src/HOL/Analysis/Linear_Algebra.thy
 author immler Thu Dec 27 21:32:36 2018 +0100 (4 months ago) changeset 69513 42e08052dae8 parent 69510 0f31dd2e540d child 69516 09bb8f470959 permissions -rw-r--r--
moved lemmas up
```     1 (*  Title:      HOL/Analysis/Linear_Algebra.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3 *)
```
```     4
```
```     5 section \<open>Elementary linear algebra on Euclidean spaces\<close>
```
```     6
```
```     7 theory Linear_Algebra
```
```     8 imports
```
```     9   Euclidean_Space
```
```    10   "HOL-Library.Infinite_Set"
```
```    11 begin
```
```    12
```
```    13 lemma linear_simps:
```
```    14   assumes "bounded_linear f"
```
```    15   shows
```
```    16     "f (a + b) = f a + f b"
```
```    17     "f (a - b) = f a - f b"
```
```    18     "f 0 = 0"
```
```    19     "f (- a) = - f a"
```
```    20     "f (s *\<^sub>R v) = s *\<^sub>R (f v)"
```
```    21 proof -
```
```    22   interpret f: bounded_linear f by fact
```
```    23   show "f (a + b) = f a + f b" by (rule f.add)
```
```    24   show "f (a - b) = f a - f b" by (rule f.diff)
```
```    25   show "f 0 = 0" by (rule f.zero)
```
```    26   show "f (- a) = - f a" by (rule f.neg)
```
```    27   show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scale)
```
```    28 qed
```
```    29
```
```    30 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x \<in> (UNIV::'a::finite set)}"
```
```    31   using finite finite_image_set by blast
```
```    32
```
```    33
```
```    34 subsection%unimportant \<open>More interesting properties of the norm\<close>
```
```    35
```
```    36 notation inner (infix "\<bullet>" 70)
```
```    37
```
```    38 text\<open>Equality of vectors in terms of @{term "(\<bullet>)"} products.\<close>
```
```    39
```
```    40 lemma linear_componentwise:
```
```    41   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
```
```    42   assumes lf: "linear f"
```
```    43   shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
```
```    44 proof -
```
```    45   interpret linear f by fact
```
```    46   have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
```
```    47     by (simp add: inner_sum_left)
```
```    48   then show ?thesis
```
```    49     by (simp add: euclidean_representation sum[symmetric] scale[symmetric])
```
```    50 qed
```
```    51
```
```    52 lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
```
```    53   (is "?lhs \<longleftrightarrow> ?rhs")
```
```    54 proof
```
```    55   assume ?lhs
```
```    56   then show ?rhs by simp
```
```    57 next
```
```    58   assume ?rhs
```
```    59   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
```
```    60     by simp
```
```    61   then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
```
```    62     by (simp add: inner_diff inner_commute)
```
```    63   then have "(x - y) \<bullet> (x - y) = 0"
```
```    64     by (simp add: field_simps inner_diff inner_commute)
```
```    65   then show "x = y" by simp
```
```    66 qed
```
```    67
```
```    68 lemma norm_triangle_half_r:
```
```    69   "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
```
```    70   using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
```
```    71
```
```    72 lemma norm_triangle_half_l:
```
```    73   assumes "norm (x - y) < e / 2"
```
```    74     and "norm (x' - y) < e / 2"
```
```    75   shows "norm (x - x') < e"
```
```    76   using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
```
```    77   unfolding dist_norm[symmetric] .
```
```    78
```
```    79 lemma abs_triangle_half_r:
```
```    80   fixes y :: "'a::linordered_field"
```
```    81   shows "abs (y - x1) < e / 2 \<Longrightarrow> abs (y - x2) < e / 2 \<Longrightarrow> abs (x1 - x2) < e"
```
```    82   by linarith
```
```    83
```
```    84 lemma abs_triangle_half_l:
```
```    85   fixes y :: "'a::linordered_field"
```
```    86   assumes "abs (x - y) < e / 2"
```
```    87     and "abs (x' - y) < e / 2"
```
```    88   shows "abs (x - x') < e"
```
```    89   using assms by linarith
```
```    90
```
```    91 lemma sum_clauses:
```
```    92   shows "sum f {} = 0"
```
```    93     and "finite S \<Longrightarrow> sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
```
```    94   by (auto simp add: insert_absorb)
```
```    95
```
```    96 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
```
```    97 proof
```
```    98   assume "\<forall>x. x \<bullet> y = x \<bullet> z"
```
```    99   then have "\<forall>x. x \<bullet> (y - z) = 0"
```
```   100     by (simp add: inner_diff)
```
```   101   then have "(y - z) \<bullet> (y - z) = 0" ..
```
```   102   then show "y = z" by simp
```
```   103 qed simp
```
```   104
```
```   105 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
```
```   106 proof
```
```   107   assume "\<forall>z. x \<bullet> z = y \<bullet> z"
```
```   108   then have "\<forall>z. (x - y) \<bullet> z = 0"
```
```   109     by (simp add: inner_diff)
```
```   110   then have "(x - y) \<bullet> (x - y) = 0" ..
