src/HOL/Analysis/Linear_Algebra.thy
 author paulson Tue May 01 23:25:00 2018 +0100 (12 months ago) changeset 68062 ee88c0fccbae parent 68058 69715dfdc286 child 68069 36209dfb981e permissions -rw-r--r--
simplified some messy proofs
```     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.minus)
```
```    27   show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scaleR)
```
```    28 qed
```
```    29
```
```    30 subsection \<open>A generic notion of "hull" (convex, affine, conic hull and closure).\<close>
```
```    31
```
```    32 definition%important hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
```
```    33   where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
```
```    34
```
```    35 lemma hull_same: "S s \<Longrightarrow> S hull s = s"
```
```    36   unfolding hull_def by auto
```
```    37
```
```    38 lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
```
```    39   unfolding hull_def Ball_def by auto
```
```    40
```
```    41 lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
```
```    42   using hull_same[of S s] hull_in[of S s] by metis
```
```    43
```
```    44 lemma hull_hull [simp]: "S hull (S hull s) = S hull s"
```
```    45   unfolding hull_def by blast
```
```    46
```
```    47 lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
```
```    48   unfolding hull_def by blast
```
```    49
```
```    50 lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
```
```    51   unfolding hull_def by blast
```
```    52
```
```    53 lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
```
```    54   unfolding hull_def by blast
```
```    55
```
```    56 lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
```
```    57   unfolding hull_def by blast
```
```    58
```
```    59 lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
```
```    60   unfolding hull_def by blast
```
```    61
```
```    62 lemma hull_UNIV [simp]: "S hull UNIV = UNIV"
```
```    63   unfolding hull_def by auto
```
```    64
```
```    65 lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
```
```    66   unfolding hull_def by auto
```
```    67
```
```    68 lemma hull_induct: "\<lbrakk>a \<in> Q hull S; \<And>x. x\<in> S \<Longrightarrow> P x; Q {x. P x}\<rbrakk> \<Longrightarrow> P a"
```
```    69   using hull_minimal[of S "{x. P x}" Q]
```
```    70   by (auto simp add: subset_eq)
```
```    71
```
```    72 lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
```
```    73   by (metis hull_subset subset_eq)
```
```    74
```
```    75 lemma hull_Un_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
```
```    76   unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
```
```    77
```
```    78 lemma hull_Un:
```
```    79   assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
```
```    80   shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
```
```    81   apply (rule equalityI)
```
```    82   apply (meson hull_mono hull_subset sup.mono)
```
```    83   by (metis hull_Un_subset hull_hull hull_mono)
```
```    84
```
```    85 lemma hull_Un_left: "P hull (S \<union> T) = P hull (P hull S \<union> T)"
```
```    86   apply (rule equalityI)
```
```    87    apply (simp add: Un_commute hull_mono hull_subset sup.coboundedI2)
```
```    88   by (metis Un_subset_iff hull_hull hull_mono hull_subset)
```
```    89
```
```    90 lemma hull_Un_right: "P hull (S \<union> T) = P hull (S \<union> P hull T)"
```
```    91   by (metis hull_Un_left sup.commute)
```
```    92
```
```    93 lemma hull_insert:
```
```    94    "P hull (insert a S) = P hull (insert a (P hull S))"
```
```    95   by (metis hull_Un_right insert_is_Un)
```
```    96
```
```    97 lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> S hull (insert a s) = S hull s"
```
```    98   unfolding hull_def by blast
```
```    99
```
```   100 lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> S hull (insert a s) = S hull s"
```
```   101   by (metis hull_redundant_eq)
```
```   102
```
```   103 subsection \<open>Linear functions.\<close>
```
```   104
```
```   105 lemma%important linear_iff:
```
```   106   "linear f \<longleftrightarrow> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>R x) = c *\<^sub>R f x)"
```
```   107   (is "linear f \<longleftrightarrow> ?rhs")
```
```   108 proof%unimportant
```
```   109   assume "linear f"
```
```   110   then interpret f: linear f .
```
```   111   show "?rhs" by (simp add: f.add f.scaleR)
```
```   112 next
```
```   113   assume "?rhs"
```
```   114   then show "linear f" by unfold_locales simp_all
```
```   115 qed
```
```   116
```
```   117 lemma linear_compose_cmul: "linear f \<Longrightarrow> linear (\<lambda>x. c *\<^sub>R f x)"
```
```   118   by (simp add: linear_iff algebra_simps)
```
```   119
```
```   120 lemma linear_compose_scaleR: "linear f \<Longrightarrow> linear (\<lambda>x. f x *\<^sub>R c)"
```
```   121   by (simp add: linear_iff scaleR_add_left)
```
```   122
```
```   123 lemma linear_compose_neg: "linear f \<Longrightarrow> linear (\<lambda>x. - f x)"
```
```   124   by (simp add: linear_iff)
```
```   125
```
```   126 lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x + g x)"
```
```   127   by (simp add: linear_iff algebra_simps)
```
```   128
```
```   129 lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x - g x)"
```
```   130   by (simp add: linear_iff algebra_simps)
```
```   131
```
```   132 lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> f)"
```
```   133   by (simp add: linear_iff)
```
```   134
```
```   135 lemma linear_id: "linear id"
```
```   136   by (simp add: linear_iff id_def)
```
```   137
```
```   138 lemma linear_zero: "linear (\<lambda>x. 0)"
```
```   139   by (simp add: linear_iff)
```
```   140
```
```   141 lemma linear_uminus: "linear uminus"
```
```   142 by (simp add: linear_iff)
```
```   143
```
```   144 lemma linear_compose_sum:
```
```   145   assumes lS: "\<forall>a \<in> S. linear (f a)"
```
```   146   shows "linear (\<lambda>x. sum (\<lambda>a. f a x) S)"
```
```   147 proof (cases "finite S")
```
```   148   case True
```
```   149   then show ?thesis
```
```   150     using lS by induct (simp_all add: linear_zero linear_compose_add)
```
```   151 next
```
```   152   case False
```
```   153   then show ?thesis
```
```   154     by (simp add: linear_zero)
```
```   155 qed
```
```   156
```
```   157 lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
```
```   158   unfolding linear_iff
```
```   159   apply clarsimp
```
```   160   apply (erule allE[where x="0::'a"])
```
```   161   apply simp
```
```   162   done
```
```   163
```
```   164 lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```   165   by (rule linear.scaleR)
```
```   166
```
```   167 lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x"
```
```   168   using linear_cmul [where c="-1"] by simp
```
```   169
```
```   170 lemma linear_add: "linear f \<Longrightarrow> f (x + y) = f x + f y"
```
```   171   by (metis linear_iff)
```
```   172
```
```   173 lemma linear_diff: "linear f \<Longrightarrow> f (x - y) = f x - f y"
```
```   174   using linear_add [of f x "- y"] by (simp add: linear_neg)
```
```   175
```
```   176 lemma linear_sum:
```
```   177   assumes f: "linear f"
```
```   178   shows "f (sum g S) = sum (f \<circ> g) S"
```
```   179 proof (cases "finite S")
```
```   180   case True
```
```   181   then show ?thesis
```
```   182     by induct (simp_all add: linear_0 [OF f] linear_add [OF f])
```
```   183 next
```
```   184   case False
```
```   185   then show ?thesis
```
```   186     by (simp add: linear_0 [OF f])
```
```   187 qed
```
```   188
```
```   189 lemma linear_sum_mul:
```
```   190   assumes lin: "linear f"
```
```   191   shows "f (sum (\<lambda>i. c i *\<^sub>R v i) S) = sum (\<lambda>i. c i *\<^sub>R f (v i)) S"
```
```   192   using linear_sum[OF lin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
```
```   193   by simp
```
```   194
```
```   195 lemma linear_injective_0:
```
```   196   assumes lin: "linear f"
```
```   197   shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
```
```   198 proof -
```
```   199   have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)"
```
```   200     by (simp add: inj_on_def)
```
```   201   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)"
```
```   202     by simp
```
```   203   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
```
```   204     by (simp add: linear_diff[OF lin])
```
```   205   also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
```
```   206     by auto
```
```   207   finally show ?thesis .
```
```   208 qed
```
```   209
```
```   210 lemma linear_scaleR  [simp]: "linear (\<lambda>x. scaleR c x)"
```
```   211   by (simp add: linear_iff scaleR_add_right)
```
```   212
```
```   213 lemma linear_scaleR_left [simp]: "linear (\<lambda>r. scaleR r x)"
```
```   214   by (simp add: linear_iff scaleR_add_left)
```
```   215
```
```   216 lemma injective_scaleR: "c \<noteq> 0 \<Longrightarrow> inj (\<lambda>x::'a::real_vector. scaleR c x)"
```
```   217   by (simp add: inj_on_def)
```
```   218
```
```   219 lemma linear_add_cmul:
```
```   220   assumes "linear f"
```
```   221   shows "f (a *\<^sub>R x + b *\<^sub>R y) = a *\<^sub>R f x +  b *\<^sub>R f y"
```
```   222   using linear_add[of f] linear_cmul[of f] assms by simp
```
```   223
```
```   224 subsection \<open>Subspaces of vector spaces\<close>
```
```   225
```
```   226 definition%important (in real_vector) subspace :: "'a set \<Rightarrow> bool"
```
```   227   where "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S)"
```
```   228
```
```   229 definition%important (in real_vector) "span S = (subspace hull S)"
```
```   230 definition%important (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))"
```
```   231 abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
```
```   232
```
```   233 text \<open>Closure properties of subspaces.\<close>
```
```   234
```
```   235 lemma subspace_UNIV[simp]: "subspace UNIV"
```
```   236   by (simp add: subspace_def)
```
```   237
```
```   238 lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
```
```   239   by (metis subspace_def)
```
```   240
```
```   241 lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
```
```   242   by (metis subspace_def)
```
```   243
```
```   244 lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
```
```   245   by (metis subspace_def)
```
```   246
```
```   247 lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
```
```   248   by (metis scaleR_minus1_left subspace_mul)
```
```   249
```
```   250 lemma subspace_diff: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
```
```   251   using subspace_add [of S x "- y"] by (simp add: subspace_neg)
```
```   252
```
```   253 lemma (in real_vector) subspace_sum:
```
```   254   assumes sA: "subspace A"
```
```   255     and f: "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A"
```
```   256   shows "sum f B \<in> A"
```
```   257 proof (cases "finite B")
```
```   258   case True
```
```   259   then show ?thesis
```
```   260     using f by induct (simp_all add: subspace_0 [OF sA] subspace_add [OF sA])
```
```   261 qed (simp add: subspace_0 [OF sA])
```
```   262
```
```   263 lemma subspace_trivial [iff]: "subspace {0}"
```
```   264   by (simp add: subspace_def)
```
```   265
```
```   266 lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
```
```   267   by (simp add: subspace_def)
```
```   268
```
```   269 lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
```
```   270   unfolding subspace_def zero_prod_def by simp
```
```   271
```
```   272 lemma subspace_sums: "\<lbrakk>subspace S; subspace T\<rbrakk> \<Longrightarrow> subspace {x + y|x y. x \<in> S \<and> y \<in> T}"
```
```   273 apply (simp add: subspace_def)
```
```   274 apply (intro conjI impI allI)
```
```   275   using add.right_neutral apply blast
```
```   276  apply clarify
```
```   277  apply (metis add.assoc add.left_commute)
```
```   278 using scaleR_add_right by blast
```
```   279
```
```   280 subsection%unimportant \<open>Properties of span\<close>
```
```   281
```
```   282 lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
```
```   283   by (metis span_def hull_mono)
```
```   284
```
```   285 lemma (in real_vector) subspace_span [iff]: "subspace (span S)"
```
```   286   unfolding span_def
```
```   287   apply (rule hull_in)
```
```   288   apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
```
```   289   apply auto
```
```   290   done
```
```   291
```
```   292 lemma (in real_vector) span_clauses:
```
```   293   "a \<in> S \<Longrightarrow> a \<in> span S"
```
```   294   "0 \<in> span S"
```
```   295   "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
```
```   296   "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
```
```   297   by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
```
```   298
```
```   299 lemma span_unique:
```
```   300   "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
```
```   301   unfolding span_def by (rule hull_unique)
```
```   302
```
```   303 lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
```
```   304   unfolding span_def by (rule hull_minimal)
```
```   305
```
```   306 lemma span_UNIV [simp]: "span UNIV = UNIV"
```
```   307   by (intro span_unique) auto
```
```   308
```
```   309 lemma (in real_vector) span_induct:
```
```   310   assumes x: "x \<in> span S"
```
```   311     and P: "subspace (Collect P)"
```
```   312     and SP: "\<And>x. x \<in> S \<Longrightarrow> P x"
```
```   313   shows "P x"
```
```   314 proof -
```
```   315   from SP have SP': "S \<subseteq> Collect P"
```
```   316     by (simp add: subset_eq)
```
```   317   from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
```
```   318   show ?thesis
```
```   319     using subset_eq by force
```
```   320 qed
```
```   321
```
```   322 lemma span_empty[simp]: "span {} = {0}"
```
```   323   apply (simp add: span_def)
```
```   324   apply (rule hull_unique)
```
```   325   apply (auto simp add: subspace_def)
```
```   326   done
```
```   327
```
```   328 lemma (in real_vector) independent_empty [iff]: "independent {}"
```
```   329   by (simp add: dependent_def)
```
```   330
```
```   331 lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
```
```   332   unfolding dependent_def by auto
```
```   333
```
```   334 lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
```
```   335   unfolding dependent_def span_mono
```
```   336   by (metis insert_Diff local.span_mono subsetCE subset_insert_iff)
```
```   337
```
```   338 lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
```
```   339   by (metis order_antisym span_def hull_minimal)
```
```   340
```
```   341 inductive_set (in real_vector) span_induct_alt_help for S :: "'a set"
```
```   342 where
```
```   343   span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
```
```   344 | span_induct_alt_help_S:
```
```   345     "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
```
```   346       (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
```
```   347
```
```   348 lemma span_induct_alt':
```
```   349   assumes h0: "h 0"
```
```   350     and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
```
```   351   shows "\<forall>x \<in> span S. h x"
```
```   352 proof -
```
```   353   {
```
```   354     fix x :: 'a
```
```   355     assume x: "x \<in> span_induct_alt_help S"
```
```   356     have "h x"
```
```   357       apply (rule span_induct_alt_help.induct[OF x])
```
```   358       apply (rule h0)
```
```   359       apply (rule hS)
```
```   360       apply assumption
```
```   361       apply assumption
```
```   362       done
```
```   363   }
```
```   364   note th0 = this
```
```   365   {
```
```   366     fix x
```
```   367     assume x: "x \<in> span S"
```
```   368     have "x \<in> span_induct_alt_help S"
```
```   369     proof (rule span_induct[where x=x and S=S])
```
```   370       show "x \<in> span S" by (rule x)
```
```   371     next
```
```   372       fix x
```
```   373       assume xS: "x \<in> S"
```
```   374       from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
```
```   375       show "x \<in> span_induct_alt_help S"
```
```   376         by simp
```
```   377     next
```
```   378       have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
```
```   379       moreover
```
```   380       {
```
```   381         fix x y
```
```   382         assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
```
```   383         from h have "(x + y) \<in> span_induct_alt_help S"
```
```   384           apply (induct rule: span_induct_alt_help.induct)
```
```   385           apply simp
```
```   386           unfolding add.assoc
```
```   387           apply (rule span_induct_alt_help_S)
```
```   388           apply assumption
```
```   389           apply simp
```
```   390           done
```
```   391       }
```
```   392       moreover
```
```   393       {
```
```   394         fix c x
```
```   395         assume xt: "x \<in> span_induct_alt_help S"
```
```   396         then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
```
```   397           apply (induct rule: span_induct_alt_help.induct)
```
```   398           apply (simp add: span_induct_alt_help_0)
```
```   399           apply (simp add: scaleR_right_distrib)
```
```   400           apply (rule span_induct_alt_help_S)
```
```   401           apply assumption
```
```   402           apply simp
```
```   403           done }
```
```   404       ultimately show "subspace {a. a \<in> span_induct_alt_help S}"
```
```   405         unfolding subspace_def Ball_def by blast
```
```   406     qed
```
```   407   }
```
```   408   with th0 show ?thesis by blast
```
```   409 qed
```
```   410
```
```   411 lemma span_induct_alt:
```
```   412   assumes h0: "h 0"
```
```   413     and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
```
```   414     and x: "x \<in> span S"
```
```   415   shows "h x"
```
```   416   using span_induct_alt'[of h S] h0 hS x by blast
```
```   417
```
```   418 text \<open>Individual closure properties.\<close>
```
```   419
```
```   420 lemma span_span: "span (span A) = span A"
```
```   421   unfolding span_def hull_hull ..
