src/HOL/Multivariate_Analysis/Linear_Algebra.thy
 author haftmann Sun Mar 16 18:09:04 2014 +0100 (2014-03-16) changeset 56166 9a241bc276cd parent 55910 0a756571c7a4 child 56196 32b7eafc5a52 permissions -rw-r--r--
normalising simp rules for compound operators
```     1 (*  Title:      HOL/Multivariate_Analysis/Linear_Algebra.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3 *)
```
```     4
```
```     5 header {* Elementary linear algebra on Euclidean spaces *}
```
```     6
```
```     7 theory Linear_Algebra
```
```     8 imports
```
```     9   Euclidean_Space
```
```    10   "~~/src/HOL/Library/Infinite_Set"
```
```    11 begin
```
```    12
```
```    13 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
```
```    14   by auto
```
```    15
```
```    16 notation inner (infix "\<bullet>" 70)
```
```    17
```
```    18 lemma square_bound_lemma:
```
```    19   fixes x :: real
```
```    20   shows "x < (1 + x) * (1 + x)"
```
```    21 proof -
```
```    22   have "(x + 1/2)\<^sup>2 + 3/4 > 0"
```
```    23     using zero_le_power2[of "x+1/2"] by arith
```
```    24   then show ?thesis
```
```    25     by (simp add: field_simps power2_eq_square)
```
```    26 qed
```
```    27
```
```    28 lemma square_continuous:
```
```    29   fixes e :: real
```
```    30   shows "e > 0 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. abs (y - x) < d \<longrightarrow> abs (y * y - x * x) < e)"
```
```    31   using isCont_power[OF isCont_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
```
```    32   apply (auto simp add: power2_eq_square)
```
```    33   apply (rule_tac x="s" in exI)
```
```    34   apply auto
```
```    35   apply (erule_tac x=y in allE)
```
```    36   apply auto
```
```    37   done
```
```    38
```
```    39 text{* Hence derive more interesting properties of the norm. *}
```
```    40
```
```    41 lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
```
```    42   by simp (* TODO: delete *)
```
```    43
```
```    44 lemma norm_triangle_sub:
```
```    45   fixes x y :: "'a::real_normed_vector"
```
```    46   shows "norm x \<le> norm y + norm (x - y)"
```
```    47   using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
```
```    48
```
```    49 lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
```
```    50   by (simp add: norm_eq_sqrt_inner)
```
```    51
```
```    52 lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
```
```    53   by (simp add: norm_eq_sqrt_inner)
```
```    54
```
```    55 lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
```
```    56   apply (subst order_eq_iff)
```
```    57   apply (auto simp: norm_le)
```
```    58   done
```
```    59
```
```    60 lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
```
```    61   by (simp add: norm_eq_sqrt_inner)
```
```    62
```
```    63 text{* Squaring equations and inequalities involving norms.  *}
```
```    64
```
```    65 lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
```
```    66   by (simp only: power2_norm_eq_inner) (* TODO: move? *)
```
```    67
```
```    68 lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
```
```    69   by (auto simp add: norm_eq_sqrt_inner)
```
```    70
```
```    71 lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)\<^sup>2 \<le> y\<^sup>2"
```
```    72 proof
```
```    73   assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
```
```    74   then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
```
```    75   then show "x\<^sup>2 \<le> y\<^sup>2" by simp
```
```    76 next
```
```    77   assume "x\<^sup>2 \<le> y\<^sup>2"
```
```    78   then have "sqrt (x\<^sup>2) \<le> sqrt (y\<^sup>2)" by (rule real_sqrt_le_mono)
```
```    79   then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
```
```    80 qed
```
```    81
```
```    82 lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2"
```
```    83   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
```
```    84   using norm_ge_zero[of x]
```
```    85   apply arith
```
```    86   done
```
```    87
```
```    88 lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2"
```
```    89   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
```
```    90   using norm_ge_zero[of x]
```
```    91   apply arith
```
```    92   done
```
```    93
```
```    94 lemma norm_lt_square: "norm x < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2"
```
```    95   by (metis not_le norm_ge_square)
```
```    96
```
```    97 lemma norm_gt_square: "norm x > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2"
```
```    98   by (metis norm_le_square not_less)
```
```    99
```
```   100 text{* Dot product in terms of the norm rather than conversely. *}
```
```   101
```
```   102 lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
```
```   103   inner_scaleR_left inner_scaleR_right
```
```   104
```
```   105 lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
```
```   106   unfolding power2_norm_eq_inner inner_simps inner_commute by auto
```
```   107
```
```   108 lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
```
```   109   unfolding power2_norm_eq_inner inner_simps inner_commute
```
```   110   by (auto simp add: algebra_simps)
```
```   111
```
```   112 text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