```
```   111   then show "x = y" by simp
```
```   112 qed simp
```
```   113
```
```   114
```
```   115 subsection \<open>Orthogonality\<close>
```
```   116
```
```   117 definition%important (in real_inner) "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
```
```   118
```
```   119 context real_inner
```
```   120 begin
```
```   121
```
```   122 lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0"
```
```   123   by (simp add: orthogonal_def)
```
```   124
```
```   125 lemma orthogonal_clauses:
```
```   126   "orthogonal a 0"
```
```   127   "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
```
```   128   "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
```
```   129   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
```
```   130   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
```
```   131   "orthogonal 0 a"
```
```   132   "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
```
```   133   "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
```
```   134   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
```
```   135   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
```
```   136   unfolding orthogonal_def inner_add inner_diff by auto
```
```   137
```
```   138 end
```
```   139
```
```   140 lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
```
```   141   by (simp add: orthogonal_def inner_commute)
```
```   142
```
```   143 lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x"
```
```   144   by (rule ext) (simp add: orthogonal_def)
```
```   145
```
```   146 lemma pairwise_ortho_scaleR:
```
```   147     "pairwise (\<lambda>i j. orthogonal (f i) (g j)) B
```
```   148     \<Longrightarrow> pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B"
```
```   149   by (auto simp: pairwise_def orthogonal_clauses)
```
```   150
```
```   151 lemma orthogonal_rvsum:
```
```   152     "\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (sum f s)"
```
```   153   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
```
```   154
```
```   155 lemma orthogonal_lvsum:
```
```   156     "\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (sum f s) y"
```
```   157   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
```
```   158
```
```   159 lemma norm_add_Pythagorean:
```
```   160   assumes "orthogonal a b"
```
```   161     shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2"
```
```   162 proof -
```
```   163   from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)"
```
```   164     by (simp add: algebra_simps orthogonal_def inner_commute)
```
```   165   then show ?thesis
```
```   166     by (simp add: power2_norm_eq_inner)
```
```   167 qed
```
```   168
```
```   169 lemma norm_sum_Pythagorean:
```
```   170   assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I"
```
```   171     shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
```
```   172 using assms
```
```   173 proof (induction I rule: finite_induct)
```
```   174   case empty then show ?case by simp
```
```   175 next
```
```   176   case (insert x I)
```
```   177   then have "orthogonal (f x) (sum f I)"
```
```   178     by (metis pairwise_insert orthogonal_rvsum)
```
```   179   with insert show ?case
```
```   180     by (simp add: pairwise_insert norm_add_Pythagorean)
```
```   181 qed
```
```   182
```
```   183
```
```   184 subsection \<open>Bilinear functions\<close>
```
```   185
```
```   186 definition%important "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
```
```   187
```
```   188 lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
```
```   189   by (simp add: bilinear_def linear_iff)
```
```   190
```
```   191 lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
```
```   192   by (simp add: bilinear_def linear_iff)
```
```   193
```
```   194 lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
```
```   195   by (simp add: bilinear_def linear_iff)
```
```   196
```
```   197 lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
```
```   198   by (simp add: bilinear_def linear_iff)
```
```   199
```
```   200 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
```
```   201   by (drule bilinear_lmul [of _ "- 1"]) simp
```
```   202
```
```   203 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
```
```   204   by (drule bilinear_rmul [of _ _ "- 1"]) simp
```
```   205
```
```   206 lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
```
```   207   using add_left_imp_eq[of x y 0] by auto
```
```   208
```
```   209 lemma bilinear_lzero:
```
```   210   assumes "bilinear h"
```
```   211   shows "h 0 x = 0"
```
```   212   using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
```
```   213
```
```   214 lemma bilinear_rzero:
```
```   215   assumes "bilinear h"
```
```   216   shows "h x 0 = 0"
```
```   217   using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
```
```   218
```
```   219 lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
```
```   220   using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
```
```   221
```
```   222 lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
```
```   223   using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
```
```   224
```
```   225 lemma bilinear_sum:
```
```   226   assumes "bilinear h"
```
```   227   shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
```
```   228 proof -
```
```   229   interpret l: linear "\<lambda>x. h x y" for y using assms by (simp add: bilinear_def)
```
```   230   interpret r: linear "\<lambda>y. h x y" for x using assms by (simp add: bilinear_def)
```
```   231   have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S"
```
```   232     by (simp add: l.sum)
```
```   233   also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S"
```
```   234     by (rule sum.cong) (simp_all add: r.sum)
```
```   235   finally show ?thesis
```
```   236     unfolding sum.cartesian_product .
```
```   237 qed
```
```   238
```
```   239
```
```   240 subsection \<open>Adjoints\<close>
```
```   241
```
```   242 definition%important "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
```
```   243
```
```   244 lemma adjoint_unique:
```
```   245   assumes "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   246   shows "adjoint f = g"
```
```   247   unfolding adjoint_def
```
```   248 proof (rule some_equality)
```
```   249   show "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   250     by (rule assms)
```
```   251 next
```
```   252   fix h
```
```   253   assume "\<forall>x y. inner (f x) y = inner x (h y)"
```
```   254   then have "\<forall>x y. inner x (g y) = inner x (h y)"
```
```   255     using assms by simp
```
```   256   then have "\<forall>x y. inner x (g y - h y) = 0"
```
```   257     by (simp add: inner_diff_right)
```
```   258   then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
```
```   259     by simp
```
```   260   then have "\<forall>y. h y = g y"
```
```   261     by simp
```
```   262   then show "h = g" by (simp add: ext)
```
```   263 qed
```
```   264
```
```   265 text \<open>TODO: The following lemmas about adjoints should hold for any
```
```   266   Hilbert space (i.e. complete inner product space).
```
```   267   (see \<^url>\<open>https://en.wikipedia.org/wiki/Hermitian_adjoint\<close>)
```
```   268 \<close>
```
```   269
```
```   270 lemma adjoint_works:
```
```   271   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   272   assumes lf: "linear f"
```
```   273   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```   274 proof -
```
```   275   interpret linear f by fact
```
```   276   have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
```
```   277   proof (intro allI exI)
```
```   278     fix y :: "'m" and x
```
```   279     let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
```
```   280     have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
```
```   281       by (simp add: euclidean_representation)
```
```   282     also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
```
```   283       by (simp add: sum scale)
```
```   284     finally show "f x \<bullet> y = x \<bullet> ?w"
```
```   285       by (simp add: inner_sum_left inner_sum_right mult.commute)
```
```   286   qed
```
```   287   then show ?thesis
```
```   288     unfolding adjoint_def choice_iff
```
```   289     by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
```
```   290 qed
```
```   291
```
```   292 lemma adjoint_clauses:
```
```   293   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   294   assumes lf: "linear f"
```
```   295   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```   296     and "adjoint f y \<bullet> x = y \<bullet> f x"
```
```   297   by (simp_all add: adjoint_works[OF lf] inner_commute)
```
```   298
```
```   299 lemma adjoint_linear:
```
```   300   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   301   assumes lf: "linear f"
```
```   302   shows "linear (adjoint f)"
```
```   303   by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
```
```   304     adjoint_clauses[OF lf] inner_distrib)
```
```   305
```
```   306 lemma adjoint_adjoint:
```
```   307   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   308   assumes lf: "linear f"
```
```   309   shows "adjoint (adjoint f) = f"
```
```   310   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
```
```   311
```
```   312
```
```   313 subsection%unimportant \<open>Interlude: Some properties of real sets\<close>
```
```   314
```
```   315 lemma seq_mono_lemma:
```
```   316   assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
```
```   317     and "\<forall>n \<ge> m. e n \<le> e m"
```
```   318   shows "\<forall>n \<ge> m. d n < e m"
```
```   319   using assms by force
```
```   320
```
```   321 lemma infinite_enumerate:
```
```   322   assumes fS: "infinite S"
```
```   323   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)"
```
```   324   unfolding strict_mono_def
```
```   325   using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
```
```   326
```
```   327 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)"
```
```   328   apply auto
```
```   329   apply (rule_tac x="d/2" in exI)
```
```   330   apply auto
```
```   331   done
```
```   332
```
```   333 lemma approachable_lt_le2:  \<comment> \<open>like the above, but pushes aside an extra formula\<close>
```
```   334     "(\<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)"
```
```   335   apply auto
```
```   336   apply (rule_tac x="d/2" in exI, auto)
```
```   337   done
```
```   338
```
```   339 lemma triangle_lemma:
```
```   340   fixes x y z :: real
```
```   341   assumes x: "0 \<le> x"
```
```   342     and y: "0 \<le> y"
```
```   343     and z: "0 \<le> z"
```
```   344     and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
```
```   345   shows "x \<le> y + z"
```
```   346 proof -
```
```   347   have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
```
```   348     using z y by simp
```
```   349   with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
```
```   350     by (simp add: power2_eq_square field_simps)
```
```   351   from y z have yz: "y + z \<ge> 0"
```
```   352     by arith
```
```   353   from power2_le_imp_le[OF th yz] show ?thesis .
