src/HOL/Analysis/Linear_Algebra.thy
 author immler Wed May 02 13:49:38 2018 +0200 (12 months ago) changeset 68072 493b818e8e10 parent 67982 7643b005b29a child 68073 fad29d2a17a5 permissions -rw-r--r--
```     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 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 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
```
```    38 proof -
```
```    39   have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
```
```    40     by auto
```
```    41   show ?thesis unfolding eq
```
```    42     apply (rule finite_imageI)
```
```    43     apply (rule finite)
```
```    44     done
```
```    45 qed
```
```    46
```
```    47
```
```    48 subsection%unimportant \<open>More interesting properties of the norm.\<close>
```
```    49
```
```    50 notation inner (infix "\<bullet>" 70)
```
```    51
```
```    52 text\<open>Equality of vectors in terms of @{term "(\<bullet>)"} products.\<close>
```
```    53
```
```    54 lemma linear_componentwise:
```
```    55   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
```
```    56   assumes lf: "linear f"
```
```    57   shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
```
```    58 proof -
```
```    59   interpret linear f by fact
```
```    60   have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
```
```    61     by (simp add: inner_sum_left)
```
```    62   then show ?thesis
```
```    63     by (simp add: euclidean_representation sum[symmetric] scale[symmetric])
```
```    64 qed
```
```    65
```
```    66 lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
```
```    67   (is "?lhs \<longleftrightarrow> ?rhs")
```
```    68 proof
```
```    69   assume ?lhs
```
```    70   then show ?rhs by simp
```
```    71 next
```
```    72   assume ?rhs
```
```    73   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
```
```    74     by simp
```
```    75   then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
```
```    76     by (simp add: inner_diff inner_commute)
```
```    77   then have "(x - y) \<bullet> (x - y) = 0"
```
```    78     by (simp add: field_simps inner_diff inner_commute)
```
```    79   then show "x = y" by simp
```
```    80 qed
```
```    81
```
```    82 lemma norm_triangle_half_r:
```
```    83   "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
```
```    84   using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
```
```    85
```
```    86 lemma norm_triangle_half_l:
```
```    87   assumes "norm (x - y) < e / 2"
```
```    88     and "norm (x' - y) < e / 2"
```
```    89   shows "norm (x - x') < e"
```
```    90   using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
```
```    91   unfolding dist_norm[symmetric] .
```
```    92
```
```    93 lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
```
```    94   by (rule norm_triangle_ineq [THEN order_trans])
```
```    95
```
```    96 lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
```
```    97   by (rule norm_triangle_ineq [THEN le_less_trans])
```
```    98
```
```    99 lemma abs_triangle_half_r:
```
```   100   fixes y :: "'a::linordered_field"
```
```   101   shows "abs (y - x1) < e / 2 \<Longrightarrow> abs (y - x2) < e / 2 \<Longrightarrow> abs (x1 - x2) < e"
```
```   102   by linarith
```
```   103
```
```   104 lemma abs_triangle_half_l:
```
```   105   fixes y :: "'a::linordered_field"
```
```   106   assumes "abs (x - y) < e / 2"
```
```   107     and "abs (x' - y) < e / 2"
```
```   108   shows "abs (x - x') < e"
```
```   109   using assms by linarith
```
```   110
```
```   111 lemma sum_clauses:
```
```   112   shows "sum f {} = 0"
```
```   113     and "finite S \<Longrightarrow> sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
```
```   114   by (auto simp add: insert_absorb)
```
```   115
```
```   116 lemma sum_norm_bound:
```
```   117   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   118   assumes K: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> K"
```
```   119   shows "norm (sum f S) \<le> of_nat (card S)*K"
```
```   120   using sum_norm_le[OF K] sum_constant[symmetric]
```
```   121   by simp
```
```   122
```
```   123 lemma sum_group:
```
```   124   assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
```
```   125   shows "sum (\<lambda>y. sum g {x. x \<in> S \<and> f x = y}) T = sum g S"
```
```   126   apply (subst sum_image_gen[OF fS, of g f])
```
```   127   apply (rule sum.mono_neutral_right[OF fT fST])
```
```   128   apply (auto intro: sum.neutral)
```
```   129   done
```
```   130
```
```   131 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
```
```   132 proof
```
```   133   assume "\<forall>x. x \<bullet> y = x \<bullet> z"
```
```   134   then have "\<forall>x. x \<bullet> (y - z) = 0"
```
```   135     by (simp add: inner_diff)
```
```   136   then have "(y - z) \<bullet> (y - z) = 0" ..
```
```   137   then show "y = z" by simp
```
```   138 qed simp
```
```   139
```
```   140 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
```
```   141 proof
```
```   142   assume "\<forall>z. x \<bullet> z = y \<bullet> z"
```
```   143   then have "\<forall>z. (x - y) \<bullet> z = 0"
```
```   144     by (simp add: inner_diff)
```
```   145   then have "(x - y) \<bullet> (x - y) = 0" ..
