src/HOL/Analysis/Linear_Algebra.thy
 author immler Wed Jan 16 18:14:02 2019 -0500 (4 months ago) changeset 69675 880ab0f27ddf parent 69674 fc252acb7100 child 69683 8b3458ca0762 permissions -rw-r--r--
Reorg, in particular Determinants as well as some linear algebra from Starlike and Change_Of_Vars
```     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 lemma substdbasis_expansion_unique:
```
```    34   includes inner_syntax
```
```    35   assumes d: "d \<subseteq> Basis"
```
```    36   shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space) \<longleftrightarrow>
```
```    37     (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
```
```    38 proof -
```
```    39   have *: "\<And>x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
```
```    40     by auto
```
```    41   have **: "finite d"
```
```    42     by (auto intro: finite_subset[OF assms])
```
```    43   have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
```
```    44     using d
```
```    45     by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
```
```    46   show ?thesis
```
```    47     unfolding euclidean_eq_iff[where 'a='a] by (auto simp: sum.delta[OF **] ***)
```
```    48 qed
```
```    49
```
```    50 lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
```
```    51   by (rule independent_mono[OF independent_Basis])
```
```    52
```
```    53 lemma sum_not_0: "sum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
```
```    54   by (rule ccontr) auto
```
```    55
```
```    56 lemma subset_translation_eq [simp]:
```
```    57     fixes a :: "'a::real_vector" shows "(+) a ` s \<subseteq> (+) a ` t \<longleftrightarrow> s \<subseteq> t"
```
```    58   by auto
```
```    59
```
```    60 lemma translate_inj_on:
```
```    61   fixes A :: "'a::ab_group_add set"
```
```    62   shows "inj_on (\<lambda>x. a + x) A"
```
```    63   unfolding inj_on_def by auto
```
```    64
```
```    65 lemma translation_assoc:
```
```    66   fixes a b :: "'a::ab_group_add"
```
```    67   shows "(\<lambda>x. b + x) ` ((\<lambda>x. a + x) ` S) = (\<lambda>x. (a + b) + x) ` S"
```
```    68   by auto
```
```    69
```
```    70 lemma translation_invert:
```
```    71   fixes a :: "'a::ab_group_add"
```
```    72   assumes "(\<lambda>x. a + x) ` A = (\<lambda>x. a + x) ` B"
```
```    73   shows "A = B"
```
```    74 proof -
```
```    75   have "(\<lambda>x. -a + x) ` ((\<lambda>x. a + x) ` A) = (\<lambda>x. - a + x) ` ((\<lambda>x. a + x) ` B)"
```
```    76     using assms by auto
```
```    77   then show ?thesis
```
```    78     using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
```
```    79 qed
```
```    80
```
```    81 lemma translation_galois:
```
```    82   fixes a :: "'a::ab_group_add"
```
```    83   shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)"
```
```    84   using translation_assoc[of "-a" a S]
```
```    85   apply auto
```
```    86   using translation_assoc[of a "-a" T]
```
```    87   apply auto
```
```    88   done
```
```    89
```
```    90 lemma translation_inverse_subset:
```
```    91   assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
```
```    92   shows "V \<le> ((\<lambda>x. a + x) ` S)"
```
```    93 proof -
```
```    94   {
```
```    95     fix x
```
```    96     assume "x \<in> V"
```
```    97     then have "x-a \<in> S" using assms by auto
```
```    98     then have "x \<in> {a + v |v. v \<in> S}"
```
```    99       apply auto
```
```   100       apply (rule exI[of _ "x-a"], simp)
```
```   101       done
```
```   102     then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
```
```   103   }
```
```   104   then show ?thesis by auto
```
```   105 qed
```
```   106
```
```   107 subsection%unimportant \<open>More interesting properties of the norm\<close>
```
```   108
```
```   109 unbundle inner_syntax
```
```   110
```
```   111 text\<open>Equality of vectors in terms of \<^term>\<open>(\<bullet>)\<close> products.\<close>
```
```   112
```
```   113 lemma linear_componentwise:
```
```   114   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
```
```   115   assumes lf: "linear f"
```
```   116   shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
```
```   117 proof -
```
```   118   interpret linear f by fact
```
```   119   have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
```
```   120     by (simp add: inner_sum_left)
```
```   121   then show ?thesis
```
```   122     by (simp add: euclidean_representation sum[symmetric] scale[symmetric])
```
```   123 qed
```
```   124
```
```   125 lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
```
```   126   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   127 proof
```
```   128   assume ?lhs
```
```   129   then show ?rhs by simp
```
```   130 next
```
```   131   assume ?rhs
```
```   132   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
```
```   133     by simp
```
```   134   then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
```
```   135     by (simp add: inner_diff inner_commute)
```
```   136   then have "(x - y) \<bullet> (x - y) = 0"
```
```   137     by (simp add: field_simps inner_diff inner_commute)
```
```   138   then show "x = y" by simp
```
```   139 qed
```
```   140
```
```   141 lemma norm_triangle_half_r:
```
```   142   "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
```
```   143   using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
```
```   144
```
```   145 lemma norm_triangle_half_l:
```
```   146   assumes "norm (x - y) < e / 2"
```
```   147     and "norm (x' - y) < e / 2"
```
```   148   shows "norm (x - x') < e"
```
```   149   using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
```
```   150   unfolding dist_norm[symmetric] .
```
```   151
```
```   152 lemma abs_triangle_half_r:
```
```   153   fixes y :: "'a::linordered_field"
```
```   154   shows "abs (y - x1) < e / 2 \<Longrightarrow> abs (y - x2) < e / 2 \<Longrightarrow> abs (x1 - x2) < e"
```
```   155   by linarith
```
```   156
```
```   157 lemma abs_triangle_half_l:
```
```   158   fixes y :: "'a::linordered_field"
```
```   159   assumes "abs (x - y) < e / 2"
```
```   160     and "abs (x' - y) < e / 2"
```
```   161   shows "abs (x - x') < e"
```
```   162   using assms by linarith
```
```   163
```
```   164 lemma sum_clauses:
```
```   165   shows "sum f {} = 0"
```
```   166     and "finite S \<Longrightarrow> sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
```
```   167   by (auto simp add: insert_absorb)
```
```   168
```
```   169 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
```
```   170 proof
```
```   171   assume "\<forall>x. x \<bullet> y = x \<bullet> z"
```
```   172   then have "\<forall>x. x \<bullet> (y - z) = 0"
```
```   173     by (simp add: inner_diff)
```
```   174   then have "(y - z) \<bullet> (y - z) = 0" ..
```
```   175   then show "y = z" by simp
```
```   176 qed simp
```
```   177
```
```   178 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
```
```   179 proof
```
```   180   assume "\<forall>z. x \<bullet> z = y \<bullet> z"
```
```   181   then have "\<forall>z. (x - y) \<bullet> z = 0"
```
```   182     by (simp add: inner_diff)
```
```   183   then have "(x - y) \<bullet> (x - y) = 0" ..
```
```   184   then show "x = y" by simp
```
```   185 qed simp
```
```   186
```
```   187 subsection \<open>Substandard Basis\<close>
```
```   188
```
```   189 lemma ex_card:
```
```   190   assumes "n \<le> card A"
```
```   191   shows "\<exists>S\<subseteq>A. card S = n"
```
```   192 proof (cases "finite A")
```
```   193   case True
```
```   194   from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" ..
```
```   195   moreover from f \<open>n \<le> card A\<close> have "{..< n} \<subseteq> {..< card A}" "inj_on f {..< n}"
```
```   196     by (auto simp: bij_betw_def intro: subset_inj_on)
```
```   197   ultimately have "f ` {..< n} \<subseteq> A" "card (f ` {..< n}) = n"
```
```   198     by (auto simp: bij_betw_def card_image)
```
```   199   then show ?thesis by blast
```
```   200 next
```
```   201   case False
```
```   202   with \<open>n \<le> card A\<close> show ?thesis by force
```
```   203 qed
```
```   204
```
```   205 lemma subspace_substandard: "subspace {x::'a::euclidean_space. (\<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0)}"
```
```   206   by (auto simp: subspace_def inner_add_left)
```
```   207
```
```   208 lemma dim_substandard:
```
```   209   assumes d: "d \<subseteq> Basis"
```
```   210   shows "dim {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0} = card d" (is "dim ?A = _")
```
```   211 proof (rule dim_unique)
```
```   212   from d show "d \<subseteq> ?A"
```
```   213     by (auto simp: inner_Basis)
```
```   214   from d show "independent d"
```
```   215     by (rule independent_mono [OF independent_Basis])
```
```   216   have "x \<in> span d" if "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0" for x
```
```   217   proof -
```
```   218     have "finite d"
```
```   219       by (rule finite_subset [OF d finite_Basis])
```
```   220     then have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) \<in> span d"
```
```   221       by (simp add: span_sum span_clauses)
```
```   222     also have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i)"
```
```   223       by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp: that)
```
```   224     finally show "x \<in> span d"
```
```   225       by (simp only: euclidean_representation)
```
```   226   qed
```
```   227   then show "?A \<subseteq> span d" by auto
```
```   228 qed simp
```
```   229
```
```   230
```
```   231 subsection \<open>Orthogonality\<close>
```
```   232
```
```   233 definition%important (in real_inner) "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
```
```   234
```
```   235 context real_inner
```
```   236 begin
```
```   237
```
```   238 lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0"
```
```   239   by (simp add: orthogonal_def)
```
```   240
```
```   241 lemma orthogonal_clauses:
```
```   242   "orthogonal a 0"
```
```   243   "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
```
```   244   "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
```
```   245   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
```
```   246   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
```
```   247   "orthogonal 0 a"
```
```   248   "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
```
```   249   "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
```
```   250   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
```
```   251   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
```
```   252   unfolding orthogonal_def inner_add inner_diff by auto
```
```   253
```
```   254 end
```
```   255
```
```   256 lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
```
```   257   by (simp add: orthogonal_def inner_commute)
```
```   258
```
```   259 lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x"
```
```   260   by (rule ext) (simp add: orthogonal_def)
```
```   261
```
```   262 lemma pairwise_ortho_scaleR:
```
```   263     "pairwise (\<lambda>i j. orthogonal (f i) (g j)) B
```
```   264     \<Longrightarrow> pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B"
```
```   265   by (auto simp: pairwise_def orthogonal_clauses)
```
```   266
```
```   267 lemma orthogonal_rvsum:
```
```   268     "\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (sum f s)"
```
```   269   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
```
```   270
```
```   271 lemma orthogonal_lvsum:
```
```   272     "\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (sum f s) y"
```
```   273   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
```
```   274
```
```   275 lemma norm_add_Pythagorean:
```
```   276   assumes "orthogonal a b"
```
```   277     shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2"
```
```   278 proof -
```
```   279   from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)"
```
```   280     by (simp add: algebra_simps orthogonal_def inner_commute)
```
```   281   then show ?thesis
```
```   282     by (simp add: power2_norm_eq_inner)
```
```   283 qed
```
```   284
```
```   285 lemma norm_sum_Pythagorean:
```
```   286   assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I"