```
```   354 qed
```
```   355
```
```   356
```
```   357
```
```   358 subsection \<open>Archimedean properties and useful consequences\<close>
```
```   359
```
```   360 text\<open>Bernoulli's inequality\<close>
```
```   361 proposition Bernoulli_inequality:
```
```   362   fixes x :: real
```
```   363   assumes "-1 \<le> x"
```
```   364     shows "1 + n * x \<le> (1 + x) ^ n"
```
```   365 proof (induct n)
```
```   366   case 0
```
```   367   then show ?case by simp
```
```   368 next
```
```   369   case (Suc n)
```
```   370   have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
```
```   371     by (simp add: algebra_simps)
```
```   372   also have "... = (1 + x) * (1 + n*x)"
```
```   373     by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
```
```   374   also have "... \<le> (1 + x) ^ Suc n"
```
```   375     using Suc.hyps assms mult_left_mono by fastforce
```
```   376   finally show ?case .
```
```   377 qed
```
```   378
```
```   379 corollary Bernoulli_inequality_even:
```
```   380   fixes x :: real
```
```   381   assumes "even n"
```
```   382     shows "1 + n * x \<le> (1 + x) ^ n"
```
```   383 proof (cases "-1 \<le> x \<or> n=0")
```
```   384   case True
```
```   385   then show ?thesis
```
```   386     by (auto simp: Bernoulli_inequality)
```
```   387 next
```
```   388   case False
```
```   389   then have "real n \<ge> 1"
```
```   390     by simp
```
```   391   with False have "n * x \<le> -1"
```
```   392     by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
```
```   393   then have "1 + n * x \<le> 0"
```
```   394     by auto
```
```   395   also have "... \<le> (1 + x) ^ n"
```
```   396     using assms
```
```   397     using zero_le_even_power by blast
```
```   398   finally show ?thesis .
```
```   399 qed
```
```   400
```
```   401 corollary real_arch_pow:
```
```   402   fixes x :: real
```
```   403   assumes x: "1 < x"
```
```   404   shows "\<exists>n. y < x^n"
```
```   405 proof -
```
```   406   from x have x0: "x - 1 > 0"
```
```   407     by arith
```
```   408   from reals_Archimedean3[OF x0, rule_format, of y]
```
```   409   obtain n :: nat where n: "y < real n * (x - 1)" by metis
```
```   410   from x0 have x00: "x- 1 \<ge> -1" by arith
```
```   411   from Bernoulli_inequality[OF x00, of n] n
```
```   412   have "y < x^n" by auto
```
```   413   then show ?thesis by metis
```
```   414 qed
```
```   415
```
```   416 corollary real_arch_pow_inv:
```
```   417   fixes x y :: real
```
```   418   assumes y: "y > 0"
```
```   419     and x1: "x < 1"
```
```   420   shows "\<exists>n. x^n < y"
```
```   421 proof (cases "x > 0")
```
```   422   case True
```
```   423   with x1 have ix: "1 < 1/x" by (simp add: field_simps)
```
```   424   from real_arch_pow[OF ix, of "1/y"]
```
```   425   obtain n where n: "1/y < (1/x)^n" by blast
```
```   426   then show ?thesis using y \<open>x > 0\<close>
```
```   427     by (auto simp add: field_simps)
```
```   428 next
```
```   429   case False
```
```   430   with y x1 show ?thesis
```
```   431     by (metis less_le_trans not_less power_one_right)
```
```   432 qed
```
```   433
```
```   434 lemma forall_pos_mono:
```
```   435   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```   436     (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
```
```   437   by (metis real_arch_inverse)
```
```   438
```
```   439 lemma forall_pos_mono_1:
```
```   440   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```   441     (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
```
```   442   apply (rule forall_pos_mono)
```
```   443   apply auto
```
```   444   apply (metis Suc_pred of_nat_Suc)
```
```   445   done
```
```   446
```
```   447
```
```   448 subsection%unimportant \<open>Euclidean Spaces as Typeclass\<close>
```
```   449
```
```   450 lemma independent_Basis: "independent Basis"
```
```   451   by (rule independent_Basis)
```
```   452
```
```   453 lemma span_Basis [simp]: "span Basis = UNIV"
```
```   454   by (rule span_Basis)
```
```   455
```
```   456 lemma in_span_Basis: "x \<in> span Basis"
```
```   457   unfolding span_Basis ..
```
```   458
```
```   459
```
```   460 subsection%unimportant \<open>Linearity and Bilinearity continued\<close>
```
```   461
```
```   462 lemma linear_bounded:
```
```   463   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   464   assumes lf: "linear f"
```
```   465   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```   466 proof
```
```   467   interpret linear f by fact
```
```   468   let ?B = "\<Sum>b\<in>Basis. norm (f b)"
```
```   469   show "\<forall>x. norm (f x) \<le> ?B * norm x"
```
```   470   proof
```
```   471     fix x :: 'a
```
```   472     let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
```
```   473     have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
```
```   474       unfolding euclidean_representation ..
```
```   475     also have "\<dots> = norm (sum ?g Basis)"
```
```   476       by (simp add: sum scale)
```
```   477     finally have th0: "norm (f x) = norm (sum ?g Basis)" .