```
```   146   then show "x = y" by simp
```
```   147 qed simp
```
```   148
```
```   149
```
```   150 subsection \<open>Orthogonality.\<close>
```
```   151
```
```   152 definition%important (in real_inner) "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
```
```   153
```
```   154 context real_inner
```
```   155 begin
```
```   156
```
```   157 lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0"
```
```   158   by (simp add: orthogonal_def)
```
```   159
```
```   160 lemma orthogonal_clauses:
```
```   161   "orthogonal a 0"
```
```   162   "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
```
```   163   "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
```
```   164   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
```
```   165   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
```
```   166   "orthogonal 0 a"
```
```   167   "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
```
```   168   "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
```
```   169   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
```
```   170   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
```
```   171   unfolding orthogonal_def inner_add inner_diff by auto
```
```   172
```
```   173 end
```
```   174
```
```   175 lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
```
```   176   by (simp add: orthogonal_def inner_commute)
```
```   177
```
```   178 lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x"
```
```   179   by (rule ext) (simp add: orthogonal_def)
```
```   180
```
```   181 lemma pairwise_ortho_scaleR:
```
```   182     "pairwise (\<lambda>i j. orthogonal (f i) (g j)) B
```
```   183     \<Longrightarrow> pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B"
```
```   184   by (auto simp: pairwise_def orthogonal_clauses)
```
```   185
```
```   186 lemma orthogonal_rvsum:
```
```   187     "\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (sum f s)"
```
```   188   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
```
```   189
```
```   190 lemma orthogonal_lvsum:
```
```   191     "\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (sum f s) y"
```
```   192   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
```
```   193
```
```   194 lemma norm_add_Pythagorean:
```
```   195   assumes "orthogonal a b"
```
```   196     shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2"
```
```   197 proof -
```
```   198   from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)"
```
```   199     by (simp add: algebra_simps orthogonal_def inner_commute)
```
```   200   then show ?thesis
```
```   201     by (simp add: power2_norm_eq_inner)
```
```   202 qed
```
```   203
```
```   204 lemma norm_sum_Pythagorean:
```
```   205   assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I"
```
```   206     shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
```
```   207 using assms
```
```   208 proof (induction I rule: finite_induct)
```
```   209   case empty then show ?case by simp
```
```   210 next
```
```   211   case (insert x I)
```
```   212   then have "orthogonal (f x) (sum f I)"
```
```   213     by (metis pairwise_insert orthogonal_rvsum)
```
```   214   with insert show ?case
```
```   215     by (simp add: pairwise_insert norm_add_Pythagorean)
```
```   216 qed
```
```   217
```
```   218
```
```   219 subsection \<open>Bilinear functions.\<close>
```
```   220
```
```   221 definition%important "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
```
```   222
```
```   223 lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
```
```   224   by (simp add: bilinear_def linear_iff)
```
```   225
```
```   226 lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
```
```   227   by (simp add: bilinear_def linear_iff)
```
```   228
```
```   229 lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
```
```   230   by (simp add: bilinear_def linear_iff)
```
```   231
```
```   232 lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
```
```   233   by (simp add: bilinear_def linear_iff)
```
```   234
```
```   235 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
```
```   236   by (drule bilinear_lmul [of _ "- 1"]) simp
```
```   237
```
```   238 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
```
```   239   by (drule bilinear_rmul [of _ _ "- 1"]) simp
```
```   240
```
```   241 lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
```
```   242   using add_left_imp_eq[of x y 0] by auto
```
```   243
```
```   244 lemma bilinear_lzero:
```
```   245   assumes "bilinear h"
```
```   246   shows "h 0 x = 0"
```
```   247   using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
```
```   248
```
```   249 lemma bilinear_rzero:
```
```   250   assumes "bilinear h"
```
```   251   shows "h x 0 = 0"
```
```   252   using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
```
```   253
```
```   254 lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
```
```   255   using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
```
```   256
```
```   257 lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
```
```   258   using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
```
```   259
```
```   260 lemma bilinear_sum:
```
```   261   assumes "bilinear h"
```
```   262   shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
```
```   263 proof -
```
```   264   interpret l: linear "\<lambda>x. h x y" for y using assms by (simp add: bilinear_def)
```
```   265   interpret r: linear "\<lambda>y. h x y" for x using assms by (simp add: bilinear_def)
```
```   266   have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S"
```
```   267     by (simp add: l.sum)
```
```   268   also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S"
```
```   269     by (rule sum.cong) (simp_all add: r.sum)
```
```   270   finally show ?thesis
```
```   271     unfolding sum.cartesian_product .
```
```   272 qed
```
```   273
```
```   274
```
```   275 subsection \<open>Adjoints.\<close>
```
```   276
```
```   277 definition%important "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
```
```   278
```
```   279 lemma adjoint_unique:
```
```   280   assumes "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   281   shows "adjoint f = g"
```
```   282   unfolding adjoint_def
```
```   283 proof (rule some_equality)
```
```   284   show "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   285     by (rule assms)
```
```   286 next
```
```   287   fix h
```
```   288   assume "\<forall>x y. inner (f x) y = inner x (h y)"
```
```   289   then have "\<forall>x y. inner x (g y) = inner x (h y)"
```
```   290     using assms by simp
```
```   291   then have "\<forall>x y. inner x (g y - h y) = 0"
```
```   292     by (simp add: inner_diff_right)
```
```   293   then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
```
```   294     by simp
```
```   295   then have "\<forall>y. h y = g y"
```
```   296     by simp
```
```   297   then show "h = g" by (simp add: ext)
```
```   298 qed
```
```   299
```
```   300 text \<open>TODO: The following lemmas about adjoints should hold for any
```
```   301   Hilbert space (i.e. complete inner product space).
```
```   302   (see \<^url>\<open>http://en.wikipedia.org/wiki/Hermitian_adjoint\<close>)
```
```   303 \<close>
```
```   304
```
```   305 lemma adjoint_works:
```
```   306   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   307   assumes lf: "linear f"
```
```   308   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```   309 proof -
```
```   310   interpret linear f by fact
```
```   311   have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
```
```   312   proof (intro allI exI)
```
```   313     fix y :: "'m" and x
```
```   314     let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
```
```   315     have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
```
```   316       by (simp add: euclidean_representation)
```
```   317     also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
```
```   318       by (simp add: sum scale)
```
```   319     finally show "f x \<bullet> y = x \<bullet> ?w"
```
```   320       by (simp add: inner_sum_left inner_sum_right mult.commute)
```
```   321   qed
```
```   322   then show ?thesis
```
```   323     unfolding adjoint_def choice_iff
```
```   324     by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
```
```   325 qed
```
```   326
```
```   327 lemma adjoint_clauses:
```
```   328   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   329   assumes lf: "linear f"
```
```   330   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```   331     and "adjoint f y \<bullet> x = y \<bullet> f x"
```
```   332   by (simp_all add: adjoint_works[OF lf] inner_commute)
```
```   333
```
```   334 lemma adjoint_linear:
```
```   335   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   336   assumes lf: "linear f"
```
```   337   shows "linear (adjoint f)"
```
```   338   by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
```
```   339     adjoint_clauses[OF lf] inner_distrib)
```
```   340
```
```   341 lemma adjoint_adjoint:
```
```   342   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   343   assumes lf: "linear f"
```
```   344   shows "adjoint (adjoint f) = f"
```
```   345   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
```
```   346
```
```   347
```
```   348 subsection%unimportant \<open>Interlude: Some properties of real sets\<close>
```
```   349
```
```   350 lemma seq_mono_lemma:
```
```   351   assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
```
```   352     and "\<forall>n \<ge> m. e n \<le> e m"
```
```   353   shows "\<forall>n \<ge> m. d n < e m"
```
```   354   using assms
```
```   355   apply auto
```
```   356   apply (erule_tac x="n" in allE)
```
```   357   apply (erule_tac x="n" in allE)
```
```   358   apply auto
```
```   359   done
```
```   360
```
```   361 lemma infinite_enumerate:
```
```   362   assumes fS: "infinite S"
```
```   363   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)"
```
```   364   unfolding strict_mono_def
```
```   365   using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
```
```   366
```
```   367 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)"
```
```   368   apply auto
```
```   369   apply (rule_tac x="d/2" in exI)
```
```   370   apply auto
```
```   371   done
```
```   372
```
```   373 lemma approachable_lt_le2:  \<comment> \<open>like the above, but pushes aside an extra formula\<close>
```
```   374     "(\<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)"
```
```   375   apply auto
```
```   376   apply (rule_tac x="d/2" in exI, auto)
```
```   377   done
```
```   378
```
```   379 lemma triangle_lemma:
```
```   380   fixes x y z :: real
```
```   381   assumes x: "0 \<le> x"
```
```   382     and y: "0 \<le> y"
```
```   383     and z: "0 \<le> z"
```
```   384     and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
```
```   385   shows "x \<le> y + z"
```
```   386 proof -
```
```   387   have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
```
```   388     using z y by simp
```
```   389   with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
```
```   390     by (simp add: power2_eq_square field_simps)
```
```   391   from y z have yz: "y + z \<ge> 0"
```
```   392     by arith
```
```   393   from power2_le_imp_le[OF th yz] show ?thesis .