```
```   422
```
```   423 lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
```
```   424   by (metis span_clauses(1))
```
```   425
```
```   426 lemma (in real_vector) span_0 [simp]: "0 \<in> span S"
```
```   427   by (metis subspace_span subspace_0)
```
```   428
```
```   429 lemma span_inc: "S \<subseteq> span S"
```
```   430   by (metis subset_eq span_superset)
```
```   431
```
```   432 lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
```
```   433   using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
```
```   434   by (auto simp add: span_span)
```
```   435
```
```   436 lemma (in real_vector) dependent_0:
```
```   437   assumes "0 \<in> A"
```
```   438   shows "dependent A"
```
```   439   unfolding dependent_def
```
```   440   using assms span_0
```
```   441   by blast
```
```   442
```
```   443 lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
```
```   444   by (metis subspace_add subspace_span)
```
```   445
```
```   446 lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
```
```   447   by (metis subspace_span subspace_mul)
```
```   448
```
```   449 lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
```
```   450   by (metis subspace_neg subspace_span)
```
```   451
```
```   452 lemma span_diff: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
```
```   453   by (metis subspace_span subspace_diff)
```
```   454
```
```   455 lemma (in real_vector) span_sum: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> span S) \<Longrightarrow> sum f A \<in> span S"
```
```   456   by (rule subspace_sum [OF subspace_span])
```
```   457
```
```   458 lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
```
```   459   by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
```
```   460
```
```   461 text \<open>The key breakdown property.\<close>
```
```   462
```
```   463 lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
```
```   464 proof (rule span_unique)
```
```   465   show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
```
```   466     by (fast intro: scaleR_one [symmetric])
```
```   467   show "subspace (range (\<lambda>k. k *\<^sub>R x))"
```
```   468     unfolding subspace_def
```
```   469     by (auto intro: scaleR_add_left [symmetric])
```
```   470 next
```
```   471   fix T
```
```   472   assume "{x} \<subseteq> T" and "subspace T"
```
```   473   then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
```
```   474     unfolding subspace_def by auto
```
```   475 qed
```
```   476
```
```   477 text \<open>Mapping under linear image.\<close>
```
```   478
```
```   479 lemma subspace_linear_image:
```
```   480   assumes lf: "linear f"
```
```   481     and sS: "subspace S"
```
```   482   shows "subspace (f ` S)"
```
```   483   using lf sS linear_0[OF lf]
```
```   484   unfolding linear_iff subspace_def
```
```   485   apply (auto simp add: image_iff)
```
```   486   apply (rule_tac x="x + y" in bexI)
```
```   487   apply auto
```
```   488   apply (rule_tac x="c *\<^sub>R x" in bexI)
```
```   489   apply auto
```
```   490   done
```
```   491
```
```   492 lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
```
```   493   by (auto simp add: subspace_def linear_iff linear_0[of f])
```
```   494
```
```   495 lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
```
```   496   by (auto simp add: subspace_def linear_iff linear_0[of f])
```
```   497
```
```   498 lemma span_linear_image:
```
```   499   assumes lf: "linear f"
```
```   500   shows "span (f ` S) = f ` span S"
```
```   501 proof (rule span_unique)
```
```   502   show "f ` S \<subseteq> f ` span S"
```
```   503     by (intro image_mono span_inc)
```
```   504   show "subspace (f ` span S)"
```
```   505     using lf subspace_span by (rule subspace_linear_image)
```
```   506 next
```
```   507   fix T
```
```   508   assume "f ` S \<subseteq> T" and "subspace T"
```
```   509   then show "f ` span S \<subseteq> T"
```
```   510     unfolding image_subset_iff_subset_vimage
```
```   511     by (intro span_minimal subspace_linear_vimage lf)
```
```   512 qed
```
```   513
```
```   514 lemma spans_image:
```
```   515   assumes lf: "linear f"
```
```   516     and VB: "V \<subseteq> span B"
```
```   517   shows "f ` V \<subseteq> span (f ` B)"
```
```   518   unfolding span_linear_image[OF lf] by (metis VB image_mono)
```
```   519
```
```   520 lemma span_Un: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
```
```   521 proof (rule span_unique)
```
```   522   show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
```
```   523     by safe (force intro: span_clauses)+
```
```   524 next
```
```   525   have "linear (\<lambda>(a, b). a + b)"
```
```   526     by (simp add: linear_iff scaleR_add_right)
```
```   527   moreover have "subspace (span A \<times> span B)"
```
```   528     by (intro subspace_Times subspace_span)
```
```   529   ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
```
```   530     by (rule subspace_linear_image)
```
```   531 next
```
```   532   fix T
```
```   533   assume "A \<union> B \<subseteq> T" and "subspace T"
```
```   534   then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
```
```   535     by (auto intro!: subspace_add elim: span_induct)
```
```   536 qed
```
```   537
```
```   538 lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
```
```   539 proof -
```
```   540   have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
```
```   541     unfolding span_Un span_singleton
```
```   542     apply safe
```
```   543     apply (rule_tac x=k in exI, simp)
```
```   544     apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
```
```   545     apply auto
```
```   546     done
```
```   547   then show ?thesis by simp
```
```   548 qed
```
```   549
```
```   550 lemma span_breakdown:
```
```   551   assumes bS: "b \<in> S"
```
```   552     and aS: "a \<in> span S"
```
```   553   shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
```
```   554   using assms span_insert [of b "S - {b}"]
```
```   555   by (simp add: insert_absorb)
```
```   556
```
```   557 lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
```
```   558   by (simp add: span_insert)
```
```   559
```
```   560 text \<open>Hence some "reversal" results.\<close>
```
```   561
```
```   562 lemma in_span_insert:
```
```   563   assumes a: "a \<in> span (insert b S)"
```
```   564     and na: "a \<notin> span S"
```
```   565   shows "b \<in> span (insert a S)"
```
```   566 proof -
```
```   567   from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
```
```   568     unfolding span_insert by fast
```
```   569   show ?thesis
```
```   570   proof (cases "k = 0")
```
```   571     case True
```
```   572     with k have "a \<in> span S" by simp
```
```   573     with na show ?thesis by simp
```
```   574   next
```
```   575     case False
```
```   576     from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
```
```   577       by (rule span_mul)
```
```   578     then have "b - inverse k *\<^sub>R a \<in> span S"
```
```   579       using \<open>k \<noteq> 0\<close> by (simp add: scaleR_diff_right)
```
```   580     then show ?thesis
```
```   581       unfolding span_insert by fast
```
```   582   qed
```
```   583 qed
```
```   584
```
```   585 lemma in_span_delete:
```
```   586   assumes a: "a \<in> span S"
```
```   587     and na: "a \<notin> span (S - {b})"
```
```   588   shows "b \<in> span (insert a (S - {b}))"
```
```   589   apply (rule in_span_insert)
```
```   590   apply (rule set_rev_mp)
```
```   591   apply (rule a)
```
```   592   apply (rule span_mono)
```
```   593   apply blast
```
```   594   apply (rule na)
```
```   595   done
```
```   596
```
```   597 text \<open>Transitivity property.\<close>
```
```   598
```
```   599 lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
```
```   600   unfolding span_def by (rule hull_redundant)
```
```   601
```
```   602 lemma span_trans:
```
```   603   assumes x: "x \<in> span S"
```
```   604     and y: "y \<in> span (insert x S)"
```
```   605   shows "y \<in> span S"
```
```   606   using assms by (simp only: span_redundant)
```
```   607
```
```   608 lemma span_insert_0[simp]: "span (insert 0 S) = span S"
```
```   609   by (simp only: span_redundant span_0)
```
```   610
```
```   611 text \<open>An explicit expansion is sometimes needed.\<close>
```
```   612
```
```   613 lemma span_explicit:
```
```   614   "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
```
```   615   (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
```
```   616 proof -
```
```   617   {
```
```   618     fix x
```
```   619     assume "?h x"
```
```   620     then obtain S u where "finite S" and "S \<subseteq> P" and "sum (\<lambda>v. u v *\<^sub>R v) S = x"
```
```   621       by blast
```
```   622     then have "x \<in> span P"
```
```   623       by (auto intro: span_sum span_mul span_superset)
```
```   624   }
```
```   625   moreover
```
```   626   have "\<forall>x \<in> span P. ?h x"
```
```   627   proof (rule span_induct_alt')
```
```   628     show "?h 0"
```
```   629       by (rule exI[where x="{}"], simp)
```
```   630   next
```
```   631     fix c x y
```
```   632     assume x: "x \<in> P"
```
```   633     assume hy: "?h y"
```
```   634     from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
```
```   635       and u: "sum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
```
```   636     let ?S = "insert x S"
```
```   637     let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
```
```   638     from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
```
```   639       by blast+
```
```   640     have "?Q ?S ?u (c*\<^sub>R x + y)"
```
```   641     proof cases
```
```   642       assume xS: "x \<in> S"
```
```   643       have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = (\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
```
```   644         using xS by (simp add: sum.remove [OF fS xS] insert_absorb)
```
```   645       also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
```
```   646         by (simp add: sum.remove [OF fS xS] algebra_simps)
```
```   647       also have "\<dots> = c*\<^sub>R x + y"
```
```   648         by (simp add: add.commute u)
```
```   649       finally have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
```
```   650       then show ?thesis using th0 by blast
```
```   651     next
```
```   652       assume xS: "x \<notin> S"
```
```   653       have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
```
```   654         unfolding u[symmetric]
```
```   655         apply (rule sum.cong)
```
```   656         using xS
```
```   657         apply auto
```
```   658         done
```
```   659       show ?thesis using fS xS th0
```
```   660         by (simp add: th00 add.commute cong del: if_weak_cong)
```
```   661     qed
```
```   662     then show "?h (c*\<^sub>R x + y)"
```
```   663       by fast
```
```   664   qed
```
```   665   ultimately show ?thesis by blast
```
```   666 qed
```
```   667
```
```   668 lemma dependent_explicit:
```
```   669   "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0))"
```
```   670   (is "?lhs = ?rhs")
```
```   671 proof -
```
```   672   {
```
```   673     assume dP: "dependent P"
```
```   674     then obtain a S u where aP: "a \<in> P" and fS: "finite S"
```
```   675       and SP: "S \<subseteq> P - {a}" and ua: "sum (\<lambda>v. u v *\<^sub>R v) S = a"
```
```   676       unfolding dependent_def span_explicit by blast
```
```   677     let ?S = "insert a S"
```
```   678     let ?u = "\<lambda>y. if y = a then - 1 else u y"
```
```   679     let ?v = a
```
```   680     from aP SP have aS: "a \<notin> S"
```
```   681       by blast
```
```   682     from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
```
```   683       by auto
```
```   684     have s0: "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
```
```   685       using fS aS
```
```   686       apply simp
```
```   687       apply (subst (2) ua[symmetric])
```
```   688       apply (rule sum.cong)
```
```   689       apply auto
```
```   690       done
```
```   691     with th0 have ?rhs by fast
```
```   692   }
```
```   693   moreover
```
```   694   {
```
```   695     fix S u v
```
```   696     assume fS: "finite S"
```
```   697       and SP: "S \<subseteq> P"
```
```   698       and vS: "v \<in> S"
```
```   699       and uv: "u v \<noteq> 0"
```
```   700       and u: "sum (\<lambda>v. u v *\<^sub>R v) S = 0"
```
```   701     let ?a = v
```
```   702     let ?S = "S - {v}"
```
```   703     let ?u = "\<lambda>i. (- u i) / u v"
```
```   704     have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
```
```   705       using fS SP vS by auto
```
```   706     have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S =
```
```   707       sum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
```
```   708       using fS vS uv by (simp add: sum_diff1 field_simps)
```
```   709     also have "\<dots> = ?a"
```
```   710       unfolding scaleR_right.sum [symmetric] u using uv by simp
```
```   711     finally have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
```
```   712     with th0 have ?lhs
```
```   713       unfolding dependent_def span_explicit
```
```   714       apply -
```
```   715       apply (rule bexI[where x= "?a"])
```
```   716       apply (simp_all del: scaleR_minus_left)
```
```   717       apply (rule exI[where x= "?S"])
```
```   718       apply (auto simp del: scaleR_minus_left)
```
```   719       done
```
```   720   }
```
```   721   ultimately show ?thesis by blast
```
```   722 qed
```
```   723
```
```   724 lemma dependent_finite:
```
```   725   assumes "finite S"
```
```   726     shows "dependent S \<longleftrightarrow> (\<exists>u. (\<exists>v \<in> S. u v \<noteq> 0) \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = 0)"
```
```   727            (is "?lhs = ?rhs")
```
```   728 proof
```
```   729   assume ?lhs
```
```   730   then obtain T u v
```
```   731          where "finite T" "T \<subseteq> S" "v\<in>T" "u v \<noteq> 0" "(\<Sum>v\<in>T. u v *\<^sub>R v) = 0"
```
```   732     by (force simp: dependent_explicit)
```
```   733   with assms show ?rhs
```
```   734     apply (rule_tac x="\<lambda>v. if v \<in> T then u v else 0" in exI)
```
```   735     apply (auto simp: sum.mono_neutral_right)
```
```   736     done
```
```   737 next
```
```   738   assume ?rhs  with assms show ?lhs
```
```   739     by (fastforce simp add: dependent_explicit)
```
```   740 qed
```
```   741
```
```   742 lemma span_alt:
```
```   743   "span B = {(\<Sum>x | f x \<noteq> 0. f x *\<^sub>R x) | f. {x. f x \<noteq> 0} \<subseteq> B \<and> finite {x. f x \<noteq> 0}}"
```
```   744   unfolding span_explicit
```
```   745   apply safe
```
```   746   subgoal for x S u
```
```   747     by (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
```
```   748         (auto intro!: sum.mono_neutral_cong_right)
```
```   749   apply auto
```
```   750   done
```
```   751
```
```   752 lemma dependent_alt:
```
```   753   "dependent B \<longleftrightarrow>
```
```   754     (\<exists>X. finite {x. X x \<noteq> 0} \<and> {x. X x \<noteq> 0} \<subseteq> B \<and> (\<Sum>x|X x \<noteq> 0. X x *\<^sub>R x) = 0 \<and> (\<exists>x. X x \<noteq> 0))"
```
```   755   unfolding dependent_explicit
```
```   756   apply safe
```
```   757   subgoal for S u v
```
```   758     apply (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
```
```   759     apply (subst sum.mono_neutral_cong_left[where T=S])
```
```   760     apply (auto intro!: sum.mono_neutral_cong_right cong: rev_conj_cong)
```
```   761     done
```
```   762   apply auto
```
```   763   done
```
```   764
```
```   765 lemma independent_alt:
```
```   766   "independent B \<longleftrightarrow>
```
```   767     (\<forall>X. finite {x. X x \<noteq> 0} \<longrightarrow> {x. X x \<noteq> 0} \<subseteq> B \<longrightarrow> (\<Sum>x|X x \<noteq> 0. X x *\<^sub>R x) = 0 \<longrightarrow> (\<forall>x. X x = 0))"
```
```   768   unfolding dependent_alt by auto
```
```   769
```
```   770 lemma independentD_alt:
```
```   771   "independent B \<Longrightarrow> finite {x. X x \<noteq> 0} \<Longrightarrow> {x. X x \<noteq> 0} \<subseteq> B \<Longrightarrow> (\<Sum>x|X x \<noteq> 0. X x *\<^sub>R x) = 0 \<Longrightarrow> X x = 0"
```
```   772   unfolding independent_alt by blast
```
```   773
```
```   774 lemma independentD_unique:
```
```   775   assumes B: "independent B"
```
```   776     and X: "finite {x. X x \<noteq> 0}" "{x. X x \<noteq> 0} \<subseteq> B"
```
```   777     and Y: "finite {x. Y x \<noteq> 0}" "{x. Y x \<noteq> 0} \<subseteq> B"
```
```   778     and "(\<Sum>x | X x \<noteq> 0. X x *\<^sub>R x) = (\<Sum>x| Y x \<noteq> 0. Y x *\<^sub>R x)"
```
```   779   shows "X = Y"
```
```   780 proof -
```
```   781   have "X x - Y x = 0" for x
```
```   782     using B
```
```   783   proof (rule independentD_alt)
```
```   784     have "{x. X x - Y x \<noteq> 0} \<subseteq> {x. X x \<noteq> 0} \<union> {x. Y x \<noteq> 0}"
```
```   785       by auto
```
```   786     then show "finite {x. X x - Y x \<noteq> 0}" "{x. X x - Y x \<noteq> 0} \<subseteq> B"
```
```   787       using X Y by (auto dest: finite_subset)
```
```   788     then have "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *\<^sub>R x) = (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. (X v - Y v) *\<^sub>R v)"
```
```   789       using X Y by (intro sum.mono_neutral_cong_left) auto
```
```   790     also have "\<dots> = (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. X v *\<^sub>R v) - (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. Y v *\<^sub>R v)"
```