```
```   287     shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
```
```   288 using assms
```
```   289 proof (induction I rule: finite_induct)
```
```   290   case empty then show ?case by simp
```
```   291 next
```
```   292   case (insert x I)
```
```   293   then have "orthogonal (f x) (sum f I)"
```
```   294     by (metis pairwise_insert orthogonal_rvsum)
```
```   295   with insert show ?case
```
```   296     by (simp add: pairwise_insert norm_add_Pythagorean)
```
```   297 qed
```
```   298
```
```   299
```
```   300 subsection%important  \<open>Orthogonality of a transformation\<close>
```
```   301
```
```   302 definition%important  "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"
```
```   303
```
```   304 lemma%unimportant  orthogonal_transformation:
```
```   305   "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v. norm (f v) = norm v)"
```
```   306   unfolding orthogonal_transformation_def
```
```   307   apply auto
```
```   308   apply (erule_tac x=v in allE)+
```
```   309   apply (simp add: norm_eq_sqrt_inner)
```
```   310   apply (simp add: dot_norm linear_add[symmetric])
```
```   311   done
```
```   312
```
```   313 lemma%unimportant  orthogonal_transformation_id [simp]: "orthogonal_transformation (\<lambda>x. x)"
```
```   314   by (simp add: linear_iff orthogonal_transformation_def)
```
```   315
```
```   316 lemma%unimportant  orthogonal_orthogonal_transformation:
```
```   317     "orthogonal_transformation f \<Longrightarrow> orthogonal (f x) (f y) \<longleftrightarrow> orthogonal x y"
```
```   318   by (simp add: orthogonal_def orthogonal_transformation_def)
```
```   319
```
```   320 lemma%unimportant  orthogonal_transformation_compose:
```
```   321    "\<lbrakk>orthogonal_transformation f; orthogonal_transformation g\<rbrakk> \<Longrightarrow> orthogonal_transformation(f \<circ> g)"
```
```   322   by (auto simp: orthogonal_transformation_def linear_compose)
```
```   323
```
```   324 lemma%unimportant  orthogonal_transformation_neg:
```
```   325   "orthogonal_transformation(\<lambda>x. -(f x)) \<longleftrightarrow> orthogonal_transformation f"
```
```   326   by (auto simp: orthogonal_transformation_def dest: linear_compose_neg)
```
```   327
```
```   328 lemma%unimportant  orthogonal_transformation_scaleR: "orthogonal_transformation f \<Longrightarrow> f (c *\<^sub>R v) = c *\<^sub>R f v"
```
```   329   by (simp add: linear_iff orthogonal_transformation_def)
```
```   330
```
```   331 lemma%unimportant  orthogonal_transformation_linear:
```
```   332    "orthogonal_transformation f \<Longrightarrow> linear f"
```
```   333   by (simp add: orthogonal_transformation_def)
```
```   334
```
```   335 lemma%unimportant  orthogonal_transformation_inj:
```
```   336   "orthogonal_transformation f \<Longrightarrow> inj f"
```
```   337   unfolding orthogonal_transformation_def inj_on_def
```
```   338   by (metis vector_eq)
```
```   339
```
```   340 lemma%unimportant  orthogonal_transformation_surj:
```
```   341   "orthogonal_transformation f \<Longrightarrow> surj f"
```
```   342   for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```   343   by (simp add: linear_injective_imp_surjective orthogonal_transformation_inj orthogonal_transformation_linear)
```
```   344
```
```   345 lemma%unimportant  orthogonal_transformation_bij:
```
```   346   "orthogonal_transformation f \<Longrightarrow> bij f"
```
```   347   for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```   348   by (simp add: bij_def orthogonal_transformation_inj orthogonal_transformation_surj)
```
```   349
```
```   350 lemma%unimportant  orthogonal_transformation_inv:
```
```   351   "orthogonal_transformation f \<Longrightarrow> orthogonal_transformation (inv f)"
```
```   352   for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```   353   by (metis (no_types, hide_lams) bijection.inv_right bijection_def inj_linear_imp_inv_linear orthogonal_transformation orthogonal_transformation_bij orthogonal_transformation_inj)
```
```   354
```
```   355 lemma%unimportant  orthogonal_transformation_norm:
```
```   356   "orthogonal_transformation f \<Longrightarrow> norm (f x) = norm x"
```
```   357   by (metis orthogonal_transformation)
```
```   358
```
```   359
```
```   360 subsection \<open>Bilinear functions\<close>
```
```   361
```
```   362 definition%important
```
```   363 bilinear :: "('a::real_vector \<Rightarrow> 'b::real_vector \<Rightarrow> 'c::real_vector) \<Rightarrow> bool" where
```
```   364 "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
```
```   365
```
```   366 lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
```
```   367   by (simp add: bilinear_def linear_iff)
```
```   368
```
```   369 lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
```
```   370   by (simp add: bilinear_def linear_iff)
```
```   371
```
```   372 lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
```
```   373   by (simp add: bilinear_def linear_iff)
```
```   374
```
```   375 lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
```
```   376   by (simp add: bilinear_def linear_iff)
```
```   377
```
```   378 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
```
```   379   by (drule bilinear_lmul [of _ "- 1"]) simp
```
```   380
```
```   381 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
```
```   382   by (drule bilinear_rmul [of _ _ "- 1"]) simp
```
```   383
```
```   384 lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
```
```   385   using add_left_imp_eq[of x y 0] by auto
```
```   386
```
```   387 lemma bilinear_lzero:
```
```   388   assumes "bilinear h"
```
```   389   shows "h 0 x = 0"
```
```   390   using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
```
```   391
```
```   392 lemma bilinear_rzero:
```
```   393   assumes "bilinear h"
```
```   394   shows "h x 0 = 0"
```
```   395   using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
```
```   396
```
```   397 lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
```
```   398   using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
```
```   399
```
```   400 lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
```
```   401   using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
```
```   402
```
```   403 lemma bilinear_sum:
```
```   404   assumes "bilinear h"
```
```   405   shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
```
```   406 proof -
```
```   407   interpret l: linear "\<lambda>x. h x y" for y using assms by (simp add: bilinear_def)
```
```   408   interpret r: linear "\<lambda>y. h x y" for x using assms by (simp add: bilinear_def)
```
```   409   have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S"
```
```   410     by (simp add: l.sum)
```
```   411   also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S"
```
```   412     by (rule sum.cong) (simp_all add: r.sum)
```
```   413   finally show ?thesis
```
```   414     unfolding sum.cartesian_product .
```
```   415 qed
```
```   416
```
```   417
```
```   418 subsection \<open>Adjoints\<close>
```
```   419
```
```   420 definition%important adjoint :: "(('a::real_inner) \<Rightarrow> ('b::real_inner)) \<Rightarrow> 'b \<Rightarrow> 'a" where
```
```   421 "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
```
```   422
```
```   423 lemma adjoint_unique:
```
```   424   assumes "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   425   shows "adjoint f = g"
```
```   426   unfolding adjoint_def
```
```   427 proof (rule some_equality)
```
```   428   show "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   429     by (rule assms)
```
```   430 next
```
```   431   fix h
```
```   432   assume "\<forall>x y. inner (f x) y = inner x (h y)"
```
```   433   then have "\<forall>x y. inner x (g y) = inner x (h y)"
```
```   434     using assms by simp
```
```   435   then have "\<forall>x y. inner x (g y - h y) = 0"
```
```   436     by (simp add: inner_diff_right)
```
```   437   then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
```
```   438     by simp
```
```   439   then have "\<forall>y. h y = g y"
```
```   440     by simp
```
```   441   then show "h = g" by (simp add: ext)
```
```   442 qed
```
```   443
```
```   444 text \<open>TODO: The following lemmas about adjoints should hold for any
```
```   445   Hilbert space (i.e. complete inner product space).
```
```   446   (see \<^url>\<open>https://en.wikipedia.org/wiki/Hermitian_adjoint\<close>)
```
```   447 \<close>
```
```   448
```
```   449 lemma adjoint_works:
```
```   450   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   451   assumes lf: "linear f"
```
```   452   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```   453 proof -
```
```   454   interpret linear f by fact
```
```   455   have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
```
```   456   proof (intro allI exI)
```
```   457     fix y :: "'m" and x
```
```   458     let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
```
```   459     have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
```
```   460       by (simp add: euclidean_representation)
```
```   461     also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
```
```   462       by (simp add: sum scale)
```
```   463     finally show "f x \<bullet> y = x \<bullet> ?w"
```
```   464       by (simp add: inner_sum_left inner_sum_right mult.commute)
```
```   465   qed
```
```   466   then show ?thesis
```
```   467     unfolding adjoint_def choice_iff
```
```   468     by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
```
```   469 qed
```
```   470
```
```   471 lemma adjoint_clauses:
```
```   472   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   473   assumes lf: "linear f"
```
```   474   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```   475     and "adjoint f y \<bullet> x = y \<bullet> f x"
```
```   476   by (simp_all add: adjoint_works[OF lf] inner_commute)
```
```   477
```
```   478 lemma adjoint_linear:
```
```   479   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   480   assumes lf: "linear f"
```
```   481   shows "linear (adjoint f)"
```
```   482   by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
```
```   483     adjoint_clauses[OF lf] inner_distrib)
```
```   484
```
```   485 lemma adjoint_adjoint:
```
```   486   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   487   assumes lf: "linear f"
```
```   488   shows "adjoint (adjoint f) = f"
```
```   489   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
```
```   490
```
```   491
```
```   492 subsection \<open>Archimedean properties and useful consequences\<close>
```
```   493
```
```   494 text\<open>Bernoulli's inequality\<close>
```
```   495 proposition Bernoulli_inequality:
```
```   496   fixes x :: real
```
```   497   assumes "-1 \<le> x"
```
```   498     shows "1 + n * x \<le> (1 + x) ^ n"
```
```   499 proof (induct n)
```
```   500   case 0
```
```   501   then show ?case by simp
```
```   502 next
```
```   503   case (Suc n)
```
```   504   have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
```
```   505     by (simp add: algebra_simps)
```
```   506   also have "... = (1 + x) * (1 + n*x)"
```
```   507     by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
```
```   508   also have "... \<le> (1 + x) ^ Suc n"
```
```   509     using Suc.hyps assms mult_left_mono by fastforce
```
```   510   finally show ?case .
```
```   511 qed
```
```   512
```
```   513 corollary Bernoulli_inequality_even:
```
```   514   fixes x :: real
```
```   515   assumes "even n"
```
```   516     shows "1 + n * x \<le> (1 + x) ^ n"
```
```   517 proof (cases "-1 \<le> x \<or> n=0")
```
```   518   case True
```
```   519   then show ?thesis
```
```   520     by (auto simp: Bernoulli_inequality)
```
```   521 next
```
```   522   case False
```
```   523   then have "real n \<ge> 1"
```
```   524     by simp
```
```   525   with False have "n * x \<le> -1"
```
```   526     by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
```
```   527   then have "1 + n * x \<le> 0"
```
```   528     by auto
```
```   529   also have "... \<le> (1 + x) ^ n"
```
```   530     using assms
```
```   531     using zero_le_even_power by blast
```
```   532   finally show ?thesis .