```
```   478     have th: "norm (?g i) \<le> norm (f i) * norm x" if "i \<in> Basis" for i
```
```   479     proof -
```
```   480       from Basis_le_norm[OF that, of x]
```
```   481       show "norm (?g i) \<le> norm (f i) * norm x"
```
```   482         unfolding norm_scaleR  by (metis mult.commute mult_left_mono norm_ge_zero)
```
```   483     qed
```
```   484     from sum_norm_le[of _ ?g, OF th]
```
```   485     show "norm (f x) \<le> ?B * norm x"
```
```   486       unfolding th0 sum_distrib_right by metis
```
```   487   qed
```
```   488 qed
```
```   489
```
```   490 lemma linear_conv_bounded_linear:
```
```   491   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   492   shows "linear f \<longleftrightarrow> bounded_linear f"
```
```   493 proof
```
```   494   assume "linear f"
```
```   495   then interpret f: linear f .
```
```   496   show "bounded_linear f"
```
```   497   proof
```
```   498     have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```   499       using \<open>linear f\<close> by (rule linear_bounded)
```
```   500     then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```   501       by (simp add: mult.commute)
```
```   502   qed
```
```   503 next
```
```   504   assume "bounded_linear f"
```
```   505   then interpret f: bounded_linear f .
```
```   506   show "linear f" ..
```
```   507 qed
```
```   508
```
```   509 lemmas linear_linear = linear_conv_bounded_linear[symmetric]
```
```   510
```
```   511 lemma linear_bounded_pos:
```
```   512   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   513   assumes lf: "linear f"
```
```   514  obtains B where "B > 0" "\<And>x. norm (f x) \<le> B * norm x"
```
```   515 proof -
```
```   516   have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
```
```   517     using lf unfolding linear_conv_bounded_linear
```
```   518     by (rule bounded_linear.pos_bounded)
```
```   519   with that show ?thesis
```
```   520     by (auto simp: mult.commute)
```
```   521 qed
```
```   522
```
```   523 lemma linear_invertible_bounded_below_pos:
```
```   524   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
```
```   525   assumes "linear f" "linear g" "g \<circ> f = id"
```
```   526   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
```
```   527 proof -
```
```   528   obtain B where "B > 0" and B: "\<And>x. norm (g x) \<le> B * norm x"
```
```   529     using linear_bounded_pos [OF \<open>linear g\<close>] by blast
```
```   530   show thesis
```
```   531   proof
```
```   532     show "0 < 1/B"
```
```   533       by (simp add: \<open>B > 0\<close>)
```
```   534     show "1/B * norm x \<le> norm (f x)" for x
```
```   535     proof -
```
```   536       have "1/B * norm x = 1/B * norm (g (f x))"
```
```   537         using assms by (simp add: pointfree_idE)
```
```   538       also have "\<dots> \<le> norm (f x)"
```
```   539         using B [of "f x"] by (simp add: \<open>B > 0\<close> mult.commute pos_divide_le_eq)
```
```   540       finally show ?thesis .
```
```   541     qed
```
```   542   qed
```
```   543 qed
```
```   544
```
```   545 lemma linear_inj_bounded_below_pos:
```
```   546   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
```
```   547   assumes "linear f" "inj f"
```
```   548   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
```
```   549   using linear_injective_left_inverse [OF assms]
```
```   550     linear_invertible_bounded_below_pos assms by blast
```
```   551
```
```   552 lemma bounded_linearI':
```
```   553   fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   554   assumes "\<And>x y. f (x + y) = f x + f y"
```
```   555     and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```   556   shows "bounded_linear f"
```
```   557   using assms linearI linear_conv_bounded_linear by blast
```
```   558
```
```   559 lemma bilinear_bounded:
```
```   560   fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
```
```   561   assumes bh: "bilinear h"
```
```   562   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```   563 proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
```
```   564   fix x :: 'm
```
```   565   fix y :: 'n
```
```   566   have "norm (h x y) = norm (h (sum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (sum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
```
```   567     by (simp add: euclidean_representation)
```
```   568   also have "\<dots> = norm (sum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
```
```   569     unfolding bilinear_sum[OF bh] ..
```
```   570   finally have th: "norm (h x y) = \<dots>" .
```
```   571   have "\<And>i j. \<lbrakk>i \<in> Basis; j \<in> Basis\<rbrakk>
```
```   572            \<Longrightarrow> \<bar>x \<bullet> i\<bar> * (\<bar>y \<bullet> j\<bar> * norm (h i j)) \<le> norm x * (norm y * norm (h i j))"
```
```   573     by (auto simp add: zero_le_mult_iff Basis_le_norm mult_mono)
```
```   574   then show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
```
```   575     unfolding sum_distrib_right th sum.cartesian_product
```
```   576     by (clarsimp simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
```
```   577       field_simps simp del: scaleR_scaleR intro!: sum_norm_le)
```
```   578 qed
```
```   579
```
```   580 lemma bilinear_conv_bounded_bilinear:
```
```   581   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```   582   shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
```
```   583 proof
```
```   584   assume "bilinear h"
```
```   585   show "bounded_bilinear h"
```
```   586   proof
```
```   587     fix x y z
```
```   588     show "h (x + y) z = h x z + h y z"
```
```   589       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
```
```   590   next
```
```   591     fix x y z
```
```   592     show "h x (y + z) = h x y + h x z"
```
```   593       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
```
```   594   next
```
```   595     show "h (scaleR r x) y = scaleR r (h x y)" "h x (scaleR r y) = scaleR r (h x y)" for r x y
```
```   596       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
```
```   597       by simp_all
```
```   598   next
```
```   599     have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```   600       using \<open>bilinear h\<close> by (rule bilinear_bounded)
```
```   601     then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
```
```   602       by (simp add: ac_simps)
```
```   603   qed
```
```   604 next
```
```   605   assume "bounded_bilinear h"
```
```   606   then interpret h: bounded_bilinear h .