```   791       by (simp add: scaleR_diff_left sum_subtractf assms)
```
```   792     also have "(\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. X v *\<^sub>R v) = (\<Sum>v\<in>{S. X S \<noteq> 0}. X v *\<^sub>R v)"
```
```   793       using X Y by (intro sum.mono_neutral_cong_right) auto
```
```   794     also have "(\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. Y v *\<^sub>R v) = (\<Sum>v\<in>{S. Y S \<noteq> 0}. Y v *\<^sub>R v)"
```
```   795       using X Y by (intro sum.mono_neutral_cong_right) auto
```
```   796     finally show "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *\<^sub>R x) = 0"
```
```   797       using assms by simp
```
```   798   qed
```
```   799   then show ?thesis
```
```   800     by auto
```
```   801 qed
```
```   802
```
```   803 text \<open>This is useful for building a basis step-by-step.\<close>
```
```   804
```
```   805 lemma independent_insert:
```
```   806   "independent (insert a S) \<longleftrightarrow>
```
```   807     (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
```
```   808   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   809 proof (cases "a \<in> S")
```
```   810   case True
```
```   811   then show ?thesis
```
```   812     using insert_absorb[OF True] by simp
```
```   813 next
```
```   814   case False
```
```   815   show ?thesis
```
```   816   proof
```
```   817     assume i: ?lhs
```
```   818     then show ?rhs
```
```   819       using False
```
```   820       apply simp
```
```   821       apply (rule conjI)
```
```   822       apply (rule independent_mono)
```
```   823       apply assumption
```
```   824       apply blast
```
```   825       apply (simp add: dependent_def)
```
```   826       done
```
```   827   next
```
```   828     assume i: ?rhs
```
```   829     show ?lhs
```
```   830       using i False
```
```   831       apply (auto simp add: dependent_def)
```
```   832       by (metis in_span_insert insert_Diff_if insert_Diff_single insert_absorb)
```
```   833   qed
```
```   834 qed
```
```   835
```
```   836 lemma independent_Union_directed:
```
```   837   assumes directed: "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
```
```   838   assumes indep: "\<And>c. c \<in> C \<Longrightarrow> independent c"
```
```   839   shows "independent (\<Union>C)"
```
```   840 proof
```
```   841   assume "dependent (\<Union>C)"
```
```   842   then obtain u v S where S: "finite S" "S \<subseteq> \<Union>C" "v \<in> S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
```
```   843     by (auto simp: dependent_explicit)
```
```   844
```
```   845   have "S \<noteq> {}"
```
```   846     using \<open>v \<in> S\<close> by auto
```
```   847   have "\<exists>c\<in>C. S \<subseteq> c"
```
```   848     using \<open>finite S\<close> \<open>S \<noteq> {}\<close> \<open>S \<subseteq> \<Union>C\<close>
```
```   849   proof (induction rule: finite_ne_induct)
```
```   850     case (insert i I)
```
```   851     then obtain c d where cd: "c \<in> C" "d \<in> C" and iI: "I \<subseteq> c" "i \<in> d"
```
```   852       by blast
```
```   853     from directed[OF cd] cd have "c \<union> d \<in> C"
```
```   854       by (auto simp: sup.absorb1 sup.absorb2)
```
```   855     with iI show ?case
```
```   856       by (intro bexI[of _ "c \<union> d"]) auto
```
```   857   qed auto
```
```   858   then obtain c where "c \<in> C" "S \<subseteq> c"
```
```   859     by auto
```
```   860   have "dependent c"
```
```   861     unfolding dependent_explicit
```
```   862     by (intro exI[of _ S] exI[of _ u] bexI[of _ v] conjI) fact+
```
```   863   with indep[OF \<open>c \<in> C\<close>] show False
```
```   864     by auto
```
```   865 qed
```
```   866
```
```   867 text \<open>Hence we can create a maximal independent subset.\<close>
```
```   868
```
```   869 lemma maximal_independent_subset_extend:
```
```   870   assumes "S \<subseteq> V" "independent S"
```
```   871   shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
```
```   872 proof -
```
```   873   let ?C = "{B. S \<subseteq> B \<and> independent B \<and> B \<subseteq> V}"
```
```   874   have "\<exists>M\<in>?C. \<forall>X\<in>?C. M \<subseteq> X \<longrightarrow> X = M"
```
```   875   proof (rule subset_Zorn)
```
```   876     fix C :: "'a set set" assume "subset.chain ?C C"
```
```   877     then have C: "\<And>c. c \<in> C \<Longrightarrow> c \<subseteq> V" "\<And>c. c \<in> C \<Longrightarrow> S \<subseteq> c" "\<And>c. c \<in> C \<Longrightarrow> independent c"
```
```   878       "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
```
```   879       unfolding subset.chain_def by blast+
```
```   880
```
```   881     show "\<exists>U\<in>?C. \<forall>X\<in>C. X \<subseteq> U"
```
```   882     proof cases
```
```   883       assume "C = {}" with assms show ?thesis
```
```   884         by (auto intro!: exI[of _ S])
```
```   885     next
```
```   886       assume "C \<noteq> {}"
```
```   887       with C(2) have "S \<subseteq> \<Union>C"
```
```   888         by auto
```
```   889       moreover have "independent (\<Union>C)"
```
```   890         by (intro independent_Union_directed C)
```
```   891       moreover have "\<Union>C \<subseteq> V"
```
```   892         using C by auto
```
```   893       ultimately show ?thesis
```
```   894         by auto
```
```   895     qed
```
```   896   qed
```
```   897   then obtain B where B: "independent B" "B \<subseteq> V" "S \<subseteq> B"
```
```   898     and max: "\<And>S. independent S \<Longrightarrow> S \<subseteq> V \<Longrightarrow> B \<subseteq> S \<Longrightarrow> S = B"
```
```   899     by auto
```
```   900   moreover
```
```   901   { assume "\<not> V \<subseteq> span B"
```
```   902     then obtain v where "v \<in> V" "v \<notin> span B"
```
```   903       by auto
```
```   904     with B have "independent (insert v B)"
```
```   905       unfolding independent_insert by auto
```
```   906     from max[OF this] \<open>v \<in> V\<close> \<open>B \<subseteq> V\<close>
```
```   907     have "v \<in> B"
```
```   908       by auto
```
```   909     with \<open>v \<notin> span B\<close> have False
```
```   910       by (auto intro: span_superset) }
```
```   911   ultimately show ?thesis
```
```   912     by (auto intro!: exI[of _ B])
```
```   913 qed
```
```   914
```
```   915
```
```   916 lemma maximal_independent_subset:
```
```   917   "\<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
```
```   918   by (metis maximal_independent_subset_extend[of "{}"] empty_subsetI independent_empty)
```
```   919
```
```   920 lemma span_finite:
```
```   921   assumes fS: "finite S"
```
```   922   shows "span S = {y. \<exists>u. sum (\<lambda>v. u v *\<^sub>R v) S = y}"
```
```   923   (is "_ = ?rhs")
```
```   924 proof -
```
```   925   {
```
```   926     fix y
```
```   927     assume y: "y \<in> span S"
```
```   928     from y obtain S' u where fS': "finite S'"
```
```   929       and SS': "S' \<subseteq> S"
```
```   930       and u: "sum (\<lambda>v. u v *\<^sub>R v) S' = y"
```
```   931       unfolding span_explicit by blast
```
```   932     let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
```
```   933     have "sum (\<lambda>v. ?u v *\<^sub>R v) S = sum (\<lambda>v. u v *\<^sub>R v) S'"
```
```   934       using SS' fS by (auto intro!: sum.mono_neutral_cong_right)
```
```   935     then have "sum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
```
```   936     then have "y \<in> ?rhs" by auto
```
```   937   }
```
```   938   moreover
```
```   939   {
```
```   940     fix y u
```
```   941     assume u: "sum (\<lambda>v. u v *\<^sub>R v) S = y"
```
```   942     then have "y \<in> span S" using fS unfolding span_explicit by auto
```
```   943   }
```
```   944   ultimately show ?thesis by blast
```
```   945 qed
```
```   946
```
```   947 lemma linear_independent_extend_subspace:
```
```   948   assumes "independent B"
```
```   949   shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x) \<and> range g = span (f`B)"
```
```   950 proof -
```
```   951   from maximal_independent_subset_extend[OF _ \<open>independent B\<close>, of UNIV]
```
```   952   obtain B' where "B \<subseteq> B'" "independent B'" "span B' = UNIV"
```
```   953     by (auto simp: top_unique)
```
```   954   have "\<forall>y. \<exists>X. {x. X x \<noteq> 0} \<subseteq> B' \<and> finite {x. X x \<noteq> 0} \<and> y = (\<Sum>x|X x \<noteq> 0. X x *\<^sub>R x)"
```
```   955     using \<open>span B' = UNIV\<close> unfolding span_alt by auto
```
```   956   then obtain X where X: "\<And>y. {x. X y x \<noteq> 0} \<subseteq> B'" "\<And>y. finite {x. X y x \<noteq> 0}"
```
```   957     "\<And>y. y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R x)"
```
```   958     unfolding choice_iff by auto
```
```   959
```
```   960   have X_add: "X (x + y) = (\<lambda>z. X x z + X y z)" for x y
```
```   961     using \<open>independent B'\<close>
```
```   962   proof (rule independentD_unique)
```
```   963     have "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R z)
```
```   964       = (\<Sum>z\<in>{z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}. (X x z + X y z) *\<^sub>R z)"
```
```   965       by (intro sum.mono_neutral_cong_left) (auto intro: X)
```
```   966     also have "\<dots> = (\<Sum>z\<in>{z. X x z \<noteq> 0}. X x z *\<^sub>R z) + (\<Sum>z\<in>{z. X y z \<noteq> 0}. X y z *\<^sub>R z)"
```
```   967       by (auto simp add: scaleR_add_left sum.distrib
```
```   968                intro!: arg_cong2[where f="(+)"]  sum.mono_neutral_cong_right X)
```
```   969     also have "\<dots> = x + y"
```
```   970       by (simp add: X(3)[symmetric])
```
```   971     also have "\<dots> = (\<Sum>z | X (x + y) z \<noteq> 0. X (x + y) z *\<^sub>R z)"
```
```   972       by (rule X(3))
```
```   973     finally show "(\<Sum>z | X (x + y) z \<noteq> 0. X (x + y) z *\<^sub>R z) = (\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R z)"
```
```   974       ..
```
```   975     have "{z. X x z + X y z \<noteq> 0} \<subseteq> {z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}"
```
```   976       by auto
```
```   977     then show "finite {z. X x z + X y z \<noteq> 0}" "{xa. X x xa + X y xa \<noteq> 0} \<subseteq> B'"
```
```   978         "finite {xa. X (x + y) xa \<noteq> 0}" "{xa. X (x + y) xa \<noteq> 0} \<subseteq> B'"
```
```   979       using X(1) by (auto dest: finite_subset intro: X)
```
```   980   qed
```
```   981
```
```   982   have X_cmult: "X (c *\<^sub>R x) = (\<lambda>z. c * X x z)" for x c
```
```   983     using \<open>independent B'\<close>
```
```   984   proof (rule independentD_unique)
```
```   985     show "finite {z. X (c *\<^sub>R x) z \<noteq> 0}" "{z. X (c *\<^sub>R x) z \<noteq> 0} \<subseteq> B'"
```
```   986       "finite {z. c * X x z \<noteq> 0}" "{z. c * X x z \<noteq> 0} \<subseteq> B' "
```
```   987       using X(1,2) by auto
```
```   988     show "(\<Sum>z | X (c *\<^sub>R x) z \<noteq> 0. X (c *\<^sub>R x) z *\<^sub>R z) = (\<Sum>z | c * X x z \<noteq> 0. (c * X x z) *\<^sub>R z)"
```
```   989       unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric]
```
```   990       by (cases "c = 0") (auto simp: X(3)[symmetric])
```
```   991   qed
```
```   992
```
```   993   have X_B': "x \<in> B' \<Longrightarrow> X x = (\<lambda>z. if z = x then 1 else 0)" for x
```
```   994     using \<open>independent B'\<close>
```
```   995     by (rule independentD_unique[OF _ X(2) X(1)]) (auto intro: X simp: X(3)[symmetric])
```
```   996
```
```   997   define f' where "f' y = (if y \<in> B then f y else 0)" for y
```
```   998   define g where "g y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R f' x)" for y
```
```   999
```
```  1000   have g_f': "x \<in> B' \<Longrightarrow> g x = f' x" for x
```
```  1001     by (auto simp: g_def X_B')
```
```  1002
```
```  1003   have "linear g"
```
```  1004   proof
```
```  1005     fix x y
```
```  1006     have *: "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R f' z)
```
```  1007       = (\<Sum>z\<in>{z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}. (X x z + X y z) *\<^sub>R f' z)"
```
```  1008       by (intro sum.mono_neutral_cong_left) (auto intro: X)
```
```  1009     show "g (x + y) = g x + g y"
```
```  1010       unfolding g_def X_add *
```
```  1011       by (auto simp add: scaleR_add_left sum.distrib
```
```  1012                intro!: arg_cong2[where f="(+)"]  sum.mono_neutral_cong_right X)
```
```  1013   next
```
```  1014     show "g (r *\<^sub>R x) = r *\<^sub>R g x" for r x
```
```  1015       by (auto simp add: g_def X_cmult scaleR_sum_right intro!: sum.mono_neutral_cong_left X)
```
```  1016   qed
```
```  1017   moreover have "\<forall>x\<in>B. g x = f x"
```
```  1018     using \<open>B \<subseteq> B'\<close> by (auto simp: g_f' f'_def)
```
```  1019   moreover have "range g = span (f`B)"
```
```  1020     unfolding \<open>span B' = UNIV\<close>[symmetric] span_linear_image[OF \<open>linear g\<close>, symmetric]
```
```  1021   proof (rule span_subspace)
```
```  1022     have "g ` B' \<subseteq> f`B \<union> {0}"
```
```  1023       by (auto simp: g_f' f'_def)
```
```  1024     also have "\<dots> \<subseteq> span (f`B)"
```
```  1025       by (auto intro: span_superset span_0)
```
```  1026     finally show "g ` B' \<subseteq> span (f`B)"
```
```  1027       by auto
```
```  1028     have "x \<in> B \<Longrightarrow> f x = g x" for x
```
```  1029       using \<open>B \<subseteq> B'\<close> by (auto simp add: g_f' f'_def)
```
```  1030     then show "span (f ` B) \<subseteq> span (g ` B')"
```
```  1031       using \<open>B \<subseteq> B'\<close> by (intro span_mono) auto
```
```  1032   qed (rule subspace_span)
```
```  1033   ultimately show ?thesis
```
```  1034     by auto
```
```  1035 qed
```
```  1036
```
```  1037 lemma linear_independent_extend:
```
```  1038   "independent B \<Longrightarrow> \<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
```
```  1039   using linear_independent_extend_subspace[of B f] by auto
```
```  1040
```
```  1041 text \<open>Linear functions are equal on a subspace if they are on a spanning set.\<close>
```
```  1042
```
```  1043 lemma subspace_kernel:
```
```  1044   assumes lf: "linear f"
```
```  1045   shows "subspace {x. f x = 0}"
```
```  1046   apply (simp add: subspace_def)
```
```  1047   apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
```
```  1048   done
```
```  1049
```
```  1050 lemma linear_eq_0_span:
```
```  1051   assumes x: "x \<in> span B" and lf: "linear f" and f0: "\<And>x. x\<in>B \<Longrightarrow> f x = 0"
```
```  1052   shows "f x = 0"
```
```  1053   using x f0 subspace_kernel[OF lf] span_induct
```
```  1054   by blast
```
```  1055
```
```  1056 lemma linear_eq_0: "\<lbrakk>x \<in> S; linear f; S \<subseteq> span B; \<And>x. x\<in>B \<Longrightarrow> f x = 0\<rbrakk> \<Longrightarrow> f x = 0"
```
```  1057   using linear_eq_0_span[of x B f] by auto
```
```  1058
```
```  1059 lemma linear_eq_span: "\<lbrakk>x \<in> span B; linear f; linear g; \<And>x. x\<in>B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> f x = g x"
```
```  1060   using linear_eq_0_span[of x B "\<lambda>x. f x - g x"]  by (auto simp: linear_compose_sub)
```
```  1061
```
```  1062 lemma linear_eq: "\<lbrakk>x \<in> S; linear f; linear g; S \<subseteq> span B; \<And>x. x\<in>B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> f x = g x"
```
```  1063   using linear_eq_span[of _ B f g] by auto
```
```  1064
```
```  1065 text \<open>The degenerate case of the Exchange Lemma.\<close>
```
```  1066
```
```  1067 lemma spanning_subset_independent:
```
```  1068   assumes BA: "B \<subseteq> A"
```
```  1069     and iA: "independent A"
```
```  1070     and AsB: "A \<subseteq> span B"
```
```  1071   shows "A = B"
```
```  1072 proof
```
```  1073   show "B \<subseteq> A" by (rule BA)
```
```  1074
```
```  1075   from span_mono[OF BA] span_mono[OF AsB]
```
```  1076   have sAB: "span A = span B" unfolding span_span by blast
```
```  1077
```
```  1078   {
```
```  1079     fix x
```
```  1080     assume x: "x \<in> A"
```
```  1081     from iA have th0: "x \<notin> span (A - {x})"
```
```  1082       unfolding dependent_def using x by blast
```
```  1083     from x have xsA: "x \<in> span A"
```
```  1084       by (blast intro: span_superset)
```
```  1085     have "A - {x} \<subseteq> A" by blast
```
```  1086     then have th1: "span (A - {x}) \<subseteq> span A"
```
```  1087       by (metis span_mono)
```
```  1088     {
```
```  1089       assume xB: "x \<notin> B"
```
```  1090       from xB BA have "B \<subseteq> A - {x}"
```
```  1091         by blast
```
```  1092       then have "span B \<subseteq> span (A - {x})"
```
```  1093         by (metis span_mono)
```
```  1094       with th1 th0 sAB have "x \<notin> span A"
```
```  1095         by blast
```
```  1096       with x have False
```
```  1097         by (metis span_superset)
```
```  1098     }
```
```  1099     then have "x \<in> B" by blast
```
```  1100   }
```
```  1101   then show "A \<subseteq> B" by blast
```
```  1102 qed
```
```  1103
```
```  1104 text \<open>Relation between bases and injectivity/surjectivity of map.\<close>
```
```  1105
```
```  1106 lemma spanning_surjective_image:
```
```  1107   assumes us: "UNIV \<subseteq> span S"
```
```  1108     and lf: "linear f"
```
```  1109     and sf: "surj f"
```
```  1110   shows "UNIV \<subseteq> span (f ` S)"
```
```  1111 proof -
```
```  1112   have "UNIV \<subseteq> f ` UNIV"
```
```  1113     using sf by (auto simp add: surj_def)
```
```  1114   also have " \<dots> \<subseteq> span (f ` S)"
```
```  1115     using spans_image[OF lf us] .
```
```  1116   finally show ?thesis .