```
```   533 qed
```
```   534
```
```   535 corollary real_arch_pow:
```
```   536   fixes x :: real
```
```   537   assumes x: "1 < x"
```
```   538   shows "\<exists>n. y < x^n"
```
```   539 proof -
```
```   540   from x have x0: "x - 1 > 0"
```
```   541     by arith
```
```   542   from reals_Archimedean3[OF x0, rule_format, of y]
```
```   543   obtain n :: nat where n: "y < real n * (x - 1)" by metis
```
```   544   from x0 have x00: "x- 1 \<ge> -1" by arith
```
```   545   from Bernoulli_inequality[OF x00, of n] n
```
```   546   have "y < x^n" by auto
```
```   547   then show ?thesis by metis
```
```   548 qed
```
```   549
```
```   550 corollary real_arch_pow_inv:
```
```   551   fixes x y :: real
```
```   552   assumes y: "y > 0"
```
```   553     and x1: "x < 1"
```
```   554   shows "\<exists>n. x^n < y"
```
```   555 proof (cases "x > 0")
```
```   556   case True
```
```   557   with x1 have ix: "1 < 1/x" by (simp add: field_simps)
```
```   558   from real_arch_pow[OF ix, of "1/y"]
```
```   559   obtain n where n: "1/y < (1/x)^n" by blast
```
```   560   then show ?thesis using y \<open>x > 0\<close>
```
```   561     by (auto simp add: field_simps)
```
```   562 next
```
```   563   case False
```
```   564   with y x1 show ?thesis
```
```   565     by (metis less_le_trans not_less power_one_right)
```
```   566 qed
```
```   567
```
```   568 lemma forall_pos_mono:
```
```   569   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```   570     (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
```
```   571   by (metis real_arch_inverse)
```
```   572
```
```   573 lemma forall_pos_mono_1:
```
```   574   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```   575     (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
```
```   576   apply (rule forall_pos_mono)
```
```   577   apply auto
```
```   578   apply (metis Suc_pred of_nat_Suc)
```
```   579   done
```
```   580
```
```   581
```
```   582 subsection%unimportant \<open>Euclidean Spaces as Typeclass\<close>
```
```   583
```
```   584 lemma independent_Basis: "independent Basis"
```
```   585   by (rule independent_Basis)
```
```   586
```
```   587 lemma span_Basis [simp]: "span Basis = UNIV"
```
```   588   by (rule span_Basis)
```
```   589
```
```   590 lemma in_span_Basis: "x \<in> span Basis"
```
```   591   unfolding span_Basis ..
```
```   592
```
```   593
```
```   594 subsection%unimportant \<open>Linearity and Bilinearity continued\<close>
```
```   595
```
```   596 lemma linear_bounded:
```
```   597   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   598   assumes lf: "linear f"
```
```   599   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```   600 proof
```
```   601   interpret linear f by fact
```
```   602   let ?B = "\<Sum>b\<in>Basis. norm (f b)"
```
```   603   show "\<forall>x. norm (f x) \<le> ?B * norm x"
```
```   604   proof
```
```   605     fix x :: 'a
```
```   606     let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
```
```   607     have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
```
```   608       unfolding euclidean_representation ..
```
```   609     also have "\<dots> = norm (sum ?g Basis)"
```
```   610       by (simp add: sum scale)
```
```   611     finally have th0: "norm (f x) = norm (sum ?g Basis)" .
```
```   612     have th: "norm (?g i) \<le> norm (f i) * norm x" if "i \<in> Basis" for i
```
```   613     proof -
```
```   614       from Basis_le_norm[OF that, of x]
```
```   615       show "norm (?g i) \<le> norm (f i) * norm x"
```
```   616         unfolding norm_scaleR  by (metis mult.commute mult_left_mono norm_ge_zero)
```
```   617     qed
```
```   618     from sum_norm_le[of _ ?g, OF th]
```
```   619     show "norm (f x) \<le> ?B * norm x"
```
```   620       unfolding th0 sum_distrib_right by metis
```
```   621   qed
```
```   622 qed
```
```   623
```
```   624 lemma linear_conv_bounded_linear:
```
```   625   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   626   shows "linear f \<longleftrightarrow> bounded_linear f"
```
```   627 proof
```
```   628   assume "linear f"
```
```   629   then interpret f: linear f .
```
```   630   show "bounded_linear f"
```
```   631   proof
```
```   632     have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```   633       using \<open>linear f\<close> by (rule linear_bounded)
```
```   634     then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```   635       by (simp add: mult.commute)
```
```   636   qed
```
```   637 next
```
```   638   assume "bounded_linear f"
```
```   639   then interpret f: bounded_linear f .
```
```   640   show "linear f" ..
```
```   641 qed
```
```   642
```
```   643 lemmas linear_linear = linear_conv_bounded_linear[symmetric]
```
```   644
```
```   645 lemma linear_bounded_pos:
```
```   646   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   647   assumes lf: "linear f"
```
```   648  obtains B where "B > 0" "\<And>x. norm (f x) \<le> B * norm x"
```
```   649 proof -
```
```   650   have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
```
```   651     using lf unfolding linear_conv_bounded_linear
```
```   652     by (rule bounded_linear.pos_bounded)
```
```   653   with that show ?thesis
```
```   654     by (auto simp: mult.commute)
```
```   655 qed
```
```   656
```
```   657 lemma linear_invertible_bounded_below_pos:
```
```   658   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
```
```   659   assumes "linear f" "linear g" "g \<circ> f = id"
```
```   660   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
```
```   661 proof -
```
```   662   obtain B where "B > 0" and B: "\<And>x. norm (g x) \<le> B * norm x"
```
```   663     using linear_bounded_pos [OF \<open>linear g\<close>] by blast
```
```   664   show thesis
```
```   665   proof
```
```   666     show "0 < 1/B"
```
```   667       by (simp add: \<open>B > 0\<close>)
```
```   668     show "1/B * norm x \<le> norm (f x)" for x
```
```   669     proof -
```
```   670       have "1/B * norm x = 1/B * norm (g (f x))"
```
```   671         using assms by (simp add: pointfree_idE)
```
```   672       also have "\<dots> \<le> norm (f x)"
```
```   673         using B [of "f x"] by (simp add: \<open>B > 0\<close> mult.commute pos_divide_le_eq)
```
```   674       finally show ?thesis .
```
```   675     qed
```
```   676   qed
```
```   677 qed
```
```   678
```
```   679 lemma linear_inj_bounded_below_pos:
```
```   680   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
```
```   681   assumes "linear f" "inj f"
```
```   682   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
```
```   683   using linear_injective_left_inverse [OF assms]
```
```   684     linear_invertible_bounded_below_pos assms by blast
```
```   685
```
```   686 lemma bounded_linearI':
```
```   687   fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   688   assumes "\<And>x y. f (x + y) = f x + f y"
```
```   689     and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```   690   shows "bounded_linear f"
```
```   691   using assms linearI linear_conv_bounded_linear by blast
```
```   692
```
```   693 lemma bilinear_bounded:
```
```   694   fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
```
```   695   assumes bh: "bilinear h"
```
```   696   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```   697 proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
```
```   698   fix x :: 'm
```
```   699   fix y :: 'n
```
```   700   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))"
```
```   701     by (simp add: euclidean_representation)
```
```   702   also have "\<dots> = norm (sum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
```
```   703     unfolding bilinear_sum[OF bh] ..
```
```   704   finally have th: "norm (h x y) = \<dots>" .
```
```   705   have "\<And>i j. \<lbrakk>i \<in> Basis; j \<in> Basis\<rbrakk>
```
```   706            \<Longrightarrow> \<bar>x \<bullet> i\<bar> * (\<bar>y \<bullet> j\<bar> * norm (h i j)) \<le> norm x * (norm y * norm (h i j))"
```
```   707     by (auto simp add: zero_le_mult_iff Basis_le_norm mult_mono)
```
```   708   then show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
```
```   709     unfolding sum_distrib_right th sum.cartesian_product
```
```   710     by (clarsimp simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
```
```   711       field_simps simp del: scaleR_scaleR intro!: sum_norm_le)
```
```   712 qed
```
```   713
```
```   714 lemma bilinear_conv_bounded_bilinear:
```
```   715   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```   716   shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
```
```   717 proof
```
```   718   assume "bilinear h"
```
```   719   show "bounded_bilinear h"
```
```   720   proof
```
```   721     fix x y z
```
```   722     show "h (x + y) z = h x z + h y z"
```
```   723       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
```
```   724   next
```
```   725     fix x y z
```
```   726     show "h x (y + z) = h x y + h x z"
```
```   727       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
```
```   728   next
```
```   729     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
```
```   730       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
```
```   731       by simp_all
```
```   732   next
```
```   733     have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```   734       using \<open>bilinear h\<close> by (rule bilinear_bounded)
```
```   735     then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
```
```   736       by (simp add: ac_simps)
```
```   737   qed
```
```   738 next
```
```   739   assume "bounded_bilinear h"
```
```   740   then interpret h: bounded_bilinear h .
