src/HOL/Analysis/Linear_Algebra.thy
 author hoelzl Fri Sep 16 13:56:51 2016 +0200 (2016-09-16) changeset 63886 685fb01256af parent 63881 b746b19197bd child 63918 6bf55e6e0b75 permissions -rw-r--r--
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
```     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   "~~/src/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 lemma bounded_linearI:
```
```    31   assumes "\<And>x y. f (x + y) = f x + f y"
```
```    32     and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x"
```
```    33     and "\<And>x. norm (f x) \<le> norm x * K"
```
```    34   shows "bounded_linear f"
```
```    35   using assms by (rule bounded_linear_intro) (* FIXME: duplicate *)
```
```    36
```
```    37 subsection \<open>A generic notion of "hull" (convex, affine, conic hull and closure).\<close>
```
```    38
```
```    39 definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
```
```    40   where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
```
```    41
```
```    42 lemma hull_same: "S s \<Longrightarrow> S hull s = s"
```
```    43   unfolding hull_def by auto
```
```    44
```
```    45 lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
```
```    46   unfolding hull_def Ball_def by auto
```
```    47
```
```    48 lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
```
```    49   using hull_same[of S s] hull_in[of S s] by metis
```
```    50
```
```    51 lemma hull_hull [simp]: "S hull (S hull s) = S hull s"
```
```    52   unfolding hull_def by blast
```
```    53
```
```    54 lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
```
```    55   unfolding hull_def by blast
```
```    56
```
```    57 lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
```
```    58   unfolding hull_def by blast
```
```    59
```
```    60 lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
```
```    61   unfolding hull_def by blast
```
```    62
```
```    63 lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
```
```    64   unfolding hull_def by blast
```
```    65
```
```    66 lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
```
```    67   unfolding hull_def by blast
```
```    68
```
```    69 lemma hull_UNIV [simp]: "S hull UNIV = UNIV"
```
```    70   unfolding hull_def by auto
```
```    71
```
```    72 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)"
```
```    73   unfolding hull_def by auto
```
```    74
```
```    75 lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
```
```    76   using hull_minimal[of S "{x. P x}" Q]
```
```    77   by (auto simp add: subset_eq)
```
```    78
```
```    79 lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
```
```    80   by (metis hull_subset subset_eq)
```
```    81
```
```    82 lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
```
```    83   unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
```
```    84
```
```    85 lemma hull_union:
```
```    86   assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
```
```    87   shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
```
```    88   apply rule
```
```    89   apply (rule hull_mono)
```
```    90   unfolding Un_subset_iff
```
```    91   apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
```
```    92   apply (rule hull_minimal)
```
```    93   apply (metis hull_union_subset)
```
```    94   apply (metis hull_in T)
```
```    95   done
```
```    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 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
```
```   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_setsum:
```
```   145   assumes lS: "\<forall>a \<in> S. linear (f a)"
```
```   146   shows "linear (\<lambda>x. setsum (\<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_setsum:
```
```   177   assumes f: "linear f"
```
```   178   shows "f (setsum g S) = setsum (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_setsum_mul:
```
```   190   assumes lin: "linear f"
```
```   191   shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
```
```   192   using linear_setsum[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 (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 (in real_vector) "span S = (subspace hull S)"
```
```   230 definition (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_setsum:
```
```   254   assumes sA: "subspace A"
```
```   255     and f: "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A"
```
```   256   shows "setsum 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 \<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: "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   apply (clarsimp simp add: dependent_def span_mono)
```
```   336   apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
```
```   337   apply force
```
```   338   apply (rule span_mono)
```
```   339   apply auto
```
```   340   done
```
```   341
```
```   342 lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
```
```   343   by (metis order_antisym span_def hull_minimal)
```
```   344
```
```   345 lemma (in real_vector) span_induct':
```
```   346   "\<forall>x \<in> S. P x \<Longrightarrow> subspace {x. P x} \<Longrightarrow> \<forall>x \<in> span S. P x"
```
```   347   unfolding span_def by (rule hull_induct) auto
```
```   348
```
```   349 inductive_set (in real_vector) span_induct_alt_help for S :: "'a set"
```
```   350 where
```
```   351   span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
```
```   352 | span_induct_alt_help_S:
```
```   353     "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
```
```   354       (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
```
```   355
```
```   356 lemma span_induct_alt':
```
```   357   assumes h0: "h 0"
```
```   358     and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
```
```   359   shows "\<forall>x \<in> span S. h x"
```
```   360 proof -
```
```   361   {
```
```   362     fix x :: 'a
```
```   363     assume x: "x \<in> span_induct_alt_help S"
```
```   364     have "h x"
```
```   365       apply (rule span_induct_alt_help.induct[OF x])
```
```   366       apply (rule h0)
```
```   367       apply (rule hS)
```
```   368       apply assumption
```
```   369       apply assumption
```
```   370       done
```
```   371   }
```
```   372   note th0 = this
```
```   373   {
```
```   374     fix x
```
```   375     assume x: "x \<in> span S"
```
```   376     have "x \<in> span_induct_alt_help S"
```
```   377     proof (rule span_induct[where x=x and S=S])
```
```   378       show "x \<in> span S" by (rule x)
```
```   379     next
```
```   380       fix x
```
```   381       assume xS: "x \<in> S"
```
```   382       from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
```
```   383       show "x \<in> span_induct_alt_help S"
```
```   384         by simp
```
```   385     next
```
```   386       have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
```
```   387       moreover
```
```   388       {
```
```   389         fix x y
```
```   390         assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
```
```   391         from h have "(x + y) \<in> span_induct_alt_help S"
```
```   392           apply (induct rule: span_induct_alt_help.induct)
```
```   393           apply simp
```
```   394           unfolding add.assoc
```
```   395           apply (rule span_induct_alt_help_S)
```
```   396           apply assumption
```
```   397           apply simp
```
```   398           done
```
```   399       }
```
```   400       moreover
```
```   401       {
```
```   402         fix c x
```
```   403         assume xt: "x \<in> span_induct_alt_help S"
```
```   404         then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
```
```   405           apply (induct rule: span_induct_alt_help.induct)
```
```   406           apply (simp add: span_induct_alt_help_0)
```
```   407           apply (simp add: scaleR_right_distrib)
```
```   408           apply (rule span_induct_alt_help_S)
```
```   409           apply assumption
```
```   410           apply simp
```
```   411           done }
```
```   412       ultimately show "subspace {a. a \<in> span_induct_alt_help S}"
```
```   413         unfolding subspace_def Ball_def by blast
```
```   414     qed
```
```   415   }
```
```   416   with th0 show ?thesis by blast
```
```   417 qed
```
```   418
```
```   419 lemma span_induct_alt:
```
```   420   assumes h0: "h 0"
```
```   421     and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
```
```   422     and x: "x \<in> span S"
```
```   423   shows "h x"
```
```   424   using span_induct_alt'[of h S] h0 hS x by blast
```
```   425
```
```   426 text \<open>Individual closure properties.\<close>
```
```   427
```
```   428 lemma span_span: "span (span A) = span A"
```
```   429   unfolding span_def hull_hull ..
```
```   430
```
```   431 lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
```
```   432   by (metis span_clauses(1))
```
```   433
```
```   434 lemma (in real_vector) span_0 [simp]: "0 \<in> span S"
```
```   435   by (metis subspace_span subspace_0)
```
```   436
```
```   437 lemma span_inc: "S \<subseteq> span S"
```
```   438   by (metis subset_eq span_superset)
```
```   439
```
```   440 lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
```
```   441   using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
```
```   442   by (auto simp add: span_span)
```
```   443
```
```   444 lemma (in real_vector) dependent_0:
```
```   445   assumes "0 \<in> A"
```
```   446   shows "dependent A"
```
```   447   unfolding dependent_def
```
```   448   using assms span_0
```
```   449   by blast
```
```   450
```
```   451 lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
```
```   452   by (metis subspace_add subspace_span)
```
```   453
```
```   454 lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
```
```   455   by (metis subspace_span subspace_mul)
```
```   456
```
```   457 lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
```
```   458   by (metis subspace_neg subspace_span)
```
```   459
```
```   460 lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
```
```   461   by (metis subspace_span subspace_diff)
```
```   462
```
```   463 lemma (in real_vector) span_setsum: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> span S) \<Longrightarrow> setsum f A \<in> span S"
```
```   464   by (rule subspace_setsum [OF subspace_span])
```
```   465
```
```   466 lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
```
```   467   by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
```
```   468
```
```   469 text \<open>The key breakdown property.\<close>
```
```   470
```
```   471 lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
```
```   472 proof (rule span_unique)
```
```   473   show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
```
```   474     by (fast intro: scaleR_one [symmetric])
```
```   475   show "subspace (range (\<lambda>k. k *\<^sub>R x))"
```
```   476     unfolding subspace_def
```
```   477     by (auto intro: scaleR_add_left [symmetric])
```
```   478 next
```
```   479   fix T
```
```   480   assume "{x} \<subseteq> T" and "subspace T"
```
```   481   then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
```
```   482     unfolding subspace_def by auto
```
```   483 qed
```
```   484
```
```   485 text \<open>Mapping under linear image.\<close>
```
```   486
```
```   487 lemma subspace_linear_image:
```
```   488   assumes lf: "linear f"
```
```   489     and sS: "subspace S"
```
```   490   shows "subspace (f ` S)"
```
```   491   using lf sS linear_0[OF lf]
```
```   492   unfolding linear_iff subspace_def
```
```   493   apply (auto simp add: image_iff)
```
```   494   apply (rule_tac x="x + y" in bexI)
```
```   495   apply auto
```
```   496   apply (rule_tac x="c *\<^sub>R x" in bexI)
```
```   497   apply auto
```
```   498   done
```
```   499
```
```   500 lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
```
```   501   by (auto simp add: subspace_def linear_iff linear_0[of f])
```
```   502
```
```   503 lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
```
```   504   by (auto simp add: subspace_def linear_iff linear_0[of f])
```
```   505
```
```   506 lemma span_linear_image:
```
```   507   assumes lf: "linear f"
```
```   508   shows "span (f ` S) = f ` span S"
```
```   509 proof (rule span_unique)
```
```   510   show "f ` S \<subseteq> f ` span S"
```
```   511     by (intro image_mono span_inc)
```
```   512   show "subspace (f ` span S)"
```
```   513     using lf subspace_span by (rule subspace_linear_image)
```
```   514 next
```
```   515   fix T
```
```   516   assume "f ` S \<subseteq> T" and "subspace T"
```
```   517   then show "f ` span S \<subseteq> T"
```
```   518     unfolding image_subset_iff_subset_vimage
```
```   519     by (intro span_minimal subspace_linear_vimage lf)
```
```   520 qed
```
```   521
```
```   522 lemma spans_image:
```
```   523   assumes lf: "linear f"
```
```   524     and VB: "V \<subseteq> span B"
```
```   525   shows "f ` V \<subseteq> span (f ` B)"
```
```   526   unfolding span_linear_image[OF lf] by (metis VB image_mono)
```
```   527
```
```   528 lemma span_Un: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
```
```   529 proof (rule span_unique)
```
```   530   show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
```
```   531     by safe (force intro: span_clauses)+
```
```   532 next
```
```   533   have "linear (\<lambda>(a, b). a + b)"
```
```   534     by (simp add: linear_iff scaleR_add_right)
```
```   535   moreover have "subspace (span A \<times> span B)"
```
```   536     by (intro subspace_Times subspace_span)
```
```   537   ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
```
```   538     by (rule subspace_linear_image)
```
```   539 next
```
```   540   fix T
```
```   541   assume "A \<union> B \<subseteq> T" and "subspace T"
```
```   542   then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
```
```   543     by (auto intro!: subspace_add elim: span_induct)
```
```   544 qed
```
```   545
```
```   546 lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
```
```   547 proof -
```
```   548   have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
```
```   549     unfolding span_Un span_singleton
```
```   550     apply safe
```
```   551     apply (rule_tac x=k in exI, simp)
```
```   552     apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
```
```   553     apply auto
```
```   554     done
```
```   555   then show ?thesis by simp
```
```   556 qed
```
```   557
```
```   558 lemma span_breakdown:
```
```   559   assumes bS: "b \<in> S"
```
```   560     and aS: "a \<in> span S"
```
```   561   shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
```
```   562   using assms span_insert [of b "S - {b}"]
```
```   563   by (simp add: insert_absorb)
```
```   564
```
```   565 lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
```
```   566   by (simp add: span_insert)
```
```   567
```
```   568 text \<open>Hence some "reversal" results.\<close>
```
```   569
```
```   570 lemma in_span_insert:
```
```   571   assumes a: "a \<in> span (insert b S)"
```
```   572     and na: "a \<notin> span S"
```
```   573   shows "b \<in> span (insert a S)"
```
```   574 proof -
```
```   575   from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
```
```   576     unfolding span_insert by fast
```
```   577   show ?thesis
```
```   578   proof (cases "k = 0")
```
```   579     case True
```
```   580     with k have "a \<in> span S" by simp
```
```   581     with na show ?thesis by simp
```
```   582   next
```
```   583     case False
```
```   584     from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
```
```   585       by (rule span_mul)
```
```   586     then have "b - inverse k *\<^sub>R a \<in> span S"
```
```   587       using \<open>k \<noteq> 0\<close> by (simp add: scaleR_diff_right)
```
```   588     then show ?thesis
```
```   589       unfolding span_insert by fast
```
```   590   qed
```
```   591 qed
```
```   592
```
```   593 lemma in_span_delete:
```
```   594   assumes a: "a \<in> span S"
```
```   595     and na: "a \<notin> span (S - {b})"
```
```   596   shows "b \<in> span (insert a (S - {b}))"
```
```   597   apply (rule in_span_insert)
```
```   598   apply (rule set_rev_mp)
```
```   599   apply (rule a)
```
```   600   apply (rule span_mono)
```
```   601   apply blast
```
```   602   apply (rule na)
```
```   603   done
```
```   604
```
```   605 text \<open>Transitivity property.\<close>
```
```   606
```
```   607 lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
```
```   608   unfolding span_def by (rule hull_redundant)
```
```   609
```
```   610 lemma span_trans:
```
```   611   assumes x: "x \<in> span S"
```
```   612     and y: "y \<in> span (insert x S)"
```
```   613   shows "y \<in> span S"
```
```   614   using assms by (simp only: span_redundant)
```
```   615
```
```   616 lemma span_insert_0[simp]: "span (insert 0 S) = span S"
```
```   617   by (simp only: span_redundant span_0)
```
```   618
```
```   619 text \<open>An explicit expansion is sometimes needed.\<close>
```
```   620
```
```   621 lemma span_explicit:
```
```   622   "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
```
```   623   (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
```
```   624 proof -
```
```   625   {
```
```   626     fix x
```
```   627     assume "?h x"
```