```
```  1117 qed
```
```  1118
```
```  1119 lemma independent_inj_on_image:
```
```  1120   assumes iS: "independent S"
```
```  1121     and lf: "linear f"
```
```  1122     and fi: "inj_on f (span S)"
```
```  1123   shows "independent (f ` S)"
```
```  1124 proof -
```
```  1125   {
```
```  1126     fix a
```
```  1127     assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
```
```  1128     have eq: "f ` S - {f a} = f ` (S - {a})"
```
```  1129       using fi \<open>a\<in>S\<close> by (auto simp add: inj_on_def span_superset)
```
```  1130     from a have "f a \<in> f ` span (S - {a})"
```
```  1131       unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
```
```  1132     then have "a \<in> span (S - {a})"
```
```  1133       by (rule inj_on_image_mem_iff_alt[OF fi, rotated])
```
```  1134          (insert span_mono[of "S - {a}" S], auto intro: span_superset \<open>a\<in>S\<close>)
```
```  1135     with a(1) iS have False
```
```  1136       by (simp add: dependent_def)
```
```  1137   }
```
```  1138   then show ?thesis
```
```  1139     unfolding dependent_def by blast
```
```  1140 qed
```
```  1141
```
```  1142 lemma independent_injective_image:
```
```  1143   "independent S \<Longrightarrow> linear f \<Longrightarrow> inj f \<Longrightarrow> independent (f ` S)"
```
```  1144   using independent_inj_on_image[of S f] by (auto simp: subset_inj_on)
```
```  1145
```
```  1146 text \<open>Detailed theorems about left and right invertibility in general case.\<close>
```
```  1147
```
```  1148 lemma linear_inj_on_left_inverse:
```
```  1149   assumes lf: "linear f" and fi: "inj_on f (span S)"
```
```  1150   shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span S. g (f x) = x)"
```
```  1151 proof -
```
```  1152   obtain B where "independent B" "B \<subseteq> S" "S \<subseteq> span B"
```
```  1153     using maximal_independent_subset[of S] by auto
```
```  1154   then have "span S = span B"
```
```  1155     unfolding span_eq by (auto simp: span_superset)
```
```  1156   with linear_independent_extend_subspace[OF independent_inj_on_image, OF \<open>independent B\<close> lf] fi
```
```  1157   obtain g where g: "linear g" "\<forall>x\<in>f ` B. g x = inv_into B f x" "range g = span (inv_into B f ` f ` B)"
```
```  1158     by fastforce
```
```  1159   have fB: "inj_on f B"
```
```  1160     using fi by (auto simp: \<open>span S = span B\<close> intro: subset_inj_on span_superset)
```
```  1161
```
```  1162   have "g (f x) = x" if "x \<in> span B" for x
```
```  1163   proof (rule linear_eq_span)
```
```  1164     show "linear (\<lambda>x. x)" "linear (\<lambda>x. g (f x))"
```
```  1165       using linear_id linear_compose[OF \<open>linear f\<close> \<open>linear g\<close>] by (auto simp: id_def comp_def)
```
```  1166     show "g (f x) = x" if "x \<in> B" for x
```
```  1167       using g fi \<open>span S = span B\<close>   by (simp add: fB that)
```
```  1168   qed (rule that)
```
```  1169   moreover
```
```  1170   have "inv_into B f ` f ` B \<subseteq> B"
```
```  1171     by (auto simp: fB)
```
```  1172   then have "range g \<subseteq> span S"
```
```  1173     unfolding g \<open>span S = span B\<close> by (intro span_mono)
```
```  1174   ultimately show ?thesis
```
```  1175     using \<open>span S = span B\<close> \<open>linear g\<close> by auto
```
```  1176 qed
```
```  1177
```
```  1178 lemma linear_injective_left_inverse: "linear f \<Longrightarrow> inj f \<Longrightarrow> \<exists>g. linear g \<and> g \<circ> f = id"
```
```  1179   using linear_inj_on_left_inverse[of f UNIV] by (auto simp: fun_eq_iff span_UNIV)
```
```  1180
```
```  1181 lemma linear_surj_right_inverse:
```
```  1182   assumes lf: "linear f" and sf: "span T \<subseteq> f`span S"
```
```  1183   shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span T. f (g x) = x)"
```
```  1184 proof -
```
```  1185   obtain B where "independent B" "B \<subseteq> T" "T \<subseteq> span B"
```
```  1186     using maximal_independent_subset[of T] by auto
```
```  1187   then have "span T = span B"
```
```  1188     unfolding span_eq by (auto simp: span_superset)
```
```  1189
```
```  1190   from linear_independent_extend_subspace[OF \<open>independent B\<close>, of "inv_into (span S) f"]
```
```  1191   obtain g where g: "linear g" "\<forall>x\<in>B. g x = inv_into (span S) f x" "range g = span (inv_into (span S) f`B)"
```
```  1192     by auto
```
```  1193   moreover have "x \<in> B \<Longrightarrow> f (inv_into (span S) f x) = x" for x
```
```  1194     using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (intro f_inv_into_f) (auto intro: span_superset)
```
```  1195   ultimately have "\<forall>x\<in>B. f (g x) = x"
```
```  1196     by auto
```
```  1197   then have "\<forall>x\<in>span B. f (g x) = x"
```
```  1198     using linear_id linear_compose[OF \<open>linear g\<close> \<open>linear f\<close>] linear_eq_span by fastforce
```
```  1199   moreover have "inv_into (span S) f ` B \<subseteq> span S"
```
```  1200     using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (auto intro: inv_into_into span_superset)
```
```  1201   then have "range g \<subseteq> span S"
```
```  1202     unfolding g by (intro span_minimal subspace_span) auto
```
```  1203   ultimately show ?thesis
```
```  1204     using \<open>linear g\<close> \<open>span T = span B\<close> by auto
```
```  1205 qed
```
```  1206
```
```  1207 lemma linear_surjective_right_inverse: "linear f \<Longrightarrow> surj f \<Longrightarrow> \<exists>g. linear g \<and> f \<circ> g = id"
```
```  1208   using linear_surj_right_inverse[of f UNIV UNIV]
```
```  1209   by (auto simp: span_UNIV fun_eq_iff)
```
```  1210
```
```  1211 text \<open>The general case of the Exchange Lemma, the key to what follows.\<close>
```
```  1212
```
```  1213 lemma exchange_lemma:
```
```  1214   assumes f:"finite t"
```
```  1215     and i: "independent s"
```
```  1216     and sp: "s \<subseteq> span t"
```
```  1217   shows "\<exists>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
```
```  1218   using f i sp
```
```  1219 proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
```
```  1220   case less
```
```  1221   note ft = \<open>finite t\<close> and s = \<open>independent s\<close> and sp = \<open>s \<subseteq> span t\<close>
```
```  1222   let ?P = "\<lambda>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
```
```  1223   let ?ths = "\<exists>t'. ?P t'"
```
```  1224   {
```
```  1225     assume "s \<subseteq> t"
```
```  1226     then have ?ths
```
```  1227       by (metis ft Un_commute sp sup_ge1)
```
```  1228   }
```
```  1229   moreover
```
```  1230   {
```
```  1231     assume st: "t \<subseteq> s"
```
```  1232     from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
```
```  1233     have ?ths
```
```  1234       by (metis Un_absorb sp)
```
```  1235   }
```
```  1236   moreover
```
```  1237   {
```
```  1238     assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
```
```  1239     from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
```
```  1240       by blast
```
```  1241     from b have "t - {b} - s \<subset> t - s"
```
```  1242       by blast
```
```  1243     then have cardlt: "card (t - {b} - s) < card (t - s)"
```
```  1244       using ft by (auto intro: psubset_card_mono)
```
```  1245     from b ft have ct0: "card t \<noteq> 0"
```
```  1246       by auto
```
```  1247     have ?ths
```
```  1248     proof cases
```
```  1249       assume stb: "s \<subseteq> span (t - {b})"
```
```  1250       from ft have ftb: "finite (t - {b})"
```
```  1251         by auto
```
```  1252       from less(1)[OF cardlt ftb s stb]
```
```  1253       obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
```
```  1254         and fu: "finite u" by blast
```
```  1255       let ?w = "insert b u"
```
```  1256       have th0: "s \<subseteq> insert b u"
```
```  1257         using u by blast
```
```  1258       from u(3) b have "u \<subseteq> s \<union> t"
```
```  1259         by blast
```
```  1260       then have th1: "insert b u \<subseteq> s \<union> t"
```
```  1261         using u b by blast
```
```  1262       have bu: "b \<notin> u"
```
```  1263         using b u by blast
```
```  1264       from u(1) ft b have "card u = (card t - 1)"
```
```  1265         by auto
```
```  1266       then have th2: "card (insert b u) = card t"
```
```  1267         using card_insert_disjoint[OF fu bu] ct0 by auto
```
```  1268       from u(4) have "s \<subseteq> span u" .
```
```  1269       also have "\<dots> \<subseteq> span (insert b u)"
```
```  1270         by (rule span_mono) blast
```
```  1271       finally have th3: "s \<subseteq> span (insert b u)" .
```
```  1272       from th0 th1 th2 th3 fu have th: "?P ?w"
```
```  1273         by blast
```
```  1274       from th show ?thesis by blast
```
```  1275     next
```
```  1276       assume stb: "\<not> s \<subseteq> span (t - {b})"
```
```  1277       from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
```
```  1278         by blast
```
```  1279       have ab: "a \<noteq> b"
```
```  1280         using a b by blast
```
```  1281       have at: "a \<notin> t"
```
```  1282         using a ab span_superset[of a "t- {b}"] by auto
```
```  1283       have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
```
```  1284         using cardlt ft a b by auto
```
```  1285       have ft': "finite (insert a (t - {b}))"
```
```  1286         using ft by auto
```
```  1287       {
```
```  1288         fix x
```
```  1289         assume xs: "x \<in> s"
```
```  1290         have t: "t \<subseteq> insert b (insert a (t - {b}))"
```
```  1291           using b by auto
```
```  1292         from b(1) have "b \<in> span t"
```
```  1293           by (simp add: span_superset)
```
```  1294         have bs: "b \<in> span (insert a (t - {b}))"
```
```  1295           apply (rule in_span_delete)
```
```  1296           using a sp unfolding subset_eq
```
```  1297           apply auto
```
```  1298           done
```
```  1299         from xs sp have "x \<in> span t"
```
```  1300           by blast
```
```  1301         with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
```
```  1302         from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
```
```  1303       }
```
```  1304       then have sp': "s \<subseteq> span (insert a (t - {b}))"
```
```  1305         by blast
```
```  1306       from less(1)[OF mlt ft' s sp'] obtain u where u:
```
```  1307         "card u = card (insert a (t - {b}))"
```
```  1308         "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
```
```  1309         "s \<subseteq> span u" by blast
```
```  1310       from u a b ft at ct0 have "?P u"
```
```  1311         by auto
```
```  1312       then show ?thesis by blast
```
```  1313     qed
```
```  1314   }
```
```  1315   ultimately show ?ths by blast
```
```  1316 qed
```
```  1317
```
```  1318 text \<open>This implies corresponding size bounds.\<close>
```
```  1319
```
```  1320 lemma independent_span_bound:
```
```  1321   assumes f: "finite t"
```
```  1322     and i: "independent s"
```
```  1323     and sp: "s \<subseteq> span t"
```
```  1324   shows "finite s \<and> card s \<le> card t"
```
```  1325   by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
```
```  1326
```
```  1327 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
```
```  1328 proof -
```
```  1329   have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
```
```  1330     by auto
```
```  1331   show ?thesis unfolding eq
```
```  1332     apply (rule finite_imageI)
```
```  1333     apply (rule finite)
```
```  1334     done
```
```  1335 qed
```
```  1336
```
```  1337
```
```  1338 subsection%unimportant \<open>More interesting properties of the norm.\<close>
```
```  1339
```
```  1340 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
```
```  1341   by auto
```
```  1342
```
```  1343 notation inner (infix "\<bullet>" 70)
```
```  1344
```
```  1345 lemma square_bound_lemma:
```
```  1346   fixes x :: real
```
```  1347   shows "x < (1 + x) * (1 + x)"
```
```  1348 proof -
```
```  1349   have "(x + 1/2)\<^sup>2 + 3/4 > 0"
```
```  1350     using zero_le_power2[of "x+1/2"] by arith
```
```  1351   then show ?thesis
```
```  1352     by (simp add: field_simps power2_eq_square)
```
```  1353 qed
```
```  1354
```
```  1355 lemma square_continuous:
```
```  1356   fixes e :: real
```
```  1357   shows "e > 0 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> \<bar>y * y - x * x\<bar> < e)"
```
```  1358   using isCont_power[OF continuous_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
```
```  1359   by (force simp add: power2_eq_square)
```
```  1360
```
```  1361
```
```  1362 lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
```
```  1363   by simp (* TODO: delete *)
```
```  1364
```
```  1365 lemma norm_triangle_sub:
```
```  1366   fixes x y :: "'a::real_normed_vector"
```
```  1367   shows "norm x \<le> norm y + norm (x - y)"
```
```  1368   using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
```
```  1369
```
```  1370 lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
```
```  1371   by (simp add: norm_eq_sqrt_inner)
```
```  1372
```
```  1373 lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
```
```  1374   by (simp add: norm_eq_sqrt_inner)
```
```  1375
```
```  1376 lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
```
```  1377   apply (subst order_eq_iff)
```
```  1378   apply (auto simp: norm_le)
```
```  1379   done
```
```  1380
```
```  1381 lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
```
```  1382   by (simp add: norm_eq_sqrt_inner)
```
```  1383
```
```  1384
```
```  1385 text\<open>Equality of vectors in terms of @{term "(\<bullet>)"} products.\<close>
```
```  1386
```
```  1387 lemma linear_componentwise:
```
```  1388   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
```
```  1389   assumes lf: "linear f"
```
```  1390   shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
```
```  1391 proof -
```
```  1392   have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
```
```  1393     by (simp add: inner_sum_left)
```
```  1394   then show ?thesis
```
```  1395     unfolding linear_sum_mul[OF lf, symmetric]
```
```  1396     unfolding euclidean_representation ..
```
```  1397 qed
```
```  1398
```
```  1399 lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
```
```  1400   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1401 proof
```
```  1402   assume ?lhs
```
```  1403   then show ?rhs by simp
```
```  1404 next
```
```  1405   assume ?rhs
```
```  1406   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
```
```  1407     by simp
```
```  1408   then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
```
```  1409     by (simp add: inner_diff inner_commute)
```
```  1410   then have "(x - y) \<bullet> (x - y) = 0"
```
```  1411     by (simp add: field_simps inner_diff inner_commute)
```
```  1412   then show "x = y" by simp
```
```  1413 qed
```
```  1414
```
```  1415 lemma norm_triangle_half_r:
```
```  1416   "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
```
```  1417   using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
```
```  1418
```
```  1419 lemma norm_triangle_half_l:
```
```  1420   assumes "norm (x - y) < e / 2"
```
```  1421     and "norm (x' - y) < e / 2"
```
```  1422   shows "norm (x - x') < e"
```
```  1423   using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
```
```  1424   unfolding dist_norm[symmetric] .
```
```  1425
```
```  1426 lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
```
```  1427   by (rule norm_triangle_ineq [THEN order_trans])
```
```  1428
```
```  1429 lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
```
```  1430   by (rule norm_triangle_ineq [THEN le_less_trans])
```
```  1431
```
```  1432 lemma abs_triangle_half_r:
```
```  1433   fixes y :: "'a::linordered_field"
```
```  1434   shows "abs (y - x1) < e / 2 \<Longrightarrow> abs (y - x2) < e / 2 \<Longrightarrow> abs (x1 - x2) < e"
```
```  1435   by linarith
```
```  1436
```
```  1437 lemma abs_triangle_half_l:
```
```  1438   fixes y :: "'a::linordered_field"
```
```  1439   assumes "abs (x - y) < e / 2"
```
```  1440     and "abs (x' - y) < e / 2"
```
```  1441   shows "abs (x - x') < e"
```
```  1442   using assms by linarith
```
```  1443
```
```  1444 lemma sum_clauses:
```
```  1445   shows "sum f {} = 0"
```
```  1446     and "finite S \<Longrightarrow> sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
```
```  1447   by (auto simp add: insert_absorb)
```
```  1448
```
```  1449 lemma sum_norm_bound:
```
```  1450   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  1451   assumes K: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> K"
```
```  1452   shows "norm (sum f S) \<le> of_nat (card S)*K"
```
```  1453   using sum_norm_le[OF K] sum_constant[symmetric]
```
```  1454   by simp
```
```  1455
```
```  1456 lemma sum_group:
```
```  1457   assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
```
```  1458   shows "sum (\<lambda>y. sum g {x. x \<in> S \<and> f x = y}) T = sum g S"
```
```  1459   apply (subst sum_image_gen[OF fS, of g f])
```
```  1460   apply (rule sum.mono_neutral_right[OF fT fST])
```
```  1461   apply (auto intro: sum.neutral)
```
```  1462   done
```
```  1463
```
```  1464 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
```
```  1465 proof
```
```  1466   assume "\<forall>x. x \<bullet> y = x \<bullet> z"
```
```  1467   then have "\<forall>x. x \<bullet> (y - z) = 0"
```
```  1468     by (simp add: inner_diff)
```
```  1469   then have "(y - z) \<bullet> (y - z) = 0" ..
```
```  1470   then show "y = z" by simp
```
```  1471 qed simp
```
```  1472
```
```  1473 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
```
```  1474 proof
```
```  1475   assume "\<forall>z. x \<bullet> z = y \<bullet> z"
```
```  1476   then have "\<forall>z. (x - y) \<bullet> z = 0"
```
```  1477     by (simp add: inner_diff)
```
```  1478   then have "(x - y) \<bullet> (x - y) = 0" ..