```
```   741   show "bilinear h"
```
```   742     unfolding bilinear_def linear_conv_bounded_linear
```
```   743     using h.bounded_linear_left h.bounded_linear_right by simp
```
```   744 qed
```
```   745
```
```   746 lemma bilinear_bounded_pos:
```
```   747   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```   748   assumes bh: "bilinear h"
```
```   749   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```   750 proof -
```
```   751   have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
```
```   752     using bh [unfolded bilinear_conv_bounded_bilinear]
```
```   753     by (rule bounded_bilinear.pos_bounded)
```
```   754   then show ?thesis
```
```   755     by (simp only: ac_simps)
```
```   756 qed
```
```   757
```
```   758 lemma bounded_linear_imp_has_derivative: "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
```
```   759   by (auto simp add: has_derivative_def linear_diff linear_linear linear_def
```
```   760       dest: bounded_linear.linear)
```
```   761
```
```   762 lemma linear_imp_has_derivative:
```
```   763   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   764   shows "linear f \<Longrightarrow> (f has_derivative f) net"
```
```   765   by (simp add: bounded_linear_imp_has_derivative linear_conv_bounded_linear)
```
```   766
```
```   767 lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net"
```
```   768   using bounded_linear_imp_has_derivative differentiable_def by blast
```
```   769
```
```   770 lemma linear_imp_differentiable:
```
```   771   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   772   shows "linear f \<Longrightarrow> f differentiable net"
```
```   773   by (metis linear_imp_has_derivative differentiable_def)
```
```   774
```
```   775
```
```   776 subsection%unimportant \<open>We continue\<close>
```
```   777
```
```   778 lemma independent_bound:
```
```   779   fixes S :: "'a::euclidean_space set"
```
```   780   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
```
```   781   by (metis dim_subset_UNIV finiteI_independent dim_span_eq_card_independent)
```
```   782
```
```   783 lemmas independent_imp_finite = finiteI_independent
```
```   784
```
```   785 corollary
```
```   786   fixes S :: "'a::euclidean_space set"
```
```   787   assumes "independent S"
```
```   788   shows independent_card_le:"card S \<le> DIM('a)"
```
```   789   using assms independent_bound by auto
```
```   790
```
```   791 lemma dependent_biggerset:
```
```   792   fixes S :: "'a::euclidean_space set"
```
```   793   shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
```
```   794   by (metis independent_bound not_less)
```
```   795
```
```   796 text \<open>Picking an orthogonal replacement for a spanning set.\<close>
```
```   797
```
```   798 lemma vector_sub_project_orthogonal:
```
```   799   fixes b x :: "'a::euclidean_space"
```
```   800   shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
```
```   801   unfolding inner_simps by auto
```
```   802
```
```   803 lemma pairwise_orthogonal_insert:
```
```   804   assumes "pairwise orthogonal S"
```
```   805     and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
```
```   806   shows "pairwise orthogonal (insert x S)"
```
```   807   using assms unfolding pairwise_def
```
```   808   by (auto simp add: orthogonal_commute)
```
```   809
```
```   810 lemma basis_orthogonal:
```
```   811   fixes B :: "'a::real_inner set"
```
```   812   assumes fB: "finite B"
```
```   813   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
```
```   814   (is " \<exists>C. ?P B C")
```
```   815   using fB
```
```   816 proof (induct rule: finite_induct)
```
```   817   case empty
```
```   818   then show ?case
```
```   819     apply (rule exI[where x="{}"])
```
```   820     apply (auto simp add: pairwise_def)
```
```   821     done
```
```   822 next
```
```   823   case (insert a B)
```
```   824   note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
```
```   825   from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
```
```   826   obtain C where C: "finite C" "card C \<le> card B"
```
```   827     "span C = span B" "pairwise orthogonal C" by blast
```
```   828   let ?a = "a - sum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
```
```   829   let ?C = "insert ?a C"
```
```   830   from C(1) have fC: "finite ?C"
```
```   831     by simp
```
```   832   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
```
```   833     by (simp add: card_insert_if)
```
```   834   {
```
```   835     fix x k
```
```   836     have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
```
```   837       by (simp add: field_simps)
```
```   838     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"
```
```   839       apply (simp only: scaleR_right_diff_distrib th0)
```
```   840       apply (rule span_add_eq)
```
```   841       apply (rule span_scale)
```
```   842       apply (rule span_sum)
```
```   843       apply (rule span_scale)
```
```   844       apply (rule span_base)
```
```   845       apply assumption
```
```   846       done
```
```   847   }
```
```   848   then have SC: "span ?C = span (insert a B)"
```
```   849     unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
```
```   850   {
```
```   851     fix y
```
```   852     assume yC: "y \<in> C"
```
```   853     then have Cy: "C = insert y (C - {y})"
```
```   854       by blast
```
```   855     have fth: "finite (C - {y})"
```
```   856       using C by simp
```
```   857     have "orthogonal ?a y"
```
```   858       unfolding orthogonal_def
```
```   859       unfolding inner_diff inner_sum_left right_minus_eq
```
```   860       unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
```
```   861       apply (clarsimp simp add: inner_commute[of y a])
```
```   862       apply (rule sum.neutral)
```
```   863       apply clarsimp
```
```   864       apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
```
```   865       using \<open>y \<in> C\<close> by auto
```
```   866   }
```
```   867   with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
```
```   868     by (rule pairwise_orthogonal_insert)
```
```   869   from fC cC SC CPO have "?P (insert a B) ?C"
```
```   870     by blast
```
```   871   then show ?case by blast
```
```   872 qed
```
```   873
```
```   874 lemma orthogonal_basis_exists:
```
```   875   fixes V :: "('a::euclidean_space) set"
```
```   876   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and>
```
```   877   (card B = dim V) \<and> pairwise orthogonal B"
```
```   878 proof -
```
```   879   from basis_exists[of V] obtain B where
```
```   880     B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
```
```   881     by force
```
```   882   from B have fB: "finite B" "card B = dim V"
```
```   883     using independent_bound by auto
```
```   884   from basis_orthogonal[OF fB(1)] obtain C where
```
```   885     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
```
```   886     by blast
```
```   887   from C B have CSV: "C \<subseteq> span V"
```
```   888     by (metis span_superset span_mono subset_trans)
```
```   889   from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
```
```   890     by (simp add: span_span)
```
```   891   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
```
```   892   have iC: "independent C"
```
```   893     by (simp add: dim_span)
```
```   894   from C fB have "card C \<le> dim V"
```
```   895     by simp
```
```   896   moreover have "dim V \<le> card C"
```
```   897     using span_card_ge_dim[OF CSV SVC C(1)]
```
```   898     by simp
```
```   899   ultimately have CdV: "card C = dim V"
```
```   900     using C(1) by simp
```
```   901   from C B CSV CdV iC show ?thesis
```
```   902     by auto
```
```   903 qed
```
```   904
```
```   905 text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
```
```   906
```
```   907 lemma span_not_univ_orthogonal:
```
```   908   fixes S :: "'a::euclidean_space set"
```
```   909   assumes sU: "span S \<noteq> UNIV"
```
```   910   shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
```
```   911 proof -
```
```   912   from sU obtain a where a: "a \<notin> span S"
```
```   913     by blast
```
```   914   from orthogonal_basis_exists obtain B where
```
```   915     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B"
```
```   916     "card B = dim S" "pairwise orthogonal B"
```
```   917     by blast
```
```   918   from B have fB: "finite B" "card B = dim S"
```
```   919     using independent_bound by auto
```
```   920   from span_mono[OF B(2)] span_mono[OF B(3)]
```
```   921   have sSB: "span S = span B"
```
```   922     by (simp add: span_span)
```
```   923   let ?a = "a - sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
```
```   924   have "sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
```
```   925     unfolding sSB
```
```   926     apply (rule span_sum)
```
```   927     apply (rule span_scale)
```
```   928     apply (rule span_base)
```
```   929     apply assumption
```
```   930     done
```
```   931   with a have a0:"?a  \<noteq> 0"
```
```   932     by auto
```
```   933   have "?a \<bullet> x = 0" if "x\<in>span B" for x
```
```   934   proof (rule span_induct [OF that])
```
```   935     show "subspace {x. ?a \<bullet> x = 0}"
```
```   936       by (auto simp add: subspace_def inner_add)
```
```   937   next
```
```   938     {
```
```   939       fix x
```
```   940       assume x: "x \<in> B"
```
```   941       from x have B': "B = insert x (B - {x})"
```
```   942         by blast
```
```   943       have fth: "finite (B - {x})"
```
```   944         using fB by simp
```
```   945       have "?a \<bullet> x = 0"
```
```   946         apply (subst B')
```
```   947         using fB fth
```
```   948         unfolding sum_clauses(2)[OF fth]
```
```   949         apply simp unfolding inner_simps
```
```   950         apply (clarsimp simp add: inner_add inner_sum_left)
```
```   951         apply (rule sum.neutral, rule ballI)
```
```   952         apply (simp only: inner_commute)
```
```   953         apply (auto simp add: x field_simps
```
```   954           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
```
```   955         done
```
```   956     }
```
```   957     then show "?a \<bullet> x = 0" if "x \<in> B" for x
```
```   958       using that by blast
```
```   959     qed
```
```   960   with a0 show ?thesis
```
```   961     unfolding sSB by (auto intro: exI[where x="?a"])
```
```   962 qed
```
```   963
```
```   964 lemma span_not_univ_subset_hyperplane:
```
```   965   fixes S :: "'a::euclidean_space set"
```
```   966   assumes SU: "span S \<noteq> UNIV"
```
```   967   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```   968   using span_not_univ_orthogonal[OF SU] by auto
```
```   969
```
```   970 lemma lowdim_subset_hyperplane:
```
```   971   fixes S :: "'a::euclidean_space set"
```
```   972   assumes d: "dim S < DIM('a)"
```
```   973   shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```   974 proof -
```
```   975   {
```
```   976     assume "span S = UNIV"
```
```   977     then have "dim (span S) = dim (UNIV :: ('a) set)"
```
```   978       by simp
```
```   979     then have "dim S = DIM('a)"
```
```   980       by (metis Euclidean_Space.dim_UNIV dim_span)
```
```   981     with d have False by arith
```
```   982   }
```
```   983   then have th: "span S \<noteq> UNIV"
```
```   984     by blast
```
```   985   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
```
```   986 qed
```
```   987
```
```   988 lemma linear_eq_stdbasis:
```
```   989   fixes f :: "'a::euclidean_space \<Rightarrow> _"
```
```   990   assumes lf: "linear f"
```
```   991     and lg: "linear g"
```
```   992     and fg: "\<And>b. b \<in> Basis \<Longrightarrow> f b = g b"
```
```   993   shows "f = g"
```
```   994   using linear_eq_on_span[OF lf lg, of Basis] fg
```
```   995   by auto
```
```   996
```
```   997
```
```   998 text \<open>Similar results for bilinear functions.\<close>
```
```   999
```
```  1000 lemma bilinear_eq:
```
```  1001   assumes bf: "bilinear f"
```
```  1002     and bg: "bilinear g"
```
```  1003     and SB: "S \<subseteq> span B"
```
```  1004     and TC: "T \<subseteq> span C"
```
```  1005     and "x\<in>S" "y\<in>T"
```
```  1006     and fg: "\<And>x y. \<lbrakk>x \<in> B; y\<in> C\<rbrakk> \<Longrightarrow> f x y = g x y"
```
```  1007   shows "f x y = g x y"
```
```  1008 proof -
```
```  1009   let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
```
```  1010   from bf bg have sp: "subspace ?P"
```
```  1011     unfolding bilinear_def linear_iff subspace_def bf bg
```
```  1012     by (auto simp add: span_zero bilinear_lzero[OF bf] bilinear_lzero[OF bg]
```
```  1013         span_add Ball_def
```
```  1014       intro: bilinear_ladd[OF bf])
```
```  1015   have sfg: "\<And>x. x \<in> B \<Longrightarrow> subspace {a. f x a = g x a}"
```
```  1016     apply (auto simp add: subspace_def)
```
```  1017     using bf bg unfolding bilinear_def linear_iff
```
```  1018       apply (auto simp add: span_zero bilinear_rzero[OF bf] bilinear_rzero[OF bg]
```
```  1019         span_add Ball_def
```
```  1020       intro: bilinear_ladd[OF bf])
```
```  1021     done
```
```  1022   have "\<forall>y\<in> span C. f x y = g x y" if "x \<in> span B" for x
```
```  1023     apply (rule span_induct [OF that sp])
```
```  1024     using fg sfg span_induct by blast
```
```  1025   then show ?thesis
```
```  1026     using SB TC assms by auto
```
```  1027 qed
```
```  1028
```
```  1029 lemma bilinear_eq_stdbasis:
```
```  1030   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
```
```  1031   assumes bf: "bilinear f"
```
```  1032     and bg: "bilinear g"
```
```  1033     and fg: "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> f i j = g i j"
```
```  1034   shows "f = g"
```
```  1035   using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis]] fg by blast
```
```  1036
```
```  1037
```
```  1038 subsection \<open>Infinity norm\<close>
```
```  1039
```
```  1040 definition%important "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
```
```  1041
```
```  1042 lemma infnorm_set_image:
```
```  1043   fixes x :: "'a::euclidean_space"
```
```  1044   shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```  1045   by blast
```
```  1046
```
```  1047 lemma infnorm_Max:
```
```  1048   fixes x :: "'a::euclidean_space"
```
```  1049   shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)"
```
```  1050   by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)
```
```  1051
```
```  1052 lemma infnorm_set_lemma:
```
```  1053   fixes x :: "'a::euclidean_space"
```
```  1054   shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
```
```  1055     and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}"
```
```  1056   unfolding infnorm_set_image
```
```  1057   by auto
```
```  1058
```
```  1059 lemma infnorm_pos_le:
```
```  1060   fixes x :: "'a::euclidean_space"
```
```  1061   shows "0 \<le> infnorm x"
```
```  1062   by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
```
```  1063
```
```  1064 lemma infnorm_triangle:
```
```  1065   fixes x :: "'a::euclidean_space"
```
```  1066   shows "infnorm (x + y) \<le> infnorm x + infnorm y"
```
```  1067 proof -
```
```  1068   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"
```
```  1069     by simp
```
```  1070   show ?thesis
```
```  1071     by (auto simp: infnorm_Max inner_add_left intro!: *)
```
```  1072 qed
```
```  1073
```
```  1074 lemma infnorm_eq_0:
```
```  1075   fixes x :: "'a::euclidean_space"
```
```  1076   shows "infnorm x = 0 \<longleftrightarrow> x = 0"
```
```  1077 proof -
```
```  1078   have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
```
```  1079     unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
```
```  1080   then show ?thesis
```
```  1081     using infnorm_pos_le[of x] by simp
```
```  1082 qed
```
```  1083
```
```  1084 lemma infnorm_0: "infnorm 0 = 0"
```
```  1085   by (simp add: infnorm_eq_0)
```
```  1086
```
```  1087 lemma infnorm_neg: "infnorm (- x) = infnorm x"
```
```  1088   unfolding infnorm_def by simp
```
```  1089
```
```  1090 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
```
```  1091   by (metis infnorm_neg minus_diff_eq)
```
```  1092
```
```  1093 lemma absdiff_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
```
```  1094 proof -
```
```  1095   have *: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
```
```  1096     by arith
```
```  1097   show ?thesis
```
```  1098   proof (rule *)
```
```  1099     from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
```
```  1100     show "infnorm x \<le> infnorm (x - y) + infnorm y" "infnorm y \<le> infnorm (x - y) + infnorm x"
```
```  1101       by (simp_all add: field_simps infnorm_neg)
```
```  1102   qed
```
```  1103 qed
```
```  1104
```
```  1105 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
```
```  1106   using infnorm_pos_le[of x] by arith
```
```  1107
```
```  1108 lemma Basis_le_infnorm:
```
```  1109   fixes x :: "'a::euclidean_space"
```
```  1110   shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
```
```  1111   by (simp add: infnorm_Max)
```
```  1112
```
```  1113 lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
```
```  1114   unfolding infnorm_Max
```
```  1115 proof (safe intro!: Max_eqI)
```
```  1116   let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```  1117   { fix b :: 'a
```
```  1118     assume "b \<in> Basis"
```
```  1119     then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
```
```  1120       by (simp add: abs_mult mult_left_mono)
```
```  1121   next
```
```  1122     from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
```
```  1123       by (auto simp del: Max_in)
```
```  1124     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"
```
```  1125       by (intro image_eqI[where x=b]) (auto simp: abs_mult)
```
```  1126   }
```
```  1127 qed simp
```
```  1128
```
```  1129 lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
```
```  1130   unfolding infnorm_mul ..