```   628     then obtain S u where "finite S" and "S \<subseteq> P" and "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
```
```   629       by blast
```
```   630     then have "x \<in> span P"
```
```   631       by (auto intro: span_setsum span_mul span_superset)
```
```   632   }
```
```   633   moreover
```
```   634   have "\<forall>x \<in> span P. ?h x"
```
```   635   proof (rule span_induct_alt')
```
```   636     show "?h 0"
```
```   637       by (rule exI[where x="{}"], simp)
```
```   638   next
```
```   639     fix c x y
```
```   640     assume x: "x \<in> P"
```
```   641     assume hy: "?h y"
```
```   642     from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
```
```   643       and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
```
```   644     let ?S = "insert x S"
```
```   645     let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
```
```   646     from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
```
```   647       by blast+
```
```   648     have "?Q ?S ?u (c*\<^sub>R x + y)"
```
```   649     proof cases
```
```   650       assume xS: "x \<in> S"
```
```   651       have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = (\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
```
```   652         using xS by (simp add: setsum.remove [OF fS xS] insert_absorb)
```
```   653       also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
```
```   654         by (simp add: setsum.remove [OF fS xS] algebra_simps)
```
```   655       also have "\<dots> = c*\<^sub>R x + y"
```
```   656         by (simp add: add.commute u)
```
```   657       finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
```
```   658       then show ?thesis using th0 by blast
```
```   659     next
```
```   660       assume xS: "x \<notin> S"
```
```   661       have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
```
```   662         unfolding u[symmetric]
```
```   663         apply (rule setsum.cong)
```
```   664         using xS
```
```   665         apply auto
```
```   666         done
```
```   667       show ?thesis using fS xS th0
```
```   668         by (simp add: th00 add.commute cong del: if_weak_cong)
```
```   669     qed
```
```   670     then show "?h (c*\<^sub>R x + y)"
```
```   671       by fast
```
```   672   qed
```
```   673   ultimately show ?thesis by blast
```
```   674 qed
```
```   675
```
```   676 lemma dependent_explicit:
```
```   677   "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))"
```
```   678   (is "?lhs = ?rhs")
```
```   679 proof -
```
```   680   {
```
```   681     assume dP: "dependent P"
```
```   682     then obtain a S u where aP: "a \<in> P" and fS: "finite S"
```
```   683       and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
```
```   684       unfolding dependent_def span_explicit by blast
```
```   685     let ?S = "insert a S"
```
```   686     let ?u = "\<lambda>y. if y = a then - 1 else u y"
```
```   687     let ?v = a
```
```   688     from aP SP have aS: "a \<notin> S"
```
```   689       by blast
```
```   690     from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
```
```   691       by auto
```
```   692     have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
```
```   693       using fS aS
```
```   694       apply simp
```
```   695       apply (subst (2) ua[symmetric])
```
```   696       apply (rule setsum.cong)
```
```   697       apply auto
```
```   698       done
```
```   699     with th0 have ?rhs by fast
```
```   700   }
```
```   701   moreover
```
```   702   {
```
```   703     fix S u v
```
```   704     assume fS: "finite S"
```
```   705       and SP: "S \<subseteq> P"
```
```   706       and vS: "v \<in> S"
```
```   707       and uv: "u v \<noteq> 0"
```
```   708       and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
```
```   709     let ?a = v
```
```   710     let ?S = "S - {v}"
```
```   711     let ?u = "\<lambda>i. (- u i) / u v"
```
```   712     have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
```
```   713       using fS SP vS by auto
```
```   714     have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
```
```   715       setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
```
```   716       using fS vS uv by (simp add: setsum_diff1 field_simps)
```
```   717     also have "\<dots> = ?a"
```
```   718       unfolding scaleR_right.setsum [symmetric] u using uv by simp
```
```   719     finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
```
```   720     with th0 have ?lhs
```
```   721       unfolding dependent_def span_explicit
```
```   722       apply -
```
```   723       apply (rule bexI[where x= "?a"])
```
```   724       apply (simp_all del: scaleR_minus_left)
```
```   725       apply (rule exI[where x= "?S"])
```
```   726       apply (auto simp del: scaleR_minus_left)
```
```   727       done
```
```   728   }
```
```   729   ultimately show ?thesis by blast
```
```   730 qed
```
```   731
```
```   732 lemma dependent_finite:
```
```   733   assumes "finite S"
```
```   734     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)"
```
```   735            (is "?lhs = ?rhs")
```
```   736 proof
```
```   737   assume ?lhs
```
```   738   then obtain T u v
```
```   739          where "finite T" "T \<subseteq> S" "v\<in>T" "u v \<noteq> 0" "(\<Sum>v\<in>T. u v *\<^sub>R v) = 0"
```
```   740     by (force simp: dependent_explicit)
```
```   741   with assms show ?rhs
```
```   742     apply (rule_tac x="\<lambda>v. if v \<in> T then u v else 0" in exI)
```
```   743     apply (auto simp: setsum.mono_neutral_right)
```
```   744     done
```
```   745 next
```
```   746   assume ?rhs  with assms show ?lhs
```
```   747     by (fastforce simp add: dependent_explicit)
```
```   748 qed
```
```   749
```
```   750 lemma span_alt:
```
```   751   "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}}"
```
```   752   unfolding span_explicit
```
```   753   apply safe
```
```   754   subgoal for x S u
```
```   755     by (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
```
```   756         (auto intro!: setsum.mono_neutral_cong_right)
```
```   757   apply auto
```
```   758   done
```
```   759
```
```   760 lemma dependent_alt:
```
```   761   "dependent B \<longleftrightarrow>
```
```   762     (\<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))"
```
```   763   unfolding dependent_explicit
```
```   764   apply safe
```
```   765   subgoal for S u v
```
```   766     apply (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
```
```   767     apply (subst setsum.mono_neutral_cong_left[where T=S])
```
```   768     apply (auto intro!: setsum.mono_neutral_cong_right cong: rev_conj_cong)
```
```   769     done
```
```   770   apply auto
```
```   771   done
```
```   772
```
```   773 lemma independent_alt:
```
```   774   "independent B \<longleftrightarrow>
```
```   775     (\<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))"
```
```   776   unfolding dependent_alt by auto
```
```   777
```
```   778 lemma independentD_alt:
```
```   779   "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"
```
```   780   unfolding independent_alt by blast
```
```   781
```
```   782 lemma independentD_unique:
```
```   783   assumes B: "independent B"
```
```   784     and X: "finite {x. X x \<noteq> 0}" "{x. X x \<noteq> 0} \<subseteq> B"
```
```   785     and Y: "finite {x. Y x \<noteq> 0}" "{x. Y x \<noteq> 0} \<subseteq> B"
```
```   786     and "(\<Sum>x | X x \<noteq> 0. X x *\<^sub>R x) = (\<Sum>x| Y x \<noteq> 0. Y x *\<^sub>R x)"
```
```   787   shows "X = Y"
```
```   788 proof -
```
```   789   have "X x - Y x = 0" for x
```
```   790     using B
```
```   791   proof (rule independentD_alt)
```
```   792     have "{x. X x - Y x \<noteq> 0} \<subseteq> {x. X x \<noteq> 0} \<union> {x. Y x \<noteq> 0}"
```
```   793       by auto
```
```   794     then show "finite {x. X x - Y x \<noteq> 0}" "{x. X x - Y x \<noteq> 0} \<subseteq> B"
```
```   795       using X Y by (auto dest: finite_subset)
```
```   796     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)"
```
```   797       using X Y by (intro setsum.mono_neutral_cong_left) auto
```
```   798     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)"
```
```   799       by (simp add: scaleR_diff_left setsum_subtractf assms)
```
```   800     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)"
```
```   801       using X Y by (intro setsum.mono_neutral_cong_right) auto
```
```   802     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)"
```
```   803       using X Y by (intro setsum.mono_neutral_cong_right) auto
```
```   804     finally show "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *\<^sub>R x) = 0"
```
```   805       using assms by simp
```
```   806   qed
```
```   807   then show ?thesis
```
```   808     by auto
```
```   809 qed
```
```   810
```
```   811 text \<open>This is useful for building a basis step-by-step.\<close>
```
```   812
```
```   813 lemma independent_insert:
```
```   814   "independent (insert a S) \<longleftrightarrow>
```
```   815     (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
```
```   816   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   817 proof (cases "a \<in> S")
```
```   818   case True
```
```   819   then show ?thesis
```
```   820     using insert_absorb[OF True] by simp
```
```   821 next
```
```   822   case False
```
```   823   show ?thesis
```
```   824   proof
```
```   825     assume i: ?lhs
```
```   826     then show ?rhs
```
```   827       using False
```
```   828       apply simp
```
```   829       apply (rule conjI)
```
```   830       apply (rule independent_mono)
```
```   831       apply assumption
```
```   832       apply blast
```
```   833       apply (simp add: dependent_def)
```
```   834       done
```
```   835   next
```
```   836     assume i: ?rhs
```
```   837     show ?lhs
```
```   838       using i False
```
```   839       apply (auto simp add: dependent_def)
```
```   840       by (metis in_span_insert insert_Diff_if insert_Diff_single insert_absorb)
```
```   841   qed
```
```   842 qed
```
```   843
```
```   844 lemma independent_Union_directed:
```
```   845   assumes directed: "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
```
```   846   assumes indep: "\<And>c. c \<in> C \<Longrightarrow> independent c"
```
```   847   shows "independent (\<Union>C)"
```
```   848 proof
```
```   849   assume "dependent (\<Union>C)"
```
```   850   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"
```
```   851     by (auto simp: dependent_explicit)
```
```   852
```
```   853   have "S \<noteq> {}"
```
```   854     using \<open>v \<in> S\<close> by auto
```
```   855   have "\<exists>c\<in>C. S \<subseteq> c"
```
```   856     using \<open>finite S\<close> \<open>S \<noteq> {}\<close> \<open>S \<subseteq> \<Union>C\<close>
```
```   857   proof (induction rule: finite_ne_induct)
```
```   858     case (insert i I)
```
```   859     then obtain c d where cd: "c \<in> C" "d \<in> C" and iI: "I \<subseteq> c" "i \<in> d"
```
```   860       by blast
```
```   861     from directed[OF cd] cd have "c \<union> d \<in> C"
```
```   862       by (auto simp: sup.absorb1 sup.absorb2)
```
```   863     with iI show ?case
```
```   864       by (intro bexI[of _ "c \<union> d"]) auto
```
```   865   qed auto
```
```   866   then obtain c where "c \<in> C" "S \<subseteq> c"
```
```   867     by auto
```
```   868   have "dependent c"
```
```   869     unfolding dependent_explicit
```
```   870     by (intro exI[of _ S] exI[of _ u] bexI[of _ v] conjI) fact+
```
```   871   with indep[OF \<open>c \<in> C\<close>] show False
```
```   872     by auto
```
```   873 qed
```
```   874
```
```   875 text \<open>Hence we can create a maximal independent subset.\<close>
```
```   876
```
```   877 lemma maximal_independent_subset_extend:
```
```   878   assumes "S \<subseteq> V" "independent S"
```
```   879   shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
```
```   880 proof -
```
```   881   let ?C = "{B. S \<subseteq> B \<and> independent B \<and> B \<subseteq> V}"
```
```   882   have "\<exists>M\<in>?C. \<forall>X\<in>?C. M \<subseteq> X \<longrightarrow> X = M"
```
```   883   proof (rule subset_Zorn)
```
```   884     fix C :: "'a set set" assume "subset.chain ?C C"
```
```   885     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"
```
```   886       "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
```
```   887       unfolding subset.chain_def by blast+
```
```   888
```
```   889     show "\<exists>U\<in>?C. \<forall>X\<in>C. X \<subseteq> U"
```
```   890     proof cases
```
```   891       assume "C = {}" with assms show ?thesis
```
```   892         by (auto intro!: exI[of _ S])
```
```   893     next
```
```   894       assume "C \<noteq> {}"
```
```   895       with C(2) have "S \<subseteq> \<Union>C"
```
```   896         by auto
```
```   897       moreover have "independent (\<Union>C)"
```
```   898         by (intro independent_Union_directed C)
```
```   899       moreover have "\<Union>C \<subseteq> V"
```
```   900         using C by auto
```
```   901       ultimately show ?thesis
```
```   902         by auto
```
```   903     qed
```
```   904   qed
```
```   905   then obtain B where B: "independent B" "B \<subseteq> V" "S \<subseteq> B"
```
```   906     and max: "\<And>S. independent S \<Longrightarrow> S \<subseteq> V \<Longrightarrow> B \<subseteq> S \<Longrightarrow> S = B"
```
```   907     by auto
```
```   908   moreover
```
```   909   { assume "\<not> V \<subseteq> span B"
```
```   910     then obtain v where "v \<in> V" "v \<notin> span B"
```
```   911       by auto
```
```   912     with B have "independent (insert v B)"
```
```   913       unfolding independent_insert by auto
```
```   914     from max[OF this] \<open>v \<in> V\<close> \<open>B \<subseteq> V\<close>
```
```   915     have "v \<in> B"
```
```   916       by auto
```
```   917     with \<open>v \<notin> span B\<close> have False
```
```   918       by (auto intro: span_superset) }
```
```   919   ultimately show ?thesis
```
```   920     by (auto intro!: exI[of _ B])
```
```   921 qed
```
```   922
```
```   923
```
```   924 lemma maximal_independent_subset:
```
```   925   "\<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
```
```   926   by (metis maximal_independent_subset_extend[of "{}"] empty_subsetI independent_empty)
```
```   927
```
```   928 lemma span_finite:
```
```   929   assumes fS: "finite S"
```
```   930   shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
```
```   931   (is "_ = ?rhs")
```
```   932 proof -
```
```   933   {
```
```   934     fix y
```
```   935     assume y: "y \<in> span S"
```
```   936     from y obtain S' u where fS': "finite S'"
```
```   937       and SS': "S' \<subseteq> S"
```
```   938       and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
```
```   939       unfolding span_explicit by blast
```
```   940     let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
```
```   941     have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
```
```   942       using SS' fS by (auto intro!: setsum.mono_neutral_cong_right)
```
```   943     then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
```
```   944     then have "y \<in> ?rhs" by auto
```
```   945   }
```
```   946   moreover
```
```   947   {
```
```   948     fix y u
```
```   949     assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
```
```   950     then have "y \<in> span S" using fS unfolding span_explicit by auto
```
```   951   }
```
```   952   ultimately show ?thesis by blast
```
```   953 qed
```
```   954
```
```   955 lemma linear_independent_extend_subspace:
```
```   956   assumes "independent B"
```
```   957   shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x) \<and> range g = span (f`B)"
```
```   958 proof -
```
```   959   from maximal_independent_subset_extend[OF _ \<open>independent B\<close>, of UNIV]
```
```   960   obtain B' where "B \<subseteq> B'" "independent B'" "span B' = UNIV"
```
```   961     by (auto simp: top_unique)
```
```   962   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)"
```
```   963     using \<open>span B' = UNIV\<close> unfolding span_alt by auto
```
```   964   then obtain X where X: "\<And>y. {x. X y x \<noteq> 0} \<subseteq> B'" "\<And>y. finite {x. X y x \<noteq> 0}"
```
```   965     "\<And>y. y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R x)"
```
```   966     unfolding choice_iff by auto
```
```   967
```
```   968   have X_add: "X (x + y) = (\<lambda>z. X x z + X y z)" for x y
```
```   969     using \<open>independent B'\<close>
```
```   970   proof (rule independentD_unique)
```
```   971     have "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R z)
```
```   972       = (\<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)"
```
```   973       by (intro setsum.mono_neutral_cong_left) (auto intro: X)
```
```   974     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)"
```
```   975       by (auto simp add: scaleR_add_left setsum.distrib
```
```   976                intro!: arg_cong2[where f="op +"]  setsum.mono_neutral_cong_right X)
```
```   977     also have "\<dots> = x + y"
```
```   978       by (simp add: X(3)[symmetric])
```
```   979     also have "\<dots> = (\<Sum>z | X (x + y) z \<noteq> 0. X (x + y) z *\<^sub>R z)"
```
```   980       by (rule X(3))
```
```   981     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)"
```
```   982       ..