```
```  1479   then show "x = y" by simp
```
```  1480 qed simp
```
```  1481
```
```  1482
```
```  1483 subsection \<open>Orthogonality.\<close>
```
```  1484
```
```  1485 definition%important (in real_inner) "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
```
```  1486
```
```  1487 context real_inner
```
```  1488 begin
```
```  1489
```
```  1490 lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0"
```
```  1491   by (simp add: orthogonal_def)
```
```  1492
```
```  1493 lemma orthogonal_clauses:
```
```  1494   "orthogonal a 0"
```
```  1495   "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
```
```  1496   "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
```
```  1497   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
```
```  1498   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
```
```  1499   "orthogonal 0 a"
```
```  1500   "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
```
```  1501   "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
```
```  1502   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
```
```  1503   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
```
```  1504   unfolding orthogonal_def inner_add inner_diff by auto
```
```  1505
```
```  1506 end
```
```  1507
```
```  1508 lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
```
```  1509   by (simp add: orthogonal_def inner_commute)
```
```  1510
```
```  1511 lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x"
```
```  1512   by (rule ext) (simp add: orthogonal_def)
```
```  1513
```
```  1514 lemma pairwise_ortho_scaleR:
```
```  1515     "pairwise (\<lambda>i j. orthogonal (f i) (g j)) B
```
```  1516     \<Longrightarrow> pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B"
```
```  1517   by (auto simp: pairwise_def orthogonal_clauses)
```
```  1518
```
```  1519 lemma orthogonal_rvsum:
```
```  1520     "\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (sum f s)"
```
```  1521   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
```
```  1522
```
```  1523 lemma orthogonal_lvsum:
```
```  1524     "\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (sum f s) y"
```
```  1525   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
```
```  1526
```
```  1527 lemma norm_add_Pythagorean:
```
```  1528   assumes "orthogonal a b"
```
```  1529     shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2"
```
```  1530 proof -
```
```  1531   from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)"
```
```  1532     by (simp add: algebra_simps orthogonal_def inner_commute)
```
```  1533   then show ?thesis
```
```  1534     by (simp add: power2_norm_eq_inner)
```
```  1535 qed
```
```  1536
```
```  1537 lemma norm_sum_Pythagorean:
```
```  1538   assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I"
```
```  1539     shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
```
```  1540 using assms
```
```  1541 proof (induction I rule: finite_induct)
```
```  1542   case empty then show ?case by simp
```
```  1543 next
```
```  1544   case (insert x I)
```
```  1545   then have "orthogonal (f x) (sum f I)"
```
```  1546     by (metis pairwise_insert orthogonal_rvsum)
```
```  1547   with insert show ?case
```
```  1548     by (simp add: pairwise_insert norm_add_Pythagorean)
```
```  1549 qed
```
```  1550
```
```  1551
```
```  1552 subsection \<open>Bilinear functions.\<close>
```
```  1553
```
```  1554 definition%important "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
```
```  1555
```
```  1556 lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
```
```  1557   by (simp add: bilinear_def linear_iff)
```
```  1558
```
```  1559 lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
```
```  1560   by (simp add: bilinear_def linear_iff)
```
```  1561
```
```  1562 lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
```
```  1563   by (simp add: bilinear_def linear_iff)
```
```  1564
```
```  1565 lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
```
```  1566   by (simp add: bilinear_def linear_iff)
```
```  1567
```
```  1568 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
```
```  1569   by (drule bilinear_lmul [of _ "- 1"]) simp
```
```  1570
```
```  1571 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
```
```  1572   by (drule bilinear_rmul [of _ _ "- 1"]) simp
```
```  1573
```
```  1574 lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
```
```  1575   using add_left_imp_eq[of x y 0] by auto
```
```  1576
```
```  1577 lemma bilinear_lzero:
```
```  1578   assumes "bilinear h"
```
```  1579   shows "h 0 x = 0"
```
```  1580   using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
```
```  1581
```
```  1582 lemma bilinear_rzero:
```
```  1583   assumes "bilinear h"
```
```  1584   shows "h x 0 = 0"
```
```  1585   using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
```
```  1586
```
```  1587 lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
```
```  1588   using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
```
```  1589
```
```  1590 lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
```
```  1591   using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
```
```  1592
```
```  1593 lemma bilinear_sum:
```
```  1594   assumes bh: "bilinear h"
```
```  1595     and fS: "finite S"
```
```  1596     and fT: "finite T"
```
```  1597   shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
```
```  1598 proof -
```
```  1599   have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S"
```
```  1600     apply (rule linear_sum[unfolded o_def])
```
```  1601     using bh fS
```
```  1602     apply (auto simp add: bilinear_def)
```
```  1603     done
```
```  1604   also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S"
```
```  1605     apply (rule sum.cong, simp)
```
```  1606     apply (rule linear_sum[unfolded o_def])
```
```  1607     using bh fT
```
```  1608     apply (auto simp add: bilinear_def)
```
```  1609     done
```
```  1610   finally show ?thesis
```
```  1611     unfolding sum.cartesian_product .
```
```  1612 qed
```
```  1613
```
```  1614
```
```  1615 subsection \<open>Adjoints.\<close>
```
```  1616
```
```  1617 definition%important "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
```
```  1618
```
```  1619 lemma adjoint_unique:
```
```  1620   assumes "\<forall>x y. inner (f x) y = inner x (g y)"
```
```  1621   shows "adjoint f = g"
```
```  1622   unfolding adjoint_def
```
```  1623 proof (rule some_equality)
```
```  1624   show "\<forall>x y. inner (f x) y = inner x (g y)"
```
```  1625     by (rule assms)
```
```  1626 next
```
```  1627   fix h
```
```  1628   assume "\<forall>x y. inner (f x) y = inner x (h y)"
```
```  1629   then have "\<forall>x y. inner x (g y) = inner x (h y)"
```
```  1630     using assms by simp
```
```  1631   then have "\<forall>x y. inner x (g y - h y) = 0"
```
```  1632     by (simp add: inner_diff_right)
```
```  1633   then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
```
```  1634     by simp
```
```  1635   then have "\<forall>y. h y = g y"
```
```  1636     by simp
```
```  1637   then show "h = g" by (simp add: ext)
```
```  1638 qed
```
```  1639
```
```  1640 text \<open>TODO: The following lemmas about adjoints should hold for any
```
```  1641   Hilbert space (i.e. complete inner product space).
```
```  1642   (see \<^url>\<open>http://en.wikipedia.org/wiki/Hermitian_adjoint\<close>)
```
```  1643 \<close>
```
```  1644
```
```  1645 lemma adjoint_works:
```
```  1646   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  1647   assumes lf: "linear f"
```
```  1648   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```  1649 proof -
```
```  1650   have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
```
```  1651   proof (intro allI exI)
```
```  1652     fix y :: "'m" and x
```
```  1653     let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
```
```  1654     have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
```
```  1655       by (simp add: euclidean_representation)
```
```  1656     also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
```
```  1657       unfolding linear_sum[OF lf]
```
```  1658       by (simp add: linear_cmul[OF lf])
```
```  1659     finally show "f x \<bullet> y = x \<bullet> ?w"
```
```  1660       by (simp add: inner_sum_left inner_sum_right mult.commute)
```
```  1661   qed
```
```  1662   then show ?thesis
```
```  1663     unfolding adjoint_def choice_iff
```
```  1664     by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
```
```  1665 qed
```
```  1666
```
```  1667 lemma adjoint_clauses:
```
```  1668   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  1669   assumes lf: "linear f"
```
```  1670   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```  1671     and "adjoint f y \<bullet> x = y \<bullet> f x"
```
```  1672   by (simp_all add: adjoint_works[OF lf] inner_commute)
```
```  1673
```
```  1674 lemma adjoint_linear:
```
```  1675   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  1676   assumes lf: "linear f"
```
```  1677   shows "linear (adjoint f)"
```
```  1678   by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
```
```  1679     adjoint_clauses[OF lf] inner_distrib)
```
```  1680
```
```  1681 lemma adjoint_adjoint:
```
```  1682   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  1683   assumes lf: "linear f"
```
```  1684   shows "adjoint (adjoint f) = f"
```
```  1685   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
```
```  1686
```
```  1687
```
```  1688 subsection%unimportant \<open>Interlude: Some properties of real sets\<close>
```
```  1689
```
```  1690 lemma seq_mono_lemma:
```
```  1691   assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
```
```  1692     and "\<forall>n \<ge> m. e n \<le> e m"
```
```  1693   shows "\<forall>n \<ge> m. d n < e m"
```
```  1694   using assms
```
```  1695   apply auto
```
```  1696   apply (erule_tac x="n" in allE)
```
```  1697   apply (erule_tac x="n" in allE)
```
```  1698   apply auto
```
```  1699   done
```
```  1700
```
```  1701 lemma infinite_enumerate:
```
```  1702   assumes fS: "infinite S"
```
```  1703   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)"
```
```  1704   unfolding strict_mono_def
```
```  1705   using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
```
```  1706
```
```  1707 lemma approachable_lt_le: "(\<exists>(d::real) > 0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
```
```  1708   apply auto
```
```  1709   apply (rule_tac x="d/2" in exI)
```
```  1710   apply auto
```
```  1711   done
```
```  1712
```
```  1713 lemma approachable_lt_le2:  \<comment> \<open>like the above, but pushes aside an extra formula\<close>
```
```  1714     "(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
```
```  1715   apply auto
```
```  1716   apply (rule_tac x="d/2" in exI, auto)
```
```  1717   done
```
```  1718
```
```  1719 lemma triangle_lemma:
```
```  1720   fixes x y z :: real
```
```  1721   assumes x: "0 \<le> x"
```
```  1722     and y: "0 \<le> y"
```
```  1723     and z: "0 \<le> z"
```
```  1724     and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
```
```  1725   shows "x \<le> y + z"
```
```  1726 proof -
```
```  1727   have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
```
```  1728     using z y by simp
```
```  1729   with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
```
```  1730     by (simp add: power2_eq_square field_simps)
```
```  1731   from y z have yz: "y + z \<ge> 0"
```
```  1732     by arith
```
```  1733   from power2_le_imp_le[OF th yz] show ?thesis .
```
```  1734 qed
```
```  1735
```
```  1736
```
```  1737
```
```  1738 subsection \<open>Archimedean properties and useful consequences\<close>
```
```  1739
```
```  1740 text\<open>Bernoulli's inequality\<close>
```
```  1741 proposition%important Bernoulli_inequality:
```
```  1742   fixes x :: real
```
```  1743   assumes "-1 \<le> x"
```
```  1744     shows "1 + n * x \<le> (1 + x) ^ n"
```
```  1745 proof%unimportant (induct n)
```
```  1746   case 0
```
```  1747   then show ?case by simp
```
```  1748 next
```
```  1749   case (Suc n)
```
```  1750   have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
```
```  1751     by (simp add: algebra_simps)
```
```  1752   also have "... = (1 + x) * (1 + n*x)"
```
```  1753     by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
```
```  1754   also have "... \<le> (1 + x) ^ Suc n"
```
```  1755     using Suc.hyps assms mult_left_mono by fastforce
```
```  1756   finally show ?case .
```
```  1757 qed
```
```  1758
```
```  1759 corollary Bernoulli_inequality_even:
```
```  1760   fixes x :: real
```
```  1761   assumes "even n"
```
```  1762     shows "1 + n * x \<le> (1 + x) ^ n"
```
```  1763 proof (cases "-1 \<le> x \<or> n=0")
```
```  1764   case True
```
```  1765   then show ?thesis
```
```  1766     by (auto simp: Bernoulli_inequality)
```
```  1767 next
```
```  1768   case False
```
```  1769   then have "real n \<ge> 1"
```
```  1770     by simp
```
```  1771   with False have "n * x \<le> -1"
```
```  1772     by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
```
```  1773   then have "1 + n * x \<le> 0"
```
```  1774     by auto
```
```  1775   also have "... \<le> (1 + x) ^ n"
```
```  1776     using assms
```
```  1777     using zero_le_even_power by blast
```
```  1778   finally show ?thesis .
```
```  1779 qed
```
```  1780
```
```  1781 corollary real_arch_pow:
```
```  1782   fixes x :: real
```
```  1783   assumes x: "1 < x"
```
```  1784   shows "\<exists>n. y < x^n"
```
```  1785 proof -
```
```  1786   from x have x0: "x - 1 > 0"
```
```  1787     by arith
```
```  1788   from reals_Archimedean3[OF x0, rule_format, of y]
```
```  1789   obtain n :: nat where n: "y < real n * (x - 1)" by metis
```
```  1790   from x0 have x00: "x- 1 \<ge> -1" by arith
```
```  1791   from Bernoulli_inequality[OF x00, of n] n
```
```  1792   have "y < x^n" by auto
```
```  1793   then show ?thesis by metis
```
```  1794 qed
```
```  1795
```
```  1796 corollary real_arch_pow_inv:
```
```  1797   fixes x y :: real
```
```  1798   assumes y: "y > 0"
```
```  1799     and x1: "x < 1"
```
```  1800   shows "\<exists>n. x^n < y"
```
```  1801 proof (cases "x > 0")
```
```  1802   case True
```
```  1803   with x1 have ix: "1 < 1/x" by (simp add: field_simps)
```
```  1804   from real_arch_pow[OF ix, of "1/y"]
```
```  1805   obtain n where n: "1/y < (1/x)^n" by blast
```
```  1806   then show ?thesis using y \<open>x > 0\<close>
```
```  1807     by (auto simp add: field_simps)
```
```  1808 next
```
```  1809   case False
```
```  1810   with y x1 show ?thesis
```
```  1811     apply auto
```
```  1812     apply (rule exI[where x=1])
```
```  1813     apply auto
```
```  1814     done
```
```  1815 qed
```
```  1816
```
```  1817 lemma forall_pos_mono:
```
```  1818   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```  1819     (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
```
```  1820   by (metis real_arch_inverse)
```
```  1821
```
```  1822 lemma forall_pos_mono_1:
```
```  1823   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```  1824     (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
```
```  1825   apply (rule forall_pos_mono)
```
```  1826   apply auto
```
```  1827   apply (metis Suc_pred of_nat_Suc)
```
```  1828   done
```
```  1829
```
```  1830
```
```  1831 subsection%unimportant \<open>Euclidean Spaces as Typeclass\<close>
```
```  1832
```
```  1833 lemma independent_Basis: "independent Basis"
```
```  1834   unfolding dependent_def
```
```  1835   apply (subst span_finite)
```
```  1836   apply simp
```
```  1837   apply clarify
```
```  1838   apply (drule_tac f="inner a" in arg_cong)
```
```  1839   apply (simp add: inner_Basis inner_sum_right eq_commute)
```
```  1840   done
```
```  1841
```
```  1842 lemma span_Basis [simp]: "span Basis = UNIV"
```
```  1843   unfolding span_finite [OF finite_Basis]
```
```  1844   by (fast intro: euclidean_representation)
```
```  1845
```
```  1846 lemma in_span_Basis: "x \<in> span Basis"
```
```  1847   unfolding span_Basis ..
```
```  1848
```
```  1849 lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
```
```  1850   by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
```
```  1851
```
```  1852 lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
```
```  1853   by (metis Basis_le_norm order_trans)
```
```  1854
```
```  1855 lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
```
```  1856   by (metis Basis_le_norm le_less_trans)
```
```  1857
```
```  1858 lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
```
```  1859   apply (subst euclidean_representation[of x, symmetric])
```
```  1860   apply (rule order_trans[OF norm_sum])
```
```  1861   apply (auto intro!: sum_mono)
```
```  1862   done
```
```  1863
```
```  1864 lemma sum_norm_allsubsets_bound:
```
```  1865   fixes f :: "'a \<Rightarrow> 'n::euclidean_space"
```
```  1866   assumes fP: "finite P"
```
```  1867     and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
```
```  1868   shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
```
```  1869 proof -
```
```  1870   have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
```
```  1871     by (rule sum_mono) (rule norm_le_l1)
```
```  1872   also have "(\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>) = (\<Sum>b\<in>Basis. \<Sum>x\<in>P. \<bar>f x \<bullet> b\<bar>)"
```
```  1873     by (rule sum.swap)
```
```  1874   also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
```
```  1875   proof (rule sum_bounded_above)
```
```  1876     fix i :: 'n
```
```  1877     assume i: "i \<in> Basis"
```
```  1878     have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
```
```  1879       norm ((\<Sum>x\<in>P \<inter> - {x. f x \<bullet> i < 0}. f x) \<bullet> i) + norm ((\<Sum>x\<in>P \<inter> {x. f x \<bullet> i < 0}. f x) \<bullet> i)"
```
```  1880       by (simp add: abs_real_def sum.If_cases[OF fP] sum_negf norm_triangle_ineq4 inner_sum_left
```
```  1881         del: real_norm_def)
```
```  1882     also have "\<dots> \<le> e + e"
```
```  1883       unfolding real_norm_def
```
```  1884       by (intro add_mono norm_bound_Basis_le i fPs) auto
```
```  1885     finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
```
```  1886   qed
```
```  1887   also have "\<dots> = 2 * real DIM('n) * e" by simp
```
```  1888   finally show ?thesis .
```
```  1889 qed
```
```  1890
```
```  1891
```
```  1892 subsection%unimportant \<open>Linearity and Bilinearity continued\<close>
```
```  1893
```
```  1894 lemma linear_bounded:
```
```  1895   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1896   assumes lf: "linear f"
```
```  1897   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```  1898 proof
```
```  1899   let ?B = "\<Sum>b\<in>Basis. norm (f b)"
```
```  1900   show "\<forall>x. norm (f x) \<le> ?B * norm x"
```
```  1901   proof
```
```  1902     fix x :: 'a
```
```  1903     let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
```
```  1904     have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
```
```  1905       unfolding euclidean_representation ..
```
```  1906     also have "\<dots> = norm (sum ?g Basis)"
```
```  1907       by (simp add: linear_sum [OF lf] linear_cmul [OF lf])
```
```  1908     finally have th0: "norm (f x) = norm (sum ?g Basis)" .
```
```  1909     have th: "norm (?g i) \<le> norm (f i) * norm x" if "i \<in> Basis" for i
```
```  1910     proof -
```
```  1911       from Basis_le_norm[OF that, of x]
```
```  1912       show "norm (?g i) \<le> norm (f i) * norm x"
```
```  1913         unfolding norm_scaleR
```
```  1914         apply (subst mult.commute)
```
```  1915         apply (rule mult_mono)
```
```  1916         apply (auto simp add: field_simps)
```
```  1917         done
```
```  1918     qed
```
```  1919     from sum_norm_le[of _ ?g, OF th]
```
```  1920     show "norm (f x) \<le> ?B * norm x"
```
```  1921       unfolding th0 sum_distrib_right by metis
```
```  1922   qed
```
```  1923 qed
```
```  1924
```
```  1925 lemma linear_conv_bounded_linear:
```
```  1926   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1927   shows "linear f \<longleftrightarrow> bounded_linear f"
```
```  1928 proof
```
```  1929   assume "linear f"
```
```  1930   then interpret f: linear f .
```
```  1931   show "bounded_linear f"
```
```  1932   proof
```
```  1933     have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```  1934       using \<open>linear f\<close> by (rule linear_bounded)
```
```  1935     then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1936       by (simp add: mult.commute)
```
```  1937   qed
```
```  1938 next
```
```  1939   assume "bounded_linear f"
```
```  1940   then interpret f: bounded_linear f .
```
```  1941   show "linear f" ..