```
```   113
```
```   114 lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
```
```   115   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   116 proof
```
```   117   assume ?lhs
```
```   118   then show ?rhs by simp
```
```   119 next
```
```   120   assume ?rhs
```
```   121   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
```
```   122     by simp
```
```   123   then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
```
```   124     by (simp add: inner_diff inner_commute)
```
```   125   then have "(x - y) \<bullet> (x - y) = 0"
```
```   126     by (simp add: field_simps inner_diff inner_commute)
```
```   127   then show "x = y" by simp
```
```   128 qed
```
```   129
```
```   130 lemma norm_triangle_half_r:
```
```   131   "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
```
```   132   using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
```
```   133
```
```   134 lemma norm_triangle_half_l:
```
```   135   assumes "norm (x - y) < e / 2"
```
```   136     and "norm (x' - y) < e / 2"
```
```   137   shows "norm (x - x') < e"
```
```   138   using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
```
```   139   unfolding dist_norm[symmetric] .
```
```   140
```
```   141 lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
```
```   142   by (rule norm_triangle_ineq [THEN order_trans])
```
```   143
```
```   144 lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
```
```   145   by (rule norm_triangle_ineq [THEN le_less_trans])
```
```   146
```
```   147 lemma setsum_clauses:
```
```   148   shows "setsum f {} = 0"
```
```   149     and "finite S \<Longrightarrow> setsum f (insert x S) = (if x \<in> S then setsum f S else f x + setsum f S)"
```
```   150   by (auto simp add: insert_absorb)
```
```   151
```
```   152 lemma setsum_norm_le:
```
```   153   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   154   assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
```
```   155   shows "norm (setsum f S) \<le> setsum g S"
```
```   156   by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)
```
```   157
```
```   158 lemma setsum_norm_bound:
```
```   159   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   160   assumes fS: "finite S"
```
```   161     and K: "\<forall>x \<in> S. norm (f x) \<le> K"
```
```   162   shows "norm (setsum f S) \<le> of_nat (card S) * K"
```
```   163   using setsum_norm_le[OF K] setsum_constant[symmetric]
```
```   164   by simp
```
```   165
```
```   166 lemma setsum_group:
```
```   167   assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
```
```   168   shows "setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) T = setsum g S"
```
```   169   apply (subst setsum_image_gen[OF fS, of g f])
```
```   170   apply (rule setsum_mono_zero_right[OF fT fST])
```
```   171   apply (auto intro: setsum_0')
```
```   172   done
```
```   173
```
```   174 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
```
```   175 proof
```
```   176   assume "\<forall>x. x \<bullet> y = x \<bullet> z"
```
```   177   then have "\<forall>x. x \<bullet> (y - z) = 0"
```
```   178     by (simp add: inner_diff)
```
```   179   then have "(y - z) \<bullet> (y - z) = 0" ..
```
```   180   then show "y = z" by simp
```
```   181 qed simp
```
```   182
```
```   183 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
```
```   184 proof
```
```   185   assume "\<forall>z. x \<bullet> z = y \<bullet> z"
```
```   186   then have "\<forall>z. (x - y) \<bullet> z = 0"
```
```   187     by (simp add: inner_diff)
```
```   188   then have "(x - y) \<bullet> (x - y) = 0" ..
```
```   189   then show "x = y" by simp
```
```   190 qed simp
```
```   191
```
```   192
```
```   193 subsection {* Orthogonality. *}
```
```   194
```
```   195 context real_inner
```
```   196 begin
```
```   197
```
```   198 definition "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
```
```   199
```
```   200 lemma orthogonal_clauses:
```
```   201   "orthogonal a 0"
```
```   202   "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
```
```   203   "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
```
```   204   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
```
```   205   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
```
```   206   "orthogonal 0 a"
```
```   207   "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
```
```   208   "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
```
```   209   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
```
```   210   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
```
```   211   unfolding orthogonal_def inner_add inner_diff by auto
```
```   212
```
```   213 end
```
```   214
```
```   215 lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
```
```   216   by (simp add: orthogonal_def inner_commute)
```
```   217
```
```   218
```
```   219 subsection {* Linear functions. *}
```
```   220
```
```   221 lemma linear_iff:
```
```   222   "linear f \<longleftrightarrow> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>R x) = c *\<^sub>R f x)"
```
```   223   (is "linear f \<longleftrightarrow> ?rhs")
```
```   224 proof
```
```   225   assume "linear f" then interpret f: linear f .