```
```   983     have "{z. X x z + X y z \<noteq> 0} \<subseteq> {z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}"
```
```   984       by auto
```
```   985     then show "finite {z. X x z + X y z \<noteq> 0}" "{xa. X x xa + X y xa \<noteq> 0} \<subseteq> B'"
```
```   986         "finite {xa. X (x + y) xa \<noteq> 0}" "{xa. X (x + y) xa \<noteq> 0} \<subseteq> B'"
```
```   987       using X(1) by (auto dest: finite_subset intro: X)
```
```   988   qed
```
```   989
```
```   990   have X_cmult: "X (c *\<^sub>R x) = (\<lambda>z. c * X x z)" for x c
```
```   991     using \<open>independent B'\<close>
```
```   992   proof (rule independentD_unique)
```
```   993     show "finite {z. X (c *\<^sub>R x) z \<noteq> 0}" "{z. X (c *\<^sub>R x) z \<noteq> 0} \<subseteq> B'"
```
```   994       "finite {z. c * X x z \<noteq> 0}" "{z. c * X x z \<noteq> 0} \<subseteq> B' "
```
```   995       using X(1,2) by auto
```
```   996     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)"
```
```   997       unfolding scaleR_scaleR[symmetric] scaleR_setsum_right[symmetric]
```
```   998       by (cases "c = 0") (auto simp: X(3)[symmetric])
```
```   999   qed
```
```  1000
```
```  1001   have X_B': "x \<in> B' \<Longrightarrow> X x = (\<lambda>z. if z = x then 1 else 0)" for x
```
```  1002     using \<open>independent B'\<close>
```
```  1003     by (rule independentD_unique[OF _ X(2) X(1)]) (auto intro: X simp: X(3)[symmetric])
```
```  1004
```
```  1005   define f' where "f' y = (if y \<in> B then f y else 0)" for y
```
```  1006   define g where "g y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R f' x)" for y
```
```  1007
```
```  1008   have g_f': "x \<in> B' \<Longrightarrow> g x = f' x" for x
```
```  1009     by (auto simp: g_def X_B')
```
```  1010
```
```  1011   have "linear g"
```
```  1012   proof
```
```  1013     fix x y
```
```  1014     have *: "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R f' z)
```
```  1015       = (\<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)"
```
```  1016       by (intro setsum.mono_neutral_cong_left) (auto intro: X)
```
```  1017     show "g (x + y) = g x + g y"
```
```  1018       unfolding g_def X_add *
```
```  1019       by (auto simp add: scaleR_add_left setsum.distrib
```
```  1020                intro!: arg_cong2[where f="op +"]  setsum.mono_neutral_cong_right X)
```
```  1021   next
```
```  1022     show "g (r *\<^sub>R x) = r *\<^sub>R g x" for r x
```
```  1023       by (auto simp add: g_def X_cmult scaleR_setsum_right intro!: setsum.mono_neutral_cong_left X)
```
```  1024   qed
```
```  1025   moreover have "\<forall>x\<in>B. g x = f x"
```
```  1026     using \<open>B \<subseteq> B'\<close> by (auto simp: g_f' f'_def)
```
```  1027   moreover have "range g = span (f`B)"
```
```  1028     unfolding \<open>span B' = UNIV\<close>[symmetric] span_linear_image[OF \<open>linear g\<close>, symmetric]
```
```  1029   proof (rule span_subspace)
```
```  1030     have "g ` B' \<subseteq> f`B \<union> {0}"
```
```  1031       by (auto simp: g_f' f'_def)
```
```  1032     also have "\<dots> \<subseteq> span (f`B)"
```
```  1033       by (auto intro: span_superset span_0)
```
```  1034     finally show "g ` B' \<subseteq> span (f`B)"
```
```  1035       by auto
```
```  1036     have "x \<in> B \<Longrightarrow> f x = g x" for x
```
```  1037       using \<open>B \<subseteq> B'\<close> by (auto simp add: g_f' f'_def)
```
```  1038     then show "span (f ` B) \<subseteq> span (g ` B')"
```
```  1039       using \<open>B \<subseteq> B'\<close> by (intro span_mono) auto
```
```  1040   qed (rule subspace_span)
```
```  1041   ultimately show ?thesis
```
```  1042     by auto
```
```  1043 qed
```
```  1044
```
```  1045 lemma linear_independent_extend:
```
```  1046   "independent B \<Longrightarrow> \<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
```
```  1047   using linear_independent_extend_subspace[of B f] by auto
```
```  1048
```
```  1049 text \<open>Linear functions are equal on a subspace if they are on a spanning set.\<close>
```
```  1050
```
```  1051 lemma subspace_kernel:
```
```  1052   assumes lf: "linear f"
```
```  1053   shows "subspace {x. f x = 0}"
```
```  1054   apply (simp add: subspace_def)
```
```  1055   apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
```
```  1056   done
```
```  1057
```
```  1058 lemma linear_eq_0_span:
```
```  1059   assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
```
```  1060   shows "\<forall>x \<in> span B. f x = 0"
```
```  1061   using f0 subspace_kernel[OF lf]
```
```  1062   by (rule span_induct')
```
```  1063
```
```  1064 lemma linear_eq_0: "linear f \<Longrightarrow> S \<subseteq> span B \<Longrightarrow> \<forall>x\<in>B. f x = 0 \<Longrightarrow> \<forall>x\<in>S. f x = 0"
```
```  1065   using linear_eq_0_span[of f B] by auto
```
```  1066
```
```  1067 lemma linear_eq_span:  "linear f \<Longrightarrow> linear g \<Longrightarrow> \<forall>x\<in>B. f x = g x \<Longrightarrow> \<forall>x \<in> span B. f x = g x"
```
```  1068   using linear_eq_0_span[of "\<lambda>x. f x - g x" B] by (auto simp: linear_compose_sub)
```
```  1069
```
```  1070 lemma linear_eq: "linear f \<Longrightarrow> linear g \<Longrightarrow> S \<subseteq> span B \<Longrightarrow> \<forall>x\<in>B. f x = g x \<Longrightarrow> \<forall>x\<in>S. f x = g x"
```
```  1071   using linear_eq_span[of f g B] by auto
```
```  1072
```
```  1073 text \<open>The degenerate case of the Exchange Lemma.\<close>
```
```  1074
```
```  1075 lemma spanning_subset_independent:
```
```  1076   assumes BA: "B \<subseteq> A"
```
```  1077     and iA: "independent A"
```
```  1078     and AsB: "A \<subseteq> span B"
```
```  1079   shows "A = B"
```
```  1080 proof
```
```  1081   show "B \<subseteq> A" by (rule BA)
```
```  1082
```
```  1083   from span_mono[OF BA] span_mono[OF AsB]
```
```  1084   have sAB: "span A = span B" unfolding span_span by blast
```
```  1085
```
```  1086   {
```
```  1087     fix x
```
```  1088     assume x: "x \<in> A"
```
```  1089     from iA have th0: "x \<notin> span (A - {x})"
```
```  1090       unfolding dependent_def using x by blast
```
```  1091     from x have xsA: "x \<in> span A"
```
```  1092       by (blast intro: span_superset)
```
```  1093     have "A - {x} \<subseteq> A" by blast
```
```  1094     then have th1: "span (A - {x}) \<subseteq> span A"
```
```  1095       by (metis span_mono)
```
```  1096     {
```
```  1097       assume xB: "x \<notin> B"
```
```  1098       from xB BA have "B \<subseteq> A - {x}"
```
```  1099         by blast
```
```  1100       then have "span B \<subseteq> span (A - {x})"
```
```  1101         by (metis span_mono)
```
```  1102       with th1 th0 sAB have "x \<notin> span A"
```
```  1103         by blast
```
```  1104       with x have False
```
```  1105         by (metis span_superset)
```
```  1106     }
```
```  1107     then have "x \<in> B" by blast
```
```  1108   }
```
```  1109   then show "A \<subseteq> B" by blast
```
```  1110 qed
```
```  1111
```
```  1112 text \<open>Relation between bases and injectivity/surjectivity of map.\<close>
```
```  1113
```
```  1114 lemma spanning_surjective_image:
```
```  1115   assumes us: "UNIV \<subseteq> span S"
```
```  1116     and lf: "linear f"
```
```  1117     and sf: "surj f"
```
```  1118   shows "UNIV \<subseteq> span (f ` S)"
```
```  1119 proof -
```
```  1120   have "UNIV \<subseteq> f ` UNIV"
```
```  1121     using sf by (auto simp add: surj_def)
```
```  1122   also have " \<dots> \<subseteq> span (f ` S)"
```
```  1123     using spans_image[OF lf us] .
```
```  1124   finally show ?thesis .
```
```  1125 qed
```
```  1126
```
```  1127 lemma independent_inj_on_image:
```
```  1128   assumes iS: "independent S"
```
```  1129     and lf: "linear f"
```
```  1130     and fi: "inj_on f (span S)"
```
```  1131   shows "independent (f ` S)"
```
```  1132 proof -
```
```  1133   {
```
```  1134     fix a
```
```  1135     assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
```
```  1136     have eq: "f ` S - {f a} = f ` (S - {a})"
```
```  1137       using fi \<open>a\<in>S\<close> by (auto simp add: inj_on_def span_superset)
```
```  1138     from a have "f a \<in> f ` span (S - {a})"
```
```  1139       unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
```
```  1140     then have "a \<in> span (S - {a})"
```
```  1141       by (rule inj_on_image_mem_iff_alt[OF fi, rotated])
```
```  1142          (insert span_mono[of "S - {a}" S], auto intro: span_superset \<open>a\<in>S\<close>)
```
```  1143     with a(1) iS have False
```
```  1144       by (simp add: dependent_def)
```
```  1145   }
```
```  1146   then show ?thesis
```
```  1147     unfolding dependent_def by blast
```
```  1148 qed
```
```  1149
```
```  1150 lemma independent_injective_image:
```
```  1151   "independent S \<Longrightarrow> linear f \<Longrightarrow> inj f \<Longrightarrow> independent (f ` S)"
```
```  1152   using independent_inj_on_image[of S f] by (auto simp: subset_inj_on)
```
```  1153
```
```  1154 text \<open>Detailed theorems about left and right invertibility in general case.\<close>
```
```  1155
```
```  1156 lemma linear_inj_on_left_inverse:
```
```  1157   assumes lf: "linear f" and fi: "inj_on f (span S)"
```
```  1158   shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span S. g (f x) = x)"
```
```  1159 proof -
```
```  1160   obtain B where "independent B" "B \<subseteq> S" "S \<subseteq> span B"
```
```  1161     using maximal_independent_subset[of S] by auto
```
```  1162   then have "span S = span B"
```
```  1163     unfolding span_eq by (auto simp: span_superset)
```
```  1164   with linear_independent_extend_subspace[OF independent_inj_on_image, OF \<open>independent B\<close> lf] fi
```
```  1165   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)"
```
```  1166     by fastforce
```
```  1167   have fB: "inj_on f B"
```
```  1168     using fi by (auto simp: \<open>span S = span B\<close> intro: subset_inj_on span_superset)
```
```  1169
```
```  1170   have "\<forall>x\<in>span B. g (f x) = x"
```
```  1171   proof (intro linear_eq_span)
```
```  1172     show "linear (\<lambda>x. x)" "linear (\<lambda>x. g (f x))"
```
```  1173       using linear_id linear_compose[OF \<open>linear f\<close> \<open>linear g\<close>] by (auto simp: id_def comp_def)
```
```  1174     show "\<forall>x \<in> B. g (f x) = x"
```
```  1175       using g fi \<open>span S = span B\<close> by (auto simp: fB)
```
```  1176   qed
```
```  1177   moreover
```
```  1178   have "inv_into B f ` f ` B \<subseteq> B"
```
```  1179     by (auto simp: fB)
```
```  1180   then have "range g \<subseteq> span S"
```
```  1181     unfolding g \<open>span S = span B\<close> by (intro span_mono)
```
```  1182   ultimately show ?thesis
```
```  1183     using \<open>span S = span B\<close> \<open>linear g\<close> by auto
```
```  1184 qed
```
```  1185
```
```  1186 lemma linear_injective_left_inverse: "linear f \<Longrightarrow> inj f \<Longrightarrow> \<exists>g. linear g \<and> g \<circ> f = id"
```
```  1187   using linear_inj_on_left_inverse[of f UNIV] by (auto simp: fun_eq_iff span_UNIV)
```
```  1188
```
```  1189 lemma linear_surj_right_inverse:
```
```  1190   assumes lf: "linear f" and sf: "span T \<subseteq> f`span S"
```
```  1191   shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span T. f (g x) = x)"
```
```  1192 proof -
```
```  1193   obtain B where "independent B" "B \<subseteq> T" "T \<subseteq> span B"
```
```  1194     using maximal_independent_subset[of T] by auto
```
```  1195   then have "span T = span B"
```
```  1196     unfolding span_eq by (auto simp: span_superset)
```
```  1197
```
```  1198   from linear_independent_extend_subspace[OF \<open>independent B\<close>, of "inv_into (span S) f"]
```
```  1199   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)"
```
```  1200     by auto
```
```  1201   moreover have "x \<in> B \<Longrightarrow> f (inv_into (span S) f x) = x" for x
```
```  1202     using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (intro f_inv_into_f) (auto intro: span_superset)
```
```  1203   ultimately have "\<forall>x\<in>B. f (g x) = x"
```
```  1204     by auto
```
```  1205   then have "\<forall>x\<in>span B. f (g x) = x"
```
```  1206     using linear_id linear_compose[OF \<open>linear g\<close> \<open>linear f\<close>]
```
```  1207     by (intro linear_eq_span) (auto simp: id_def comp_def)
```
```  1208   moreover have "inv_into (span S) f ` B \<subseteq> span S"
```
```  1209     using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (auto intro: inv_into_into span_superset)
```
```  1210   then have "range g \<subseteq> span S"
```
```  1211     unfolding g by (intro span_minimal subspace_span) auto
```
```  1212   ultimately show ?thesis
```
```  1213     using \<open>linear g\<close> \<open>span T = span B\<close> by auto
```
```  1214 qed
```
```  1215
```
```  1216 lemma linear_surjective_right_inverse: "linear f \<Longrightarrow> surj f \<Longrightarrow> \<exists>g. linear g \<and> f \<circ> g = id"
```
```  1217   using linear_surj_right_inverse[of f UNIV UNIV]
```
```  1218   by (auto simp: span_UNIV fun_eq_iff)
```
```  1219
```
```  1220 text \<open>The general case of the Exchange Lemma, the key to what follows.\<close>
```
```  1221
```
```  1222 lemma exchange_lemma:
```
```  1223   assumes f:"finite t"
```
```  1224     and i: "independent s"
```
```  1225     and sp: "s \<subseteq> span t"
```
```  1226   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'"
```
```  1227   using f i sp
```
```  1228 proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
```
```  1229   case less
```
```  1230   note ft = \<open>finite t\<close> and s = \<open>independent s\<close> and sp = \<open>s \<subseteq> span t\<close>
```
```  1231   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'"
```
```  1232   let ?ths = "\<exists>t'. ?P t'"
```
```  1233   {
```
```  1234     assume "s \<subseteq> t"
```
```  1235     then have ?ths
```
```  1236       by (metis ft Un_commute sp sup_ge1)
```
```  1237   }
```
```  1238   moreover
```
```  1239   {
```
```  1240     assume st: "t \<subseteq> s"
```
```  1241     from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
```
```  1242     have ?ths
```
```  1243       by (metis Un_absorb sp)
```
```  1244   }
```
```  1245   moreover
```
```  1246   {
```
```  1247     assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
```
```  1248     from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
```
```  1249       by blast
```
```  1250     from b have "t - {b} - s \<subset> t - s"
```
```  1251       by blast
```
```  1252     then have cardlt: "card (t - {b} - s) < card (t - s)"
```
```  1253       using ft by (auto intro: psubset_card_mono)
```
```  1254     from b ft have ct0: "card t \<noteq> 0"
```
```  1255       by auto
```
```  1256     have ?ths
```
```  1257     proof cases
```
```  1258       assume stb: "s \<subseteq> span (t - {b})"
```
```  1259       from ft have ftb: "finite (t - {b})"
```
```  1260         by auto
```
```  1261       from less(1)[OF cardlt ftb s stb]
```
```  1262       obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
```
```  1263         and fu: "finite u" by blast
```
```  1264       let ?w = "insert b u"
```
```  1265       have th0: "s \<subseteq> insert b u"
```
```  1266         using u by blast
```
```  1267       from u(3) b have "u \<subseteq> s \<union> t"
```
```  1268         by blast
```
```  1269       then have th1: "insert b u \<subseteq> s \<union> t"
```
```  1270         using u b by blast
```
```  1271       have bu: "b \<notin> u"
```
```  1272         using b u by blast
```
```  1273       from u(1) ft b have "card u = (card t - 1)"
```
```  1274         by auto
```
```  1275       then have th2: "card (insert b u) = card t"
```
```  1276         using card_insert_disjoint[OF fu bu] ct0 by auto
```
```  1277       from u(4) have "s \<subseteq> span u" .
```
```  1278       also have "\<dots> \<subseteq> span (insert b u)"
```
```  1279         by (rule span_mono) blast
```
```  1280       finally have th3: "s \<subseteq> span (insert b u)" .