```
```   394 qed
```
```   395
```
```   396
```
```   397
```
```   398 subsection \<open>Archimedean properties and useful consequences\<close>
```
```   399
```
```   400 text\<open>Bernoulli's inequality\<close>
```
```   401 proposition%important Bernoulli_inequality:
```
```   402   fixes x :: real
```
```   403   assumes "-1 \<le> x"
```
```   404     shows "1 + n * x \<le> (1 + x) ^ n"
```
```   405 proof%unimportant (induct n)
```
```   406   case 0
```
```   407   then show ?case by simp
```
```   408 next
```
```   409   case (Suc n)
```
```   410   have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
```
```   411     by (simp add: algebra_simps)
```
```   412   also have "... = (1 + x) * (1 + n*x)"
```
```   413     by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
```
```   414   also have "... \<le> (1 + x) ^ Suc n"
```
```   415     using Suc.hyps assms mult_left_mono by fastforce
```
```   416   finally show ?case .
```
```   417 qed
```
```   418
```
```   419 corollary Bernoulli_inequality_even:
```
```   420   fixes x :: real
```
```   421   assumes "even n"
```
```   422     shows "1 + n * x \<le> (1 + x) ^ n"
```
```   423 proof (cases "-1 \<le> x \<or> n=0")
```
```   424   case True
```
```   425   then show ?thesis
```
```   426     by (auto simp: Bernoulli_inequality)
```
```   427 next
```
```   428   case False
```
```   429   then have "real n \<ge> 1"
```
```   430     by simp
```
```   431   with False have "n * x \<le> -1"
```
```   432     by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
```
```   433   then have "1 + n * x \<le> 0"
```
```   434     by auto
```
```   435   also have "... \<le> (1 + x) ^ n"
```
```   436     using assms
```
```   437     using zero_le_even_power by blast
```
```   438   finally show ?thesis .
```
```   439 qed
```
```   440
```
```   441 corollary real_arch_pow:
```
```   442   fixes x :: real
```
```   443   assumes x: "1 < x"
```
```   444   shows "\<exists>n. y < x^n"
```
```   445 proof -
```
```   446   from x have x0: "x - 1 > 0"
```
```   447     by arith
```
```   448   from reals_Archimedean3[OF x0, rule_format, of y]
```
```   449   obtain n :: nat where n: "y < real n * (x - 1)" by metis
```
```   450   from x0 have x00: "x- 1 \<ge> -1" by arith
```
```   451   from Bernoulli_inequality[OF x00, of n] n
```
```   452   have "y < x^n" by auto
```
```   453   then show ?thesis by metis
```
```   454 qed
```
```   455
```
```   456 corollary real_arch_pow_inv:
```
```   457   fixes x y :: real
```
```   458   assumes y: "y > 0"
```
```   459     and x1: "x < 1"
```
```   460   shows "\<exists>n. x^n < y"
```
```   461 proof (cases "x > 0")
```
```   462   case True
```
```   463   with x1 have ix: "1 < 1/x" by (simp add: field_simps)
```
```   464   from real_arch_pow[OF ix, of "1/y"]
```
```   465   obtain n where n: "1/y < (1/x)^n" by blast
```
```   466   then show ?thesis using y \<open>x > 0\<close>
```
```   467     by (auto simp add: field_simps)
```
```   468 next
```
```   469   case False
```
```   470   with y x1 show ?thesis
```
```   471     apply auto
```
```   472     apply (rule exI[where x=1])
```
```   473     apply auto
```
```   474     done
```
```   475 qed
```
```   476
```
```   477 lemma forall_pos_mono:
```
```   478   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```   479     (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
```
```   480   by (metis real_arch_inverse)
```
```   481
```
```   482 lemma forall_pos_mono_1:
```
```   483   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```   484     (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
```
```   485   apply (rule forall_pos_mono)
```
```   486   apply auto
```
```   487   apply (metis Suc_pred of_nat_Suc)
```
```   488   done
```
```   489
```
```   490
```
```   491 subsection%unimportant \<open>Euclidean Spaces as Typeclass\<close>
```
```   492
```
```   493 lemma independent_Basis: "independent Basis"
```
```   494   by (rule independent_Basis)
```
```   495
```
```   496 lemma span_Basis [simp]: "span Basis = UNIV"
```
```   497   by (rule span_Basis)
```
```   498
```
```   499 lemma in_span_Basis: "x \<in> span Basis"
```
```   500   unfolding span_Basis ..
```
```   501
```
```   502
```
```   503 subsection%unimportant \<open>Linearity and Bilinearity continued\<close>
```
```   504
```
```   505 lemma linear_bounded:
```
```   506   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   507   assumes lf: "linear f"
```
```   508   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```   509 proof
```
```   510   interpret linear f by fact
```
```   511   let ?B = "\<Sum>b\<in>Basis. norm (f b)"
```
```   512   show "\<forall>x. norm (f x) \<le> ?B * norm x"
```
```   513   proof
```
```   514     fix x :: 'a
```
```   515     let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
```
```   516     have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
```
```   517       unfolding euclidean_representation ..
```
```   518     also have "\<dots> = norm (sum ?g Basis)"
```
```   519       by (simp add: sum scale)
```
```   520     finally have th0: "norm (f x) = norm (sum ?g Basis)" .
```
```   521     have th: "norm (?g i) \<le> norm (f i) * norm x" if "i \<in> Basis" for i
```
```   522     proof -
```
```   523       from Basis_le_norm[OF that, of x]
```
```   524       show "norm (?g i) \<le> norm (f i) * norm x"
```
```   525         unfolding norm_scaleR
```
```   526         apply (subst mult.commute)
```
```   527         apply (rule mult_mono)
```
```   528         apply (auto simp add: field_simps)
```
```   529         done
```
```   530     qed
```
```   531     from sum_norm_le[of _ ?g, OF th]
```
```   532     show "norm (f x) \<le> ?B * norm x"
```
```   533       unfolding th0 sum_distrib_right by metis
```
```   534   qed
```
```   535 qed
```
```   536
```
```   537 lemma linear_conv_bounded_linear:
```
```   538   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   539   shows "linear f \<longleftrightarrow> bounded_linear f"
```
```   540 proof
```
```   541   assume "linear f"
```
```   542   then interpret f: linear f .