```
```  1131
```
```  1132 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
```
```  1133   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
```
```  1134
```
```  1135 text \<open>Prove that it differs only up to a bound from Euclidean norm.\<close>
```
```  1136
```
```  1137 lemma infnorm_le_norm: "infnorm x \<le> norm x"
```
```  1138   by (simp add: Basis_le_norm infnorm_Max)
```
```  1139
```
```  1140 lemma norm_le_infnorm:
```
```  1141   fixes x :: "'a::euclidean_space"
```
```  1142   shows "norm x \<le> sqrt DIM('a) * infnorm x"
```
```  1143   unfolding norm_eq_sqrt_inner id_def
```
```  1144 proof (rule real_le_lsqrt[OF inner_ge_zero])
```
```  1145   show "sqrt DIM('a) * infnorm x \<ge> 0"
```
```  1146     by (simp add: zero_le_mult_iff infnorm_pos_le)
```
```  1147   have "x \<bullet> x \<le> (\<Sum>b\<in>Basis. x \<bullet> b * (x \<bullet> b))"
```
```  1148     by (metis euclidean_inner order_refl)
```
```  1149   also have "... \<le> DIM('a) * \<bar>infnorm x\<bar>\<^sup>2"
```
```  1150     by (rule sum_bounded_above) (metis Basis_le_infnorm abs_le_square_iff power2_eq_square real_abs_infnorm)
```
```  1151   also have "... \<le> (sqrt DIM('a) * infnorm x)\<^sup>2"
```
```  1152     by (simp add: power_mult_distrib)
```
```  1153   finally show "x \<bullet> x \<le> (sqrt DIM('a) * infnorm x)\<^sup>2" .
```
```  1154 qed
```
```  1155
```
```  1156 lemma tendsto_infnorm [tendsto_intros]:
```
```  1157   assumes "(f \<longlongrightarrow> a) F"
```
```  1158   shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F"
```
```  1159 proof (rule tendsto_compose [OF LIM_I assms])
```
```  1160   fix r :: real
```
```  1161   assume "r > 0"
```
```  1162   then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
```
```  1163     by (metis real_norm_def le_less_trans absdiff_infnorm infnorm_le_norm)
```
```  1164 qed
```
```  1165
```
```  1166 text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close>
```
```  1167
```
```  1168 lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  1169   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1170 proof (cases "x=0")
```
```  1171   case True
```
```  1172   then show ?thesis
```
```  1173     by auto
```
```  1174 next
```
```  1175   case False
```
```  1176   from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
```
```  1177   have "?rhs \<longleftrightarrow>
```
```  1178       (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
```
```  1179         norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
```
```  1180     using False unfolding inner_simps
```
```  1181     by (auto simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
```
```  1182   also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)"
```
```  1183     using False  by (simp add: field_simps inner_commute)
```
```  1184   also have "\<dots> \<longleftrightarrow> ?lhs"
```
```  1185     using False by auto
```
```  1186   finally show ?thesis by metis
```
```  1187 qed
```
```  1188
```
```  1189 lemma norm_cauchy_schwarz_abs_eq:
```
```  1190   "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow>
```
```  1191     norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
```
```  1192   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1193 proof -
```
```  1194   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a"
```
```  1195     by arith
```
```  1196   have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
```
```  1197     by simp
```
```  1198   also have "\<dots> \<longleftrightarrow> (x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
```
```  1199     unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  1200     unfolding norm_minus_cancel norm_scaleR ..
```
```  1201   also have "\<dots> \<longleftrightarrow> ?lhs"
```
```  1202     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
```
```  1203     by auto
```
```  1204   finally show ?thesis ..
```
```  1205 qed
```
```  1206
```
```  1207 lemma norm_triangle_eq:
```
```  1208   fixes x y :: "'a::real_inner"
```
```  1209   shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  1210 proof (cases "x = 0 \<or> y = 0")
```
```  1211   case True
```
```  1212   then show ?thesis
```
```  1213     by force
```
```  1214 next
```
```  1215   case False
```
```  1216   then have n: "norm x > 0" "norm y > 0"
```
```  1217     by auto
```
```  1218   have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
```
```  1219     by simp
```
```  1220   also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  1221     unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  1222     unfolding power2_norm_eq_inner inner_simps
```
```  1223     by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
```
```  1224   finally show ?thesis .
```
```  1225 qed
```
```  1226
```
```  1227
```
```  1228 subsection \<open>Collinearity\<close>
```
```  1229
```
```  1230 definition%important collinear :: "'a::real_vector set \<Rightarrow> bool"
```
```  1231   where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
```
```  1232
```
```  1233 lemma collinear_alt:
```
```  1234      "collinear S \<longleftrightarrow> (\<exists>u v. \<forall>x \<in> S. \<exists>c. x = u + c *\<^sub>R v)" (is "?lhs = ?rhs")
```
```  1235 proof
```
```  1236   assume ?lhs
```
```  1237   then show ?rhs
```
```  1238     unfolding collinear_def by (metis Groups.add_ac(2) diff_add_cancel)
```
```  1239 next
```
```  1240   assume ?rhs
```
```  1241   then obtain u v where *: "\<And>x. x \<in> S \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
```
```  1242     by (auto simp: )
```
```  1243   have "\<exists>c. x - y = c *\<^sub>R v" if "x \<in> S" "y \<in> S" for x y
```
```  1244         by (metis *[OF \<open>x \<in> S\<close>] *[OF \<open>y \<in> S\<close>] scaleR_left.diff add_diff_cancel_left)
```
```  1245   then show ?lhs
```
```  1246     using collinear_def by blast
```
```  1247 qed
```
```  1248
```
```  1249 lemma collinear:
```
```  1250   fixes S :: "'a::{perfect_space,real_vector} set"
```
```  1251   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))"
```
```  1252 proof -
```
```  1253   have "\<exists>v. v \<noteq> 0 \<and> (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v)"
```
```  1254     if "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R u" "u=0" for u
```
```  1255   proof -
```
```  1256     have "\<forall>x\<in>S. \<forall>y\<in>S. x = y"
```
```  1257       using that by auto
```
```  1258     moreover
```
```  1259     obtain v::'a where "v \<noteq> 0"
```
```  1260       using UNIV_not_singleton [of 0] by auto
```
```  1261     ultimately have "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v"
```
```  1262       by auto
```
```  1263     then show ?thesis
```
```  1264       using \<open>v \<noteq> 0\<close> by blast
```
```  1265   qed
```
```  1266   then show ?thesis
```
```  1267     apply (clarsimp simp: collinear_def)
```
```  1268     by (metis scaleR_zero_right vector_fraction_eq_iff)
```
```  1269 qed
```
```  1270
```
```  1271 lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S"
```
```  1272   by (meson collinear_def subsetCE)
```
```  1273
```
```  1274 lemma collinear_empty [iff]: "collinear {}"
```
```  1275   by (simp add: collinear_def)
```
```  1276
```
```  1277 lemma collinear_sing [iff]: "collinear {x}"
```
```  1278   by (simp add: collinear_def)
```
```  1279
```
```  1280 lemma collinear_2 [iff]: "collinear {x, y}"
```
```  1281   apply (simp add: collinear_def)
```
```  1282   apply (rule exI[where x="x - y"])
```
```  1283   by (metis minus_diff_eq scaleR_left.minus scaleR_one)
```
```  1284
```
```  1285 lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
```
```  1286   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1287 proof (cases "x = 0 \<or> y = 0")
```
```  1288   case True
```
```  1289   then show ?thesis
```
```  1290     by (auto simp: insert_commute)
```
```  1291 next
```
```  1292   case False
```
```  1293   show ?thesis
```
```  1294   proof
```
```  1295     assume h: "?lhs"
```
```  1296     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"
```
```  1297       unfolding collinear_def by blast
```
```  1298     from u[rule_format, of x 0] u[rule_format, of y 0]
```
```  1299     obtain cx and cy where
```
```  1300       cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
```
```  1301       by auto
```
```  1302     from cx cy False have cx0: "cx \<noteq> 0" and cy0: "cy \<noteq> 0" by auto
```
```  1303     let ?d = "cy / cx"
```
```  1304     from cx cy cx0 have "y = ?d *\<^sub>R x"
```
```  1305       by simp
```
```  1306     then show ?rhs using False by blast
```
```  1307   next
```
```  1308     assume h: "?rhs"
```
```  1309     then obtain c where c: "y = c *\<^sub>R x"
```
```  1310       using False by blast
```
```  1311     show ?lhs
```
```  1312       unfolding collinear_def c
```
```  1313       apply (rule exI[where x=x])
```
```  1314       apply auto
```
```  1315           apply (rule exI[where x="- 1"], simp)
```
```  1316          apply (rule exI[where x= "-c"], simp)
```
```  1317         apply (rule exI[where x=1], simp)
```
```  1318        apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
```
```  1319       apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
```
```  1320       done
```
```  1321   qed
```
```  1322 qed
```
```  1323
```
```  1324 lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
```
```  1325 proof (cases "x=0")
```
```  1326   case True
```
```  1327   then show ?thesis
```
```  1328     by (auto simp: insert_commute)
```
```  1329 next
```
```  1330   case False
```
```  1331   then have nnz: "norm x \<noteq> 0"
```
```  1332     by auto
```
```  1333   show ?thesis
```
```  1334   proof
```
```  1335     assume "\<bar>x \<bullet> y\<bar> = norm x * norm y"
```
```  1336     then show "collinear {0, x, y}"
```
```  1337       unfolding norm_cauchy_schwarz_abs_eq collinear_lemma
```
```  1338       by (meson eq_vector_fraction_iff nnz)
```
```  1339   next
```
```  1340     assume "collinear {0, x, y}"
```
```  1341     with False show "\<bar>x \<bullet> y\<bar> = norm x * norm y"
```
```  1342       unfolding norm_cauchy_schwarz_abs_eq collinear_lemma  by (auto simp: abs_if)
```
```  1343   qed
```
```  1344 qed
```
```  1345
```
```  1346
```
```  1347 subsection\<open>Properties of special hyperplanes\<close>
```
```  1348
```
```  1349 lemma subspace_hyperplane: "subspace {x. a \<bullet> x = 0}"
```
```  1350   by (simp add: subspace_def inner_right_distrib)
```
```  1351
```
```  1352 lemma subspace_hyperplane2: "subspace {x. x \<bullet> a = 0}"
```
```  1353   by (simp add: inner_commute inner_right_distrib subspace_def)
```
```  1354
```
```  1355 lemma special_hyperplane_span:
```
```  1356   fixes S :: "'n::euclidean_space set"
```
```  1357   assumes "k \<in> Basis"
```
```  1358   shows "{x. k \<bullet> x = 0} = span (Basis - {k})"
```
```  1359 proof -
```
```  1360   have *: "x \<in> span (Basis - {k})" if "k \<bullet> x = 0" for x
```
```  1361   proof -
```
```  1362     have "x = (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b)"
```
```  1363       by (simp add: euclidean_representation)
```
```  1364     also have "... = (\<Sum>b \<in> Basis - {k}. (x \<bullet> b) *\<^sub>R b)"
```
```  1365       by (auto simp: sum.remove [of _ k] inner_commute assms that)
```
```  1366     finally have "x = (\<Sum>b\<in>Basis - {k}. (x \<bullet> b) *\<^sub>R b)" .