```
```  1281       from th0 th1 th2 th3 fu have th: "?P ?w"
```
```  1282         by blast
```
```  1283       from th show ?thesis by blast
```
```  1284     next
```
```  1285       assume stb: "\<not> s \<subseteq> span (t - {b})"
```
```  1286       from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
```
```  1287         by blast
```
```  1288       have ab: "a \<noteq> b"
```
```  1289         using a b by blast
```
```  1290       have at: "a \<notin> t"
```
```  1291         using a ab span_superset[of a "t- {b}"] by auto
```
```  1292       have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
```
```  1293         using cardlt ft a b by auto
```
```  1294       have ft': "finite (insert a (t - {b}))"
```
```  1295         using ft by auto
```
```  1296       {
```
```  1297         fix x
```
```  1298         assume xs: "x \<in> s"
```
```  1299         have t: "t \<subseteq> insert b (insert a (t - {b}))"
```
```  1300           using b by auto
```
```  1301         from b(1) have "b \<in> span t"
```
```  1302           by (simp add: span_superset)
```
```  1303         have bs: "b \<in> span (insert a (t - {b}))"
```
```  1304           apply (rule in_span_delete)
```
```  1305           using a sp unfolding subset_eq
```
```  1306           apply auto
```
```  1307           done
```
```  1308         from xs sp have "x \<in> span t"
```
```  1309           by blast
```
```  1310         with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
```
```  1311         from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
```
```  1312       }
```
```  1313       then have sp': "s \<subseteq> span (insert a (t - {b}))"
```
```  1314         by blast
```
```  1315       from less(1)[OF mlt ft' s sp'] obtain u where u:
```
```  1316         "card u = card (insert a (t - {b}))"
```
```  1317         "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
```
```  1318         "s \<subseteq> span u" by blast
```
```  1319       from u a b ft at ct0 have "?P u"
```
```  1320         by auto
```
```  1321       then show ?thesis by blast
```
```  1322     qed
```
```  1323   }
```
```  1324   ultimately show ?ths by blast
```
```  1325 qed
```
```  1326
```
```  1327 text \<open>This implies corresponding size bounds.\<close>
```
```  1328
```
```  1329 lemma independent_span_bound:
```
```  1330   assumes f: "finite t"
```
```  1331     and i: "independent s"
```
```  1332     and sp: "s \<subseteq> span t"
```
```  1333   shows "finite s \<and> card s \<le> card t"
```
```  1334   by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
```
```  1335
```
```  1336 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
```
```  1337 proof -
```
```  1338   have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
```
```  1339     by auto
```
```  1340   show ?thesis unfolding eq
```
```  1341     apply (rule finite_imageI)
```
```  1342     apply (rule finite)
```
```  1343     done
```
```  1344 qed
```
```  1345
```
```  1346
```
```  1347 subsection \<open>More interesting properties of the norm.\<close>
```
```  1348
```
```  1349 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
```
```  1350   by auto
```
```  1351
```
```  1352 notation inner (infix "\<bullet>" 70)
```
```  1353
```
```  1354 lemma square_bound_lemma:
```
```  1355   fixes x :: real
```
```  1356   shows "x < (1 + x) * (1 + x)"
```
```  1357 proof -
```
```  1358   have "(x + 1/2)\<^sup>2 + 3/4 > 0"
```
```  1359     using zero_le_power2[of "x+1/2"] by arith
```
```  1360   then show ?thesis
```
```  1361     by (simp add: field_simps power2_eq_square)
```
```  1362 qed
```
```  1363
```
```  1364 lemma square_continuous:
```
```  1365   fixes e :: real
```
```  1366   shows "e > 0 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> \<bar>y * y - x * x\<bar> < e)"
```
```  1367   using isCont_power[OF continuous_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
```
```  1368   by (force simp add: power2_eq_square)
```
```  1369
```
```  1370
```
```  1371 lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
```
```  1372   by simp (* TODO: delete *)
```
```  1373
```
```  1374 lemma norm_triangle_sub:
```
```  1375   fixes x y :: "'a::real_normed_vector"
```
```  1376   shows "norm x \<le> norm y + norm (x - y)"
```
```  1377   using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
```
```  1378
```
```  1379 lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
```
```  1380   by (simp add: norm_eq_sqrt_inner)
```
```  1381
```
```  1382 lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
```
```  1383   by (simp add: norm_eq_sqrt_inner)
```
```  1384
```
```  1385 lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
```
```  1386   apply (subst order_eq_iff)
```
```  1387   apply (auto simp: norm_le)
```
```  1388   done
```
```  1389
```
```  1390 lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
```
```  1391   by (simp add: norm_eq_sqrt_inner)
```
```  1392
```
```  1393 text\<open>Squaring equations and inequalities involving norms.\<close>
```
```  1394
```
```  1395 lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
```
```  1396   by (simp only: power2_norm_eq_inner) (* TODO: move? *)
```
```  1397
```
```  1398 lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
```
```  1399   by (auto simp add: norm_eq_sqrt_inner)
```
```  1400
```
```  1401 lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2"
```
```  1402   apply (simp add: dot_square_norm abs_le_square_iff[symmetric])
```
```  1403   using norm_ge_zero[of x]
```
```  1404   apply arith
```
```  1405   done
```
```  1406
```
```  1407 lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2"
```
```  1408   apply (simp add: dot_square_norm abs_le_square_iff[symmetric])
```
```  1409   using norm_ge_zero[of x]
```
```  1410   apply arith
```
```  1411   done
```
```  1412
```
```  1413 lemma norm_lt_square: "norm x < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2"
```
```  1414   by (metis not_le norm_ge_square)
```
```  1415
```
```  1416 lemma norm_gt_square: "norm x > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2"
```
```  1417   by (metis norm_le_square not_less)
```
```  1418
```
```  1419 text\<open>Dot product in terms of the norm rather than conversely.\<close>
```
```  1420
```
```  1421 lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
```
```  1422   inner_scaleR_left inner_scaleR_right
```
```  1423
```
```  1424 lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
```
```  1425   by (simp only: power2_norm_eq_inner inner_simps inner_commute) auto
```
```  1426
```
```  1427 lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
```
```  1428   by (simp only: power2_norm_eq_inner inner_simps inner_commute)
```
```  1429     (auto simp add: algebra_simps)
```
```  1430
```
```  1431 text\<open>Equality of vectors in terms of @{term "op \<bullet>"} products.\<close>
```
```  1432
```
```  1433 lemma linear_componentwise:
```
```  1434   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
```
```  1435   assumes lf: "linear f"
```
```  1436   shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
```
```  1437 proof -
```
```  1438   have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
```
```  1439     by (simp add: inner_setsum_left)
```
```  1440   then show ?thesis
```
```  1441     unfolding linear_setsum_mul[OF lf, symmetric]
```
```  1442     unfolding euclidean_representation ..
```
```  1443 qed
```
```  1444
```
```  1445 lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
```
```  1446   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1447 proof
```
```  1448   assume ?lhs
```
```  1449   then show ?rhs by simp
```
```  1450 next
```
```  1451   assume ?rhs
```
```  1452   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
```
```  1453     by simp
```
```  1454   then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
```
```  1455     by (simp add: inner_diff inner_commute)
```
```  1456   then have "(x - y) \<bullet> (x - y) = 0"
```
```  1457     by (simp add: field_simps inner_diff inner_commute)
```
```  1458   then show "x = y" by simp
```
```  1459 qed
```
```  1460
```
```  1461 lemma norm_triangle_half_r:
```
```  1462   "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
```
```  1463   using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
```
```  1464
```
```  1465 lemma norm_triangle_half_l:
```
```  1466   assumes "norm (x - y) < e / 2"
```
```  1467     and "norm (x' - y) < e / 2"
```
```  1468   shows "norm (x - x') < e"
```
```  1469   using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
```
```  1470   unfolding dist_norm[symmetric] .
```
```  1471
```
```  1472 lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
```
```  1473   by (rule norm_triangle_ineq [THEN order_trans])
```
```  1474
```
```  1475 lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
```
```  1476   by (rule norm_triangle_ineq [THEN le_less_trans])
```
```  1477
```
```  1478 lemma setsum_clauses:
```
```  1479   shows "setsum f {} = 0"
```
```  1480     and "finite S \<Longrightarrow> setsum f (insert x S) = (if x \<in> S then setsum f S else f x + setsum f S)"
```
```  1481   by (auto simp add: insert_absorb)
```
```  1482
```
```  1483 lemma setsum_norm_le:
```
```  1484   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  1485   assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
```
```  1486   shows "norm (setsum f S) \<le> setsum g S"
```
```  1487   by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)
```
```  1488
```
```  1489 lemma setsum_norm_bound:
```
```  1490   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  1491   assumes K: "\<forall>x \<in> S. norm (f x) \<le> K"
```
```  1492   shows "norm (setsum f S) \<le> of_nat (card S) * K"
```
```  1493   using setsum_norm_le[OF K] setsum_constant[symmetric]
```
```  1494   by simp
```
```  1495
```
```  1496 lemma setsum_group:
```
```  1497   assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
```
```  1498   shows "setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) T = setsum g S"
```
```  1499   apply (subst setsum_image_gen[OF fS, of g f])
```
```  1500   apply (rule setsum.mono_neutral_right[OF fT fST])
```
```  1501   apply (auto intro: setsum.neutral)
```
```  1502   done
```
```  1503
```
```  1504 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
```
```  1505 proof
```
```  1506   assume "\<forall>x. x \<bullet> y = x \<bullet> z"
```
```  1507   then have "\<forall>x. x \<bullet> (y - z) = 0"
```
```  1508     by (simp add: inner_diff)
```
```  1509   then have "(y - z) \<bullet> (y - z) = 0" ..
```
```  1510   then show "y = z" by simp
```
```  1511 qed simp
```
```  1512
```
```  1513 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
```
```  1514 proof
```
```  1515   assume "\<forall>z. x \<bullet> z = y \<bullet> z"
```
```  1516   then have "\<forall>z. (x - y) \<bullet> z = 0"
```
```  1517     by (simp add: inner_diff)
```
```  1518   then have "(x - y) \<bullet> (x - y) = 0" ..
```
```  1519   then show "x = y" by simp
```
```  1520 qed simp
```
```  1521
```
```  1522
```
```  1523 subsection \<open>Orthogonality.\<close>
```
```  1524
```
```  1525 context real_inner
```
```  1526 begin
```
```  1527
```
```  1528 definition "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
```
```  1529
```
```  1530 lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0"
```
```  1531   by (simp add: orthogonal_def)
```
```  1532
```
```  1533 lemma orthogonal_clauses:
```
```  1534   "orthogonal a 0"
```
```  1535   "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
```
```  1536   "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
```
```  1537   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
```
```  1538   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
```
```  1539   "orthogonal 0 a"
```
```  1540   "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
```
```  1541   "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
```
```  1542   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
```
```  1543   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
```
```  1544   unfolding orthogonal_def inner_add inner_diff by auto
```
```  1545
```
```  1546 end
```
```  1547
```
```  1548 lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
```
```  1549   by (simp add: orthogonal_def inner_commute)
```
```  1550
```
```  1551 lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x"
```
```  1552   by (rule ext) (simp add: orthogonal_def)
```
```  1553
```
```  1554 lemma pairwise_ortho_scaleR:
```
```  1555     "pairwise (\<lambda>i j. orthogonal (f i) (g j)) B
```
```  1556     \<Longrightarrow> pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B"
```
```  1557   by (auto simp: pairwise_def orthogonal_clauses)
```
```  1558
```
```  1559 lemma orthogonal_rvsum:
```
```  1560     "\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (setsum f s)"
```
```  1561   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
```
```  1562
```
```  1563 lemma orthogonal_lvsum:
```
```  1564     "\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (setsum f s) y"
```
```  1565   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
```
```  1566
```
```  1567 lemma norm_add_Pythagorean:
```
```  1568   assumes "orthogonal a b"
```
```  1569     shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2"
```
```  1570 proof -
```
```  1571   from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)"
```
```  1572     by (simp add: algebra_simps orthogonal_def inner_commute)
```
```  1573   then show ?thesis
```
```  1574     by (simp add: power2_norm_eq_inner)
```
```  1575 qed
```
```  1576
```
```  1577 lemma norm_setsum_Pythagorean:
```
```  1578   assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I"
```
```  1579     shows "(norm (setsum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
```
```  1580 using assms
```
```  1581 proof (induction I rule: finite_induct)
```
```  1582   case empty then show ?case by simp
```
```  1583 next
```
```  1584   case (insert x I)
```
```  1585   then have "orthogonal (f x) (setsum f I)"
```
```  1586     by (metis pairwise_insert orthogonal_rvsum)
```
```  1587   with insert show ?case
```
```  1588     by (simp add: pairwise_insert norm_add_Pythagorean)
```
```  1589 qed
```
```  1590
```
```  1591
```
```  1592 subsection \<open>Bilinear functions.\<close>
```
```  1593
```
```  1594 definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
```
```  1595
```
```  1596 lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
```
```  1597   by (simp add: bilinear_def linear_iff)
```
```  1598
```
```  1599 lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
```
```  1600   by (simp add: bilinear_def linear_iff)
```
```  1601
```
```  1602 lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
```
```  1603   by (simp add: bilinear_def linear_iff)
```
```  1604
```
```  1605 lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
```
```  1606   by (simp add: bilinear_def linear_iff)
```
```  1607
```
```  1608 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
```
```  1609   by (drule bilinear_lmul [of _ "- 1"]) simp
```
```  1610
```
```  1611 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
```
```  1612   by (drule bilinear_rmul [of _ _ "- 1"]) simp
```
```  1613
```
```  1614 lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
```
```  1615   using add_left_imp_eq[of x y 0] by auto
```
```  1616
```
```  1617 lemma bilinear_lzero:
```
```  1618   assumes "bilinear h"
```
```  1619   shows "h 0 x = 0"
```
```  1620   using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
```
```  1621
```
```  1622 lemma bilinear_rzero:
```
```  1623   assumes "bilinear h"
```
```  1624   shows "h x 0 = 0"
```
```  1625   using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
```
```  1626
```
```  1627 lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
```
```  1628   using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
```
```  1629
```
```  1630 lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
```
```  1631   using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
```
```  1632
```
```  1633 lemma bilinear_setsum:
```
```  1634   assumes bh: "bilinear h"
```
```  1635     and fS: "finite S"
```
```  1636     and fT: "finite T"
```
```  1637   shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
```
```  1638 proof -
```
```  1639   have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
```
```  1640     apply (rule linear_setsum[unfolded o_def])
```
```  1641     using bh fS
```
```  1642     apply (auto simp add: bilinear_def)
```
```  1643     done
```
```  1644   also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
```
```  1645     apply (rule setsum.cong, simp)
```
```  1646     apply (rule linear_setsum[unfolded o_def])
```
```  1647     using bh fT
```
```  1648     apply (auto simp add: bilinear_def)
```
```  1649     done
```
```  1650   finally show ?thesis
```
```  1651     unfolding setsum.cartesian_product .
```
```  1652 qed
```
```  1653
```
```  1654
```
```  1655 subsection \<open>Adjoints.\<close>
```
```  1656
```
```  1657 definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
```
```  1658
```
```  1659 lemma adjoint_unique:
```
```  1660   assumes "\<forall>x y. inner (f x) y = inner x (g y)"
```
```  1661   shows "adjoint f = g"
```
```  1662   unfolding adjoint_def
```
```  1663 proof (rule some_equality)
```
```  1664   show "\<forall>x y. inner (f x) y = inner x (g y)"
```
```  1665     by (rule assms)
```
```  1666 next
```
```  1667   fix h
```
```  1668   assume "\<forall>x y. inner (f x) y = inner x (h y)"
```
```  1669   then have "\<forall>x y. inner x (g y) = inner x (h y)"
```
```  1670     using assms by simp
```
```  1671   then have "\<forall>x y. inner x (g y - h y) = 0"
```
```  1672     by (simp add: inner_diff_right)
```
```  1673   then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
```
```  1674     by simp
```
```  1675   then have "\<forall>y. h y = g y"
```
```  1676     by simp
```
```  1677   then show "h = g" by (simp add: ext)
```
```  1678 qed
```
```  1679
```
```  1680 text \<open>TODO: The following lemmas about adjoints should hold for any
```
```  1681   Hilbert space (i.e. complete inner product space).