```
```   543   show "bounded_linear f"
```
```   544   proof
```
```   545     have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```   546       using \<open>linear f\<close> by (rule linear_bounded)
```
```   547     then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```   548       by (simp add: mult.commute)
```
```   549   qed
```
```   550 next
```
```   551   assume "bounded_linear f"
```
```   552   then interpret f: bounded_linear f .
```
```   553   show "linear f" ..
```
```   554 qed
```
```   555
```
```   556 lemmas linear_linear = linear_conv_bounded_linear[symmetric]
```
```   557
```
```   558 lemma linear_bounded_pos:
```
```   559   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   560   assumes lf: "linear f"
```
```   561  obtains B where "B > 0" "\<And>x. norm (f x) \<le> B * norm x"
```
```   562 proof -
```
```   563   have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
```
```   564     using lf unfolding linear_conv_bounded_linear
```
```   565     by (rule bounded_linear.pos_bounded)
```
```   566   with that show ?thesis
```
```   567     by (auto simp: mult.commute)
```
```   568 qed
```
```   569
```
```   570 lemma linear_invertible_bounded_below_pos:
```
```   571   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
```
```   572   assumes "linear f" "linear g" "g \<circ> f = id"
```
```   573   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
```
```   574 proof -
```
```   575   obtain B where "B > 0" and B: "\<And>x. norm (g x) \<le> B * norm x"
```
```   576     using linear_bounded_pos [OF \<open>linear g\<close>] by blast
```
```   577   show thesis
```
```   578   proof
```
```   579     show "0 < 1/B"
```
```   580       by (simp add: \<open>B > 0\<close>)
```
```   581     show "1/B * norm x \<le> norm (f x)" for x
```
```   582     proof -
```
```   583       have "1/B * norm x = 1/B * norm (g (f x))"
```
```   584         using assms by (simp add: pointfree_idE)
```
```   585       also have "\<dots> \<le> norm (f x)"
```
```   586         using B [of "f x"] by (simp add: \<open>B > 0\<close> mult.commute pos_divide_le_eq)
```
```   587       finally show ?thesis .
```
```   588     qed
```
```   589   qed
```
```   590 qed
```
```   591
```
```   592 lemma linear_inj_bounded_below_pos:
```
```   593   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
```
```   594   assumes "linear f" "inj f"
```
```   595   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
```
```   596   using linear_injective_left_inverse [OF assms]
```
```   597     linear_invertible_bounded_below_pos assms by blast
```
```   598
```
```   599 lemma bounded_linearI':
```
```   600   fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   601   assumes "\<And>x y. f (x + y) = f x + f y"
```
```   602     and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```   603   shows "bounded_linear f"
```
```   604   using assms linearI linear_conv_bounded_linear by blast
```
```   605
```
```   606 lemma bilinear_bounded:
```
```   607   fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
```
```   608   assumes bh: "bilinear h"
```
```   609   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```   610 proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
```
```   611   fix x :: 'm
```
```   612   fix y :: 'n
```
```   613   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))"
```
```   614     apply (subst euclidean_representation[where 'a='m])
```
```   615     apply (subst euclidean_representation[where 'a='n])
```
```   616     apply rule
```
```   617     done
```
```   618   also have "\<dots> = norm (sum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
```
```   619     unfolding bilinear_sum[OF bh] ..
```
```   620   finally have th: "norm (h x y) = \<dots>" .
```
```   621   show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
```
```   622     apply (auto simp add: sum_distrib_right th sum.cartesian_product)
```
```   623     apply (rule sum_norm_le)
```
```   624     apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
```
```   625       field_simps simp del: scaleR_scaleR)
```
```   626     apply (rule mult_mono)
```
```   627     apply (auto simp add: zero_le_mult_iff Basis_le_norm)
```
```   628     apply (rule mult_mono)
```
```   629     apply (auto simp add: zero_le_mult_iff Basis_le_norm)
```
```   630     done
```
```   631 qed
```
```   632
```
```   633 lemma bilinear_conv_bounded_bilinear:
```
```   634   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```   635   shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
```
```   636 proof
```
```   637   assume "bilinear h"
```
```   638   show "bounded_bilinear h"
```
```   639   proof
```
```   640     fix x y z
```
```   641     show "h (x + y) z = h x z + h y z"
```
```   642       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
```
```   643   next
```
```   644     fix x y z
```
```   645     show "h x (y + z) = h x y + h x z"
```
```   646       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
```
```   647   next
```
```   648     fix r x y
```
```   649     show "h (scaleR r x) y = scaleR r (h x y)"
```
```   650       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
```
```   651       by simp
```
```   652   next
```
```   653     fix r x y
```
```   654     show "h x (scaleR r y) = scaleR r (h x y)"
```
```   655       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
```
```   656       by simp
```
```   657   next
```
```   658     have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```   659       using \<open>bilinear h\<close> by (rule bilinear_bounded)
```
```   660     then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
```
```   661       by (simp add: ac_simps)
```
```   662   qed
```
```   663 next
```
```   664   assume "bounded_bilinear h"
```
```   665   then interpret h: bounded_bilinear h .