```
```  1367     then show ?thesis
```
```  1368       by (simp add: span_finite)
```
```  1369   qed
```
```  1370   show ?thesis
```
```  1371     apply (rule span_subspace [symmetric])
```
```  1372     using assms
```
```  1373     apply (auto simp: inner_not_same_Basis intro: * subspace_hyperplane)
```
```  1374     done
```
```  1375 qed
```
```  1376
```
```  1377 lemma dim_special_hyperplane:
```
```  1378   fixes k :: "'n::euclidean_space"
```
```  1379   shows "k \<in> Basis \<Longrightarrow> dim {x. k \<bullet> x = 0} = DIM('n) - 1"
```
```  1380 apply (simp add: special_hyperplane_span)
```
```  1381 apply (rule dim_unique [OF subset_refl])
```
```  1382 apply (auto simp: independent_substdbasis)
```
```  1383 apply (metis member_remove remove_def span_base)
```
```  1384 done
```
```  1385
```
```  1386 proposition dim_hyperplane:
```
```  1387   fixes a :: "'a::euclidean_space"
```
```  1388   assumes "a \<noteq> 0"
```
```  1389     shows "dim {x. a \<bullet> x = 0} = DIM('a) - 1"
```
```  1390 proof -
```
```  1391   have span0: "span {x. a \<bullet> x = 0} = {x. a \<bullet> x = 0}"
```
```  1392     by (rule span_unique) (auto simp: subspace_hyperplane)
```
```  1393   then obtain B where "independent B"
```
```  1394               and Bsub: "B \<subseteq> {x. a \<bullet> x = 0}"
```
```  1395               and subspB: "{x. a \<bullet> x = 0} \<subseteq> span B"
```
```  1396               and card0: "(card B = dim {x. a \<bullet> x = 0})"
```
```  1397               and ortho: "pairwise orthogonal B"
```
```  1398     using orthogonal_basis_exists by metis
```
```  1399   with assms have "a \<notin> span B"
```
```  1400     by (metis (mono_tags, lifting) span_eq inner_eq_zero_iff mem_Collect_eq span0)
```
```  1401   then have ind: "independent (insert a B)"
```
```  1402     by (simp add: \<open>independent B\<close> independent_insert)
```
```  1403   have "finite B"
```
```  1404     using \<open>independent B\<close> independent_bound by blast
```
```  1405   have "UNIV \<subseteq> span (insert a B)"
```
```  1406   proof fix y::'a
```
```  1407     obtain r z where z: "y = r *\<^sub>R a + z" "a \<bullet> z = 0"
```
```  1408       apply (rule_tac r="(a \<bullet> y) / (a \<bullet> a)" and z = "y - ((a \<bullet> y) / (a \<bullet> a)) *\<^sub>R a" in that)
```
```  1409       using assms
```
```  1410       by (auto simp: algebra_simps)
```
```  1411     show "y \<in> span (insert a B)"
```
```  1412       by (metis (mono_tags, lifting) z Bsub span_eq_iff
```
```  1413          add_diff_cancel_left' mem_Collect_eq span0 span_breakdown_eq span_subspace subspB)
```
```  1414   qed
```
```  1415   then have dima: "DIM('a) = dim(insert a B)"
```
```  1416     by (metis independent_Basis span_Basis dim_eq_card top.extremum_uniqueI)
```
```  1417   then show ?thesis
```
```  1418     by (metis (mono_tags, lifting) Bsub Diff_insert_absorb \<open>a \<notin> span B\<close> ind card0
```
```  1419         card_Diff_singleton dim_span indep_card_eq_dim_span insertI1 subsetCE
```
```  1420         subspB)
```
```  1421 qed
```
```  1422
```
```  1423 lemma lowdim_eq_hyperplane:
```
```  1424   fixes S :: "'a::euclidean_space set"
```
```  1425   assumes "dim S = DIM('a) - 1"
```
```  1426   obtains a where "a \<noteq> 0" and "span S = {x. a \<bullet> x = 0}"
```
```  1427 proof -
```
```  1428   have dimS: "dim S < DIM('a)"
```
```  1429     by (simp add: assms)
```
```  1430   then obtain b where b: "b \<noteq> 0" "span S \<subseteq> {a. b \<bullet> a = 0}"
```
```  1431     using lowdim_subset_hyperplane [of S] by fastforce
```
```  1432   show ?thesis
```
```  1433     apply (rule that[OF b(1)])
```
```  1434     apply (rule subspace_dim_equal)
```
```  1435     by (auto simp: assms b dim_hyperplane dim_span subspace_hyperplane
```
```  1436         subspace_span)
```
```  1437 qed
```
```  1438
```
```  1439 lemma dim_eq_hyperplane:
```
```  1440   fixes S :: "'n::euclidean_space set"
```
```  1441   shows "dim S = DIM('n) - 1 \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> span S = {x. a \<bullet> x = 0})"
```
```  1442 by (metis One_nat_def dim_hyperplane dim_span lowdim_eq_hyperplane)
```
```  1443
```
```  1444
```
```  1445 subsection\<open> Orthogonal bases, Gram-Schmidt process, and related theorems\<close>
```
```  1446
```
```  1447 lemma pairwise_orthogonal_independent:
```
```  1448   assumes "pairwise orthogonal S" and "0 \<notin> S"
```
```  1449     shows "independent S"
```
```  1450 proof -
```
```  1451   have 0: "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
```
```  1452     using assms by (simp add: pairwise_def orthogonal_def)
```
```  1453   have "False" if "a \<in> S" and a: "a \<in> span (S - {a})" for a
```
```  1454   proof -
```
```  1455     obtain T U where "T \<subseteq> S - {a}" "a = (\<Sum>v\<in>T. U v *\<^sub>R v)"
```
```  1456       using a by (force simp: span_explicit)
```
```  1457     then have "a \<bullet> a = a \<bullet> (\<Sum>v\<in>T. U v *\<^sub>R v)"
```
```  1458       by simp
```
```  1459     also have "... = 0"
```
```  1460       apply (simp add: inner_sum_right)
```
```  1461       apply (rule comm_monoid_add_class.sum.neutral)
```
```  1462       by (metis "0" DiffE \<open>T \<subseteq> S - {a}\<close> mult_not_zero singletonI subsetCE \<open>a \<in> S\<close>)
```
```  1463     finally show ?thesis
```
```  1464       using \<open>0 \<notin> S\<close> \<open>a \<in> S\<close> by auto
```
```  1465   qed
```
```  1466   then show ?thesis
```
```  1467     by (force simp: dependent_def)
```
```  1468 qed
```
```  1469
```
```  1470 lemma pairwise_orthogonal_imp_finite:
```
```  1471   fixes S :: "'a::euclidean_space set"
```
```  1472   assumes "pairwise orthogonal S"
```
```  1473     shows "finite S"
```
```  1474 proof -
```
```  1475   have "independent (S - {0})"
```
```  1476     apply (rule pairwise_orthogonal_independent)
```
```  1477      apply (metis Diff_iff assms pairwise_def)
```
```  1478     by blast
```
```  1479   then show ?thesis
```
```  1480     by (meson independent_imp_finite infinite_remove)
```
```  1481 qed
```
```  1482
```
```  1483 lemma subspace_orthogonal_to_vector: "subspace {y. orthogonal x y}"
```
```  1484   by (simp add: subspace_def orthogonal_clauses)
```
```  1485
```
```  1486 lemma subspace_orthogonal_to_vectors: "subspace {y. \<forall>x \<in> S. orthogonal x y}"
```
```  1487   by (simp add: subspace_def orthogonal_clauses)
```
```  1488
```
```  1489 lemma orthogonal_to_span:
```
```  1490   assumes a: "a \<in> span S" and x: "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
```
```  1491     shows "orthogonal x a"
```
```  1492   by (metis a orthogonal_clauses(1,2,4)
```
```  1493       span_induct_alt x)
```
```  1494
```
```  1495 proposition Gram_Schmidt_step:
```
```  1496   fixes S :: "'a::euclidean_space set"
```
```  1497   assumes S: "pairwise orthogonal S" and x: "x \<in> span S"
```
```  1498     shows "orthogonal x (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b))"
```
```  1499 proof -
```
```  1500   have "finite S"
```
```  1501     by (simp add: S pairwise_orthogonal_imp_finite)
```
```  1502   have "orthogonal (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)) x"
```
```  1503        if "x \<in> S" for x
```
```  1504   proof -
```
```  1505     have "a \<bullet> x = (\<Sum>y\<in>S. if y = x then y \<bullet> a else 0)"
```
```  1506       by (simp add: \<open>finite S\<close> inner_commute sum.delta that)
```
```  1507     also have "... =  (\<Sum>b\<in>S. b \<bullet> a * (b \<bullet> x) / (b \<bullet> b))"
```
```  1508       apply (rule sum.cong [OF refl], simp)
```
```  1509       by (meson S orthogonal_def pairwise_def that)
```
```  1510    finally show ?thesis
```
```  1511      by (simp add: orthogonal_def algebra_simps inner_sum_left)
```
```  1512   qed
```
```  1513   then show ?thesis
```
```  1514     using orthogonal_to_span orthogonal_commute x by blast
```
```  1515 qed
```
```  1516
```
```  1517
```
```  1518 lemma orthogonal_extension_aux:
```
```  1519   fixes S :: "'a::euclidean_space set"
```
```  1520   assumes "finite T" "finite S" "pairwise orthogonal S"
```
```  1521     shows "\<exists>U. pairwise orthogonal (S \<union> U) \<and> span (S \<union> U) = span (S \<union> T)"
```
```  1522 using assms
```
```  1523 proof (induction arbitrary: S)
```
```  1524   case empty then show ?case
```
```  1525     by simp (metis sup_bot_right)
```
```  1526 next
```
```  1527   case (insert a T)
```
```  1528   have 0: "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
```
```  1529     using insert by (simp add: pairwise_def orthogonal_def)
```
```  1530   define a' where "a' = a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)"
```
```  1531   obtain U where orthU: "pairwise orthogonal (S \<union> insert a' U)"
```
```  1532              and spanU: "span (insert a' S \<union> U) = span (insert a' S \<union> T)"
```
```  1533     by (rule exE [OF insert.IH [of "insert a' S"]])
```
```  1534       (auto simp: Gram_Schmidt_step a'_def insert.prems orthogonal_commute
```
```  1535         pairwise_orthogonal_insert span_clauses)
```
```  1536   have orthS: "\<And>x. x \<in> S \<Longrightarrow> a' \<bullet> x = 0"
```
```  1537     apply (simp add: a'_def)
```
```  1538     using Gram_Schmidt_step [OF \<open>pairwise orthogonal S\<close>]
```
```  1539     apply (force simp: orthogonal_def inner_commute span_superset [THEN subsetD])
```
```  1540     done
```
```  1541   have "span (S \<union> insert a' U) = span (insert a' (S \<union> T))"
```
```  1542     using spanU by simp
```
```  1543   also have "... = span (insert a (S \<union> T))"
```
```  1544     apply (rule eq_span_insert_eq)
```
```  1545     apply (simp add: a'_def span_neg span_sum span_base span_mul)
```
```  1546     done
```