```
```  1682   (see \<^url>\<open>http://en.wikipedia.org/wiki/Hermitian_adjoint\<close>)
```
```  1683 \<close>
```
```  1684
```
```  1685 lemma adjoint_works:
```
```  1686   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  1687   assumes lf: "linear f"
```
```  1688   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```  1689 proof -
```
```  1690   have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
```
```  1691   proof (intro allI exI)
```
```  1692     fix y :: "'m" and x
```
```  1693     let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
```
```  1694     have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
```
```  1695       by (simp add: euclidean_representation)
```
```  1696     also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
```
```  1697       unfolding linear_setsum[OF lf]
```
```  1698       by (simp add: linear_cmul[OF lf])
```
```  1699     finally show "f x \<bullet> y = x \<bullet> ?w"
```
```  1700       by (simp add: inner_setsum_left inner_setsum_right mult.commute)
```
```  1701   qed
```
```  1702   then show ?thesis
```
```  1703     unfolding adjoint_def choice_iff
```
```  1704     by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
```
```  1705 qed
```
```  1706
```
```  1707 lemma adjoint_clauses:
```
```  1708   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  1709   assumes lf: "linear f"
```
```  1710   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```  1711     and "adjoint f y \<bullet> x = y \<bullet> f x"
```
```  1712   by (simp_all add: adjoint_works[OF lf] inner_commute)
```
```  1713
```
```  1714 lemma adjoint_linear:
```
```  1715   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  1716   assumes lf: "linear f"
```
```  1717   shows "linear (adjoint f)"
```
```  1718   by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
```
```  1719     adjoint_clauses[OF lf] inner_distrib)
```
```  1720
```
```  1721 lemma adjoint_adjoint:
```
```  1722   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  1723   assumes lf: "linear f"
```
```  1724   shows "adjoint (adjoint f) = f"
```
```  1725   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
```
```  1726
```
```  1727
```
```  1728 subsection \<open>Interlude: Some properties of real sets\<close>
```
```  1729
```
```  1730 lemma seq_mono_lemma:
```
```  1731   assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
```
```  1732     and "\<forall>n \<ge> m. e n \<le> e m"
```
```  1733   shows "\<forall>n \<ge> m. d n < e m"
```
```  1734   using assms
```
```  1735   apply auto
```
```  1736   apply (erule_tac x="n" in allE)
```
```  1737   apply (erule_tac x="n" in allE)
```
```  1738   apply auto
```
```  1739   done
```
```  1740
```
```  1741 lemma infinite_enumerate:
```
```  1742   assumes fS: "infinite S"
```
```  1743   shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
```
```  1744   unfolding subseq_def
```
```  1745   using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
```
```  1746
```
```  1747 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)"
```
```  1748   apply auto
```
```  1749   apply (rule_tac x="d/2" in exI)
```
```  1750   apply auto
```
```  1751   done
```
```  1752
```
```  1753 lemma approachable_lt_le2:  \<comment>\<open>like the above, but pushes aside an extra formula\<close>
```
```  1754     "(\<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)"
```
```  1755   apply auto
```
```  1756   apply (rule_tac x="d/2" in exI, auto)
```
```  1757   done
```
```  1758
```
```  1759 lemma triangle_lemma:
```
```  1760   fixes x y z :: real
```
```  1761   assumes x: "0 \<le> x"
```
```  1762     and y: "0 \<le> y"
```
```  1763     and z: "0 \<le> z"
```
```  1764     and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
```
```  1765   shows "x \<le> y + z"
```
```  1766 proof -
```
```  1767   have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
```
```  1768     using z y by simp
```
```  1769   with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
```
```  1770     by (simp add: power2_eq_square field_simps)
```
```  1771   from y z have yz: "y + z \<ge> 0"
```
```  1772     by arith
```
```  1773   from power2_le_imp_le[OF th yz] show ?thesis .
```
```  1774 qed
```
```  1775
```
```  1776
```
```  1777
```
```  1778 subsection \<open>Archimedean properties and useful consequences\<close>
```
```  1779
```
```  1780 text\<open>Bernoulli's inequality\<close>
```
```  1781 proposition Bernoulli_inequality:
```
```  1782   fixes x :: real
```
```  1783   assumes "-1 \<le> x"
```
```  1784     shows "1 + n * x \<le> (1 + x) ^ n"
```
```  1785 proof (induct n)
```
```  1786   case 0
```
```  1787   then show ?case by simp
```
```  1788 next
```
```  1789   case (Suc n)
```
```  1790   have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
```
```  1791     by (simp add: algebra_simps)
```
```  1792   also have "... = (1 + x) * (1 + n*x)"
```
```  1793     by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
```
```  1794   also have "... \<le> (1 + x) ^ Suc n"
```
```  1795     using Suc.hyps assms mult_left_mono by fastforce
```
```  1796   finally show ?case .
```
```  1797 qed
```
```  1798
```
```  1799 corollary Bernoulli_inequality_even:
```
```  1800   fixes x :: real
```
```  1801   assumes "even n"
```
```  1802     shows "1 + n * x \<le> (1 + x) ^ n"
```
```  1803 proof (cases "-1 \<le> x \<or> n=0")
```
```  1804   case True
```
```  1805   then show ?thesis
```
```  1806     by (auto simp: Bernoulli_inequality)
```
```  1807 next
```
```  1808   case False
```
```  1809   then have "real n \<ge> 1"
```
```  1810     by simp
```
```  1811   with False have "n * x \<le> -1"
```
```  1812     by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
```
```  1813   then have "1 + n * x \<le> 0"
```
```  1814     by auto
```
```  1815   also have "... \<le> (1 + x) ^ n"
```
```  1816     using assms
```
```  1817     using zero_le_even_power by blast
```
```  1818   finally show ?thesis .
```
```  1819 qed
```
```  1820
```
```  1821 corollary real_arch_pow:
```
```  1822   fixes x :: real
```
```  1823   assumes x: "1 < x"
```
```  1824   shows "\<exists>n. y < x^n"
```
```  1825 proof -
```
```  1826   from x have x0: "x - 1 > 0"
```
```  1827     by arith
```
```  1828   from reals_Archimedean3[OF x0, rule_format, of y]
```
```  1829   obtain n :: nat where n: "y < real n * (x - 1)" by metis
```
```  1830   from x0 have x00: "x- 1 \<ge> -1" by arith
```
```  1831   from Bernoulli_inequality[OF x00, of n] n
```
```  1832   have "y < x^n" by auto
```
```  1833   then show ?thesis by metis
```
```  1834 qed
```
```  1835
```
```  1836 corollary real_arch_pow_inv:
```
```  1837   fixes x y :: real
```
```  1838   assumes y: "y > 0"
```
```  1839     and x1: "x < 1"
```
```  1840   shows "\<exists>n. x^n < y"
```
```  1841 proof (cases "x > 0")
```
```  1842   case True
```
```  1843   with x1 have ix: "1 < 1/x" by (simp add: field_simps)
```
```  1844   from real_arch_pow[OF ix, of "1/y"]
```
```  1845   obtain n where n: "1/y < (1/x)^n" by blast
```
```  1846   then show ?thesis using y \<open>x > 0\<close>
```
```  1847     by (auto simp add: field_simps)
```
```  1848 next
```
```  1849   case False
```
```  1850   with y x1 show ?thesis
```
```  1851     apply auto
```
```  1852     apply (rule exI[where x=1])
```
```  1853     apply auto
```
```  1854     done
```
```  1855 qed
```
```  1856
```
```  1857 lemma forall_pos_mono:
```
```  1858   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```  1859     (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
```
```  1860   by (metis real_arch_inverse)
```
```  1861
```
```  1862 lemma forall_pos_mono_1:
```
```  1863   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```  1864     (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
```
```  1865   apply (rule forall_pos_mono)
```
```  1866   apply auto
```
```  1867   apply (metis Suc_pred of_nat_Suc)
```
```  1868   done
```
```  1869
```
```  1870
```
```  1871 subsection \<open>Euclidean Spaces as Typeclass\<close>
```
```  1872
```
```  1873 lemma independent_Basis: "independent Basis"
```
```  1874   unfolding dependent_def
```
```  1875   apply (subst span_finite)
```
```  1876   apply simp
```
```  1877   apply clarify
```
```  1878   apply (drule_tac f="inner a" in arg_cong)
```
```  1879   apply (simp add: inner_Basis inner_setsum_right eq_commute)
```
```  1880   done
```
```  1881
```
```  1882 lemma span_Basis [simp]: "span Basis = UNIV"
```
```  1883   unfolding span_finite [OF finite_Basis]
```
```  1884   by (fast intro: euclidean_representation)
```
```  1885
```
```  1886 lemma in_span_Basis: "x \<in> span Basis"
```
```  1887   unfolding span_Basis ..
```
```  1888
```
```  1889 lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
```
```  1890   by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
```
```  1891
```
```  1892 lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
```
```  1893   by (metis Basis_le_norm order_trans)
```
```  1894
```
```  1895 lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
```
```  1896   by (metis Basis_le_norm le_less_trans)
```
```  1897
```
```  1898 lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
```
```  1899   apply (subst euclidean_representation[of x, symmetric])
```
```  1900   apply (rule order_trans[OF norm_setsum])
```
```  1901   apply (auto intro!: setsum_mono)
```
```  1902   done
```
```  1903
```
```  1904 lemma setsum_norm_allsubsets_bound:
```
```  1905   fixes f :: "'a \<Rightarrow> 'n::euclidean_space"
```
```  1906   assumes fP: "finite P"
```
```  1907     and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
```
```  1908   shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
```
```  1909 proof -
```
```  1910   have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
```
```  1911     by (rule setsum_mono) (rule norm_le_l1)
```
```  1912   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>)"
```
```  1913     by (rule setsum.commute)
```
```  1914   also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
```
```  1915   proof (rule setsum_bounded_above)
```
```  1916     fix i :: 'n
```
```  1917     assume i: "i \<in> Basis"
```
```  1918     have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
```
```  1919       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)"
```
```  1920       by (simp add: abs_real_def setsum.If_cases[OF fP] setsum_negf norm_triangle_ineq4 inner_setsum_left
```
```  1921         del: real_norm_def)
```
```  1922     also have "\<dots> \<le> e + e"
```
```  1923       unfolding real_norm_def
```
```  1924       by (intro add_mono norm_bound_Basis_le i fPs) auto
```
```  1925     finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
```
```  1926   qed
```
```  1927   also have "\<dots> = 2 * real DIM('n) * e" by simp
```
```  1928   finally show ?thesis .
```
```  1929 qed
```
```  1930
```
```  1931
```
```  1932 subsection \<open>Linearity and Bilinearity continued\<close>
```
```  1933
```
```  1934 lemma linear_bounded:
```
```  1935   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1936   assumes lf: "linear f"
```
```  1937   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```  1938 proof
```
```  1939   let ?B = "\<Sum>b\<in>Basis. norm (f b)"
```
```  1940   show "\<forall>x. norm (f x) \<le> ?B * norm x"
```
```  1941   proof
```
```  1942     fix x :: 'a
```
```  1943     let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
```
```  1944     have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
```
```  1945       unfolding euclidean_representation ..
```
```  1946     also have "\<dots> = norm (setsum ?g Basis)"
```
```  1947       by (simp add: linear_setsum [OF lf] linear_cmul [OF lf])
```
```  1948     finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
```
```  1949     have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
```
```  1950     proof
```
```  1951       fix i :: 'a
```
```  1952       assume i: "i \<in> Basis"
```
```  1953       from Basis_le_norm[OF i, of x]
```
```  1954       show "norm (?g i) \<le> norm (f i) * norm x"
```
```  1955         unfolding norm_scaleR
```
```  1956         apply (subst mult.commute)
```
```  1957         apply (rule mult_mono)
```
```  1958         apply (auto simp add: field_simps)
```
```  1959         done
```
```  1960     qed
```
```  1961     from setsum_norm_le[of _ ?g, OF th]
```
```  1962     show "norm (f x) \<le> ?B * norm x"
```
```  1963       unfolding th0 setsum_left_distrib by metis
```
```  1964   qed
```
```  1965 qed
```
```  1966
```
```  1967 lemma linear_conv_bounded_linear:
```
```  1968   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1969   shows "linear f \<longleftrightarrow> bounded_linear f"
```
```  1970 proof
```
```  1971   assume "linear f"
```
```  1972   then interpret f: linear f .
```
```  1973   show "bounded_linear f"
```
```  1974   proof
```
```  1975     have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```  1976       using \<open>linear f\<close> by (rule linear_bounded)
```
```  1977     then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1978       by (simp add: mult.commute)
```
```  1979   qed
```
```  1980 next
```
```  1981   assume "bounded_linear f"
```
```  1982   then interpret f: bounded_linear f .
```
```  1983   show "linear f" ..
```
```  1984 qed
```
```  1985
```
```  1986 lemmas linear_linear = linear_conv_bounded_linear[symmetric]
```
```  1987
```
```  1988 lemma linear_bounded_pos:
```
```  1989   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1990   assumes lf: "linear f"
```
```  1991   shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
```
```  1992 proof -
```
```  1993   have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
```
```  1994     using lf unfolding linear_conv_bounded_linear
```
```  1995     by (rule bounded_linear.pos_bounded)
```
```  1996   then show ?thesis
```
```  1997     by (simp only: mult.commute)
```
```  1998 qed
```
```  1999
```
```  2000 lemma bounded_linearI':
```
```  2001   fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  2002   assumes "\<And>x y. f (x + y) = f x + f y"
```
```  2003     and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```  2004   shows "bounded_linear f"
```
```  2005   unfolding linear_conv_bounded_linear[symmetric]
```
```  2006   by (rule linearI[OF assms])
```
```  2007
```
```  2008 lemma bilinear_bounded:
```
```  2009   fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
```
```  2010   assumes bh: "bilinear h"
```
```  2011   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```  2012 proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
```
```  2013   fix x :: 'm
```
```  2014   fix y :: 'n
```
```  2015   have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
```
```  2016     apply (subst euclidean_representation[where 'a='m])
```
```  2017     apply (subst euclidean_representation[where 'a='n])
```
```  2018     apply rule
```
```  2019     done
```
```  2020   also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
```
```  2021     unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
```
```  2022   finally have th: "norm (h x y) = \<dots>" .
```
```  2023   show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
```
```  2024     apply (auto simp add: setsum_left_distrib th setsum.cartesian_product)
```
```  2025     apply (rule setsum_norm_le)
```
```  2026     apply simp
```
```  2027     apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
```
```  2028       field_simps simp del: scaleR_scaleR)
```
```  2029     apply (rule mult_mono)
```
```  2030     apply (auto simp add: zero_le_mult_iff Basis_le_norm)
```
```  2031     apply (rule mult_mono)
```
```  2032     apply (auto simp add: zero_le_mult_iff Basis_le_norm)
```
```  2033     done
```
```  2034 qed
```
```  2035
```
```  2036 lemma bilinear_conv_bounded_bilinear:
```
```  2037   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```  2038   shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
```
```  2039 proof
```
```  2040   assume "bilinear h"
```
```  2041   show "bounded_bilinear h"
```
```  2042   proof
```
```  2043     fix x y z
```
```  2044     show "h (x + y) z = h x z + h y z"
```
```  2045       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
```
```  2046   next
```
```  2047     fix x y z
```
```  2048     show "h x (y + z) = h x y + h x z"
```
```  2049       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
```
```  2050   next
```
```  2051     fix r x y
```
```  2052     show "h (scaleR r x) y = scaleR r (h x y)"
```
```  2053       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
```
```  2054       by simp
```
```  2055   next
```
```  2056     fix r x y
```
```  2057     show "h x (scaleR r y) = scaleR r (h x y)"
```
```  2058       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
```
```  2059       by simp
```
```  2060   next
```
```  2061     have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```  2062       using \<open>bilinear h\<close> by (rule bilinear_bounded)
```
```  2063     then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
```
```  2064       by (simp add: ac_simps)
```
```  2065   qed
```
```  2066 next
```
```  2067   assume "bounded_bilinear h"
```
```  2068   then interpret h: bounded_bilinear h .
