteI_independent
```
```   669
```
```   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
```
```   675
```
```   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)
```
```   680
```
```   681 text \<open>Picking an orthogonal replacement for a spanning set.\<close>
```
```   682
```
```   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
```
```   687
```
```   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)
```
```   694
```
```   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)"
```
```   718     by (simp add: card_insert_if)
```
```   719   {
```
```   720     fix x k
```
```   721     have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
```
```   722       by (simp add: field_simps)
```
```   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)
```
```   725       apply (rule span_add_eq)
```
```   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
```
```   758
```
```   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"
```
```   775     by (simp add: span_span)
```
```   776   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
```
```   777   have iC: "independent C"
```
```   778     by (simp add: dim_span)
```
```   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
```
```   789
```
```   790 text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
```
```   791
```
```   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"
```
```   807     by (simp add: span_span)
```
```   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}"
```
```   821       by (auto simp add: subspace_def inner_add)
```
```   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
```
```   835         apply (clarsimp simp add: inner_add inner_sum_left)
```
```   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
```
```   848
```
```   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
```
```   854
```
```   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
```
```   872
```
```   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
```
```   881
```
```   882
```
```   883 text \<open>Similar results for bilinear functions.\<close>
```
```   884
```
```   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]
```
```   898         span_add Ball_def
```
```   899       intro: bilinear_ladd[OF bf])
```
```   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]
```
```   904         span_add Ball_def
```
```   905       intro: bilinear_ladd[OF bf])
```
```   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
```
```   913
```
```   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
```
```   921
```
```   922 subsection \<open>Infinity norm\<close>
```
```   923
```
```   924 definition%important "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
```
```   925
```
```   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
```
```   930
```
```   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)
```
```   935
```
```   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
```
```   942
```
```   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)
```
```   947
```
```   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
```
```   957
```
```   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
```
```   967
```
```   968 lemma infnorm_0: "infnorm 0 = 0"
```
```   969   by (simp add: infnorm_eq_0)
```
```   970
```
```   971 lemma infnorm_neg: "infnorm (- x) = infnorm x"
```
```   972   unfolding infnorm_def by simp
```
```   973
```
```   974 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
```
```   975   by (metis infnorm_neg minus_diff_eq)
```
```   976
```
```   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
```
```   988
```
```   989 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
```
```   990   using infnorm_pos_le[of x] by arith
```
```   991
```
```   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"
```
```   995   by (simp add: infnorm_Max)
```
```   996
```
```   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
```
```  1012
```
```  1013 lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
```
```  1014   unfolding infnorm_mul ..
```
```  1015
```
```  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
```
```  1018
```
```  1019 text \<open>Prove that it differs only up to a bound from Euclidean norm.\<close>
```
```  1020
```
```  1021 lemma infnorm_le_norm: "infnorm x \<le> norm x"
```
```  1022   by (simp add: Basis_le_norm infnorm_Max)
```
```  1023
```
```  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"
```
```  1036     by (simp add: power_mult_distrib)
```
```  1037   finally show "x \<bullet> x \<le> (sqrt DIM('a) * infnorm x)\<^sup>2" .
```
```  1038 qed
```
```  1039
```
```  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
```
```  1049
```
```  1050 text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close>
```
```  1051
```
```  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
```
```  1072
```
```  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
```
```  1090
```
```  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
```
```  1110
```
```  1111
```
```  1112 subsection \<open>Collinearity\<close>
```
```  1113
```
```  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)"
```
```  1116
```
```  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
```
```  1122     unfolding collinear_def by (metis Groups.add_ac(2) diff_add_cancel)
```
```  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
```
```  1132
```
```  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
```
```  1154
```
```  1155 lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S"
```
```  1156   by (meson collinear_def subsetCE)
```
```  1157
```
```  1158 lemma collinear_empty [iff]: "collinear {}"
```
```  1159   by (simp add: collinear_def)
```
```  1160
```
```  1161 lemma collinear_sing [iff]: "collinear {x}"
```
```  1162   by (simp add: collinear_def)
```
```  1163
```
```  1164 lemma collinear_2 [iff]: "collinear {x, y}"
```
```  1165   apply (simp add: collinear_def)
```
```  1166   apply (rule exI[where x="x - y"])
```
```  1167   by (metis minus_diff_eq scaleR_left.minus scaleR_one)
```
```  1168
```
```  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
```
```  1207
```
```  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
```
```  1229
```
```  1230 end
```