```
```   226   show "?rhs" by (simp add: f.add f.scaleR)
```
```   227 next
```
```   228   assume "?rhs" then show "linear f" by unfold_locales simp_all
```
```   229 qed
```
```   230
```
```   231 lemma linear_compose_cmul: "linear f \<Longrightarrow> linear (\<lambda>x. c *\<^sub>R f x)"
```
```   232   by (simp add: linear_iff algebra_simps)
```
```   233
```
```   234 lemma linear_compose_neg: "linear f \<Longrightarrow> linear (\<lambda>x. - f x)"
```
```   235   by (simp add: linear_iff)
```
```   236
```
```   237 lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x + g x)"
```
```   238   by (simp add: linear_iff algebra_simps)
```
```   239
```
```   240 lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x - g x)"
```
```   241   by (simp add: linear_iff algebra_simps)
```
```   242
```
```   243 lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> f)"
```
```   244   by (simp add: linear_iff)
```
```   245
```
```   246 lemma linear_id: "linear id"
```
```   247   by (simp add: linear_iff id_def)
```
```   248
```
```   249 lemma linear_zero: "linear (\<lambda>x. 0)"
```
```   250   by (simp add: linear_iff)
```
```   251
```
```   252 lemma linear_compose_setsum:
```
```   253   assumes fS: "finite S"
```
```   254     and lS: "\<forall>a \<in> S. linear (f a)"
```
```   255   shows "linear (\<lambda>x. setsum (\<lambda>a. f a x) S)"
```
```   256   using lS
```
```   257   apply (induct rule: finite_induct[OF fS])
```
```   258   apply (auto simp add: linear_zero intro: linear_compose_add)
```
```   259   done
```
```   260
```
```   261 lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
```
```   262   unfolding linear_iff
```
```   263   apply clarsimp
```
```   264   apply (erule allE[where x="0::'a"])
```
```   265   apply simp
```
```   266   done
```
```   267
```
```   268 lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```   269   by (simp add: linear_iff)
```
```   270
```
```   271 lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x"
```
```   272   using linear_cmul [where c="-1"] by simp
```
```   273
```
```   274 lemma linear_add: "linear f \<Longrightarrow> f (x + y) = f x + f y"
```
```   275   by (metis linear_iff)
```
```   276
```
```   277 lemma linear_sub: "linear f \<Longrightarrow> f (x - y) = f x - f y"
```
```   278   using linear_add [of f x "- y"] by (simp add: linear_neg)
```
```   279
```
```   280 lemma linear_setsum:
```
```   281   assumes lin: "linear f"
```
```   282     and fin: "finite S"
```
```   283   shows "f (setsum g S) = setsum (f \<circ> g) S"
```
```   284   using fin
```
```   285 proof induct
```
```   286   case empty
```
```   287   then show ?case
```
```   288     by (simp add: linear_0[OF lin])
```
```   289 next
```
```   290   case (insert x F)
```
```   291   have "f (setsum g (insert x F)) = f (g x + setsum g F)"
```
```   292     using insert.hyps by simp
```
```   293   also have "\<dots> = f (g x) + f (setsum g F)"
```
```   294     using linear_add[OF lin] by simp
```
```   295   also have "\<dots> = setsum (f \<circ> g) (insert x F)"
```
```   296     using insert.hyps by simp
```
```   297   finally show ?case .
```
```   298 qed
```
```   299
```
```   300 lemma linear_setsum_mul:
```
```   301   assumes lin: "linear f"
```
```   302     and fin: "finite S"
```
```   303   shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
```
```   304   using linear_setsum[OF lin fin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
```
```   305   by simp
```
```   306
```
```   307 lemma linear_injective_0:
```
```   308   assumes lin: "linear f"
```
```   309   shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
```
```   310 proof -
```
```   311   have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)"
```
```   312     by (simp add: inj_on_def)
```
```   313   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)"
```
```   314     by simp
```
```   315   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
```
```   316     by (simp add: linear_sub[OF lin])
```
```   317   also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
```
```   318     by auto
```
```   319   finally show ?thesis .