```
```  2069   show "bilinear h"
```
```  2070     unfolding bilinear_def linear_conv_bounded_linear
```
```  2071     using h.bounded_linear_left h.bounded_linear_right by simp
```
```  2072 qed
```
```  2073
```
```  2074 lemma bilinear_bounded_pos:
```
```  2075   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```  2076   assumes bh: "bilinear h"
```
```  2077   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```  2078 proof -
```
```  2079   have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
```
```  2080     using bh [unfolded bilinear_conv_bounded_bilinear]
```
```  2081     by (rule bounded_bilinear.pos_bounded)
```
```  2082   then show ?thesis
```
```  2083     by (simp only: ac_simps)
```
```  2084 qed
```
```  2085
```
```  2086 lemma bounded_linear_imp_has_derivative:
```
```  2087      "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
```
```  2088   by (simp add: has_derivative_def bounded_linear.linear linear_diff)
```
```  2089
```
```  2090 lemma linear_imp_has_derivative:
```
```  2091   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  2092   shows "linear f \<Longrightarrow> (f has_derivative f) net"
```
```  2093 by (simp add: has_derivative_def linear_conv_bounded_linear linear_diff)
```
```  2094
```
```  2095 lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net"
```
```  2096   using bounded_linear_imp_has_derivative differentiable_def by blast
```
```  2097
```
```  2098 lemma linear_imp_differentiable:
```
```  2099   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  2100   shows "linear f \<Longrightarrow> f differentiable net"
```
```  2101 by (metis linear_imp_has_derivative differentiable_def)
```
```  2102
```
```  2103
```
```  2104 subsection \<open>We continue.\<close>
```
```  2105
```
```  2106 lemma independent_bound:
```
```  2107   fixes S :: "'a::euclidean_space set"
```
```  2108   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
```
```  2109   using independent_span_bound[OF finite_Basis, of S] by auto
```
```  2110
```
```  2111 corollary
```
```  2112   fixes S :: "'a::euclidean_space set"
```
```  2113   assumes "independent S"
```
```  2114   shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)"
```
```  2115 using assms independent_bound by auto
```
```  2116
```
```  2117 lemma independent_explicit:
```
```  2118   fixes B :: "'a::euclidean_space set"
```
```  2119   shows "independent B \<longleftrightarrow>
```
```  2120          finite B \<and> (\<forall>c. (\<Sum>v\<in>B. c v *\<^sub>R v) = 0 \<longrightarrow> (\<forall>v \<in> B. c v = 0))"
```
```  2121 apply (cases "finite B")
```
```  2122  apply (force simp: dependent_finite)
```
```  2123 using independent_bound
```
```  2124 apply auto
```
```  2125 done
```
```  2126
```
```  2127 lemma dependent_biggerset:
```
```  2128   fixes S :: "'a::euclidean_space set"
```
```  2129   shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
```
```  2130   by (metis independent_bound not_less)
```
```  2131
```
```  2132 text \<open>Notion of dimension.\<close>
```
```  2133
```
```  2134 definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
```
```  2135
```
```  2136 lemma basis_exists:
```
```  2137   "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
```
```  2138   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)"]
```
```  2139   using maximal_independent_subset[of V] independent_bound
```
```  2140   by auto
```
```  2141
```
```  2142 corollary dim_le_card:
```
```  2143   fixes s :: "'a::euclidean_space set"
```
```  2144   shows "finite s \<Longrightarrow> dim s \<le> card s"
```
```  2145 by (metis basis_exists card_mono)
```
```  2146
```
```  2147 text \<open>Consequences of independence or spanning for cardinality.\<close>
```
```  2148
```
```  2149 lemma independent_card_le_dim:
```
```  2150   fixes B :: "'a::euclidean_space set"
```
```  2151   assumes "B \<subseteq> V"
```
```  2152     and "independent B"
```
```  2153   shows "card B \<le> dim V"
```
```  2154 proof -
```
```  2155   from basis_exists[of V] \<open>B \<subseteq> V\<close>
```
```  2156   obtain B' where "independent B'"
```
```  2157     and "B \<subseteq> span B'"
```
```  2158     and "card B' = dim V"
```
```  2159     by blast
```
```  2160   with independent_span_bound[OF _ \<open>independent B\<close> \<open>B \<subseteq> span B'\<close>] independent_bound[of B']
```
```  2161   show ?thesis by auto
```
```  2162 qed
```
```  2163
```
```  2164 lemma span_card_ge_dim:
```
```  2165   fixes B :: "'a::euclidean_space set"
```
```  2166   shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
```
```  2167   by (metis basis_exists[of V] independent_span_bound subset_trans)
```
```  2168
```
```  2169 lemma basis_card_eq_dim:
```
```  2170   fixes V :: "'a::euclidean_space set"
```
```  2171   shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
```
```  2172   by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
```
```  2173
```
```  2174 lemma dim_unique:
```
```  2175   fixes B :: "'a::euclidean_space set"
```
```  2176   shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
```
```  2177   by (metis basis_card_eq_dim)
```
```  2178
```
```  2179 text \<open>More lemmas about dimension.\<close>
```
```  2180
```
```  2181 lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
```
```  2182   using independent_Basis
```
```  2183   by (intro dim_unique[of Basis]) auto
```
```  2184
```
```  2185 lemma dim_subset:
```
```  2186   fixes S :: "'a::euclidean_space set"
```
```  2187   shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
```
```  2188   using basis_exists[of T] basis_exists[of S]
```
```  2189   by (metis independent_card_le_dim subset_trans)
```
```  2190
```
```  2191 lemma dim_subset_UNIV:
```
```  2192   fixes S :: "'a::euclidean_space set"
```
```  2193   shows "dim S \<le> DIM('a)"
```
```  2194   by (metis dim_subset subset_UNIV dim_UNIV)
```
```  2195
```
```  2196 text \<open>Converses to those.\<close>
```
```  2197
```
```  2198 lemma card_ge_dim_independent:
```
```  2199   fixes B :: "'a::euclidean_space set"
```
```  2200   assumes BV: "B \<subseteq> V"
```
```  2201     and iB: "independent B"
```
```  2202     and dVB: "dim V \<le> card B"
```
```  2203   shows "V \<subseteq> span B"
```
```  2204 proof
```
```  2205   fix a
```
```  2206   assume aV: "a \<in> V"
```
```  2207   {
```
```  2208     assume aB: "a \<notin> span B"
```
```  2209     then have iaB: "independent (insert a B)"
```
```  2210       using iB aV BV by (simp add: independent_insert)
```
```  2211     from aV BV have th0: "insert a B \<subseteq> V"
```
```  2212       by blast
```
```  2213     from aB have "a \<notin>B"
```
```  2214       by (auto simp add: span_superset)
```
```  2215     with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
```
```  2216     have False by auto
```
```  2217   }
```
```  2218   then show "a \<in> span B" by blast
```
```  2219 qed
```
```  2220
```
```  2221 lemma card_le_dim_spanning:
```
```  2222   assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
```
```  2223     and VB: "V \<subseteq> span B"
```
```  2224     and fB: "finite B"
```
```  2225     and dVB: "dim V \<ge> card B"
```
```  2226   shows "independent B"
```
```  2227 proof -
```
```  2228   {
```
```  2229     fix a
```
```  2230     assume a: "a \<in> B" "a \<in> span (B - {a})"
```
```  2231     from a fB have c0: "card B \<noteq> 0"
```
```  2232       by auto
```
```  2233     from a fB have cb: "card (B - {a}) = card B - 1"
```
```  2234       by auto
```
```  2235     from BV a have th0: "B - {a} \<subseteq> V"
```
```  2236       by blast
```
```  2237     {
```
```  2238       fix x
```
```  2239       assume x: "x \<in> V"
```
```  2240       from a have eq: "insert a (B - {a}) = B"
```
```  2241         by blast
```
```  2242       from x VB have x': "x \<in> span B"
```
```  2243         by blast
```
```  2244       from span_trans[OF a(2), unfolded eq, OF x']
```
```  2245       have "x \<in> span (B - {a})" .
```
```  2246     }
```
```  2247     then have th1: "V \<subseteq> span (B - {a})"
```
```  2248       by blast
```
```  2249     have th2: "finite (B - {a})"
```
```  2250       using fB by auto
```
```  2251     from span_card_ge_dim[OF th0 th1 th2]
```
```  2252     have c: "dim V \<le> card (B - {a})" .
```
```  2253     from c c0 dVB cb have False by simp
```
```  2254   }
```
```  2255   then show ?thesis
```
```  2256     unfolding dependent_def by blast
```
```  2257 qed
```
```  2258
```
```  2259 lemma card_eq_dim:
```
```  2260   fixes B :: "'a::euclidean_space set"
```
```  2261   shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
```
```  2262   by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
```
```  2263
```
```  2264 text \<open>More general size bound lemmas.\<close>
```
```  2265
```
```  2266 lemma independent_bound_general:
```
```  2267   fixes S :: "'a::euclidean_space set"
```
```  2268   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
```
```  2269   by (metis independent_card_le_dim independent_bound subset_refl)
```
```  2270
```
```  2271 lemma dependent_biggerset_general:
```
```  2272   fixes S :: "'a::euclidean_space set"
```
```  2273   shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
```
```  2274   using independent_bound_general[of S] by (metis linorder_not_le)
```
```  2275
```
```  2276 lemma dim_span [simp]:
```
```  2277   fixes S :: "'a::euclidean_space set"
```
```  2278   shows "dim (span S) = dim S"
```
```  2279 proof -
```
```  2280   have th0: "dim S \<le> dim (span S)"
```
```  2281     by (auto simp add: subset_eq intro: dim_subset span_superset)
```
```  2282   from basis_exists[of S]
```
```  2283   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
```
```  2284     by blast
```
```  2285   from B have fB: "finite B" "card B = dim S"
```
```  2286     using independent_bound by blast+
```
```  2287   have bSS: "B \<subseteq> span S"
```
```  2288     using B(1) by (metis subset_eq span_inc)
```
```  2289   have sssB: "span S \<subseteq> span B"
```
```  2290     using span_mono[OF B(3)] by (simp add: span_span)
```
```  2291   from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
```
```  2292     using fB(2) by arith
```
```  2293 qed
```
```  2294
```
```  2295 lemma subset_le_dim:
```
```  2296   fixes S :: "'a::euclidean_space set"
```
```  2297   shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
```
```  2298   by (metis dim_span dim_subset)
```
```  2299
```
```  2300 lemma span_eq_dim:
```
```  2301   fixes S :: "'a::euclidean_space set"
```
```  2302   shows "span S = span T \<Longrightarrow> dim S = dim T"
```
```  2303   by (metis dim_span)
```
```  2304
```
```  2305 lemma dim_image_le:
```
```  2306   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  2307   assumes lf: "linear f"
```
```  2308   shows "dim (f ` S) \<le> dim (S)"
```
```  2309 proof -
```
```  2310   from basis_exists[of S] obtain B where
```
```  2311     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
```
```  2312   from B have fB: "finite B" "card B = dim S"
```
```  2313     using independent_bound by blast+
```
```  2314   have "dim (f ` S) \<le> card (f ` B)"
```
```  2315     apply (rule span_card_ge_dim)
```
```  2316     using lf B fB
```
```  2317     apply (auto simp add: span_linear_image spans_image subset_image_iff)
```
```  2318     done
```
```  2319   also have "\<dots> \<le> dim S"
```
```  2320     using card_image_le[OF fB(1)] fB by simp
```
```  2321   finally show ?thesis .
```
```  2322 qed
```
```  2323
```
```  2324 text \<open>Picking an orthogonal replacement for a spanning set.\<close>
```
```  2325
```
```  2326 lemma vector_sub_project_orthogonal:
```
```  2327   fixes b x :: "'a::euclidean_space"
```
```  2328   shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
```
```  2329   unfolding inner_simps by auto
```
```  2330
```
```  2331 lemma pairwise_orthogonal_insert:
```
```  2332   assumes "pairwise orthogonal S"
```
```  2333     and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
```
```  2334   shows "pairwise orthogonal (insert x S)"
```
```  2335   using assms unfolding pairwise_def
```
```  2336   by (auto simp add: orthogonal_commute)
```
```  2337
```
```  2338 lemma basis_orthogonal:
```
```  2339   fixes B :: "'a::real_inner set"
```
```  2340   assumes fB: "finite B"
```
```  2341   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
```
```  2342   (is " \<exists>C. ?P B C")
```
```  2343   using fB
```
```  2344 proof (induct rule: finite_induct)
```
```  2345   case empty
```
```  2346   then show ?case
```
```  2347     apply (rule exI[where x="{}"])
```
```  2348     apply (auto simp add: pairwise_def)
```
```  2349     done
```
```  2350 next
```
```  2351   case (insert a B)
```
```  2352   note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
```
```  2353   from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
```
```  2354   obtain C where C: "finite C" "card C \<le> card B"
```
```  2355     "span C = span B" "pairwise orthogonal C" by blast
```
```  2356   let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
```
```  2357   let ?C = "insert ?a C"
```
```  2358   from C(1) have fC: "finite ?C"
```
```  2359     by simp
```
```  2360   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
```
```  2361     by (simp add: card_insert_if)
```
```  2362   {
```
```  2363     fix x k
```
```  2364     have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
```
```  2365       by (simp add: field_simps)
```
```  2366     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"
```
```  2367       apply (simp only: scaleR_right_diff_distrib th0)
```
```  2368       apply (rule span_add_eq)
```
```  2369       apply (rule span_mul)
```
```  2370       apply (rule span_setsum)
```
```  2371       apply (rule span_mul)
```
```  2372       apply (rule span_superset)
```
```  2373       apply assumption
```
```  2374       done
```
```  2375   }
```
```  2376   then have SC: "span ?C = span (insert a B)"
```
```  2377     unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
```
```  2378   {
```
```  2379     fix y
```
```  2380     assume yC: "y \<in> C"
```
```  2381     then have Cy: "C = insert y (C - {y})"
```
```  2382       by blast
```
```  2383     have fth: "finite (C - {y})"
```
```  2384       using C by simp
```
```  2385     have "orthogonal ?a y"
```
```  2386       unfolding orthogonal_def
```
```  2387       unfolding inner_diff inner_setsum_left right_minus_eq
```
```  2388       unfolding setsum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
```
```  2389       apply (clarsimp simp add: inner_commute[of y a])
```
```  2390       apply (rule setsum.neutral)
```
```  2391       apply clarsimp
```
```  2392       apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
```
```  2393       using \<open>y \<in> C\<close> by auto
```
```  2394   }
```
```  2395   with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
```
```  2396     by (rule pairwise_orthogonal_insert)
```
```  2397   from fC cC SC CPO have "?P (insert a B) ?C"
```
```  2398     by blast
```
```  2399   then show ?case by blast
```
```  2400 qed
```
```  2401
```
```  2402 lemma orthogonal_basis_exists:
```
```  2403   fixes V :: "('a::euclidean_space) set"
```
```  2404   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
```
```  2405 proof -
```
```  2406   from basis_exists[of V] obtain B where
```
```  2407     B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
```
```  2408     by blast
```
```  2409   from B have fB: "finite B" "card B = dim V"
```
```  2410     using independent_bound by auto
```
```  2411   from basis_orthogonal[OF fB(1)] obtain C where
```
```  2412     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
```
```  2413     by blast
```
```  2414   from C B have CSV: "C \<subseteq> span V"
```
```  2415     by (metis span_inc span_mono subset_trans)
```
```  2416   from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
```
```  2417     by (simp add: span_span)
```
```  2418   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
```
```  2419   have iC: "independent C"
```
```  2420     by (simp add: dim_span)
```
```  2421   from C fB have "card C \<le> dim V"
```
```  2422     by simp
```
```  2423   moreover have "dim V \<le> card C"
```
```  2424     using span_card_ge_dim[OF CSV SVC C(1)]
```
```  2425     by (simp add: dim_span)
```
```  2426   ultimately have CdV: "card C = dim V"
```
```  2427     using C(1) by simp
```
```  2428   from C B CSV CdV iC show ?thesis
```
```  2429     by auto
```
```  2430 qed
```
```  2431
```
```  2432 text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
```
```  2433
```
```  2434 lemma span_not_univ_orthogonal:
```
```  2435   fixes S :: "'a::euclidean_space set"
```
```  2436   assumes sU: "span S \<noteq> UNIV"
```
```  2437   shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
```
```  2438 proof -
```
```  2439   from sU obtain a where a: "a \<notin> span S"
```
```  2440     by blast
```
```  2441   from orthogonal_basis_exists obtain B where
```
```  2442     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
```
```  2443     by blast
```
```  2444   from B have fB: "finite B" "card B = dim S"
```
```  2445     using independent_bound by auto
```
```  2446   from span_mono[OF B(2)] span_mono[OF B(3)]
```
```  2447   have sSB: "span S = span B"
```
```  2448     by (simp add: span_span)
```
```  2449   let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
```
```  2450   have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
```
```  2451     unfolding sSB
```
```  2452     apply (rule span_setsum)
```
```  2453     apply (rule span_mul)
```
```  2454     apply (rule span_superset)
```
```  2455     apply assumption
```
```  2456     done
```
```  2457   with a have a0:"?a  \<noteq> 0"
```
```  2458     by auto
```
```  2459   have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
```
```  2460   proof (rule span_induct')
```
```  2461     show "subspace {x. ?a \<bullet> x = 0}"
```
```  2462       by (auto simp add: subspace_def inner_add)
```
```  2463   next
```
```  2464     {
```
```  2465       fix x
```
```  2466       assume x: "x \<in> B"
```
```  2467       from x have B': "B = insert x (B - {x})"
```
```  2468         by blast
```
```  2469       have fth: "finite (B - {x})"
```
```  2470         using fB by simp
```
```  2471       have "?a \<bullet> x = 0"
```
```  2472         apply (subst B')
```
```  2473         using fB fth
```
```  2474         unfolding setsum_clauses(2)[OF fth]
```
```  2475         apply simp unfolding inner_simps
```
```  2476         apply (clarsimp simp add: inner_add inner_setsum_left)
```
```  2477         apply (rule setsum.neutral, rule ballI)
```
```  2478         apply (simp only: inner_commute)
```
```  2479         apply (auto simp add: x field_simps
```
```  2480           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
```
```  2481         done
```
```  2482     }
```
```  2483     then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
```
```  2484       by blast
```
```  2485   qed
```
```  2486   with a0 show ?thesis
```
```  2487     unfolding sSB by (auto intro: exI[where x="?a"])
```
```  2488 qed
```
```  2489
```
```  2490 lemma span_not_univ_subset_hyperplane:
```
```  2491   fixes S :: "'a::euclidean_space set"
```
```  2492   assumes SU: "span S \<noteq> UNIV"
```
```  2493   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```  2494   using span_not_univ_orthogonal[OF SU] by auto
```
```  2495
```
```  2496 lemma lowdim_subset_hyperplane:
```
```  2497   fixes S :: "'a::euclidean_space set"
```
```  2498   assumes d: "dim S < DIM('a)"
```
```  2499   shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```  2500 proof -
```
```  2501   {
```
```  2502     assume "span S = UNIV"
```
```  2503     then have "dim (span S) = dim (UNIV :: ('a) set)"
```
```  2504       by simp
```
```  2505     then have "dim S = DIM('a)"
```
```  2506       by (simp add: dim_span dim_UNIV)
```
```  2507     with d have False by arith
```
```  2508   }
```
```  2509   then have th: "span S \<noteq> UNIV"
```
```  2510     by blast
```
```  2511   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
```
```  2512 qed
```
```  2513
```
```  2514 text \<open>We can extend a linear basis-basis injection to the whole set.\<close>
```
```  2515
```
```  2516 lemma linear_indep_image_lemma:
```
```  2517   assumes lf: "linear f"
```
```  2518     and fB: "finite B"
```
```  2519     and ifB: "independent (f ` B)"
```
```  2520     and fi: "inj_on f B"
```
```  2521     and xsB: "x \<in> span B"
```
```  2522     and fx: "f x = 0"
```
```  2523   shows "x = 0"
```
```  2524   using fB ifB fi xsB fx
```
```  2525 proof (induct arbitrary: x rule: finite_induct[OF fB])
```
```  2526   case 1
```
```  2527   then show ?case by auto
```
```  2528 next
```
```  2529   case (2 a b x)
```
```  2530   have fb: "finite b" using "2.prems" by simp
```
```  2531   have th0: "f ` b \<subseteq> f ` (insert a b)"
```
```  2532     apply (rule image_mono)
```
```  2533     apply blast
```
```  2534     done
```
```  2535   from independent_mono[ OF "2.prems"(2) th0]
```
```  2536   have ifb: "independent (f ` b)"  .