```
```   607   show "bilinear h"
```
```   608     unfolding bilinear_def linear_conv_bounded_linear
```
```   609     using h.bounded_linear_left h.bounded_linear_right by simp
```
```   610 qed
```
```   611
```
```   612 lemma bilinear_bounded_pos:
```
```   613   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```   614   assumes bh: "bilinear h"
```
```   615   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```   616 proof -
```
```   617   have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
```
```   618     using bh [unfolded bilinear_conv_bounded_bilinear]
```
```   619     by (rule bounded_bilinear.pos_bounded)
```
```   620   then show ?thesis
```
```   621     by (simp only: ac_simps)
```
```   622 qed
```
```   623
```
```   624 lemma bounded_linear_imp_has_derivative: "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
```
```   625   by (auto simp add: has_derivative_def linear_diff linear_linear linear_def
```
```   626       dest: bounded_linear.linear)
```
```   627
```
```   628 lemma linear_imp_has_derivative:
```
```   629   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   630   shows "linear f \<Longrightarrow> (f has_derivative f) net"
```
```   631   by (simp add: bounded_linear_imp_has_derivative linear_conv_bounded_linear)
```
```   632
```
```   633 lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net"
```
```   634   using bounded_linear_imp_has_derivative differentiable_def by blast
```
```   635
```
```   636 lemma linear_imp_differentiable:
```
```   637   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   638   shows "linear f \<Longrightarrow> f differentiable net"
```
```   639   by (metis linear_imp_has_derivative differentiable_def)
```
```   640
```
```   641
```
```   642 subsection%unimportant \<open>We continue\<close>
```
```   643
```
```   644 lemma independent_bound:
```
```   645   fixes S :: "'a::euclidean_space set"
```
```   646   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
```
```   647   by (metis dim_subset_UNIV finiteI_independent dim_span_eq_card_independent)
```
```   648
```
```   649 lemmas independent_imp_finite = finiteI_independent
```
```   650
```
```   651 corollary
```
```   652   fixes S :: "'a::euclidean_space set"
```
```   653   assumes "independent S"
```
```   654   shows independent_card_le:"card S \<le> DIM('a)"
```
```   655   using assms independent_bound by auto
```
```   656
```
```   657 lemma dependent_biggerset:
```
```   658   fixes S :: "'a::euclidean_space set"
```
```   659   shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
```
```   660   by (metis independent_bound not_less)
```
```   661
```
```   662 text \<open>Picking an orthogonal replacement for a spanning set.\<close>
```
```   663
```
```   664 lemma vector_sub_project_orthogonal:
```
```   665   fixes b x :: "'a::euclidean_space"
```
```   666   shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
```
```   667   unfolding inner_simps by auto
```
```   668
```
```   669 lemma pairwise_orthogonal_insert:
```
```   670   assumes "pairwise orthogonal S"
```
```   671     and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
```
```   672   shows "pairwise orthogonal (insert x S)"
```
```   673   using assms unfolding pairwise_def
```
```   674   by (auto simp add: orthogonal_commute)
```
```   675
```
```   676 lemma basis_orthogonal:
```
```   677   fixes B :: "'a::real_inner set"
```
```   678   assumes fB: "finite B"
```
```   679   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
```
```   680   (is " \<exists>C. ?P B C")
```
```   681   using fB
```
```   682 proof (induct rule: finite_induct)
```
```   683   case empty
```
```   684   then show ?case
```
```   685     apply (rule exI[where x="{}"])
```
```   686     apply (auto simp add: pairwise_def)
```
```   687     done
```
```   688 next
```
```   689   case (insert a B)
```
```   690   note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
```
```   691   from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
```
```   692   obtain C where C: "finite C" "card C \<le> card B"
```
```   693     "span C = span B" "pairwise orthogonal C" by blast
```
```   694   let ?a = "a - sum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
```
```   695   let ?C = "insert ?a C"
```
```   696   from C(1) have fC: "finite ?C"
```
```   697     by simp
```
```   698   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
```
```   699     by (simp add: card_insert_if)
```
```   700   {
```
```   701     fix x k
```
```   702     have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
```
```   703       by (simp add: field_simps)
```
```   704     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"
```
```   705       apply (simp only: scaleR_right_diff_distrib th0)
```
```   706       apply (rule span_add_eq)
```
```   707       apply (rule span_scale)
```
```   708       apply (rule span_sum)
```
```   709       apply (rule span_scale)
```
```   710       apply (rule span_base)
```
```   711       apply assumption
```
```   712       done
```
```   713   }
```
```   714   then have SC: "span ?C = span (insert a B)"
```
```   715     unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
```
```   716   {
```
```   717     fix y
```
```   718     assume yC: "y \<in> C"
```
```   719     then have Cy: "C = insert y (C - {y})"
```
```   720       by blast
```
```   721     have fth: "finite (C - {y})"
```
```   722       using C by simp
```
```   723     have "orthogonal ?a y"
```
```   724       unfolding orthogonal_def
```
```   725       unfolding inner_diff inner_sum_left right_minus_eq
```
```   726       unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
```
```   727       apply (clarsimp simp add: inner_commute[of y a])
```
```   728       apply (rule sum.neutral)
```
```   729       apply clarsimp
```
```   730       apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
```
```   731       using \<open>y \<in> C\<close> by auto
```
```   732   }
```
```   733   with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
```
```   734     by (rule pairwise_orthogonal_insert)
```
```   735   from fC cC SC CPO have "?P (insert a B) ?C"
```
```   736     by blast
```
```   737   then show ?case by blast
```
```   738 qed
```
```   739
```
```   740 lemma orthogonal_basis_exists:
```
```   741   fixes V :: "('a::euclidean_space) set"
```
```   742   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and>
```
```   743   (card B = dim V) \<and> pairwise orthogonal B"
```
```   744 proof -
```
```   745   from basis_exists[of V] obtain B where
```
```   746     B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
```
```   747     by force
```
```   748   from B have fB: "finite B" "card B = dim V"
```
```   749     using independent_bound by auto
```
```   750   from basis_orthogonal[OF fB(1)] obtain C where
```
```   751     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
```
```   752     by blast
```
```   753   from C B have CSV: "C \<subseteq> span V"
```
```   754     by (metis span_superset span_mono subset_trans)
```
```   755   from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
```
```   756     by (simp add: span_span)
```
```   757   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
```
```   758   have iC: "independent C"
```