```
```   666   show "bilinear h"
```
```   667     unfolding bilinear_def linear_conv_bounded_linear
```
```   668     using h.bounded_linear_left h.bounded_linear_right by simp
```
```   669 qed
```
```   670
```
```   671 lemma bilinear_bounded_pos:
```
```   672   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```   673   assumes bh: "bilinear h"
```
```   674   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```   675 proof -
```
```   676   have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
```
```   677     using bh [unfolded bilinear_conv_bounded_bilinear]
```
```   678     by (rule bounded_bilinear.pos_bounded)
```
```   679   then show ?thesis
```
```   680     by (simp only: ac_simps)
```
```   681 qed
```
```   682
```
```   683 lemma bounded_linear_imp_has_derivative: "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
```
```   684   by (auto simp add: has_derivative_def linear_diff linear_linear linear_def
```
```   685       dest: bounded_linear.linear)
```
```   686
```
```   687 lemma linear_imp_has_derivative:
```
```   688   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   689   shows "linear f \<Longrightarrow> (f has_derivative f) net"
```
```   690   by (simp add: bounded_linear_imp_has_derivative linear_conv_bounded_linear)
```
```   691
```
```   692 lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net"
```
```   693   using bounded_linear_imp_has_derivative differentiable_def by blast
```
```   694
```
```   695 lemma linear_imp_differentiable:
```
```   696   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   697   shows "linear f \<Longrightarrow> f differentiable net"
```
```   698   by (metis linear_imp_has_derivative differentiable_def)
```
```   699
```
```   700
```
```   701 subsection%unimportant \<open>We continue.\<close>
```
```   702
```
```   703 lemma independent_bound:
```
```   704   fixes S :: "'a::euclidean_space set"
```
```   705   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
```
```   706   by (metis dim_subset_UNIV finiteI_independent dim_span_eq_card_independent)
```
```   707
```
```   708 lemmas independent_imp_finite = finiteI_independent
```
```   709
```
```   710 corollary
```
```   711   fixes S :: "'a::euclidean_space set"
```
```   712   assumes "independent S"
```
```   713   shows independent_card_le:"card S \<le> DIM('a)"
```
```   714   using assms independent_bound by auto
```
```   715
```
```   716 lemma dependent_biggerset:
```
```   717   fixes S :: "'a::euclidean_space set"
```
```   718   shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
```
```   719   by (metis independent_bound not_less)
```
```   720
```
```   721 text \<open>Picking an orthogonal replacement for a spanning set.\<close>
```
```   722
```
```   723 lemma vector_sub_project_orthogonal:
```
```   724   fixes b x :: "'a::euclidean_space"
```
```   725   shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
```
```   726   unfolding inner_simps by auto
```
```   727
```
```   728 lemma pairwise_orthogonal_insert:
```
```   729   assumes "pairwise orthogonal S"
```
```   730     and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
```
```   731   shows "pairwise orthogonal (insert x S)"
```
```   732   using assms unfolding pairwise_def
```
```   733   by (auto simp add: orthogonal_commute)
```
```   734
```
```   735 lemma basis_orthogonal:
```
```   736   fixes B :: "'a::real_inner set"
```
```   737   assumes fB: "finite B"
```
```   738   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
```
```   739   (is " \<exists>C. ?P B C")
```
```   740   using fB
```
```   741 proof (induct rule: finite_induct)
```
```   742   case empty
```
```   743   then show ?case
```
```   744     apply (rule exI[where x="{}"])
```
```   745     apply (auto simp add: pairwise_def)
```
```   746     done
```
```   747 next
```
```   748   case (insert a B)
```
```   749   note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
```
```   750   from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
```
```   751   obtain C where C: "finite C" "card C \<le> card B"
```
```   752     "span C = span B" "pairwise orthogonal C" by blast
```
```   753   let ?a = "a - sum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
```
```   754   let ?C = "insert ?a C"
```
```   755   from C(1) have fC: "finite ?C"
```
```   756     by simp
```
```   757   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
```
```   758     by (simp add: card_insert_if)
```
```   759   {
```
```   760     fix x k
```
```   761     have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
```
```   762       by (simp add: field_simps)
```
```   763     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"
```
```   764       apply (simp only: scaleR_right_diff_distrib th0)
```
```   765       apply (rule span_add_eq)
```
```   766       apply (rule span_scale)
```
```   767       apply (rule span_sum)
```
```   768       apply (rule span_scale)
```
```   769       apply (rule span_base)
```
```   770       apply assumption
```
```   771       done
```
```   772   }
```
```   773   then have SC: "span ?C = span (insert a B)"
```
```   774     unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
```
```   775   {
```
```   776     fix y
```
```   777     assume yC: "y \<in> C"
```
```   778     then have Cy: "C = insert y (C - {y})"
```
```   779       by blast
```
```   780     have fth: "finite (C - {y})"
```
```   781       using C by simp
```
```   782     have "orthogonal ?a y"
```
```   783       unfolding orthogonal_def
```
```   784       unfolding inner_diff inner_sum_left right_minus_eq
```
```   785       unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
```
```   786       apply (clarsimp simp add: inner_commute[of y a])
```
```   787       apply (rule sum.neutral)
```
```   788       apply clarsimp
```
```   789       apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
```
```   790       using \<open>y \<in> C\<close> by auto
```
```   791   }
```
```   792   with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
```
```   793     by (rule pairwise_orthogonal_insert)
```
```   794   from fC cC SC CPO have "?P (insert a B) ?C"
```
```   795     by blast
```
```   796   then show ?case by blast
```
```   797 qed
```
```   798
```
```   799 lemma orthogonal_basis_exists:
```
```   800   fixes V :: "('a::euclidean_space) set"
```
```   801   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and>
```
```   802   (card B = dim V) \<and> pairwise orthogonal B"
```
```   803 proof -
```
```   804   from basis_exists[of V] obtain B where
```
```   805     B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
```
```   806     by blast
```
```   807   from B have fB: "finite B" "card B = dim V"
```
```   808     using independent_bound by auto
```
```   809   from basis_orthogonal[OF fB(1)] obtain C where
```
```   810     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
```
```   811     by blast
```
```   812   from C B have CSV: "C \<subseteq> span V"
```
```   813     by (metis span_superset span_mono subset_trans)
```
```   814   from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
```
```   815     by (simp add: span_span)
```
```   816   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
```
```   817   have iC: "independent C"
```
```   818     by (simp add: dim_span)
```
```   819   from C fB have "card C \<le> dim V"
```
```   820     by simp
```
```   821   moreover have "dim V \<le> card C"
```
```   822     using span_card_ge_dim[OF CSV SVC C(1)]
```
```   823     by simp
```
```   824   ultimately have CdV: "card C = dim V"
```
```   825     using C(1) by simp
```
```   826   from C B CSV CdV iC show ?thesis
```
```   827     by auto
```
```   828 qed
```
```   829
```
```   830 text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
```
```   831
```
```   832 lemma span_not_univ_orthogonal:
```
```   833   fixes S :: "'a::euclidean_space set"
```
```   834   assumes sU: "span S \<noteq> UNIV"
```
```   835   shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
```
```   836 proof -
```
```   837   from sU obtain a where a: "a \<notin> span S"
```
```   838     by blast
```
```   839   from orthogonal_basis_exists obtain B where
```
```   840     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B"
```
```   841     "card B = dim S" "pairwise orthogonal B"
```
```   842     by blast
```
```   843   from B have fB: "finite B" "card B = dim S"