```
```  2537   have fib: "inj_on f b"
```
```  2538     apply (rule subset_inj_on [OF "2.prems"(3)])
```
```  2539     apply blast
```
```  2540     done
```
```  2541   from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
```
```  2542   obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
```
```  2543     by blast
```
```  2544   have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
```
```  2545     unfolding span_linear_image[OF lf]
```
```  2546     apply (rule imageI)
```
```  2547     using k span_mono[of "b - {a}" b]
```
```  2548     apply blast
```
```  2549     done
```
```  2550   then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
```
```  2551     by (simp add: linear_diff[OF lf] linear_cmul[OF lf])
```
```  2552   then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
```
```  2553     using "2.prems"(5) by simp
```
```  2554   have xsb: "x \<in> span b"
```
```  2555   proof (cases "k = 0")
```
```  2556     case True
```
```  2557     with k have "x \<in> span (b - {a})" by simp
```
```  2558     then show ?thesis using span_mono[of "b - {a}" b]
```
```  2559       by blast
```
```  2560   next
```
```  2561     case False
```
```  2562     with span_mul[OF th, of "- 1/ k"]
```
```  2563     have th1: "f a \<in> span (f ` b)"
```
```  2564       by auto
```
```  2565     from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
```
```  2566     have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
```
```  2567     from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
```
```  2568     have "f a \<notin> span (f ` b)" using tha
```
```  2569       using "2.hyps"(2)
```
```  2570       "2.prems"(3) by auto
```
```  2571     with th1 have False by blast
```
```  2572     then show ?thesis by blast
```
```  2573   qed
```
```  2574   from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
```
```  2575 qed
```
```  2576
```
```  2577 text \<open>Can construct an isomorphism between spaces of same dimension.\<close>
```
```  2578
```
```  2579 lemma subspace_isomorphism:
```
```  2580   fixes S :: "'a::euclidean_space set"
```
```  2581     and T :: "'b::euclidean_space set"
```
```  2582   assumes s: "subspace S"
```
```  2583     and t: "subspace T"
```
```  2584     and d: "dim S = dim T"
```
```  2585   shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
```
```  2586 proof -
```
```  2587   from basis_exists[of S] independent_bound
```
```  2588   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
```
```  2589     by blast
```
```  2590   from basis_exists[of T] independent_bound
```
```  2591   obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C"
```
```  2592     by blast
```
```  2593   from B(4) C(4) card_le_inj[of B C] d
```
```  2594   obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using \<open>finite B\<close> \<open>finite C\<close>
```
```  2595     by auto
```
```  2596   from linear_independent_extend[OF B(2)]
```
```  2597   obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x"
```
```  2598     by blast
```
```  2599   from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
```
```  2600     by simp
```
```  2601   with B(4) C(4) have ceq: "card (f ` B) = card C"
```
```  2602     using d by simp
```
```  2603   have "g ` B = f ` B"
```
```  2604     using g(2) by (auto simp add: image_iff)
```
```  2605   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
```
```  2606   finally have gBC: "g ` B = C" .
```
```  2607   have gi: "inj_on g B"
```
```  2608     using f(2) g(2) by (auto simp add: inj_on_def)
```
```  2609   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
```
```  2610   {
```
```  2611     fix x y
```
```  2612     assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
```
```  2613     from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
```
```  2614       by blast+
```
```  2615     from gxy have th0: "g (x - y) = 0"
```
```  2616       by (simp add: linear_diff[OF g(1)])
```
```  2617     have th1: "x - y \<in> span B"
```
```  2618       using x' y' by (metis span_sub)
```
```  2619     have "x = y"
```
```  2620       using g0[OF th1 th0] by simp
```
```  2621   }
```
```  2622   then have giS: "inj_on g S"
```
```  2623     unfolding inj_on_def by blast
```
```  2624   from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)"
```
```  2625     by (simp add: span_linear_image[OF g(1)])
```
```  2626   also have "\<dots> = span C" unfolding gBC ..
```
```  2627   also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
```
```  2628   finally have gS: "g ` S = T" .
```
```  2629   from g(1) gS giS show ?thesis
```
```  2630     by blast
```
```  2631 qed
```
```  2632
```
```  2633 lemma linear_eq_stdbasis:
```
```  2634   fixes f :: "'a::euclidean_space \<Rightarrow> _"
```
```  2635   assumes lf: "linear f"
```
```  2636     and lg: "linear g"
```
```  2637     and fg: "\<forall>b\<in>Basis. f b = g b"
```
```  2638   shows "f = g"
```
```  2639   using linear_eq[OF lf lg, of _ Basis] fg by auto
```
```  2640
```
```  2641 text \<open>Similar results for bilinear functions.\<close>
```
```  2642
```
```  2643 lemma bilinear_eq:
```
```  2644   assumes bf: "bilinear f"
```
```  2645     and bg: "bilinear g"
```
```  2646     and SB: "S \<subseteq> span B"
```
```  2647     and TC: "T \<subseteq> span C"
```
```  2648     and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
```
```  2649   shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
```
```  2650 proof -
```
```  2651   let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
```
```  2652   from bf bg have sp: "subspace ?P"
```
```  2653     unfolding bilinear_def linear_iff subspace_def bf bg
```
```  2654     by (auto simp add: span_0 bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def
```
```  2655       intro: bilinear_ladd[OF bf])
```
```  2656
```
```  2657   have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
```
```  2658     apply (rule span_induct' [OF _ sp])
```
```  2659     apply (rule ballI)
```
```  2660     apply (rule span_induct')
```
```  2661     apply (simp add: fg)
```
```  2662     apply (auto simp add: subspace_def)
```
```  2663     using bf bg unfolding bilinear_def linear_iff
```
```  2664     apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
```
```  2665       intro: bilinear_ladd[OF bf])
```
```  2666     done
```
```  2667   then show ?thesis
```
```  2668     using SB TC by auto
```
```  2669 qed
```
```  2670
```
```  2671 lemma bilinear_eq_stdbasis:
```
```  2672   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
```
```  2673   assumes bf: "bilinear f"
```
```  2674     and bg: "bilinear g"
```
```  2675     and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
```
```  2676   shows "f = g"
```
```  2677   using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast
```
```  2678
```
```  2679 text \<open>An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective.\<close>
```
```  2680
```
```  2681 lemma linear_injective_imp_surjective:
```
```  2682   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2683   assumes lf: "linear f"
```
```  2684     and fi: "inj f"
```
```  2685   shows "surj f"
```
```  2686 proof -
```
```  2687   let ?U = "UNIV :: 'a set"
```
```  2688   from basis_exists[of ?U] obtain B
```
```  2689     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
```
```  2690     by blast
```
```  2691   from B(4) have d: "dim ?U = card B"
```
```  2692     by simp
```
```  2693   have th: "?U \<subseteq> span (f ` B)"
```
```  2694     apply (rule card_ge_dim_independent)
```
```  2695     apply blast
```
```  2696     apply (rule independent_injective_image[OF B(2) lf fi])
```
```  2697     apply (rule order_eq_refl)
```
```  2698     apply (rule sym)
```
```  2699     unfolding d
```
```  2700     apply (rule card_image)
```
```  2701     apply (rule subset_inj_on[OF fi])
```
```  2702     apply blast
```
```  2703     done
```
```  2704   from th show ?thesis
```
```  2705     unfolding span_linear_image[OF lf] surj_def
```
```  2706     using B(3) by blast
```
```  2707 qed
```
```  2708
```
```  2709 text \<open>And vice versa.\<close>
```
```  2710
```
```  2711 lemma surjective_iff_injective_gen:
```
```  2712   assumes fS: "finite S"
```
```  2713     and fT: "finite T"
```
```  2714     and c: "card S = card T"
```
```  2715     and ST: "f ` S \<subseteq> T"
```
```  2716   shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S"
```
```  2717   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2718 proof
```
```  2719   assume h: "?lhs"
```
```  2720   {
```
```  2721     fix x y
```
```  2722     assume x: "x \<in> S"
```
```  2723     assume y: "y \<in> S"
```
```  2724     assume f: "f x = f y"
```
```  2725     from x fS have S0: "card S \<noteq> 0"
```
```  2726       by auto
```
```  2727     have "x = y"
```
```  2728     proof (rule ccontr)
```
```  2729       assume xy: "\<not> ?thesis"
```
```  2730       have th: "card S \<le> card (f ` (S - {y}))"
```
```  2731         unfolding c
```
```  2732         apply (rule card_mono)
```
```  2733         apply (rule finite_imageI)
```
```  2734         using fS apply simp
```
```  2735         using h xy x y f unfolding subset_eq image_iff
```
```  2736         apply auto
```
```  2737         apply (case_tac "xa = f x")
```
```  2738         apply (rule bexI[where x=x])
```
```  2739         apply auto
```
```  2740         done
```
```  2741       also have " \<dots> \<le> card (S - {y})"
```
```  2742         apply (rule card_image_le)
```
```  2743         using fS by simp
```
```  2744       also have "\<dots> \<le> card S - 1" using y fS by simp
```
```  2745       finally show False using S0 by arith
```
```  2746     qed
```
```  2747   }
```
```  2748   then show ?rhs
```
```  2749     unfolding inj_on_def by blast
```
```  2750 next
```
```  2751   assume h: ?rhs
```
```  2752   have "f ` S = T"
```
```  2753     apply (rule card_subset_eq[OF fT ST])
```
```  2754     unfolding card_image[OF h]
```
```  2755     apply (rule c)
```
```  2756     done
```
```  2757   then show ?lhs by blast
```
```  2758 qed
```
```  2759
```
```  2760 lemma linear_surjective_imp_injective:
```
```  2761   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2762   assumes lf: "linear f"
```
```  2763     and sf: "surj f"
```
```  2764   shows "inj f"
```
```  2765 proof -
```
```  2766   let ?U = "UNIV :: 'a set"
```
```  2767   from basis_exists[of ?U] obtain B
```
```  2768     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
```
```  2769     by blast
```
```  2770   {
```
```  2771     fix x
```
```  2772     assume x: "x \<in> span B"
```
```  2773     assume fx: "f x = 0"
```
```  2774     from B(2) have fB: "finite B"
```
```  2775       using independent_bound by auto
```
```  2776     have fBi: "independent (f ` B)"
```
```  2777       apply (rule card_le_dim_spanning[of "f ` B" ?U])
```
```  2778       apply blast
```
```  2779       using sf B(3)
```
```  2780       unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
```
```  2781       apply blast
```
```  2782       using fB apply blast
```
```  2783       unfolding d[symmetric]
```
```  2784       apply (rule card_image_le)
```
```  2785       apply (rule fB)
```
```  2786       done
```
```  2787     have th0: "dim ?U \<le> card (f ` B)"
```
```  2788       apply (rule span_card_ge_dim)
```
```  2789       apply blast
```
```  2790       unfolding span_linear_image[OF lf]
```
```  2791       apply (rule subset_trans[where B = "f ` UNIV"])
```
```  2792       using sf unfolding surj_def
```
```  2793       apply blast
```
```  2794       apply (rule image_mono)
```
```  2795       apply (rule B(3))
```
```  2796       apply (metis finite_imageI fB)
```
```  2797       done
```
```  2798     moreover have "card (f ` B) \<le> card B"
```
```  2799       by (rule card_image_le, rule fB)
```
```  2800     ultimately have th1: "card B = card (f ` B)"
```
```  2801       unfolding d by arith
```
```  2802     have fiB: "inj_on f B"
```
```  2803       unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric]
```
```  2804       by blast
```
```  2805     from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
```
```  2806     have "x = 0" by blast
```
```  2807   }
```
```  2808   then show ?thesis
```
```  2809     unfolding linear_injective_0[OF lf]
```
```  2810     using B(3)
```
```  2811     by blast
```
```  2812 qed
```
```  2813
```
```  2814 text \<open>Hence either is enough for isomorphism.\<close>
```
```  2815
```
```  2816 lemma left_right_inverse_eq:
```
```  2817   assumes fg: "f \<circ> g = id"
```
```  2818     and gh: "g \<circ> h = id"
```
```  2819   shows "f = h"
```
```  2820 proof -
```
```  2821   have "f = f \<circ> (g \<circ> h)"
```
```  2822     unfolding gh by simp
```
```  2823   also have "\<dots> = (f \<circ> g) \<circ> h"
```
```  2824     by (simp add: o_assoc)
```
```  2825   finally show "f = h"
```
```  2826     unfolding fg by simp
```
```  2827 qed
```
```  2828
```
```  2829 lemma isomorphism_expand:
```
```  2830   "f \<circ> g = id \<and> g \<circ> f = id \<longleftrightarrow> (\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x)"
```
```  2831   by (simp add: fun_eq_iff o_def id_def)
```
```  2832
```
```  2833 lemma linear_injective_isomorphism:
```
```  2834   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2835   assumes lf: "linear f"
```
```  2836     and fi: "inj f"
```
```  2837   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
```
```  2838   unfolding isomorphism_expand[symmetric]
```
```  2839   using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]]
```
```  2840     linear_injective_left_inverse[OF lf fi]
```
```  2841   by (metis left_right_inverse_eq)
```
```  2842
```
```  2843 lemma linear_surjective_isomorphism:
```
```  2844   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2845   assumes lf: "linear f"
```
```  2846     and sf: "surj f"
```
```  2847   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
```
```  2848   unfolding isomorphism_expand[symmetric]
```
```  2849   using linear_surjective_right_inverse[OF lf sf]
```
```  2850     linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
```
```  2851   by (metis left_right_inverse_eq)
```
```  2852
```
```  2853 text \<open>Left and right inverses are the same for
```
```  2854   @{typ "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"}.\<close>
```
```  2855
```
```  2856 lemma linear_inverse_left:
```
```  2857   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2858   assumes lf: "linear f"
```
```  2859     and lf': "linear f'"
```
```  2860   shows "f \<circ> f' = id \<longleftrightarrow> f' \<circ> f = id"
```
```  2861 proof -
```
```  2862   {
```
```  2863     fix f f':: "'a \<Rightarrow> 'a"
```
```  2864     assume lf: "linear f" "linear f'"
```
```  2865     assume f: "f \<circ> f' = id"
```
```  2866     from f have sf: "surj f"
```
```  2867       apply (auto simp add: o_def id_def surj_def)
```
```  2868       apply metis
```
```  2869       done
```
```  2870     from linear_surjective_isomorphism[OF lf(1) sf] lf f
```
```  2871     have "f' \<circ> f = id"
```
```  2872       unfolding fun_eq_iff o_def id_def by metis
```
```  2873   }
```
```  2874   then show ?thesis
```
```  2875     using lf lf' by metis
```
```  2876 qed
```
```  2877
```
```  2878 text \<open>Moreover, a one-sided inverse is automatically linear.\<close>
```
```  2879
```
```  2880 lemma left_inverse_linear:
```
```  2881   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2882   assumes lf: "linear f"
```
```  2883     and gf: "g \<circ> f = id"
```
```  2884   shows "linear g"
```
```  2885 proof -
```
```  2886   from gf have fi: "inj f"
```
```  2887     apply (auto simp add: inj_on_def o_def id_def fun_eq_iff)
```
```  2888     apply metis