```
```  1942 qed
```
```  1943
```
```  1944 lemmas linear_linear = linear_conv_bounded_linear[symmetric]
```
```  1945
```
```  1946 lemma linear_bounded_pos:
```
```  1947   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1948   assumes lf: "linear f"
```
```  1949  obtains B where "B > 0" "\<And>x. norm (f x) \<le> B * norm x"
```
```  1950 proof -
```
```  1951   have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
```
```  1952     using lf unfolding linear_conv_bounded_linear
```
```  1953     by (rule bounded_linear.pos_bounded)
```
```  1954   with that show ?thesis
```
```  1955     by (auto simp: mult.commute)
```
```  1956 qed
```
```  1957
```
```  1958 lemma linear_invertible_bounded_below_pos:
```
```  1959   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
```
```  1960   assumes "linear f" "linear g" "g \<circ> f = id"
```
```  1961   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
```
```  1962 proof -
```
```  1963   obtain B where "B > 0" and B: "\<And>x. norm (g x) \<le> B * norm x"
```
```  1964     using linear_bounded_pos [OF \<open>linear g\<close>] by blast
```
```  1965   show thesis
```
```  1966   proof
```
```  1967     show "0 < 1/B"
```
```  1968       by (simp add: \<open>B > 0\<close>)
```
```  1969     show "1/B * norm x \<le> norm (f x)" for x
```
```  1970     proof -
```
```  1971       have "1/B * norm x = 1/B * norm (g (f x))"
```
```  1972         using assms by (simp add: pointfree_idE)
```
```  1973       also have "\<dots> \<le> norm (f x)"
```
```  1974         using B [of "f x"] by (simp add: \<open>B > 0\<close> mult.commute pos_divide_le_eq)
```
```  1975       finally show ?thesis .
```
```  1976     qed
```
```  1977   qed
```
```  1978 qed
```
```  1979
```
```  1980 lemma linear_inj_bounded_below_pos:
```
```  1981   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
```
```  1982   assumes "linear f" "inj f"
```
```  1983   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
```
```  1984   using linear_injective_left_inverse [OF assms] linear_invertible_bounded_below_pos assms by blast
```
```  1985
```
```  1986 lemma bounded_linearI':
```
```  1987   fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1988   assumes "\<And>x y. f (x + y) = f x + f y"
```
```  1989     and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```  1990   shows "bounded_linear f"
```
```  1991   unfolding linear_conv_bounded_linear[symmetric]
```
```  1992   by (rule linearI[OF assms])
```
```  1993
```
```  1994 lemma bilinear_bounded:
```
```  1995   fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
```
```  1996   assumes bh: "bilinear h"
```
```  1997   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```  1998 proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
```
```  1999   fix x :: 'm
```
```  2000   fix y :: 'n
```
```  2001   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))"
```
```  2002     apply (subst euclidean_representation[where 'a='m])
```
```  2003     apply (subst euclidean_representation[where 'a='n])
```
```  2004     apply rule
```
```  2005     done
```
```  2006   also have "\<dots> = norm (sum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
```
```  2007     unfolding bilinear_sum[OF bh finite_Basis finite_Basis] ..
```
```  2008   finally have th: "norm (h x y) = \<dots>" .
```
```  2009   show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
```
```  2010     apply (auto simp add: sum_distrib_right th sum.cartesian_product)
```
```  2011     apply (rule sum_norm_le)
```
```  2012     apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
```
```  2013       field_simps simp del: scaleR_scaleR)
```
```  2014     apply (rule mult_mono)
```
```  2015     apply (auto simp add: zero_le_mult_iff Basis_le_norm)
```
```  2016     apply (rule mult_mono)
```
```  2017     apply (auto simp add: zero_le_mult_iff Basis_le_norm)
```
```  2018     done
```
```  2019 qed
```
```  2020
```
```  2021 lemma bilinear_conv_bounded_bilinear:
```
```  2022   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```  2023   shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
```
```  2024 proof
```
```  2025   assume "bilinear h"
```
```  2026   show "bounded_bilinear h"
```
```  2027   proof
```
```  2028     fix x y z
```
```  2029     show "h (x + y) z = h x z + h y z"
```
```  2030       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
```
```  2031   next
```
```  2032     fix x y z
```
```  2033     show "h x (y + z) = h x y + h x z"
```
```  2034       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
```
```  2035   next
```
```  2036     fix r x y
```
```  2037     show "h (scaleR r x) y = scaleR r (h x y)"
```
```  2038       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
```
```  2039       by simp
```
```  2040   next
```
```  2041     fix r x y
```
```  2042     show "h x (scaleR r y) = scaleR r (h x y)"
```
```  2043       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
```
```  2044       by simp
```
```  2045   next
```
```  2046     have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```  2047       using \<open>bilinear h\<close> by (rule bilinear_bounded)
```
```  2048     then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
```
```  2049       by (simp add: ac_simps)
```
```  2050   qed
```
```  2051 next
```
```  2052   assume "bounded_bilinear h"
```
```  2053   then interpret h: bounded_bilinear h .
```
```  2054   show "bilinear h"
```
```  2055     unfolding bilinear_def linear_conv_bounded_linear
```
```  2056     using h.bounded_linear_left h.bounded_linear_right by simp
```
```  2057 qed
```
```  2058
```
```  2059 lemma bilinear_bounded_pos:
```
```  2060   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```  2061   assumes bh: "bilinear h"
```
```  2062   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```  2063 proof -
```
```  2064   have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
```
```  2065     using bh [unfolded bilinear_conv_bounded_bilinear]
```
```  2066     by (rule bounded_bilinear.pos_bounded)
```
```  2067   then show ?thesis
```
```  2068     by (simp only: ac_simps)
```
```  2069 qed
```
```  2070
```
```  2071 lemma bounded_linear_imp_has_derivative:
```
```  2072      "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
```
```  2073   by (simp add: has_derivative_def bounded_linear.linear linear_diff)
```
```  2074
```
```  2075 lemma linear_imp_has_derivative:
```
```  2076   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  2077   shows "linear f \<Longrightarrow> (f has_derivative f) net"
```
```  2078 by (simp add: has_derivative_def linear_conv_bounded_linear linear_diff)
```
```  2079
```
```  2080 lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net"
```
```  2081   using bounded_linear_imp_has_derivative differentiable_def by blast
```
```  2082
```
```  2083 lemma linear_imp_differentiable:
```
```  2084   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  2085   shows "linear f \<Longrightarrow> f differentiable net"
```
```  2086 by (metis linear_imp_has_derivative differentiable_def)
```
```  2087
```
```  2088
```
```  2089 subsection%unimportant \<open>We continue.\<close>
```
```  2090
```
```  2091 lemma independent_bound:
```
```  2092   fixes S :: "'a::euclidean_space set"
```
```  2093   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
```
```  2094   using independent_span_bound[OF finite_Basis, of S] by auto
```
```  2095
```
```  2096 corollary
```
```  2097   fixes S :: "'a::euclidean_space set"
```
```  2098   assumes "independent S"
```
```  2099   shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)"
```
```  2100 using assms independent_bound by auto
```
```  2101
```
```  2102 lemma independent_explicit:
```
```  2103   fixes B :: "'a::euclidean_space set"
```
```  2104   shows "independent B \<longleftrightarrow>
```
```  2105          finite B \<and> (\<forall>c. (\<Sum>v\<in>B. c v *\<^sub>R v) = 0 \<longrightarrow> (\<forall>v \<in> B. c v = 0))"
```
```  2106 apply (cases "finite B")
```
```  2107  apply (force simp: dependent_finite)
```
```  2108 using independent_bound
```
```  2109 apply auto
```
```  2110 done
```
```  2111
```
```  2112 lemma dependent_biggerset:
```
```  2113   fixes S :: "'a::euclidean_space set"
```
```  2114   shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
```
```  2115   by (metis independent_bound not_less)
```
```  2116
```
```  2117 text \<open>Notion of dimension.\<close>
```
```  2118
```
```  2119 definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
```
```  2120
```
```  2121 lemma basis_exists:
```
```  2122   "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
```
```  2123   unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n)"]
```
```  2124   using maximal_independent_subset[of V] independent_bound
```
```  2125   by auto
```
```  2126
```
```  2127 corollary dim_le_card:
```
```  2128   fixes s :: "'a::euclidean_space set"
```
```  2129   shows "finite s \<Longrightarrow> dim s \<le> card s"
```
```  2130 by (metis basis_exists card_mono)
```
```  2131
```
```  2132 text \<open>Consequences of independence or spanning for cardinality.\<close>
```
```  2133
```
```  2134 lemma independent_card_le_dim:
```
```  2135   fixes B :: "'a::euclidean_space set"
```
```  2136   assumes "B \<subseteq> V"
```
```  2137     and "independent B"
```
```  2138   shows "card B \<le> dim V"
```
```  2139 proof -
```
```  2140   from basis_exists[of V] \<open>B \<subseteq> V\<close>
```
```  2141   obtain B' where "independent B'"
```
```  2142     and "B \<subseteq> span B'"
```
```  2143     and "card B' = dim V"
```
```  2144     by blast
```
```  2145   with independent_span_bound[OF _ \<open>independent B\<close> \<open>B \<subseteq> span B'\<close>] independent_bound[of B']
```
```  2146   show ?thesis by auto
```
```  2147 qed
```
```  2148
```
```  2149 lemma span_card_ge_dim:
```
```  2150   fixes B :: "'a::euclidean_space set"
```
```  2151   shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
```
```  2152   by (metis basis_exists[of V] independent_span_bound subset_trans)
```
```  2153
```
```  2154 lemma basis_card_eq_dim:
```
```  2155   fixes V :: "'a::euclidean_space set"
```
```  2156   shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
```
```  2157   by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
```
```  2158
```
```  2159 lemma dim_unique:
```
```  2160   fixes B :: "'a::euclidean_space set"
```
```  2161   shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
```
```  2162   by (metis basis_card_eq_dim)
```
```  2163
```
```  2164 text \<open>More lemmas about dimension.\<close>
```
```  2165
```
```  2166 lemma dim_UNIV [simp]: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
```
```  2167   using independent_Basis
```
```  2168   by (intro dim_unique[of Basis]) auto
```
```  2169
```
```  2170 lemma dim_subset:
```
```  2171   fixes S :: "'a::euclidean_space set"
```
```  2172   shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
```
```  2173   using basis_exists[of T] basis_exists[of S]
```
```  2174   by (metis independent_card_le_dim subset_trans)
```
```  2175
```
```  2176 lemma dim_subset_UNIV:
```
```  2177   fixes S :: "'a::euclidean_space set"
```
```  2178   shows "dim S \<le> DIM('a)"
```
```  2179   by (metis dim_subset subset_UNIV dim_UNIV)
```
```  2180
```
```  2181 text \<open>Converses to those.\<close>
```
```  2182
```
```  2183 lemma card_ge_dim_independent:
```
```  2184   fixes B :: "'a::euclidean_space set"
```
```  2185   assumes BV: "B \<subseteq> V"
```
```  2186     and iB: "independent B"
```
```  2187     and dVB: "dim V \<le> card B"
```
```  2188   shows "V \<subseteq> span B"
```
```  2189 proof
```
```  2190   fix a
```
```  2191   assume aV: "a \<in> V"
```
```  2192   {
```
```  2193     assume aB: "a \<notin> span B"
```
```  2194     then have iaB: "independent (insert a B)"
```
```  2195       using iB aV BV by (simp add: independent_insert)
```
```  2196     from aV BV have th0: "insert a B \<subseteq> V"
```
```  2197       by blast
```
```  2198     from aB have "a \<notin>B"
```
```  2199       by (auto simp add: span_superset)
```
```  2200     with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
```
```  2201     have False by auto
```
```  2202   }
```
```  2203   then show "a \<in> span B" by blast
```
```  2204 qed
```
```  2205
```
```  2206 lemma card_le_dim_spanning:
```
```  2207   assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
```
```  2208     and VB: "V \<subseteq> span B"
```
```  2209     and fB: "finite B"
```
```  2210     and dVB: "dim V \<ge> card B"
```
```  2211   shows "independent B"
```
```  2212 proof -
```
```  2213   {
```
```  2214     fix a
```
```  2215     assume a: "a \<in> B" "a \<in> span (B - {a})"
```
```  2216     from a fB have c0: "card B \<noteq> 0"
```
```  2217       by auto
```
```  2218     from a fB have cb: "card (B - {a}) = card B - 1"
```
```  2219       by auto
```
```  2220     from BV a have th0: "B - {a} \<subseteq> V"
```
```  2221       by blast
```
```  2222     {
```
```  2223       fix x
```
```  2224       assume x: "x \<in> V"
```
```  2225       from a have eq: "insert a (B - {a}) = B"
```
```  2226         by blast
```
```  2227       from x VB have x': "x \<in> span B"
```
```  2228         by blast
```
```  2229       from span_trans[OF a(2), unfolded eq, OF x']
```
```  2230       have "x \<in> span (B - {a})" .
```
```  2231     }
```
```  2232     then have th1: "V \<subseteq> span (B - {a})"
```
```  2233       by blast
```
```  2234     have th2: "finite (B - {a})"
```
```  2235       using fB by auto
```
```  2236     from span_card_ge_dim[OF th0 th1 th2]
```
```  2237     have c: "dim V \<le> card (B - {a})" .
```
```  2238     from c c0 dVB cb have False by simp
```
```  2239   }
```
```  2240   then show ?thesis
```
```  2241     unfolding dependent_def by blast
```
```  2242 qed
```
```  2243
```
```  2244 lemma card_eq_dim:
```
```  2245   fixes B :: "'a::euclidean_space set"
```
```  2246   shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
```
```  2247   by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
```
```  2248
```
```  2249 text \<open>More general size bound lemmas.\<close>
```
```  2250
```
```  2251 lemma independent_bound_general:
```
```  2252   fixes S :: "'a::euclidean_space set"
```
```  2253   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
```
```  2254   by (metis independent_card_le_dim independent_bound subset_refl)
```
```  2255
```
```  2256 lemma dependent_biggerset_general:
```
```  2257   fixes S :: "'a::euclidean_space set"
```
```  2258   shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
```
```  2259   using independent_bound_general[of S] by (metis linorder_not_le)
```
```  2260
```
```  2261 lemma dim_span [simp]:
```
```  2262   fixes S :: "'a::euclidean_space set"
```
```  2263   shows "dim (span S) = dim S"
```
```  2264 proof -
```
```  2265   have th0: "dim S \<le> dim (span S)"
```
```  2266     by (auto simp add: subset_eq intro: dim_subset span_superset)
```
```  2267   from basis_exists[of S]
```
```  2268   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
```
```  2269     by blast
```
```  2270   from B have fB: "finite B" "card B = dim S"
```
```  2271     using independent_bound by blast+
```
```  2272   have bSS: "B \<subseteq> span S"
```
```  2273     using B(1) by (metis subset_eq span_inc)
```
```  2274   have sssB: "span S \<subseteq> span B"
```
```  2275     using span_mono[OF B(3)] by (simp add: span_span)
```
```  2276   from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
```
```  2277     using fB(2) by arith
```
```  2278 qed
```
```  2279
```
```  2280 lemma subset_le_dim:
```
```  2281   fixes S :: "'a::euclidean_space set"
```
```  2282   shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
```
```  2283   by (metis dim_span dim_subset)
```
```  2284
```
```  2285 lemma span_eq_dim:
```
```  2286   fixes S :: "'a::euclidean_space set"
```
```  2287   shows "span S = span T \<Longrightarrow> dim S = dim T"
```
```  2288   by (metis dim_span)
```
```  2289
```
```  2290 lemma dim_image_le:
```
```  2291   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  2292   assumes lf: "linear f"
```
```  2293   shows "dim (f ` S) \<le> dim (S)"
```
```  2294 proof -
```
```  2295   from basis_exists[of S] obtain B where
```
```  2296     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
```
```  2297   from B have fB: "finite B" "card B = dim S"
```
```  2298     using independent_bound by blast+
```
```  2299   have "dim (f ` S) \<le> card (f ` B)"
```
```  2300     apply (rule span_card_ge_dim)
```
```  2301     using lf B fB
```
```  2302     apply (auto simp add: span_linear_image spans_image subset_image_iff)
```
```  2303     done
```
```  2304   also have "\<dots> \<le> dim S"
```
```  2305     using card_image_le[OF fB(1)] fB by simp
```
```  2306   finally show ?thesis .