```
```   844     using independent_bound by auto
```
```   845   from span_mono[OF B(2)] span_mono[OF B(3)]
```
```   846   have sSB: "span S = span B"
```
```   847     by (simp add: span_span)
```
```   848   let ?a = "a - sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
```
```   849   have "sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
```
```   850     unfolding sSB
```
```   851     apply (rule span_sum)
```
```   852     apply (rule span_scale)
```
```   853     apply (rule span_base)
```
```   854     apply assumption
```
```   855     done
```
```   856   with a have a0:"?a  \<noteq> 0"
```
```   857     by auto
```
```   858   have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
```
```   859   proof (rule span_induct')
```
```   860     show "subspace {x. ?a \<bullet> x = 0}"
```
```   861       by (auto simp add: subspace_def inner_add)
```
```   862   next
```
```   863     {
```
```   864       fix x
```
```   865       assume x: "x \<in> B"
```
```   866       from x have B': "B = insert x (B - {x})"
```
```   867         by blast
```
```   868       have fth: "finite (B - {x})"
```
```   869         using fB by simp
```
```   870       have "?a \<bullet> x = 0"
```
```   871         apply (subst B')
```
```   872         using fB fth
```
```   873         unfolding sum_clauses(2)[OF fth]
```
```   874         apply simp unfolding inner_simps
```
```   875         apply (clarsimp simp add: inner_add inner_sum_left)
```
```   876         apply (rule sum.neutral, rule ballI)
```
```   877         apply (simp only: inner_commute)
```
```   878         apply (auto simp add: x field_simps
```
```   879           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
```
```   880         done
```
```   881     }
```
```   882     then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
```
```   883       by blast
```
```   884   qed
```
```   885   with a0 show ?thesis
```
```   886     unfolding sSB by (auto intro: exI[where x="?a"])
```
```   887 qed
```
```   888
```
```   889 lemma span_not_univ_subset_hyperplane:
```
```   890   fixes S :: "'a::euclidean_space set"
```
```   891   assumes SU: "span S \<noteq> UNIV"
```
```   892   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```   893   using span_not_univ_orthogonal[OF SU] by auto
```
```   894
```
```   895 lemma lowdim_subset_hyperplane:
```
```   896   fixes S :: "'a::euclidean_space set"
```
```   897   assumes d: "dim S < DIM('a)"
```
```   898   shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```   899 proof -
```
```   900   {
```
```   901     assume "span S = UNIV"
```
```   902     then have "dim (span S) = dim (UNIV :: ('a) set)"
```
```   903       by simp
```
```   904     then have "dim S = DIM('a)"
```
```   905       by (metis Euclidean_Space.dim_UNIV dim_span)
```
```   906     with d have False by arith
```
```   907   }
```
```   908   then have th: "span S \<noteq> UNIV"
```
```   909     by blast
```
```   910   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
```
```   911 qed
```
```   912
```
```   913 lemma linear_eq_stdbasis:
```
```   914   fixes f :: "'a::euclidean_space \<Rightarrow> _"
```
```   915   assumes lf: "linear f"
```
```   916     and lg: "linear g"
```
```   917     and fg: "\<And>b. b \<in> Basis \<Longrightarrow> f b = g b"
```
```   918   shows "f = g"
```
```   919   using linear_eq_on_span[OF lf lg, of Basis] fg
```
```   920   by auto
```
```   921
```
```   922
```
```   923 text \<open>Similar results for bilinear functions.\<close>
```
```   924
```
```   925 lemma bilinear_eq:
```
```   926   assumes bf: "bilinear f"
```
```   927     and bg: "bilinear g"
```
```   928     and SB: "S \<subseteq> span B"
```
```   929     and TC: "T \<subseteq> span C"
```
```   930     and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
```
```   931   shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
```
```   932 proof -
```
```   933   let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
```
```   934   from bf bg have sp: "subspace ?P"
```
```   935     unfolding bilinear_def linear_iff subspace_def bf bg
```
```   936     by (auto simp add: span_zero bilinear_lzero[OF bf] bilinear_lzero[OF bg]
```
```   937         span_add Ball_def
```
```   938       intro: bilinear_ladd[OF bf])
```
```   939
```
```   940   have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
```
```   941     apply (rule span_induct' [OF _ sp])
```
```   942     apply (rule ballI)
```
```   943     apply (rule span_induct')
```
```   944     apply (simp add: fg)
```
```   945     apply (auto simp add: subspace_def)
```
```   946     using bf bg unfolding bilinear_def linear_iff
```
```   947       apply (auto simp add: span_zero bilinear_rzero[OF bf] bilinear_rzero[OF bg]
```
```   948         span_add Ball_def
```
```   949       intro: bilinear_ladd[OF bf])
```
```   950     done
```
```   951   then show ?thesis
```
```   952     using SB TC by auto
```
```   953 qed
```
```   954
```
```   955 lemma bilinear_eq_stdbasis:
```
```   956   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
```
```   957   assumes bf: "bilinear f"
```
```   958     and bg: "bilinear g"
```
```   959     and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
```
```   960   shows "f = g"
```
```   961   using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast
```
```   962
```
```   963 subsection \<open>Infinity norm\<close>
```
```   964
```
```   965 definition%important "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
```
```   966
```
```   967 lemma infnorm_set_image:
```
```   968   fixes x :: "'a::euclidean_space"
```
```   969   shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```   970   by blast
```
```   971
```
```   972 lemma infnorm_Max:
```
```   973   fixes x :: "'a::euclidean_space"
```
```   974   shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)"
```
```   975   by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)
```
```   976
```
```   977 lemma infnorm_set_lemma:
```
```   978   fixes x :: "'a::euclidean_space"
```
```   979   shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
```
```   980     and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}"
```
```   981   unfolding infnorm_set_image
```
```   982   by auto
```
```   983
```
```   984 lemma infnorm_pos_le:
```
```   985   fixes x :: "'a::euclidean_space"
```
```   986   shows "0 \<le> infnorm x"
```
```   987   by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
```
```   988
```
```   989 lemma infnorm_triangle:
```
```   990   fixes x :: "'a::euclidean_space"
```
```   991   shows "infnorm (x + y) \<le> infnorm x + infnorm y"
```
```   992 proof -
```
```   993   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"
```
```   994     by simp
```
```   995   show ?thesis
```
```   996     by (auto simp: infnorm_Max inner_add_left intro!: *)
```
```   997 qed
```
```   998
```
```   999 lemma infnorm_eq_0:
```
```  1000   fixes x :: "'a::euclidean_space"
```
```  1001   shows "infnorm x = 0 \<longleftrightarrow> x = 0"
```
```  1002 proof -
```
```  1003   have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
```
```  1004     unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
```
```  1005   then show ?thesis
```
```  1006     using infnorm_pos_le[of x] by simp
```
```  1007 qed
```
```  1008
```
```  1009 lemma infnorm_0: "infnorm 0 = 0"
```
```  1010   by (simp add: infnorm_eq_0)
```
```  1011
```
```  1012 lemma infnorm_neg: "infnorm (- x) = infnorm x"
```
```  1013   unfolding infnorm_def
```
```  1014   apply (rule cong[of "Sup" "Sup"])
```
```  1015   apply blast
```
```  1016   apply auto
```
```  1017   done
```
```  1018
```
```  1019 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
```
```  1020 proof -
```
```  1021   have "y - x = - (x - y)" by simp
```
```  1022   then show ?thesis
```
```  1023     by (metis infnorm_neg)
```
```  1024 qed
```
```  1025
```
```  1026 lemma real_abs_sub_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
```
```  1027 proof -
```
```  1028   have th: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
```
```  1029     by arith
```
```  1030   from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
```
```  1031   have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
```
```  1032     "infnorm y \<le> infnorm (x - y) + infnorm x"
```
```  1033     by (simp_all add: field_simps infnorm_neg)
```
```  1034   from th[OF ths] show ?thesis .