```
```  2889     done
```
```  2890   from linear_injective_isomorphism[OF lf fi]
```
```  2891   obtain h :: "'a \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x"
```
```  2892     by blast
```
```  2893   have "h = g"
```
```  2894     apply (rule ext) using gf h(2,3)
```
```  2895     apply (simp add: o_def id_def fun_eq_iff)
```
```  2896     apply metis
```
```  2897     done
```
```  2898   with h(1) show ?thesis by blast
```
```  2899 qed
```
```  2900
```
```  2901 lemma inj_linear_imp_inv_linear:
```
```  2902   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2903   assumes "linear f" "inj f" shows "linear (inv f)"
```
```  2904 using assms inj_iff left_inverse_linear by blast
```
```  2905
```
```  2906
```
```  2907 subsection \<open>Infinity norm\<close>
```
```  2908
```
```  2909 definition "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
```
```  2910
```
```  2911 lemma infnorm_set_image:
```
```  2912   fixes x :: "'a::euclidean_space"
```
```  2913   shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```  2914   by blast
```
```  2915
```
```  2916 lemma infnorm_Max:
```
```  2917   fixes x :: "'a::euclidean_space"
```
```  2918   shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)"
```
```  2919   by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)
```
```  2920
```
```  2921 lemma infnorm_set_lemma:
```
```  2922   fixes x :: "'a::euclidean_space"
```
```  2923   shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
```
```  2924     and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}"
```
```  2925   unfolding infnorm_set_image
```
```  2926   by auto
```
```  2927
```
```  2928 lemma infnorm_pos_le:
```
```  2929   fixes x :: "'a::euclidean_space"
```
```  2930   shows "0 \<le> infnorm x"
```
```  2931   by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
```
```  2932
```
```  2933 lemma infnorm_triangle:
```
```  2934   fixes x :: "'a::euclidean_space"
```
```  2935   shows "infnorm (x + y) \<le> infnorm x + infnorm y"
```
```  2936 proof -
```
```  2937   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"
```
```  2938     by simp
```
```  2939   show ?thesis
```
```  2940     by (auto simp: infnorm_Max inner_add_left intro!: *)
```
```  2941 qed
```
```  2942
```
```  2943 lemma infnorm_eq_0:
```
```  2944   fixes x :: "'a::euclidean_space"
```
```  2945   shows "infnorm x = 0 \<longleftrightarrow> x = 0"
```
```  2946 proof -
```
```  2947   have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
```
```  2948     unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
```
```  2949   then show ?thesis
```
```  2950     using infnorm_pos_le[of x] by simp
```
```  2951 qed
```
```  2952
```
```  2953 lemma infnorm_0: "infnorm 0 = 0"
```
```  2954   by (simp add: infnorm_eq_0)
```
```  2955
```
```  2956 lemma infnorm_neg: "infnorm (- x) = infnorm x"
```
```  2957   unfolding infnorm_def
```
```  2958   apply (rule cong[of "Sup" "Sup"])
```
```  2959   apply blast
```
```  2960   apply auto
```
```  2961   done
```
```  2962
```
```  2963 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
```
```  2964 proof -
```
```  2965   have "y - x = - (x - y)" by simp
```
```  2966   then show ?thesis
```
```  2967     by (metis infnorm_neg)
```
```  2968 qed
```
```  2969
```
```  2970 lemma real_abs_sub_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
```
```  2971 proof -
```
```  2972   have th: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
```
```  2973     by arith
```
```  2974   from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
```
```  2975   have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
```
```  2976     "infnorm y \<le> infnorm (x - y) + infnorm x"
```
```  2977     by (simp_all add: field_simps infnorm_neg)
```
```  2978   from th[OF ths] show ?thesis .
```
```  2979 qed
```
```  2980
```
```  2981 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
```
```  2982   using infnorm_pos_le[of x] by arith
```
```  2983
```
```  2984 lemma Basis_le_infnorm:
```
```  2985   fixes x :: "'a::euclidean_space"
```
```  2986   shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
```
```  2987   by (simp add: infnorm_Max)
```
```  2988
```
```  2989 lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
```
```  2990   unfolding infnorm_Max
```
```  2991 proof (safe intro!: Max_eqI)
```
```  2992   let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```  2993   {
```
```  2994     fix b :: 'a
```
```  2995     assume "b \<in> Basis"
```
```  2996     then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
```
```  2997       by (simp add: abs_mult mult_left_mono)
```
```  2998   next
```
```  2999     from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
```
```  3000       by (auto simp del: Max_in)
```
```  3001     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"
```
```  3002       by (intro image_eqI[where x=b]) (auto simp: abs_mult)
```
```  3003   }
```
```  3004 qed simp
```
```  3005
```
```  3006 lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
```
```  3007   unfolding infnorm_mul ..
```
```  3008
```
```  3009 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
```
```  3010   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
```
```  3011
```
```  3012 text \<open>Prove that it differs only up to a bound from Euclidean norm.\<close>
```
```  3013
```
```  3014 lemma infnorm_le_norm: "infnorm x \<le> norm x"
```
```  3015   by (simp add: Basis_le_norm infnorm_Max)
```
```  3016
```
```  3017 lemma (in euclidean_space) euclidean_inner: "inner x y = (\<Sum>b\<in>Basis. (x \<bullet> b) * (y \<bullet> b))"
```
```  3018   by (subst (1 2) euclidean_representation [symmetric])
```
```  3019     (simp add: inner_setsum_right inner_Basis ac_simps)
```
```  3020
```
```  3021 lemma norm_le_infnorm:
```
```  3022   fixes x :: "'a::euclidean_space"
```
```  3023   shows "norm x \<le> sqrt DIM('a) * infnorm x"
```
```  3024 proof -
```
```  3025   let ?d = "DIM('a)"
```
```  3026   have "real ?d \<ge> 0"
```
```  3027     by simp
```
```  3028   then have d2: "(sqrt (real ?d))\<^sup>2 = real ?d"
```
```  3029     by (auto intro: real_sqrt_pow2)
```
```  3030   have th: "sqrt (real ?d) * infnorm x \<ge> 0"
```
```  3031     by (simp add: zero_le_mult_iff infnorm_pos_le)
```
```  3032   have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)\<^sup>2"
```
```  3033     unfolding power_mult_distrib d2
```
```  3034     apply (subst euclidean_inner)
```
```  3035     apply (subst power2_abs[symmetric])
```
```  3036     apply (rule order_trans[OF setsum_bounded_above[where K="\<bar>infnorm x\<bar>\<^sup>2"]])
```
```  3037     apply (auto simp add: power2_eq_square[symmetric])
```
```  3038     apply (subst power2_abs[symmetric])
```
```  3039     apply (rule power_mono)
```
```  3040     apply (auto simp: infnorm_Max)
```
```  3041     done
```
```  3042   from real_le_lsqrt[OF inner_ge_zero th th1]
```
```  3043   show ?thesis
```
```  3044     unfolding norm_eq_sqrt_inner id_def .
```
```  3045 qed
```
```  3046
```
```  3047 lemma tendsto_infnorm [tendsto_intros]:
```
```  3048   assumes "(f \<longlongrightarrow> a) F"
```
```  3049   shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F"
```
```  3050 proof (rule tendsto_compose [OF LIM_I assms])
```
```  3051   fix r :: real
```
```  3052   assume "r > 0"
```
```  3053   then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
```
```  3054     by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm)
```
```  3055 qed
```
```  3056
```
```  3057 text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close>
```
```  3058
```
```  3059 lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  3060   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  3061 proof -
```
```  3062   {
```
```  3063     assume h: "x = 0"
```
```  3064     then have ?thesis by simp
```
```  3065   }
```
```  3066   moreover
```
```  3067   {
```
```  3068     assume h: "y = 0"
```
```  3069     then have ?thesis by simp
```
```  3070   }
```
```  3071   moreover
```
```  3072   {
```
```  3073     assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
```
```  3074     from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
```
```  3075     have "?rhs \<longleftrightarrow>
```
```  3076       (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
```
```  3077         norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
```
```  3078       using x y
```
```  3079       unfolding inner_simps
```
```  3080       unfolding power2_norm_eq_inner[symmetric] power2_eq_square right_minus_eq
```
```  3081       apply (simp add: inner_commute)
```
```  3082       apply (simp add: field_simps)
```
```  3083       apply metis
```
```  3084       done
```
```  3085     also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
```
```  3086       by (simp add: field_simps inner_commute)
```
```  3087     also have "\<dots> \<longleftrightarrow> ?lhs" using x y
```
```  3088       apply simp
```
```  3089       apply metis
```
```  3090       done
```
```  3091     finally have ?thesis by blast
```
```  3092   }
```
```  3093   ultimately show ?thesis by blast
```
```  3094 qed
```
```  3095
```
```  3096 lemma norm_cauchy_schwarz_abs_eq:
```
```  3097   "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow>
```
```  3098     norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
```
```  3099   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  3100 proof -
```
```  3101   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a"
```
```  3102     by arith
```
```  3103   have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
```
```  3104     by simp
```
```  3105   also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
```
```  3106     unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  3107     unfolding norm_minus_cancel norm_scaleR ..
```
```  3108   also have "\<dots> \<longleftrightarrow> ?lhs"
```
```  3109     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
```
```  3110     by auto
```
```  3111   finally show ?thesis ..
```
```  3112 qed
```
```  3113
```
```  3114 lemma norm_triangle_eq:
```
```  3115   fixes x y :: "'a::real_inner"
```
```  3116   shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  3117 proof -
```
```  3118   {
```
```  3119     assume x: "x = 0 \<or> y = 0"
```
```  3120     then have ?thesis
```
```  3121       by (cases "x = 0") simp_all
```
```  3122   }
```
```  3123   moreover
```
```  3124   {
```
```  3125     assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
```
```  3126     then have "norm x \<noteq> 0" "norm y \<noteq> 0"
```
```  3127       by simp_all
```
```  3128     then have n: "norm x > 0" "norm y > 0"
```
```  3129       using norm_ge_zero[of x] norm_ge_zero[of y] by arith+
```
```  3130     have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 \<Longrightarrow> a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2"
```
```  3131       by algebra
```
```  3132     have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
```
```  3133       apply (rule th)
```
```  3134       using n norm_ge_zero[of "x + y"]
```
```  3135       apply arith
```
```  3136       done
```
```  3137     also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  3138       unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  3139       unfolding power2_norm_eq_inner inner_simps
```
```  3140       by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
```
```  3141     finally have ?thesis .
```
```  3142   }
```
```  3143   ultimately show ?thesis by blast
```
```  3144 qed
```
```  3145
```
```  3146
```
```  3147 subsection \<open>Collinearity\<close>
```
```  3148
```
```  3149 definition collinear :: "'a::real_vector set \<Rightarrow> bool"
```
```  3150   where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
```
```  3151
```
```  3152 lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S"
```
```  3153   by (meson collinear_def subsetCE)
```
```  3154
```
```  3155 lemma collinear_empty [iff]: "collinear {}"
```
```  3156   by (simp add: collinear_def)
```
```  3157
```
```  3158 lemma collinear_sing [iff]: "collinear {x}"
```
```  3159   by (simp add: collinear_def)
```
```  3160
```
```  3161 lemma collinear_2 [iff]: "collinear {x, y}"
```
```  3162   apply (simp add: collinear_def)
```
```  3163   apply (rule exI[where x="x - y"])
```
```  3164   apply auto
```
```  3165   apply (rule exI[where x=1], simp)
```
```  3166   apply (rule exI[where x="- 1"], simp)
```
```  3167   done
```
```  3168
```
```  3169 lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
```
```  3170   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  3171 proof -
```
```  3172   {
```
```  3173     assume "x = 0 \<or> y = 0"
```
```  3174     then have ?thesis
```
```  3175       by (cases "x = 0") (simp_all add: collinear_2 insert_commute)
```
```  3176   }
```
```  3177   moreover
```
```  3178   {
```
```  3179     assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
```
```  3180     have ?thesis
```
```  3181     proof
```
```  3182       assume h: "?lhs"
```
```  3183       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"
```
```  3184         unfolding collinear_def by blast
```
```  3185       from u[rule_format, of x 0] u[rule_format, of y 0]
```
```  3186       obtain cx and cy where
```
```  3187         cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
```
```  3188         by auto
```
```  3189       from cx x have cx0: "cx \<noteq> 0" by auto
```
```  3190       from cy y have cy0: "cy \<noteq> 0" by auto
```
```  3191       let ?d = "cy / cx"
```
```  3192       from cx cy cx0 have "y = ?d *\<^sub>R x"
```
```  3193         by simp
```
```  3194       then show ?rhs using x y by blast
```
```  3195     next
```
```  3196       assume h: "?rhs"
```
```  3197       then obtain c where c: "y = c *\<^sub>R x"
```
```  3198         using x y by blast
```
```  3199       show ?lhs
```
```  3200         unfolding collinear_def c
```
```  3201         apply (rule exI[where x=x])
```
```  3202         apply auto
```
```  3203         apply (rule exI[where x="- 1"], simp)
```
```  3204         apply (rule exI[where x= "-c"], simp)
```
```  3205         apply (rule exI[where x=1], simp)
```
```  3206         apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
```
```  3207         apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
```
```  3208         done
```
```  3209     qed
```
```  3210   }
```
```  3211   ultimately show ?thesis by blast
```
```  3212 qed
```
```  3213
```
```  3214 lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
```
```  3215   unfolding norm_cauchy_schwarz_abs_eq
```
```  3216   apply (cases "x=0", simp_all)
```
```  3217   apply (cases "y=0", simp_all add: insert_commute)
```
```  3218   unfolding collinear_lemma
```
```  3219   apply simp
```
```  3220   apply (subgoal_tac "norm x \<noteq> 0")
```
```  3221   apply (subgoal_tac "norm y \<noteq> 0")
```
```  3222   apply (rule iffI)
```
```  3223   apply (cases "norm x *\<^sub>R y = norm y *\<^sub>R x")
```
```  3224   apply (rule exI[where x="(1/norm x) * norm y"])
```
```  3225   apply (drule sym)
```
```  3226   unfolding scaleR_scaleR[symmetric]
```
```  3227   apply (simp add: field_simps)
```
```  3228   apply (rule exI[where x="(1/norm x) * - norm y"])
```
```  3229   apply clarify
```
```  3230   apply (drule sym)
```
```  3231   unfolding scaleR_scaleR[symmetric]
```
```  3232   apply (simp add: field_simps)
```
```  3233   apply (erule exE)
```
```  3234   apply (erule ssubst)
```
```  3235   unfolding scaleR_scaleR
```
```  3236   unfolding norm_scaleR
```
```  3237   apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
```
```  3238   apply (auto simp add: field_simps)
```
```  3239   done
```
```  3240
```
```  3241 end
```