```
```  2307 qed
```
```  2308
```
```  2309 text \<open>Picking an orthogonal replacement for a spanning set.\<close>
```
```  2310
```
```  2311 lemma vector_sub_project_orthogonal:
```
```  2312   fixes b x :: "'a::euclidean_space"
```
```  2313   shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
```
```  2314   unfolding inner_simps by auto
```
```  2315
```
```  2316 lemma pairwise_orthogonal_insert:
```
```  2317   assumes "pairwise orthogonal S"
```
```  2318     and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
```
```  2319   shows "pairwise orthogonal (insert x S)"
```
```  2320   using assms unfolding pairwise_def
```
```  2321   by (auto simp add: orthogonal_commute)
```
```  2322
```
```  2323 lemma basis_orthogonal:
```
```  2324   fixes B :: "'a::real_inner set"
```
```  2325   assumes fB: "finite B"
```
```  2326   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
```
```  2327   (is " \<exists>C. ?P B C")
```
```  2328   using fB
```
```  2329 proof (induct rule: finite_induct)
```
```  2330   case empty
```
```  2331   then show ?case
```
```  2332     apply (rule exI[where x="{}"])
```
```  2333     apply (auto simp add: pairwise_def)
```
```  2334     done
```
```  2335 next
```
```  2336   case (insert a B)
```
```  2337   note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
```
```  2338   from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
```
```  2339   obtain C where C: "finite C" "card C \<le> card B"
```
```  2340     "span C = span B" "pairwise orthogonal C" by blast
```
```  2341   let ?a = "a - sum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
```
```  2342   let ?C = "insert ?a C"
```
```  2343   from C(1) have fC: "finite ?C"
```
```  2344     by simp
```
```  2345   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
```
```  2346     by (simp add: card_insert_if)
```
```  2347   {
```
```  2348     fix x k
```
```  2349     have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
```
```  2350       by (simp add: field_simps)
```
```  2351     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"
```
```  2352       apply (simp only: scaleR_right_diff_distrib th0)
```
```  2353       apply (rule span_add_eq)
```
```  2354       apply (rule span_mul)
```
```  2355       apply (rule span_sum)
```
```  2356       apply (rule span_mul)
```
```  2357       apply (rule span_superset)
```
```  2358       apply assumption
```
```  2359       done
```
```  2360   }
```
```  2361   then have SC: "span ?C = span (insert a B)"
```
```  2362     unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
```
```  2363   {
```
```  2364     fix y
```
```  2365     assume yC: "y \<in> C"
```
```  2366     then have Cy: "C = insert y (C - {y})"
```
```  2367       by blast
```
```  2368     have fth: "finite (C - {y})"
```
```  2369       using C by simp
```
```  2370     have "orthogonal ?a y"
```
```  2371       unfolding orthogonal_def
```
```  2372       unfolding inner_diff inner_sum_left right_minus_eq
```
```  2373       unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
```
```  2374       apply (clarsimp simp add: inner_commute[of y a])
```
```  2375       apply (rule sum.neutral)
```
```  2376       apply clarsimp
```
```  2377       apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
```
```  2378       using \<open>y \<in> C\<close> by auto
```
```  2379   }
```
```  2380   with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
```
```  2381     by (rule pairwise_orthogonal_insert)
```
```  2382   from fC cC SC CPO have "?P (insert a B) ?C"
```
```  2383     by blast
```
```  2384   then show ?case by blast
```
```  2385 qed
```
```  2386
```
```  2387 lemma orthogonal_basis_exists:
```
```  2388   fixes V :: "('a::euclidean_space) set"
```
```  2389   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
```
```  2390 proof -
```
```  2391   from basis_exists[of V] obtain B where
```
```  2392     B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
```
```  2393     by blast
```
```  2394   from B have fB: "finite B" "card B = dim V"
```
```  2395     using independent_bound by auto
```
```  2396   from basis_orthogonal[OF fB(1)] obtain C where
```
```  2397     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
```
```  2398     by blast
```
```  2399   from C B have CSV: "C \<subseteq> span V"
```
```  2400     by (metis span_inc span_mono subset_trans)
```
```  2401   from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
```
```  2402     by (simp add: span_span)
```
```  2403   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
```
```  2404   have iC: "independent C"
```
```  2405     by (simp add: dim_span)
```
```  2406   from C fB have "card C \<le> dim V"
```
```  2407     by simp
```
```  2408   moreover have "dim V \<le> card C"
```
```  2409     using span_card_ge_dim[OF CSV SVC C(1)]
```
```  2410     by (simp add: dim_span)
```
```  2411   ultimately have CdV: "card C = dim V"
```
```  2412     using C(1) by simp
```
```  2413   from C B CSV CdV iC show ?thesis
```
```  2414     by auto
```
```  2415 qed
```
```  2416
```
```  2417 text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
```
```  2418
```
```  2419 lemma span_not_univ_orthogonal:
```
```  2420   fixes S :: "'a::euclidean_space set"
```
```  2421   assumes sU: "span S \<noteq> UNIV"
```
```  2422   shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
```
```  2423 proof -
```
```  2424   from sU obtain a where a: "a \<notin> span S"
```
```  2425     by blast
```
```  2426   from orthogonal_basis_exists obtain B where
```
```  2427     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
```
```  2428     by blast
```
```  2429   from B have fB: "finite B" "card B = dim S"
```
```  2430     using independent_bound by auto
```
```  2431   from span_mono[OF B(2)] span_mono[OF B(3)]
```
```  2432   have sSB: "span S = span B"
```
```  2433     by (simp add: span_span)
```
```  2434   let ?a = "a - sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
```
```  2435   have "sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
```
```  2436     unfolding sSB
```
```  2437     apply (rule span_sum)
```
```  2438     apply (rule span_mul)
```
```  2439     apply (rule span_superset)
```
```  2440     apply assumption
```
```  2441     done
```
```  2442   with a have a0:"?a  \<noteq> 0"
```
```  2443     by auto
```
```  2444   have "?a \<bullet> x = 0" if "x\<in>span B" for x
```
```  2445   proof (rule span_induct [OF that])
```
```  2446     show "subspace {x. ?a \<bullet> x = 0}"
```
```  2447       by (auto simp add: subspace_def inner_add)
```
```  2448   next
```
```  2449     {
```
```  2450       fix x
```
```  2451       assume x: "x \<in> B"
```
```  2452       from x have B': "B = insert x (B - {x})"
```
```  2453         by blast
```
```  2454       have fth: "finite (B - {x})"
```
```  2455         using fB by simp
```
```  2456       have "?a \<bullet> x = 0"
```
```  2457         apply (subst B')
```
```  2458         using fB fth
```
```  2459         unfolding sum_clauses(2)[OF fth]
```
```  2460         apply simp unfolding inner_simps
```
```  2461         apply (clarsimp simp add: inner_add inner_sum_left)
```
```  2462         apply (rule sum.neutral, rule ballI)
```
```  2463         apply (simp only: inner_commute)
```
```  2464         apply (auto simp add: x field_simps
```
```  2465           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
```
```  2466         done
```
```  2467     }
```
```  2468     then show "?a \<bullet> x = 0" if "x \<in> B" for x
```
```  2469       using that by blast
```
```  2470     qed
```
```  2471   with a0 show ?thesis
```
```  2472     unfolding sSB by (auto intro: exI[where x="?a"])
```
```  2473 qed
```
```  2474
```
```  2475 lemma span_not_univ_subset_hyperplane:
```
```  2476   fixes S :: "'a::euclidean_space set"
```
```  2477   assumes SU: "span S \<noteq> UNIV"
```
```  2478   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```  2479   using span_not_univ_orthogonal[OF SU] by auto
```
```  2480
```
```  2481 lemma lowdim_subset_hyperplane:
```
```  2482   fixes S :: "'a::euclidean_space set"
```
```  2483   assumes d: "dim S < DIM('a)"
```
```  2484   shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```  2485 proof -
```
```  2486   {
```
```  2487     assume "span S = UNIV"
```
```  2488     then have "dim (span S) = dim (UNIV :: ('a) set)"
```
```  2489       by simp
```
```  2490     then have "dim S = DIM('a)"
```
```  2491       by (simp add: dim_span dim_UNIV)
```
```  2492     with d have False by arith
```
```  2493   }
```
```  2494   then have th: "span S \<noteq> UNIV"
```
```  2495     by blast
```
```  2496   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
```
```  2497 qed
```
```  2498
```
```  2499 text \<open>We can extend a linear basis-basis injection to the whole set.\<close>
```
```  2500
```
```  2501 lemma linear_indep_image_lemma:
```
```  2502   assumes lf: "linear f"
```
```  2503     and fB: "finite B"
```
```  2504     and ifB: "independent (f ` B)"
```
```  2505     and fi: "inj_on f B"
```
```  2506     and xsB: "x \<in> span B"
```
```  2507     and fx: "f x = 0"
```
```  2508   shows "x = 0"
```
```  2509   using fB ifB fi xsB fx
```
```  2510 proof (induction B arbitrary: x rule: finite_induct)
```
```  2511   case empty
```
```  2512   then show ?case by auto
```
```  2513 next
```
```  2514   case (insert a b x)
```
```  2515   have th0: "f ` b \<subseteq> f ` (insert a b)"
```
```  2516     by (simp add: subset_insertI)
```
```  2517   have ifb: "independent (f ` b)"
```
```  2518     using independent_mono insert.prems(1) th0 by blast
```
```  2519   have fib: "inj_on f b"
```
```  2520     using insert.prems(2) by blast
```
```  2521   from span_breakdown[of a "insert a b", simplified, OF insert.prems(3)]
```
```  2522   obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
```
```  2523     by blast
```
```  2524   have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
```
```  2525     unfolding span_linear_image[OF lf]
```
```  2526     using "insert.hyps"(2) k by auto
```
```  2527   then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
```
```  2528     by (simp add: linear_diff[OF lf] linear_cmul[OF lf])
```
```  2529   then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
```
```  2530     using insert.prems(4) by simp
```
```  2531   have xsb: "x \<in> span b"
```
```  2532   proof (cases "k = 0")
```
```  2533     case True
```
```  2534     with k have "x \<in> span (b - {a})" by simp
```
```  2535     then show ?thesis using span_mono[of "b - {a}" b]
```
```  2536       by blast
```
```  2537   next
```
```  2538     case False
```
```  2539     from inj_on_image_set_diff[OF insert.prems(2), of "insert a b " "{a}", symmetric]
```
```  2540     have "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
```
```  2541     then have "f a \<notin> span (f ` b)"
```
```  2542       using dependent_def insert.hyps(2) insert.prems(1) by fastforce
```
```  2543     moreover have "f a \<in> span (f ` b)"
```
```  2544       using False span_mul[OF th, of "- 1/ k"] by auto
```
```  2545     ultimately have False
```
```  2546       by blast
```
```  2547     then show ?thesis by blast
```
```  2548   qed
```
```  2549   show "x = 0"
```
```  2550     using ifb fib xsb insert.IH insert.prems(4) by blast
```
```  2551 qed
```
```  2552
```
```  2553 text \<open>Can construct an isomorphism between spaces of same dimension.\<close>
```
```  2554
```
```  2555 lemma subspace_isomorphism:
```
```  2556   fixes S :: "'a::euclidean_space set"
```
```  2557     and T :: "'b::euclidean_space set"
```
```  2558   assumes s: "subspace S"
```
```  2559     and t: "subspace T"
```
```  2560     and d: "dim S = dim T"
```
```  2561   shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
```
```  2562 proof -
```
```  2563   from basis_exists[of S] independent_bound
```
```  2564   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
```
```  2565     by blast
```
```  2566   from basis_exists[of T] independent_bound
```
```  2567   obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C"
```
```  2568     by blast
```
```  2569   from B(4) C(4) card_le_inj[of B C] d
```
```  2570   obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using \<open>finite B\<close> \<open>finite C\<close>
```
```  2571     by auto
```
```  2572   from linear_independent_extend[OF B(2)]
```
```  2573   obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x"
```
```  2574     by blast
```
```  2575   from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
```
```  2576     by simp
```
```  2577   with B(4) C(4) have ceq: "card (f ` B) = card C"
```
```  2578     using d by simp
```
```  2579   have "g ` B = f ` B"
```
```  2580     using g(2) by (auto simp add: image_iff)
```
```  2581   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
```
```  2582   finally have gBC: "g ` B = C" .
```
```  2583   have gi: "inj_on g B"
```
```  2584     using f(2) g(2) by (auto simp add: inj_on_def)
```
```  2585   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
```
```  2586   {
```
```  2587     fix x y
```
```  2588     assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
```
```  2589     from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
```
```  2590       by blast+
```
```  2591     from gxy have th0: "g (x - y) = 0"
```
```  2592       by (simp add: linear_diff[OF g(1)])
```
```  2593     have th1: "x - y \<in> span B"
```
```  2594       using x' y' by (metis span_diff)
```
```  2595     have "x = y"
```
```  2596       using g0[OF th1 th0] by simp
```
```  2597   }
```
```  2598   then have giS: "inj_on g S"
```
```  2599     unfolding inj_on_def by blast
```
```  2600   from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)"
```
```  2601     by (simp add: span_linear_image[OF g(1)])
```
```  2602   also have "\<dots> = span C" unfolding gBC ..
```
```  2603   also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
```
```  2604   finally have gS: "g ` S = T" .
```
```  2605   from g(1) gS giS show ?thesis
```
```  2606     by blast
```
```  2607 qed
```
```  2608
```
```  2609 lemma linear_eq_stdbasis:
```
```  2610   fixes f :: "'a::euclidean_space \<Rightarrow> _"
```
```  2611   assumes lf: "linear f"
```
```  2612     and lg: "linear g"
```
```  2613     and fg: "\<And>b. b \<in> Basis \<Longrightarrow> f b = g b"
```
```  2614   shows "f = g"
```
```  2615   using linear_eq[OF _ lf lg, of _ _ Basis] fg by auto
```
```  2616
```
```  2617 text \<open>Similar results for bilinear functions.\<close>
```
```  2618
```
```  2619 lemma bilinear_eq:
```
```  2620   assumes bf: "bilinear f"
```
```  2621     and bg: "bilinear g"
```
```  2622     and SB: "S \<subseteq> span B"
```
```  2623     and TC: "T \<subseteq> span C"
```
```  2624     and "x\<in>S" "y\<in>T"
```
```  2625     and fg: "\<And>x y. \<lbrakk>x \<in> B; y\<in> C\<rbrakk> \<Longrightarrow> f x y = g x y"
```
```  2626   shows "f x y = g x y"
```
```  2627 proof -
```
```  2628   let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
```
```  2629   from bf bg have sp: "subspace ?P"
```
```  2630     unfolding bilinear_def linear_iff subspace_def bf bg
```
```  2631     by (auto simp add: bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add
```
```  2632       intro: bilinear_ladd[OF bf])
```
```  2633   have sfg: "\<And>x. x \<in> B \<Longrightarrow> subspace {a. f x a = g x a}"
```
```  2634     apply (auto simp add: subspace_def)
```
```  2635     using bf bg unfolding bilinear_def linear_iff
```
```  2636     apply (auto simp add: bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add
```
```  2637       intro: bilinear_ladd[OF bf])
```
```  2638     done
```
```  2639   have "\<forall>y\<in> span C. f x y = g x y" if "x \<in> span B" for x
```
```  2640     apply (rule span_induct [OF that sp])
```
```  2641     using fg sfg span_induct by blast
```
```  2642   then show ?thesis
```
```  2643     using SB TC assms by auto
```
```  2644 qed
```
```  2645
```
```  2646 lemma bilinear_eq_stdbasis:
```
```  2647   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
```
```  2648   assumes bf: "bilinear f"
```
```  2649     and bg: "bilinear g"
```
```  2650     and fg: "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> f i j = g i j"
```
```  2651   shows "f = g"
```
```  2652   using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis]] fg
```
```  2653   by blast
```
```  2654
```
```  2655 text \<open>An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective.\<close>
```
```  2656
```
```  2657 lemma linear_injective_imp_surjective:
```
```  2658   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2659   assumes lf: "linear f"
```
```  2660     and fi: "inj f"
```
```  2661   shows "surj f"
```
```  2662 proof -
```
```  2663   let ?U = "UNIV :: 'a set"
```
```  2664   from basis_exists[of ?U] obtain B
```
```  2665     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
```
```  2666     by blast
```
```  2667   from B(4) have d: "dim ?U = card B"
```
```  2668     by simp
```
```  2669   have th: "?U \<subseteq> span (f ` B)"
```
```  2670     apply (rule card_ge_dim_independent)
```
```  2671     apply blast
```
```  2672     apply (rule independent_injective_image[OF B(2) lf fi])
```
```  2673     by (metis card_image d fi inj_on_subset order_refl top_greatest)
```
```  2674   from th show ?thesis
```
```  2675     unfolding span_linear_image[OF lf] surj_def
```
```  2676     using B(3) by blast
```
```  2677 qed
```
```  2678
```
```  2679 text \<open>And vice versa.\<close>
```
```  2680
```
```  2681 lemma surjective_iff_injective_gen:
```
```  2682   assumes fS: "finite S"
```
```  2683     and fT: "finite T"
```
```  2684     and c: "card S = card T"
```
```  2685     and ST: "f ` S \<subseteq> T"
```
```  2686   shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S"
```
```  2687   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2688 proof
```
```  2689   assume h: "?lhs"
```
```  2690   { fix x y
```
```  2691     assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
```
```  2692     from x fS have S0: "card S \<noteq> 0"
```
```  2693       by auto
```
```  2694     have "x = y"
```
```  2695     proof (rule ccontr)
```
```  2696       assume xy: "\<not> ?thesis"
```
```  2697       have th: "card S \<le> card (f ` (S - {y}))"
```
```  2698         unfolding c
```
```  2699       proof (rule card_mono)
```
```  2700         show "finite (f ` (S - {y}))"
```
```  2701           by (simp add: fS)
```
```  2702         show "T \<subseteq> f ` (S - {y})"
```
```  2703           using h xy x y f unfolding subset_eq image_iff
```
```  2704           by (metis member_remove remove_def)
```
```  2705       qed
```
```  2706       also have " \<dots> \<le> card (S - {y})"
```
```  2707         apply (rule card_image_le)
```
```  2708         using fS by simp
```
```  2709       also have "\<dots> \<le> card S - 1" using y fS by simp
```
```  2710       finally show False using S0 by arith
```
```  2711     qed
```
```  2712   }
```
```  2713   then show ?rhs
```
```  2714     unfolding inj_on_def by blast
```
```  2715 next
```
```  2716   assume h: ?rhs
```
```  2717   have "f ` S = T"
```
```  2718     by (simp add: ST c card_image card_subset_eq fT h)
```
```  2719   then show ?lhs by blast
```
```  2720 qed
```
```  2721
```
```  2722 lemma linear_surjective_imp_injective:
```
```  2723   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2724   assumes lf: "linear f" and sf: "surj f"
```
```  2725   shows "inj f"
```
```  2726 proof -
```
```  2727   let ?U = "UNIV :: 'a set"
```
```  2728   from basis_exists[of ?U] obtain B
```
```  2729     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