```
```  1035 qed
```
```  1036
```
```  1037 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
```
```  1038   using infnorm_pos_le[of x] by arith
```
```  1039
```
```  1040 lemma Basis_le_infnorm:
```
```  1041   fixes x :: "'a::euclidean_space"
```
```  1042   shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
```
```  1043   by (simp add: infnorm_Max)
```
```  1044
```
```  1045 lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
```
```  1046   unfolding infnorm_Max
```
```  1047 proof (safe intro!: Max_eqI)
```
```  1048   let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```  1049   {
```
```  1050     fix b :: 'a
```
```  1051     assume "b \<in> Basis"
```
```  1052     then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
```
```  1053       by (simp add: abs_mult mult_left_mono)
```
```  1054   next
```
```  1055     from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
```
```  1056       by (auto simp del: Max_in)
```
```  1057     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"
```
```  1058       by (intro image_eqI[where x=b]) (auto simp: abs_mult)
```
```  1059   }
```
```  1060 qed simp
```
```  1061
```
```  1062 lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
```
```  1063   unfolding infnorm_mul ..
```
```  1064
```
```  1065 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
```
```  1066   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
```
```  1067
```
```  1068 text \<open>Prove that it differs only up to a bound from Euclidean norm.\<close>
```
```  1069
```
```  1070 lemma infnorm_le_norm: "infnorm x \<le> norm x"
```
```  1071   by (simp add: Basis_le_norm infnorm_Max)
```
```  1072
```
```  1073 lemma norm_le_infnorm:
```
```  1074   fixes x :: "'a::euclidean_space"
```
```  1075   shows "norm x \<le> sqrt DIM('a) * infnorm x"
```
```  1076 proof -
```
```  1077   let ?d = "DIM('a)"
```
```  1078   have "real ?d \<ge> 0"
```
```  1079     by simp
```
```  1080   then have d2: "(sqrt (real ?d))\<^sup>2 = real ?d"
```
```  1081     by (auto intro: real_sqrt_pow2)
```
```  1082   have th: "sqrt (real ?d) * infnorm x \<ge> 0"
```
```  1083     by (simp add: zero_le_mult_iff infnorm_pos_le)
```
```  1084   have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)\<^sup>2"
```
```  1085     unfolding power_mult_distrib d2
```
```  1086     apply (subst euclidean_inner)
```
```  1087     apply (subst power2_abs[symmetric])
```
```  1088     apply (rule order_trans[OF sum_bounded_above[where K="\<bar>infnorm x\<bar>\<^sup>2"]])
```
```  1089     apply (auto simp add: power2_eq_square[symmetric])
```
```  1090     apply (subst power2_abs[symmetric])
```
```  1091     apply (rule power_mono)
```
```  1092     apply (auto simp: infnorm_Max)
```
```  1093     done
```
```  1094   from real_le_lsqrt[OF inner_ge_zero th th1]
```
```  1095   show ?thesis
```
```  1096     unfolding norm_eq_sqrt_inner id_def .
```
```  1097 qed
```
```  1098
```
```  1099 lemma tendsto_infnorm [tendsto_intros]:
```
```  1100   assumes "(f \<longlongrightarrow> a) F"
```
```  1101   shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F"
```
```  1102 proof (rule tendsto_compose [OF LIM_I assms])
```
```  1103   fix r :: real
```
```  1104   assume "r > 0"
```
```  1105   then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
```
```  1106     by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm)
```
```  1107 qed
```
```  1108
```
```  1109 text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close>
```
```  1110
```
```  1111 lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  1112   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1113 proof -
```
```  1114   {
```
```  1115     assume h: "x = 0"
```
```  1116     then have ?thesis by simp
```
```  1117   }
```
```  1118   moreover
```
```  1119   {
```
```  1120     assume h: "y = 0"
```
```  1121     then have ?thesis by simp
```
```  1122   }
```
```  1123   moreover
```
```  1124   {
```
```  1125     assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
```
```  1126     from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
```
```  1127     have "?rhs \<longleftrightarrow>
```
```  1128       (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
```
```  1129         norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
```
```  1130       using x y
```
```  1131       unfolding inner_simps
```
```  1132       unfolding power2_norm_eq_inner[symmetric] power2_eq_square right_minus_eq
```
```  1133       apply (simp add: inner_commute)
```
```  1134       apply (simp add: field_simps)
```
```  1135       apply metis
```
```  1136       done
```
```  1137     also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
```
```  1138       by (simp add: field_simps inner_commute)
```
```  1139     also have "\<dots> \<longleftrightarrow> ?lhs" using x y
```
```  1140       apply simp
```
```  1141       apply metis
```
```  1142       done
```
```  1143     finally have ?thesis by blast
```
```  1144   }
```
```  1145   ultimately show ?thesis by blast
```
```  1146 qed
```
```  1147
```
```  1148 lemma norm_cauchy_schwarz_abs_eq:
```
```  1149   "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow>
```
```  1150     norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
```
```  1151   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1152 proof -
```
```  1153   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a"
```
```  1154     by arith
```
```  1155   have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
```
```  1156     by simp
```
```  1157   also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
```
```  1158     unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  1159     unfolding norm_minus_cancel norm_scaleR ..
```
```  1160   also have "\<dots> \<longleftrightarrow> ?lhs"
```
```  1161     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
```
```  1162     by auto
```
```  1163   finally show ?thesis ..
```
```  1164 qed
```
```  1165
```
```  1166 lemma norm_triangle_eq:
```
```  1167   fixes x y :: "'a::real_inner"
```
```  1168   shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  1169 proof -
```
```  1170   {
```
```  1171     assume x: "x = 0 \<or> y = 0"
```
```  1172     then have ?thesis
```
```  1173       by (cases "x = 0") simp_all
```
```  1174   }
```
```  1175   moreover
```
```  1176   {
```
```  1177     assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
```
```  1178     then have "norm x \<noteq> 0" "norm y \<noteq> 0"
```
```  1179       by simp_all
```
```  1180     then have n: "norm x > 0" "norm y > 0"
```
```  1181       using norm_ge_zero[of x] norm_ge_zero[of y] by arith+
```
```  1182     have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 \<Longrightarrow> a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2"
```
```  1183       by algebra
```
```  1184     have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
```
```  1185       apply (rule th)
```
```  1186       using n norm_ge_zero[of "x + y"]
```
```  1187       apply arith
```
```  1188       done
```
```  1189     also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  1190       unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  1191       unfolding power2_norm_eq_inner inner_simps
```
```  1192       by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
```
```  1193     finally have ?thesis .