```   759     by (simp add: dim_span)
```
```   760   from C fB have "card C \<le> dim V"
```
```   761     by simp
```
```   762   moreover have "dim V \<le> card C"
```
```   763     using span_card_ge_dim[OF CSV SVC C(1)]
```
```   764     by simp
```
```   765   ultimately have CdV: "card C = dim V"
```
```   766     using C(1) by simp
```
```   767   from C B CSV CdV iC show ?thesis
```
```   768     by auto
```
```   769 qed
```
```   770
```
```   771 text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
```
```   772
```
```   773 lemma span_not_univ_orthogonal:
```
```   774   fixes S :: "'a::euclidean_space set"
```
```   775   assumes sU: "span S \<noteq> UNIV"
```
```   776   shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
```
```   777 proof -
```
```   778   from sU obtain a where a: "a \<notin> span S"
```
```   779     by blast
```
```   780   from orthogonal_basis_exists obtain B where
```
```   781     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B"
```
```   782     "card B = dim S" "pairwise orthogonal B"
```
```   783     by blast
```
```   784   from B have fB: "finite B" "card B = dim S"
```
```   785     using independent_bound by auto
```
```   786   from span_mono[OF B(2)] span_mono[OF B(3)]
```
```   787   have sSB: "span S = span B"
```
```   788     by (simp add: span_span)
```
```   789   let ?a = "a - sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
```
```   790   have "sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
```
```   791     unfolding sSB
```
```   792     apply (rule span_sum)
```
```   793     apply (rule span_scale)
```
```   794     apply (rule span_base)
```
```   795     apply assumption
```
```   796     done
```
```   797   with a have a0:"?a  \<noteq> 0"
```
```   798     by auto
```
```   799   have "?a \<bullet> x = 0" if "x\<in>span B" for x
```
```   800   proof (rule span_induct [OF that])
```
```   801     show "subspace {x. ?a \<bullet> x = 0}"
```
```   802       by (auto simp add: subspace_def inner_add)
```
```   803   next
```
```   804     {
```
```   805       fix x
```
```   806       assume x: "x \<in> B"
```
```   807       from x have B': "B = insert x (B - {x})"
```
```   808         by blast
```
```   809       have fth: "finite (B - {x})"
```
```   810         using fB by simp
```
```   811       have "?a \<bullet> x = 0"
```
```   812         apply (subst B')
```
```   813         using fB fth
```
```   814         unfolding sum_clauses(2)[OF fth]
```
```   815         apply simp unfolding inner_simps
```
```   816         apply (clarsimp simp add: inner_add inner_sum_left)
```
```   817         apply (rule sum.neutral, rule ballI)
```
```   818         apply (simp only: inner_commute)
```
```   819         apply (auto simp add: x field_simps
```
```   820           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
```
```   821         done
```
```   822     }
```
```   823     then show "?a \<bullet> x = 0" if "x \<in> B" for x
```
```   824       using that by blast
```
```   825     qed
```
```   826   with a0 show ?thesis
```
```   827     unfolding sSB by (auto intro: exI[where x="?a"])
```
```   828 qed
```
```   829
```
```   830 lemma span_not_univ_subset_hyperplane:
```
```   831   fixes S :: "'a::euclidean_space set"
```
```   832   assumes SU: "span S \<noteq> UNIV"
```
```   833   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```   834   using span_not_univ_orthogonal[OF SU] by auto
```
```   835
```
```   836 lemma lowdim_subset_hyperplane:
```
```   837   fixes S :: "'a::euclidean_space set"
```
```   838   assumes d: "dim S < DIM('a)"
```
```   839   shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```   840 proof -
```
```   841   {
```
```   842     assume "span S = UNIV"
```
```   843     then have "dim (span S) = dim (UNIV :: ('a) set)"
```
```   844       by simp
```
```   845     then have "dim S = DIM('a)"
```
```   846       by (metis Euclidean_Space.dim_UNIV dim_span)
```
```   847     with d have False by arith
```
```   848   }
```
```   849   then have th: "span S \<noteq> UNIV"
```
```   850     by blast
```
```   851   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
```
```   852 qed
```
```   853
```
```   854 lemma linear_eq_stdbasis:
```
```   855   fixes f :: "'a::euclidean_space \<Rightarrow> _"
```
```   856   assumes lf: "linear f"
```
```   857     and lg: "linear g"
```
```   858     and fg: "\<And>b. b \<in> Basis \<Longrightarrow> f b = g b"
```
```   859   shows "f = g"
```
```   860   using linear_eq_on_span[OF lf lg, of Basis] fg
```
```   861   by auto
```
```   862
```
```   863
```
```   864 text \<open>Similar results for bilinear functions.\<close>
```
```   865
```
```   866 lemma bilinear_eq:
```
```   867   assumes bf: "bilinear f"
```
```   868     and bg: "bilinear g"
```
```   869     and SB: "S \<subseteq> span B"
```
```   870     and TC: "T \<subseteq> span C"
```
```   871     and "x\<in>S" "y\<in>T"
```
```   872     and fg: "\<And>x y. \<lbrakk>x \<in> B; y\<in> C\<rbrakk> \<Longrightarrow> f x y = g x y"
```
```   873   shows "f x y = g x y"
```
```   874 proof -
```
```   875   let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
```
```   876   from bf bg have sp: "subspace ?P"
```
```   877     unfolding bilinear_def linear_iff subspace_def bf bg
```
```   878     by (auto simp add: span_zero bilinear_lzero[OF bf] bilinear_lzero[OF bg]
```
```   879         span_add Ball_def
```
```   880       intro: bilinear_ladd[OF bf])
```
```   881   have sfg: "\<And>x. x \<in> B \<Longrightarrow> subspace {a. f x a = g x a}"
```
```   882     apply (auto simp add: subspace_def)
```
```   883     using bf bg unfolding bilinear_def linear_iff
```
```   884       apply (auto simp add: span_zero bilinear_rzero[OF bf] bilinear_rzero[OF bg]
```
```   885         span_add Ball_def
```
```   886       intro: bilinear_ladd[OF bf])
```
```   887     done
```
```   888   have "\<forall>y\<in> span C. f x y = g x y" if "x \<in> span B" for x
```
```   889     apply (rule span_induct [OF that sp])
```
```   890     using fg sfg span_induct by blast
```
```   891   then show ?thesis
```
```   892     using SB TC assms by auto
```
```   893 qed
```
```   894
```
```   895 lemma bilinear_eq_stdbasis:
```
```   896   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
```
```   897   assumes bf: "bilinear f"
```
```   898     and bg: "bilinear g"
```
```   899     and fg: "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> f i j = g i j"
```
```   900   shows "f = g"
```
```   901   using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis]] fg by blast
```
```   902
```
```   903 subsection \<open>Infinity norm\<close>
```
```   904
```
```   905 definition%important "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
```
```   906
```
```   907 lemma infnorm_set_image:
```
```   908   fixes x :: "'a::euclidean_space"
```
```   909   shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```   910   by blast
```
```   911
```
```   912 lemma infnorm_Max:
```
```   913   fixes x :: "'a::euclidean_space"
```
```   914   shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)"
```
```   915   by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)
```
```   916
```
```   917 lemma infnorm_set_lemma:
```
```   918   fixes x :: "'a::euclidean_space"
```
