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