```
```  1194   }
```
```  1195   ultimately show ?thesis by blast
```
```  1196 qed
```
```  1197
```
```  1198
```
```  1199 subsection \<open>Collinearity\<close>
```
```  1200
```
```  1201 definition%important collinear :: "'a::real_vector set \<Rightarrow> bool"
```
```  1202   where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
```
```  1203
```
```  1204 lemma collinear_alt:
```
```  1205      "collinear S \<longleftrightarrow> (\<exists>u v. \<forall>x \<in> S. \<exists>c. x = u + c *\<^sub>R v)" (is "?lhs = ?rhs")
```
```  1206 proof
```
```  1207   assume ?lhs
```
```  1208   then show ?rhs
```
```  1209     unfolding collinear_def by (metis Groups.add_ac(2) diff_add_cancel)
```
```  1210 next
```
```  1211   assume ?rhs
```
```  1212   then obtain u v where *: "\<And>x. x \<in> S \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
```
```  1213     by (auto simp: )
```
```  1214   have "\<exists>c. x - y = c *\<^sub>R v" if "x \<in> S" "y \<in> S" for x y
```
```  1215         by (metis *[OF \<open>x \<in> S\<close>] *[OF \<open>y \<in> S\<close>] scaleR_left.diff add_diff_cancel_left)
```
```  1216   then show ?lhs
```
```  1217     using collinear_def by blast
```
```  1218 qed
```
```  1219
```
```  1220 lemma collinear:
```
```  1221   fixes S :: "'a::{perfect_space,real_vector} set"
```
```  1222   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))"
```
```  1223 proof -
```
```  1224   have "\<exists>v. v \<noteq> 0 \<and> (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v)"
```
```  1225     if "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R u" "u=0" for u
```
```  1226   proof -
```
```  1227     have "\<forall>x\<in>S. \<forall>y\<in>S. x = y"
```
```  1228       using that by auto
```
```  1229     moreover
```
```  1230     obtain v::'a where "v \<noteq> 0"
```
```  1231       using UNIV_not_singleton [of 0] by auto
```
```  1232     ultimately have "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v"
```
```  1233       by auto
```
```  1234     then show ?thesis
```
```  1235       using \<open>v \<noteq> 0\<close> by blast
```
```  1236   qed
```
```  1237   then show ?thesis
```
```  1238     apply (clarsimp simp: collinear_def)
```
```  1239     by (metis scaleR_zero_right vector_fraction_eq_iff)
```
```  1240 qed
```
```  1241
```
```  1242 lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S"
```
```  1243   by (meson collinear_def subsetCE)
```
```  1244
```
```  1245 lemma collinear_empty [iff]: "collinear {}"
```
```  1246   by (simp add: collinear_def)
```
```  1247
```
```  1248 lemma collinear_sing [iff]: "collinear {x}"
```
```  1249   by (simp add: collinear_def)
```
```  1250
```
```  1251 lemma collinear_2 [iff]: "collinear {x, y}"
```
```  1252   apply (simp add: collinear_def)
```
```  1253   apply (rule exI[where x="x - y"])
```
```  1254   apply auto
```
```  1255   apply (rule exI[where x=1], simp)
```
```  1256   apply (rule exI[where x="- 1"], simp)
```
```  1257   done
```
```  1258
```
```  1259 lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
```
```  1260   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1261 proof -
```
```  1262   {
```
```  1263     assume "x = 0 \<or> y = 0"
```
```  1264     then have ?thesis
```
```  1265       by (cases "x = 0") (simp_all add: collinear_2 insert_commute)
```
```  1266   }
```
```  1267   moreover
```
```  1268   {
```
```  1269     assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
```
```  1270     have ?thesis
```
```  1271     proof
```
```  1272       assume h: "?lhs"
```
```  1273       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"
```
```  1274         unfolding collinear_def by blast
```
```  1275       from u[rule_format, of x 0] u[rule_format, of y 0]
```
```  1276       obtain cx and cy where
```
```  1277         cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
```
```  1278         by auto
```
```  1279       from cx x have cx0: "cx \<noteq> 0" by auto
```
```  1280       from cy y have cy0: "cy \<noteq> 0" by auto
```
```  1281       let ?d = "cy / cx"
```
```  1282       from cx cy cx0 have "y = ?d *\<^sub>R x"
```
```  1283         by simp
```
```  1284       then show ?rhs using x y by blast
```
```  1285     next
```
```  1286       assume h: "?rhs"
```
```  1287       then obtain c where c: "y = c *\<^sub>R x"
```
```  1288         using x y by blast
```
```  1289       show ?lhs
```
```  1290         unfolding collinear_def c
```
```  1291         apply (rule exI[where x=x])
```
```  1292         apply auto
```
```  1293         apply (rule exI[where x="- 1"], simp)
```
```  1294         apply (rule exI[where x= "-c"], simp)
```
```  1295         apply (rule exI[where x=1], simp)
```
```  1296         apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
```
```  1297         apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
```
```  1298         done
```
```  1299     qed
```
```  1300   }
```
```  1301   ultimately show ?thesis by blast
```
```  1302 qed
```
```  1303
```
```  1304 lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
```
```  1305   unfolding norm_cauchy_schwarz_abs_eq
```
```  1306   apply (cases "x=0", simp_all)
```
```  1307   apply (cases "y=0", simp_all add: insert_commute)
```
```  1308   unfolding collinear_lemma
```
```  1309   apply simp
```
```  1310   apply (subgoal_tac "norm x \<noteq> 0")
```
```  1311   apply (subgoal_tac "norm y \<noteq> 0")
```
```  1312   apply (rule iffI)
```
```  1313   apply (cases "norm x *\<^sub>R y = norm y *\<^sub>R x")
```
```  1314   apply (rule exI[where x="(1/norm x) * norm y"])
```
```  1315   apply (drule sym)
```
```  1316   unfolding scaleR_scaleR[symmetric]
```
```  1317   apply (simp add: field_simps)
```
```  1318   apply (rule exI[where x="(1/norm x) * - norm y"])
```
```  1319   apply clarify
```
```  1320   apply (drule sym)
```
```  1321   unfolding scaleR_scaleR[symmetric]
```
```  1322   apply (simp add: field_simps)
```
```  1323   apply (erule exE)
```
```  1324   apply (erule ssubst)
```
```  1325   unfolding scaleR_scaleR
```
```  1326   unfolding norm_scaleR
```
```  1327   apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
```
```  1328   apply (auto simp add: field_simps)
```
```  1329   done
```
```  1330
```
```  1331 end
```