```   919   shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
```
```   920     and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}"
```
```   921   unfolding infnorm_set_image
```
```   922   by auto
```
```   923
```
```   924 lemma infnorm_pos_le:
```
```   925   fixes x :: "'a::euclidean_space"
```
```   926   shows "0 \<le> infnorm x"
```
```   927   by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
```
```   928
```
```   929 lemma infnorm_triangle:
```
```   930   fixes x :: "'a::euclidean_space"
```
```   931   shows "infnorm (x + y) \<le> infnorm x + infnorm y"
```
```   932 proof -
```
```   933   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"
```
```   934     by simp
```
```   935   show ?thesis
```
```   936     by (auto simp: infnorm_Max inner_add_left intro!: *)
```
```   937 qed
```
```   938
```
```   939 lemma infnorm_eq_0:
```
```   940   fixes x :: "'a::euclidean_space"
```
```   941   shows "infnorm x = 0 \<longleftrightarrow> x = 0"
```
```   942 proof -
```
```   943   have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
```
```   944     unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
```
```   945   then show ?thesis
```
```   946     using infnorm_pos_le[of x] by simp
```
```   947 qed
```
```   948
```
```   949 lemma infnorm_0: "infnorm 0 = 0"
```
```   950   by (simp add: infnorm_eq_0)
```
```   951
```
```   952 lemma infnorm_neg: "infnorm (- x) = infnorm x"
```
```   953   unfolding infnorm_def by simp
```
```   954
```
```   955 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
```
```   956   by (metis infnorm_neg minus_diff_eq)
```
```   957
```
```   958 lemma absdiff_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
```
```   959 proof -
```
```   960   have *: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
```
```   961     by arith
```
```   962   show ?thesis
```
```   963   proof (rule *)
```
```   964     from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
```
```   965     show "infnorm x \<le> infnorm (x - y) + infnorm y" "infnorm y \<le> infnorm (x - y) + infnorm x"
```
```   966       by (simp_all add: field_simps infnorm_neg)
```
```   967   qed
```
```   968 qed
```
```   969
```
```   970 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
```
```   971   using infnorm_pos_le[of x] by arith
```
```   972
```
```   973 lemma Basis_le_infnorm:
```
```   974   fixes x :: "'a::euclidean_space"
```
```   975   shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
```
```   976   by (simp add: infnorm_Max)
```
```   977
```
```   978 lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
```
```   979   unfolding infnorm_Max
```
```   980 proof (safe intro!: Max_eqI)
```
```   981   let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```   982   { fix b :: 'a
```
```   983     assume "b \<in> Basis"
```
```   984     then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
```
```   985       by (simp add: abs_mult mult_left_mono)
```
```   986   next
```
```   987     from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
```
```   988       by (auto simp del: Max_in)
```
```   989     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"
```
```   990       by (intro image_eqI[where x=b]) (auto simp: abs_mult)
```
```   991   }
```
```   992 qed simp
```
```   993
```
```   994 lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
```
```   995   unfolding infnorm_mul ..
```
```   996
```
```   997 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
```
```   998   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
```
```   999
```
```  1000 text \<open>Prove that it differs only up to a bound from Euclidean norm.\<close>
```
```  1001
```
```  1002 lemma infnorm_le_norm: "infnorm x \<le> norm x"
```
```  1003   by (simp add: Basis_le_norm infnorm_Max)
```
```  1004
```
```  1005 lemma norm_le_infnorm:
```
```  1006   fixes x :: "'a::euclidean_space"
```
```  1007   shows "norm x \<le> sqrt DIM('a) * infnorm x"
```
```  1008   unfolding norm_eq_sqrt_inner id_def
```
```  1009 proof (rule real_le_lsqrt[OF inner_ge_zero])
```
```  1010   show "sqrt DIM('a) * infnorm x \<ge> 0"
```
```  1011     by (simp add: zero_le_mult_iff infnorm_pos_le)
```
```  1012   have "x \<bullet> x \<le> (\<Sum>b\<in>Basis. x \<bullet> b * (x \<bullet> b))"
```
```  1013     by (metis euclidean_inner order_refl)
```
```  1014   also have "... \<le> DIM('a) * \<bar>infnorm x\<bar>\<^sup>2"
```
```  1015     by (rule sum_bounded_above) (metis Basis_le_infnorm abs_le_square_iff power2_eq_square real_abs_infnorm)
```
```  1016   also have "... \<le> (sqrt DIM('a) * infnorm x)\<^sup>2"
```
```  1017     by (simp add: power_mult_distrib)
```
```  1018   finally show "x \<bullet> x \<le> (sqrt DIM('a) * infnorm x)\<^sup>2" .
```
```  1019 qed
```
```  1020
```
```  1021 lemma tendsto_infnorm [tendsto_intros]:
```
```  1022   assumes "(f \<longlongrightarrow> a) F"
```
```  1023   shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F"
```
```  1024 proof (rule tendsto_compose [OF LIM_I assms])
```
```  1025   fix r :: real
```
```  1026   assume "r > 0"
```
```  1027   then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
```
```  1028     by (metis real_norm_def le_less_trans absdiff_infnorm infnorm_le_norm)
```
```  1029 qed
```
```  1030
```
```  1031 text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close>
```
```  1032
```
```  1033 lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  1034   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1035 proof (cases "x=0")
```
```  1036   case True
```
```  1037   then show ?thesis
```
```  1038     by auto
```
```  1039 next
```
```  1040   case False
```
```  1041   from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
```
```  1042   have "?rhs \<longleftrightarrow>
```
```  1043       (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
```
```  1044         norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
```
```  1045     using False unfolding inner_simps
```
```  1046     by (auto simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
```
```  1047   also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)"
```
```  1048     using False  by (simp add: field_simps inner_commute)
```
```  1049   also have "\<dots> \<longleftrightarrow> ?lhs"
```
```  1050     using False by auto
```
```  1051   finally show ?thesis by metis
```
```  1052 qed
```
```  1053
```
```  1054 lemma norm_cauchy_schwarz_abs_eq:
```
```  1055   "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow>
```
```  1056     norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
```
```  1057   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1058 proof -
```
```  1059   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a"
```
```  1060     by arith
```
```  1061   have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
```
```  1062     by simp
```
```  1063   also have "\<dots> \<longleftrightarrow> (x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
```
```  1064     unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  1065     unfolding norm_minus_cancel norm_scaleR ..
```
```  1066   also have "\<dots> \<longleftrightarrow> ?lhs"
```
```  1067     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
```
```  1068     by auto
```
```  1069   finally show ?thesis ..