```  1547   also have "... = span (S \<union> insert a T)"
```
```  1548     by simp
```
```  1549   finally show ?case
```
```  1550     by (rule_tac x="insert a' U" in exI) (use orthU in auto)
```
```  1551 qed
```
```  1552
```
```  1553
```
```  1554 proposition orthogonal_extension:
```
```  1555   fixes S :: "'a::euclidean_space set"
```
```  1556   assumes S: "pairwise orthogonal S"
```
```  1557   obtains U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
```
```  1558 proof -
```
```  1559   obtain B where "finite B" "span B = span T"
```
```  1560     using basis_subspace_exists [of "span T"] subspace_span by metis
```
```  1561   with orthogonal_extension_aux [of B S]
```
```  1562   obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> B)"
```
```  1563     using assms pairwise_orthogonal_imp_finite by auto
```
```  1564   with \<open>span B = span T\<close> show ?thesis
```
```  1565     by (rule_tac U=U in that) (auto simp: span_Un)
```
```  1566 qed
```
```  1567
```
```  1568 corollary%unimportant orthogonal_extension_strong:
```
```  1569   fixes S :: "'a::euclidean_space set"
```
```  1570   assumes S: "pairwise orthogonal S"
```
```  1571   obtains U where "U \<inter> (insert 0 S) = {}" "pairwise orthogonal (S \<union> U)"
```
```  1572                   "span (S \<union> U) = span (S \<union> T)"
```
```  1573 proof -
```
```  1574   obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
```
```  1575     using orthogonal_extension assms by blast
```
```  1576   then show ?thesis
```
```  1577     apply (rule_tac U = "U - (insert 0 S)" in that)
```
```  1578       apply blast
```
```  1579      apply (force simp: pairwise_def)
```
```  1580     apply (metis Un_Diff_cancel Un_insert_left span_redundant span_zero)
```
```  1581     done
```
```  1582 qed
```
```  1583
```
```  1584 subsection\<open>Decomposing a vector into parts in orthogonal subspaces\<close>
```
```  1585
```
```  1586 text\<open>existence of orthonormal basis for a subspace.\<close>
```
```  1587
```
```  1588 lemma orthogonal_spanningset_subspace:
```
```  1589   fixes S :: "'a :: euclidean_space set"
```
```  1590   assumes "subspace S"
```
```  1591   obtains B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
```
```  1592 proof -
```
```  1593   obtain B where "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
```
```  1594     using basis_exists by blast
```
```  1595   with orthogonal_extension [of "{}" B]
```
```  1596   show ?thesis
```
```  1597     by (metis Un_empty_left assms pairwise_empty span_superset span_subspace that)
```
```  1598 qed
```
```  1599
```
```  1600 lemma orthogonal_basis_subspace:
```
```  1601   fixes S :: "'a :: euclidean_space set"
```
```  1602   assumes "subspace S"
```
```  1603   obtains B where "0 \<notin> B" "B \<subseteq> S" "pairwise orthogonal B" "independent B"
```
```  1604                   "card B = dim S" "span B = S"
```
```  1605 proof -
```
```  1606   obtain B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
```
```  1607     using assms orthogonal_spanningset_subspace by blast
```
```  1608   then show ?thesis
```
```  1609     apply (rule_tac B = "B - {0}" in that)
```
```  1610     apply (auto simp: indep_card_eq_dim_span pairwise_subset pairwise_orthogonal_independent elim: pairwise_subset)
```
```  1611     done
```
```  1612 qed
```
```  1613
```
```  1614 proposition orthonormal_basis_subspace:
```
```  1615   fixes S :: "'a :: euclidean_space set"
```
```  1616   assumes "subspace S"
```
```  1617   obtains B where "B \<subseteq> S" "pairwise orthogonal B"
```
```  1618               and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
```
```  1619               and "independent B" "card B = dim S" "span B = S"
```
```  1620 proof -
```
```  1621   obtain B where "0 \<notin> B" "B \<subseteq> S"
```
```  1622              and orth: "pairwise orthogonal B"
```
```  1623              and "independent B" "card B = dim S" "span B = S"
```
```  1624     by (blast intro: orthogonal_basis_subspace [OF assms])
```
```  1625   have 1: "(\<lambda>x. x /\<^sub>R norm x) ` B \<subseteq> S"
```
```  1626     using \<open>span B = S\<close> span_superset span_mul by fastforce
```
```  1627   have 2: "pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` B)"
```
```  1628     using orth by (force simp: pairwise_def orthogonal_clauses)
```
```  1629   have 3: "\<And>x. x \<in> (\<lambda>x. x /\<^sub>R norm x) ` B \<Longrightarrow> norm x = 1"
```
```  1630     by (metis (no_types, lifting) \<open>0 \<notin> B\<close> image_iff norm_sgn sgn_div_norm)
```
```  1631   have 4: "independent ((\<lambda>x. x /\<^sub>R norm x) ` B)"
```
```  1632     by (metis "2" "3" norm_zero pairwise_orthogonal_independent zero_neq_one)
```
```  1633   have "inj_on (\<lambda>x. x /\<^sub>R norm x) B"
```
```  1634   proof
```
```  1635     fix x y
```
```  1636     assume "x \<in> B" "y \<in> B" "x /\<^sub>R norm x = y /\<^sub>R norm y"
```
```  1637     moreover have "\<And>i. i \<in> B \<Longrightarrow> norm (i /\<^sub>R norm i) = 1"
```
```  1638       using 3 by blast
```
```  1639     ultimately show "x = y"
```
```  1640       by (metis norm_eq_1 orth orthogonal_clauses(7) orthogonal_commute orthogonal_def pairwise_def zero_neq_one)
```
```  1641   qed
```
```  1642   then have 5: "card ((\<lambda>x. x /\<^sub>R norm x) ` B) = dim S"
```
```  1643     by (metis \<open>card B = dim S\<close> card_image)
```
```  1644   have 6: "span ((\<lambda>x. x /\<^sub>R norm x) ` B) = S"
```
```  1645     by (metis "1" "4" "5" assms card_eq_dim independent_imp_finite span_subspace)
```
```  1646   show ?thesis
```
```  1647     by (rule that [OF 1 2 3 4 5 6])
```
```  1648 qed
```
```  1649
```
```  1650
```
```  1651 proposition%unimportant orthogonal_to_subspace_exists_gen:
```
```  1652   fixes S :: "'a :: euclidean_space set"
```
```  1653   assumes "span S \<subset> span T"
```
```  1654   obtains x where "x \<noteq> 0" "x \<in> span T" "\<And>y. y \<in> span S \<Longrightarrow> orthogonal x y"
```
```  1655 proof -
```
```  1656   obtain B where "B \<subseteq> span S" and orthB: "pairwise orthogonal B"
```
```  1657              and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
```
```  1658              and "independent B" "card B = dim S" "span B = span S"
```
```  1659     by (rule orthonormal_basis_subspace [of "span S", OF subspace_span])
```
```  1660       (auto simp: dim_span)
```
```  1661   with assms obtain u where spanBT: "span B \<subseteq> span T" and "u \<notin> span B" "u \<in> span T"
```
```  1662     by auto
```
```  1663   obtain C where orthBC: "pairwise orthogonal (B \<union> C)" and spanBC: "span (B \<union> C) = span (B \<union> {u})"
```
```  1664     by (blast intro: orthogonal_extension [OF orthB])
```
```  1665   show thesis
```
```  1666   proof (cases "C \<subseteq> insert 0 B")
```
```  1667     case True
```
```  1668     then have "C \<subseteq> span B"
```
```  1669       using span_eq
```
```  1670       by (metis span_insert_0 subset_trans)
```
```  1671     moreover have "u \<in> span (B \<union> C)"
```
```  1672       using \<open>span (B \<union> C) = span (B \<union> {u})\<close> span_superset by force
```
```  1673     ultimately show ?thesis
```
```  1674       using True \<open>u \<notin> span B\<close>
```
```  1675       by (metis Un_insert_left span_insert_0 sup.orderE)
```
```  1676   next
```
```  1677     case False
```
```  1678     then obtain x where "x \<in> C" "x \<noteq> 0" "x \<notin> B"
```
```  1679       by blast
```
```  1680     then have "x \<in> span T"
```
```  1681       by (metis (no_types, lifting) Un_insert_right Un_upper2 \<open>u \<in> span T\<close> spanBT spanBC
```
```  1682           \<open>u \<in> span T\<close> insert_subset span_superset span_mono
```
```  1683           span_span subsetCE subset_trans sup_bot.comm_neutral)
```
```  1684     moreover have "orthogonal x y" if "y \<in> span B" for y
```
```  1685       using that
```
```  1686     proof (rule span_induct)
```
```  1687       show "subspace {a. orthogonal x a}"
```
```  1688         by (simp add: subspace_orthogonal_to_vector)
```
```  1689       show "\<And>b. b \<in> B \<Longrightarrow> orthogonal x b"
```
```  1690         by (metis Un_iff \<open>x \<in> C\<close> \<open>x \<notin> B\<close> orthBC pairwise_def)
```
```  1691     qed
```
```  1692     ultimately show ?thesis
```
```  1693       using \<open>x \<noteq> 0\<close> that \<open>span B = span S\<close> by auto
```
```  1694   qed
```
```  1695 qed
```
```  1696
```
```  1697 corollary%unimportant orthogonal_to_subspace_exists:
```
```  1698   fixes S :: "'a :: euclidean_space set"
```
```  1699   assumes "dim S < DIM('a)"
```
```  1700   obtains x where "x \<noteq> 0" "\<And>y. y \<in> span S \<Longrightarrow> orthogonal x y"
```
```  1701 proof -
```
```  1702 have "span S \<subset> UNIV"
```
```  1703   by (metis (mono_tags) UNIV_I assms inner_eq_zero_iff less_le lowdim_subset_hyperplane
```
```  1704       mem_Collect_eq top.extremum_strict top.not_eq_extremum)
```
```  1705   with orthogonal_to_subspace_exists_gen [of S UNIV] that show ?thesis
```
```  1706     by (auto simp: span_UNIV)
```
```  1707 qed
```
```  1708
```
```  1709 corollary%unimportant orthogonal_to_vector_exists:
```
```  1710   fixes x :: "'a :: euclidean_space"
```
```  1711   assumes "2 \<le> DIM('a)"
```
```  1712   obtains y where "y \<noteq> 0" "orthogonal x y"
```
```  1713 proof -
```
```  1714   have "dim {x} < DIM('a)"
```
```  1715     using assms by auto
```
```  1716   then show thesis
```
```  1717     by (rule orthogonal_to_subspace_exists) (simp add: orthogonal_commute span_base that)
```
```  1718 qed
```
```  1719
```
```  1720 proposition%unimportant orthogonal_subspace_decomp_exists:
```
```  1721   fixes S :: "'a :: euclidean_space set"
```
```  1722   obtains y z