```
```  2730     by blast
```
```  2731   {
```
```  2732     fix x
```
```  2733     assume x: "x \<in> span B" and fx: "f x = 0"
```
```  2734     from B(2) have fB: "finite B"
```
```  2735       using independent_bound by auto
```
```  2736     have Uspan: "UNIV \<subseteq> span (f ` B)"
```
```  2737       by (simp add: B(3) lf sf spanning_surjective_image)
```
```  2738     have fBi: "independent (f ` B)"
```
```  2739     proof (rule card_le_dim_spanning)
```
```  2740       show "card (f ` B) \<le> dim ?U"
```
```  2741         using card_image_le d fB by fastforce
```
```  2742     qed (use fB Uspan in auto)
```
```  2743     have th0: "dim ?U \<le> card (f ` B)"
```
```  2744       by (rule span_card_ge_dim) (use Uspan fB in auto)
```
```  2745     moreover have "card (f ` B) \<le> card B"
```
```  2746       by (rule card_image_le, rule fB)
```
```  2747     ultimately have th1: "card B = card (f ` B)"
```
```  2748       unfolding d by arith
```
```  2749     have fiB: "inj_on f B"
```
```  2750       by (simp add: eq_card_imp_inj_on fB th1)
```
```  2751     from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
```
```  2752     have "x = 0" by blast
```
```  2753   }
```
```  2754   then show ?thesis
```
```  2755     unfolding linear_injective_0[OF lf] using B(3) by blast
```
```  2756 qed
```
```  2757
```
```  2758 text \<open>Hence either is enough for isomorphism.\<close>
```
```  2759
```
```  2760 lemma left_right_inverse_eq:
```
```  2761   assumes fg: "f \<circ> g = id"
```
```  2762     and gh: "g \<circ> h = id"
```
```  2763   shows "f = h"
```
```  2764 proof -
```
```  2765   have "f = f \<circ> (g \<circ> h)"
```
```  2766     unfolding gh by simp
```
```  2767   also have "\<dots> = (f \<circ> g) \<circ> h"
```
```  2768     by (simp add: o_assoc)
```
```  2769   finally show "f = h"
```
```  2770     unfolding fg by simp
```
```  2771 qed
```
```  2772
```
```  2773 lemma isomorphism_expand:
```
```  2774   "f \<circ> g = id \<and> g \<circ> f = id \<longleftrightarrow> (\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x)"
```
```  2775   by (simp add: fun_eq_iff o_def id_def)
```
```  2776
```
```  2777 lemma linear_injective_isomorphism:
```
```  2778   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2779   assumes lf: "linear f"
```
```  2780     and fi: "inj f"
```
```  2781   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
```
```  2782   unfolding isomorphism_expand[symmetric]
```
```  2783   using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]]
```
```  2784     linear_injective_left_inverse[OF lf fi]
```
```  2785   by (metis left_right_inverse_eq)
```
```  2786
```
```  2787 lemma linear_surjective_isomorphism:
```
```  2788   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2789   assumes lf: "linear f"
```
```  2790     and sf: "surj f"
```
```  2791   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
```
```  2792   unfolding isomorphism_expand[symmetric]
```
```  2793   using linear_surjective_right_inverse[OF lf sf]
```
```  2794     linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
```
```  2795   by (metis left_right_inverse_eq)
```
```  2796
```
```  2797 text \<open>Left and right inverses are the same for
```
```  2798   @{typ "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"}.\<close>
```
```  2799
```
```  2800 lemma linear_inverse_left:
```
```  2801   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2802   assumes lf: "linear f"
```
```  2803     and lf': "linear f'"
```
```  2804   shows "f \<circ> f' = id \<longleftrightarrow> f' \<circ> f = id"
```
```  2805 proof -
```
```  2806   {
```
```  2807     fix f f':: "'a \<Rightarrow> 'a"
```
```  2808     assume lf: "linear f" "linear f'"
```
```  2809     assume f: "f \<circ> f' = id"
```
```  2810     from f have sf: "surj f"
```
```  2811       by (auto simp add: o_def id_def surj_def) metis
```
```  2812     from linear_surjective_isomorphism[OF lf(1) sf] lf f
```
```  2813     have "f' \<circ> f = id"
```
```  2814       unfolding fun_eq_iff o_def id_def by metis
```
```  2815   }
```
```  2816   then show ?thesis
```
```  2817     using lf lf' by metis
```
```  2818 qed
```
```  2819
```
```  2820 text \<open>Moreover, a one-sided inverse is automatically linear.\<close>
```
```  2821
```
```  2822 lemma left_inverse_linear:
```
```  2823   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2824   assumes lf: "linear f"
```
```  2825     and gf: "g \<circ> f = id"
```
```  2826   shows "linear g"
```
```  2827 proof -
```
```  2828   from gf have fi: "inj f"
```
```  2829     by (auto simp add: inj_on_def o_def id_def fun_eq_iff) metis
```
```  2830   from linear_injective_isomorphism[OF lf fi]
```
```  2831   obtain h :: "'a \<Rightarrow> 'a" where "linear h" and h: "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x"
```
```  2832     by blast
```
```  2833   have "h = g"
```
```  2834     by (metis gf h isomorphism_expand left_right_inverse_eq)
```
```  2835   with \<open>linear h\<close> show ?thesis by blast
```
```  2836 qed
```
```  2837
```
```  2838 lemma inj_linear_imp_inv_linear:
```
```  2839   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2840   assumes "linear f" "inj f" shows "linear (inv f)"
```
```  2841 using assms inj_iff left_inverse_linear by blast
```
```  2842
```
```  2843
```
```  2844 subsection \<open>Infinity norm\<close>
```
```  2845
```
```  2846 definition%important "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
```
```  2847
```
```  2848 lemma infnorm_set_image:
```
```  2849   fixes x :: "'a::euclidean_space"
```
```  2850   shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```  2851   by blast
```
```  2852
```
```  2853 lemma infnorm_Max:
```
```  2854   fixes x :: "'a::euclidean_space"
```
```  2855   shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)"
```
```  2856   by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)
```
```  2857
```
```  2858 lemma infnorm_set_lemma:
```
```  2859   fixes x :: "'a::euclidean_space"
```
```  2860   shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
```
```  2861     and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}"
```
```  2862   unfolding infnorm_set_image
```
```  2863   by auto
```
```  2864
```
```  2865 lemma infnorm_pos_le:
```
```  2866   fixes x :: "'a::euclidean_space"
```
```  2867   shows "0 \<le> infnorm x"
```
```  2868   by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
```
```  2869
```
```  2870 lemma infnorm_triangle:
```
```  2871   fixes x :: "'a::euclidean_space"
```
```  2872   shows "infnorm (x + y) \<le> infnorm x + infnorm y"
```
```  2873 proof -
```
```  2874   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"
```
```  2875     by simp
```
```  2876   show ?thesis
```
```  2877     by (auto simp: infnorm_Max inner_add_left intro!: *)
```
```  2878 qed
```
```  2879
```
```  2880 lemma infnorm_eq_0:
```
```  2881   fixes x :: "'a::euclidean_space"
```
```  2882   shows "infnorm x = 0 \<longleftrightarrow> x = 0"
```
```  2883 proof -
```
```  2884   have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
```
```  2885     unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
```
```  2886   then show ?thesis
```
```  2887     using infnorm_pos_le[of x] by simp
```
```  2888 qed
```
```  2889
```
```  2890 lemma infnorm_0: "infnorm 0 = 0"
```
```  2891   by (simp add: infnorm_eq_0)
```
```  2892
```
```  2893 lemma infnorm_neg: "infnorm (- x) = infnorm x"
```
```  2894   unfolding infnorm_def by simp
```
```  2895
```
```  2896 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
```
```  2897   by (metis infnorm_neg minus_diff_eq)
```
```  2898
```
```  2899 lemma absdiff_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
```
```  2900 proof -
```
```  2901   have *: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
```
```  2902     by arith
```
```  2903   show ?thesis
```
```  2904   proof (rule *)
```
```  2905     from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
```
```  2906     show "infnorm x \<le> infnorm (x - y) + infnorm y" "infnorm y \<le> infnorm (x - y) + infnorm x"
```
```  2907       by (simp_all add: field_simps infnorm_neg)
```
```  2908   qed
```
```  2909 qed
```
```  2910
```
```  2911 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
```
```  2912   using infnorm_pos_le[of x] by arith
```
```  2913
```
```  2914 lemma Basis_le_infnorm:
```
```  2915   fixes x :: "'a::euclidean_space"
```
```  2916   shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
```
```  2917   by (simp add: infnorm_Max)
```
```  2918
```
```  2919 lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
```
```  2920   unfolding infnorm_Max
```
```  2921 proof (safe intro!: Max_eqI)
```
```  2922   let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```  2923   { fix b :: 'a
```
```  2924     assume "b \<in> Basis"
```
```  2925     then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
```
```  2926       by (simp add: abs_mult mult_left_mono)
```
```  2927   next
```
```  2928     from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
```
```  2929       by (auto simp del: Max_in)
```
```  2930     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"
```
```  2931       by (intro image_eqI[where x=b]) (auto simp: abs_mult)
```
```  2932   }
```
```  2933 qed simp
```
```  2934
```
```  2935 lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
```
```  2936   unfolding infnorm_mul ..
```
```  2937
```
```  2938 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
```
```  2939   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
```
```  2940
```
```  2941 text \<open>Prove that it differs only up to a bound from Euclidean norm.\<close>
```
```  2942
```
```  2943 lemma infnorm_le_norm: "infnorm x \<le> norm x"
```
```  2944   by (simp add: Basis_le_norm infnorm_Max)
```
```  2945
```
```  2946 lemma norm_le_infnorm:
```
```  2947   fixes x :: "'a::euclidean_space"
```
```  2948   shows "norm x \<le> sqrt DIM('a) * infnorm x"
```
```  2949   unfolding norm_eq_sqrt_inner id_def
```
```  2950 proof (rule real_le_lsqrt[OF inner_ge_zero])
```
```  2951   show "sqrt DIM('a) * infnorm x \<ge> 0"
```
```  2952     by (simp add: zero_le_mult_iff infnorm_pos_le)
```
```  2953   have "x \<bullet> x \<le> (\<Sum>b\<in>Basis. x \<bullet> b * (x \<bullet> b))"
```
```  2954     by (metis euclidean_inner order_refl)
```
```  2955   also have "... \<le> DIM('a) * \<bar>infnorm x\<bar>\<^sup>2"
```
```  2956     by (rule sum_bounded_above) (metis Basis_le_infnorm abs_le_square_iff power2_eq_square real_abs_infnorm)
```
```  2957   also have "... \<le> (sqrt DIM('a) * infnorm x)\<^sup>2"
```
```  2958     by (simp add: power_mult_distrib)
```
```  2959   finally show "x \<bullet> x \<le> (sqrt DIM('a) * infnorm x)\<^sup>2" .
```
```  2960 qed
```
```  2961
```
```  2962 lemma tendsto_infnorm [tendsto_intros]:
```
```  2963   assumes "(f \<longlongrightarrow> a) F"
```
```  2964   shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F"
```
```  2965 proof (rule tendsto_compose [OF LIM_I assms])
```
```  2966   fix r :: real
```
```  2967   assume "r > 0"
```
```  2968   then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
```
```  2969     by (metis real_norm_def le_less_trans absdiff_infnorm infnorm_le_norm)
```
```  2970 qed
```
```  2971
```
```  2972 text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close>
```
```  2973
```
```  2974 lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  2975   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2976 proof (cases "x=0")
```
```  2977   case True
```
```  2978   then show ?thesis
```
```  2979     by auto
```
```  2980 next
```
```  2981   case False
```
```  2982   from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
```
```  2983   have "?rhs \<longleftrightarrow>
```
```  2984       (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
```
```  2985         norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
```
```  2986     using False unfolding inner_simps
```
```  2987     by (auto simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
```
```  2988   also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)"
```
```  2989     using False  by (simp add: field_simps inner_commute)
```
```  2990   also have "\<dots> \<longleftrightarrow> ?lhs"
```
```  2991     using False by auto
```
```  2992   finally show ?thesis by metis
```
```  2993 qed
```
```  2994
```
```  2995 lemma norm_cauchy_schwarz_abs_eq:
```
```  2996   "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow>
```
```  2997     norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
```
```  2998   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2999 proof -
```
```  3000   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a"
```
```  3001     by arith
```
```  3002   have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
```
```  3003     by simp
```
```  3004   also have "\<dots> \<longleftrightarrow> (x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
```
```  3005     unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  3006     unfolding norm_minus_cancel norm_scaleR ..
```
```  3007   also have "\<dots> \<longleftrightarrow> ?lhs"
```
```  3008     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
```
```  3009     by auto
```
```  3010   finally show ?thesis ..
```
```  3011 qed
```
```  3012
```
```  3013 lemma norm_triangle_eq:
```
```  3014   fixes x y :: "'a::real_inner"
```
```  3015   shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  3016 proof (cases "x = 0 \<or> y = 0")
```
```  3017   case True
```
```  3018   then show ?thesis
```
```  3019     by force
```
```  3020 next
```
```  3021   case False
```
```  3022   then have n: "norm x > 0" "norm y > 0"
```
```  3023     by auto
```
```  3024   have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
```
```  3025     by simp
```
```  3026   also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  3027     unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  3028     unfolding power2_norm_eq_inner inner_simps
```
```  3029     by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
```
```  3030   finally show ?thesis .
```
```  3031 qed
```
```  3032
```
```  3033
```
```  3034 subsection \<open>Collinearity\<close>
```
```  3035
```
```  3036 definition%important collinear :: "'a::real_vector set \<Rightarrow> bool"
```
```  3037   where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
```
```  3038
```
```  3039 lemma collinear_alt:
```
```  3040      "collinear S \<longleftrightarrow> (\<exists>u v. \<forall>x \<in> S. \<exists>c. x = u + c *\<^sub>R v)" (is "?lhs = ?rhs")
```
```  3041 proof
```
```  3042   assume ?lhs
```
```  3043   then show ?rhs
```
```  3044     unfolding collinear_def by (metis Groups.add_ac(2) diff_add_cancel)
```
```  3045 next
```
```  3046   assume ?rhs
```
```  3047   then obtain u v where *: "\<And>x. x \<in> S \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
```
```  3048     by (auto simp: )
```
```  3049   have "\<exists>c. x - y = c *\<^sub>R v" if "x \<in> S" "y \<in> S" for x y
```
```  3050         by (metis *[OF \<open>x \<in> S\<close>] *[OF \<open>y \<in> S\<close>] scaleR_left.diff add_diff_cancel_left)
```
```  3051   then show ?lhs
```
```  3052     using collinear_def by blast
```
```  3053 qed
```
```  3054
```
```  3055 lemma collinear:
```
```  3056   fixes S :: "'a::{perfect_space,real_vector} set"
```
```  3057   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))"
```
```  3058 proof -
```
```  3059   have "\<exists>v. v \<noteq> 0 \<and> (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v)"
```
```  3060     if "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R u" "u=0" for u
```
```  3061   proof -
```
```  3062     have "\<forall>x\<in>S. \<forall>y\<in>S. x = y"
```
```  3063       using that by auto
```
```  3064     moreover
```
```  3065     obtain v::'a where "v \<noteq> 0"
```
```  3066       using UNIV_not_singleton [of 0] by auto
```
```  3067     ultimately have "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v"
```
```  3068       by auto
```
```  3069     then show ?thesis
```
```  3070       using \<open>v \<noteq> 0\<close> by blast
```
```  3071   qed
```
```  3072   then show ?thesis
```
```  3073     apply (clarsimp simp: collinear_def)
```
```  3074     by (metis real_vector.scale_zero_right vector_fraction_eq_iff)
```
```  3075 qed
```
```  3076
```
```  3077 lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S"
```
```  3078   by (meson collinear_def subsetCE)
```
```  3079
```
```  3080 lemma collinear_empty [iff]: "collinear {}"
```
```  3081   by (simp add: collinear_def)
```
```  3082
```
```  3083 lemma collinear_sing [iff]: "collinear {x}"
```
```  3084   by (simp add: collinear_def)
```
```  3085
```
```  3086 lemma collinear_2 [iff]: "collinear {x, y}"
```
```  3087   apply (simp add: collinear_def)
```
```  3088   apply (rule exI[where x="x - y"])
```
```  3089   by (metis minus_diff_eq scaleR_left.minus scaleR_one)
```
```  3090
```
```  3091 lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
```
```  3092   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  3093 proof (cases "x = 0 \<or> y = 0")
```
```  3094   case True
```
```  3095   then show ?thesis
```
```  3096     by (auto simp: insert_commute)
```
```  3097 next
```
```  3098   case False
```
```  3099   show ?thesis
```
```  3100   proof
```
```  3101     assume h: "?lhs"
```
```  3102     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"
```
```  3103       unfolding collinear_def by blast
```
```  3104     from u[rule_format, of x 0] u[rule_format, of y 0]
```
```  3105     obtain cx and cy where
```
```  3106       cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
```
```  3107       by auto
```
```  3108     from cx cy False have cx0: "cx \<noteq> 0" and cy0: "cy \<noteq> 0" by auto
```
```  3109     let ?d = "cy / cx"
```
```  3110     from cx cy cx0 have "y = ?d *\<^sub>R x"
```
```  3111       by simp
```
```  3112     then show ?rhs using False by blast
```
```  3113   next
```
```  3114     assume h: "?rhs"
```
```  3115     then obtain c where c: "y = c *\<^sub>R x"
```
```  3116       using False by blast
```
```  3117     show ?lhs
```
```  3118       unfolding collinear_def c
```
```  3119       apply (rule exI[where x=x])
```
```  3120       apply auto
```
```  3121           apply (rule exI[where x="- 1"], simp)
```
```  3122          apply (rule exI[where x= "-c"], simp)
```
```  3123         apply (rule exI[where x=1], simp)
```
```  3124        apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
```
```  3125       apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
```
```  3126       done
```
```  3127   qed
```
```  3128 qed
```
```  3129
```
```  3130 lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
```
```  3131 proof (cases "x=0")
```
```  3132   case True
```
```  3133   then show ?thesis
```
```  3134     by (auto simp: insert_commute)
```
```  3135 next
```
```  3136   case False
```
```  3137   then have nnz: "norm x \<noteq> 0"
```
```  3138     by auto
```
```  3139   show ?thesis
```
```  3140   proof
```
```  3141     assume "\<bar>x \<bullet> y\<bar> = norm x * norm y"
```
```  3142     then show "collinear {0, x, y}"
```
```  3143       unfolding norm_cauchy_schwarz_abs_eq collinear_lemma
```
```  3144       by (meson eq_vector_fraction_iff nnz)
```
```  3145   next
```
```  3146     assume "collinear {0, x, y}"
```
```  3147     with False show "\<bar>x \<bullet> y\<bar> = norm x * norm y"
```
```  3148       unfolding norm_cauchy_schwarz_abs_eq collinear_lemma  by (auto simp: abs_if)
```
```  3149   qed
```
```  3150 qed
```
```  3151
```
```  3152 end
```