```
```  1070 qed
```
```  1071
```
```  1072 lemma norm_triangle_eq:
```
```  1073   fixes x y :: "'a::real_inner"
```
```  1074   shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  1075 proof (cases "x = 0 \<or> y = 0")
```
```  1076   case True
```
```  1077   then show ?thesis
```
```  1078     by force
```
```  1079 next
```
```  1080   case False
```
```  1081   then have n: "norm x > 0" "norm y > 0"
```
```  1082     by auto
```
```  1083   have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
```
```  1084     by simp
```
```  1085   also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  1086     unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  1087     unfolding power2_norm_eq_inner inner_simps
```
```  1088     by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
```
```  1089   finally show ?thesis .
```
```  1090 qed
```
```  1091
```
```  1092
```
```  1093 subsection \<open>Collinearity\<close>
```
```  1094
```
```  1095 definition%important collinear :: "'a::real_vector set \<Rightarrow> bool"
```
```  1096   where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
```
```  1097
```
```  1098 lemma collinear_alt:
```
```  1099      "collinear S \<longleftrightarrow> (\<exists>u v. \<forall>x \<in> S. \<exists>c. x = u + c *\<^sub>R v)" (is "?lhs = ?rhs")
```
```  1100 proof
```
```  1101   assume ?lhs
```
```  1102   then show ?rhs
```
```  1103     unfolding collinear_def by (metis Groups.add_ac(2) diff_add_cancel)
```
```  1104 next
```
```  1105   assume ?rhs
```
```  1106   then obtain u v where *: "\<And>x. x \<in> S \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
```
```  1107     by (auto simp: )
```
```  1108   have "\<exists>c. x - y = c *\<^sub>R v" if "x \<in> S" "y \<in> S" for x y
```
```  1109         by (metis *[OF \<open>x \<in> S\<close>] *[OF \<open>y \<in> S\<close>] scaleR_left.diff add_diff_cancel_left)
```
```  1110   then show ?lhs
```
```  1111     using collinear_def by blast
```
```  1112 qed
```
```  1113
```
```  1114 lemma collinear:
```
```  1115   fixes S :: "'a::{perfect_space,real_vector} set"
```
```  1116   shows "collinear S \<longleftrightarrow> (\<exists>u. u \<noteq> 0 \<and> (\<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u))"
```
```  1117 proof -
```
```  1118   have "\<exists>v. v \<noteq> 0 \<and> (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v)"
```
```  1119     if "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R u" "u=0" for u
```
```  1120   proof -
```
```  1121     have "\<forall>x\<in>S. \<forall>y\<in>S. x = y"
```
```  1122       using that by auto
```
```  1123     moreover
```
```  1124     obtain v::'a where "v \<noteq> 0"
```
```  1125       using UNIV_not_singleton [of 0] by auto
```
```  1126     ultimately have "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v"
```
```  1127       by auto
```
```  1128     then show ?thesis
```
```  1129       using \<open>v \<noteq> 0\<close> by blast
```
```  1130   qed
```
```  1131   then show ?thesis
```
```  1132     apply (clarsimp simp: collinear_def)
```
```  1133     by (metis scaleR_zero_right vector_fraction_eq_iff)
```
```  1134 qed
```
```  1135
```
```  1136 lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S"
```
```  1137   by (meson collinear_def subsetCE)
```
```  1138
```
```  1139 lemma collinear_empty [iff]: "collinear {}"
```
```  1140   by (simp add: collinear_def)
```
```  1141
```
```  1142 lemma collinear_sing [iff]: "collinear {x}"
```
```  1143   by (simp add: collinear_def)
```
```  1144
```
```  1145 lemma collinear_2 [iff]: "collinear {x, y}"
```
```  1146   apply (simp add: collinear_def)
```
```  1147   apply (rule exI[where x="x - y"])
```
```  1148   by (metis minus_diff_eq scaleR_left.minus scaleR_one)
```
```  1149
```
```  1150 lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
```
```  1151   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1152 proof (cases "x = 0 \<or> y = 0")
```
```  1153   case True
```
```  1154   then show ?thesis
```
```  1155     by (auto simp: insert_commute)
```
```  1156 next
```
```  1157   case False
```
```  1158   show ?thesis
```
```  1159   proof
```
```  1160     assume h: "?lhs"
```
```  1161     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"
```
```  1162       unfolding collinear_def by blast
```
```  1163     from u[rule_format, of x 0] u[rule_format, of y 0]
```
```  1164     obtain cx and cy where
```
```  1165       cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
```
```  1166       by auto
```
```  1167     from cx cy False have cx0: "cx \<noteq> 0" and cy0: "cy \<noteq> 0" by auto
```
```  1168     let ?d = "cy / cx"
```
```  1169     from cx cy cx0 have "y = ?d *\<^sub>R x"
```
```  1170       by simp
```
```  1171     then show ?rhs using False by blast
```
```  1172   next
```
```  1173     assume h: "?rhs"
```
```  1174     then obtain c where c: "y = c *\<^sub>R x"
```
```  1175       using False by blast
```
```  1176     show ?lhs
```
```  1177       unfolding collinear_def c
```
```  1178       apply (rule exI[where x=x])
```
```  1179       apply auto
```
```  1180           apply (rule exI[where x="- 1"], simp)
```
```  1181          apply (rule exI[where x= "-c"], simp)
```
```  1182         apply (rule exI[where x=1], simp)
```
```  1183        apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
```
```  1184       apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
```
```  1185       done
```
```  1186   qed
```
```  1187 qed
```
```  1188
```
```  1189 lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
```
```  1190 proof (cases "x=0")
```
```  1191   case True
```
```  1192   then show ?thesis
```
```  1193     by (auto simp: insert_commute)
```
```  1194 next
```
```  1195   case False
```
```  1196   then have nnz: "norm x \<noteq> 0"
```
```  1197     by auto
```
```  1198   show ?thesis
```
```  1199   proof
```
```  1200     assume "\<bar>x \<bullet> y\<bar> = norm x * norm y"
```
```  1201     then show "collinear {0, x, y}"
```
```  1202       unfolding norm_cauchy_schwarz_abs_eq collinear_lemma
```
```  1203       by (meson eq_vector_fraction_iff nnz)
```
```  1204   next
```
```  1205     assume "collinear {0, x, y}"
```
```  1206     with False show "\<bar>x \<bullet> y\<bar> = norm x * norm y"
```
```  1207       unfolding norm_cauchy_schwarz_abs_eq collinear_lemma  by (auto simp: abs_if)
```
```  1208   qed
```
```  1209 qed
```
```  1210
```
```  1211 end
```