```
```  1723   where "y \<in> span S"
```
```  1724     and "\<And>w. w \<in> span S \<Longrightarrow> orthogonal z w"
```
```  1725     and "x = y + z"
```
```  1726 proof -
```
```  1727   obtain T where "0 \<notin> T" "T \<subseteq> span S" "pairwise orthogonal T" "independent T"
```
```  1728     "card T = dim (span S)" "span T = span S"
```
```  1729     using orthogonal_basis_subspace subspace_span by blast
```
```  1730   let ?a = "\<Sum>b\<in>T. (b \<bullet> x / (b \<bullet> b)) *\<^sub>R b"
```
```  1731   have orth: "orthogonal (x - ?a) w" if "w \<in> span S" for w
```
```  1732     by (simp add: Gram_Schmidt_step \<open>pairwise orthogonal T\<close> \<open>span T = span S\<close>
```
```  1733         orthogonal_commute that)
```
```  1734   show ?thesis
```
```  1735     apply (rule_tac y = "?a" and z = "x - ?a" in that)
```
```  1736       apply (meson \<open>T \<subseteq> span S\<close> span_scale span_sum subsetCE)
```
```  1737      apply (fact orth, simp)
```
```  1738     done
```
```  1739 qed
```
```  1740
```
```  1741 lemma orthogonal_subspace_decomp_unique:
```
```  1742   fixes S :: "'a :: euclidean_space set"
```
```  1743   assumes "x + y = x' + y'"
```
```  1744       and ST: "x \<in> span S" "x' \<in> span S" "y \<in> span T" "y' \<in> span T"
```
```  1745       and orth: "\<And>a b. \<lbrakk>a \<in> S; b \<in> T\<rbrakk> \<Longrightarrow> orthogonal a b"
```
```  1746   shows "x = x' \<and> y = y'"
```
```  1747 proof -
```
```  1748   have "x + y - y' = x'"
```
```  1749     by (simp add: assms)
```
```  1750   moreover have "\<And>a b. \<lbrakk>a \<in> span S; b \<in> span T\<rbrakk> \<Longrightarrow> orthogonal a b"
```
```  1751     by (meson orth orthogonal_commute orthogonal_to_span)
```
```  1752   ultimately have "0 = x' - x"
```
```  1753     by (metis (full_types) add_diff_cancel_left' ST diff_right_commute orthogonal_clauses(10) orthogonal_clauses(5) orthogonal_self)
```
```  1754   with assms show ?thesis by auto
```
```  1755 qed
```
```  1756
```
```  1757 lemma vector_in_orthogonal_spanningset:
```
```  1758   fixes a :: "'a::euclidean_space"
```
```  1759   obtains S where "a \<in> S" "pairwise orthogonal S" "span S = UNIV"
```
```  1760   by (metis UNIV_I Un_iff empty_iff insert_subset orthogonal_extension pairwise_def
```
```  1761       pairwise_orthogonal_insert span_UNIV subsetI subset_antisym)
```
```  1762
```
```  1763 lemma vector_in_orthogonal_basis:
```
```  1764   fixes a :: "'a::euclidean_space"
```
```  1765   assumes "a \<noteq> 0"
```
```  1766   obtains S where "a \<in> S" "0 \<notin> S" "pairwise orthogonal S" "independent S" "finite S"
```
```  1767                   "span S = UNIV" "card S = DIM('a)"
```
```  1768 proof -
```
```  1769   obtain S where S: "a \<in> S" "pairwise orthogonal S" "span S = UNIV"
```
```  1770     using vector_in_orthogonal_spanningset .
```
```  1771   show thesis
```
```  1772   proof
```
```  1773     show "pairwise orthogonal (S - {0})"
```
```  1774       using pairwise_mono S(2) by blast
```
```  1775     show "independent (S - {0})"
```
```  1776       by (simp add: \<open>pairwise orthogonal (S - {0})\<close> pairwise_orthogonal_independent)
```
```  1777     show "finite (S - {0})"
```
```  1778       using \<open>independent (S - {0})\<close> independent_imp_finite by blast
```
```  1779     show "card (S - {0}) = DIM('a)"
```
```  1780       using span_delete_0 [of S] S
```
```  1781       by (simp add: \<open>independent (S - {0})\<close> indep_card_eq_dim_span dim_UNIV)
```
```  1782   qed (use S \<open>a \<noteq> 0\<close> in auto)
```
```  1783 qed
```
```  1784
```
```  1785 lemma vector_in_orthonormal_basis:
```
```  1786   fixes a :: "'a::euclidean_space"
```
```  1787   assumes "norm a = 1"
```
```  1788   obtains S where "a \<in> S" "pairwise orthogonal S" "\<And>x. x \<in> S \<Longrightarrow> norm x = 1"
```
```  1789     "independent S" "card S = DIM('a)" "span S = UNIV"
```
```  1790 proof -
```
```  1791   have "a \<noteq> 0"
```
```  1792     using assms by auto
```
```  1793   then obtain S where "a \<in> S" "0 \<notin> S" "finite S"
```
```  1794           and S: "pairwise orthogonal S" "independent S" "span S = UNIV" "card S = DIM('a)"
```
```  1795     by (metis vector_in_orthogonal_basis)
```
```  1796   let ?S = "(\<lambda>x. x /\<^sub>R norm x) ` S"
```
```  1797   show thesis
```
```  1798   proof
```
```  1799     show "a \<in> ?S"
```
```  1800       using \<open>a \<in> S\<close> assms image_iff by fastforce
```
```  1801   next
```
```  1802     show "pairwise orthogonal ?S"
```
```  1803       using \<open>pairwise orthogonal S\<close> by (auto simp: pairwise_def orthogonal_def)
```
```  1804     show "\<And>x. x \<in> (\<lambda>x. x /\<^sub>R norm x) ` S \<Longrightarrow> norm x = 1"
```
```  1805       using \<open>0 \<notin> S\<close> by (auto simp: divide_simps)
```
```  1806     then show "independent ?S"
```
```  1807       by (metis \<open>pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` S)\<close> norm_zero pairwise_orthogonal_independent zero_neq_one)
```
```  1808     have "inj_on (\<lambda>x. x /\<^sub>R norm x) S"
```
```  1809       unfolding inj_on_def
```
```  1810       by (metis (full_types) S(1) \<open>0 \<notin> S\<close> inverse_nonzero_iff_nonzero norm_eq_zero orthogonal_scaleR orthogonal_self pairwise_def)
```
```  1811     then show "card ?S = DIM('a)"
```
```  1812       by (simp add: card_image S)
```
```  1813     show "span ?S = UNIV"
```
```  1814       by (metis (no_types) \<open>0 \<notin> S\<close> \<open>finite S\<close> \<open>span S = UNIV\<close>
```
```  1815           field_class.field_inverse_zero inverse_inverse_eq less_irrefl span_image_scale
```
```  1816           zero_less_norm_iff)
```
```  1817   qed
```
```  1818 qed
```
```  1819
```
```  1820 proposition dim_orthogonal_sum:
```
```  1821   fixes A :: "'a::euclidean_space set"
```
```  1822   assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
```
```  1823     shows "dim(A \<union> B) = dim A + dim B"
```
```  1824 proof -
```
```  1825   have 1: "\<And>x y. \<lbrakk>x \<in> span A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
```
```  1826     by (erule span_induct [OF _ subspace_hyperplane2]; simp add: assms)
```
```  1827   have "\<And>x y. \<lbrakk>x \<in> span A; y \<in> span B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
```
```  1828     using 1 by (simp add: span_induct [OF _ subspace_hyperplane])
```
```  1829   then have 0: "\<And>x y. \<lbrakk>x \<in> span A; y \<in> span B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
```
```  1830     by simp
```
```  1831   have "dim(A \<union> B) = dim (span (A \<union> B))"
```
```  1832     by (simp add: dim_span)
```
```  1833   also have "span (A \<union> B) = ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
```
```  1834     by (auto simp add: span_Un image_def)
```
```  1835   also have "dim \<dots> = dim {x + y |x y. x \<in> span A \<and> y \<in> span B}"
```
```  1836     by (auto intro!: arg_cong [where f=dim])
```
```  1837   also have "... = dim {x + y |x y. x \<in> span A \<and> y \<in> span B} + dim(span A \<inter> span B)"
```
```  1838     by (auto simp: dest: 0)
```
```  1839   also have "... = dim (span A) + dim (span B)"
```
```  1840     by (rule dim_sums_Int) (auto simp: subspace_span)
```
```  1841   also have "... = dim A + dim B"
```
```  1842     by (simp add: dim_span)
```
```  1843   finally show ?thesis .
```
```  1844 qed
```
```  1845
```
```  1846 lemma dim_subspace_orthogonal_to_vectors:
```
```  1847   fixes A :: "'a::euclidean_space set"
```
```  1848   assumes "subspace A" "subspace B" "A \<subseteq> B"
```
```  1849     shows "dim {y \<in> B. \<forall>x \<in> A. orthogonal x y} + dim A = dim B"
```
```  1850 proof -
```
```  1851   have "dim (span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)) = dim (span B)"
```
```  1852   proof (rule arg_cong [where f=dim, OF subset_antisym])
```
```  1853     show "span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A) \<subseteq> span B"
```
```  1854       by (simp add: \<open>A \<subseteq> B\<close> Collect_restrict span_mono)
```
```  1855   next
```
```  1856     have *: "x \<in> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
```
```  1857          if "x \<in> B" for x
```
```  1858     proof -
```
```  1859       obtain y z where "x = y + z" "y \<in> span A" and orth: "\<And>w. w \<in> span A \<Longrightarrow> orthogonal z w"
```
```  1860         using orthogonal_subspace_decomp_exists [of A x] that by auto
```
```  1861       have "y \<in> span B"
```
```  1862         using \<open>y \<in> span A\<close> assms(3) span_mono by blast
```
```  1863       then have "z \<in> {a \<in> B. \<forall>x. x \<in> A \<longrightarrow> orthogonal x a}"
```
```  1864         apply simp
```
```  1865         using \<open>x = y + z\<close> assms(1) assms(2) orth orthogonal_commute span_add_eq
```
```  1866           span_eq_iff that by blast
```
```  1867       then have z: "z \<in> span {y \<in> B. \<forall>x\<in>A. orthogonal x y}"
```
```  1868         by (meson span_superset subset_iff)
```
```  1869       then show ?thesis
```
```  1870         apply (auto simp: span_Un image_def  \<open>x = y + z\<close> \<open>y \<in> span A\<close>)
```
```  1871         using \<open>y \<in> span A\<close> add.commute by blast
```
```  1872     qed
```
```  1873     show "span B \<subseteq> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
```
```  1874       by (rule span_minimal)
```
```  1875         (auto intro: * span_minimal simp: subspace_span)
```
```  1876   qed
```
```  1877   then show ?thesis
```
```  1878     by (metis (no_types, lifting) dim_orthogonal_sum dim_span mem_Collect_eq
```
```  1879         orthogonal_commute orthogonal_def)
```
```  1880 qed
```
```  1881
```
```  1882 end
```