```
```   320 qed
```
```   321
```
```   322
```
```   323 subsection {* Bilinear functions. *}
```
```   324
```
```   325 definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
```
```   326
```
```   327 lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
```
```   328   by (simp add: bilinear_def linear_iff)
```
```   329
```
```   330 lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
```
```   331   by (simp add: bilinear_def linear_iff)
```
```   332
```
```   333 lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
```
```   334   by (simp add: bilinear_def linear_iff)
```
```   335
```
```   336 lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
```
```   337   by (simp add: bilinear_def linear_iff)
```
```   338
```
```   339 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
```
```   340   by (drule bilinear_lmul [of _ "- 1"]) simp
```
```   341
```
```   342 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
```
```   343   by (drule bilinear_rmul [of _ _ "- 1"]) simp
```
```   344
```
```   345 lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
```
```   346   using add_imp_eq[of x y 0] by auto
```
```   347
```
```   348 lemma bilinear_lzero:
```
```   349   assumes "bilinear h"
```
```   350   shows "h 0 x = 0"
```
```   351   using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
```
```   352
```
```   353 lemma bilinear_rzero:
```
```   354   assumes "bilinear h"
```
```   355   shows "h x 0 = 0"
```
```   356   using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
```
```   357
```
```   358 lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
```
```   359   using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
```
```   360
```
```   361 lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
```
```   362   using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
```
```   363
```
```   364 lemma bilinear_setsum:
```
```   365   assumes bh: "bilinear h"
```
```   366     and fS: "finite S"
```
```   367     and fT: "finite T"
```
```   368   shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
```
```   369 proof -
```
```   370   have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
```
```   371     apply (rule linear_setsum[unfolded o_def])
```
```   372     using bh fS
```
```   373     apply (auto simp add: bilinear_def)
```
```   374     done
```
```   375   also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
```
```   376     apply (rule setsum_cong, simp)
```
```   377     apply (rule linear_setsum[unfolded o_def])
```
```   378     using bh fT
```
```   379     apply (auto simp add: bilinear_def)
```
```   380     done
```
```   381   finally show ?thesis
```
```   382     unfolding setsum_cartesian_product .
```
```   383 qed
```
```   384
```
```   385
```
```   386 subsection {* Adjoints. *}
```
```   387
```
```   388 definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
```
```   389
```
```   390 lemma adjoint_unique:
```
```   391   assumes "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   392   shows "adjoint f = g"
```
```   393   unfolding adjoint_def
```
```   394 proof (rule some_equality)
```
```   395   show "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   396     by (rule assms)
```
```   397 next
```
```   398   fix h
```
```   399   assume "\<forall>x y. inner (f x) y = inner x (h y)"
```
```   400   then have "\<forall>x y. inner x (g y) = inner x (h y)"
```
```   401     using assms by simp
```
```   402   then have "\<forall>x y. inner x (g y - h y) = 0"
```
```   403     by (simp add: inner_diff_right)
```
```   404   then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
```
```   405     by simp
```
```   406   then have "\<forall>y. h y = g y"
```
```   407     by simp
```
```   408   then show "h = g" by (simp add: ext)
```
```   409 qed
```
```   410
```
```   411 text {* TODO: The following lemmas about adjoints should hold for any
```
```   412 Hilbert space (i.e. complete inner product space).
```
```   413 (see @{url "http://en.wikipedia.org/wiki/Hermitian_adjoint"})
```
```   414 *}
```
```   415
```
```   416 lemma adjoint_works:
```
```   417   fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   418   assumes lf: "linear f"
```
```   419   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```   420 proof -
```
```   421   have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
```
```   422   proof (intro allI exI)
```
```   423     fix y :: "'m" and x
```
```   424     let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
```
```   425     have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
```
```   426       by (simp add: euclidean_representation)
```
```   427     also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
```
```   428       unfolding linear_setsum[OF lf finite_Basis]
```
```   429       by (simp add: linear_cmul[OF lf])
```
```   430     finally show "f x \<bullet> y = x \<bullet> ?w"
```
```   431       by (simp add: inner_setsum_left inner_setsum_right mult_commute)
```
```   432   qed
```
```   433   then show ?thesis
```
```   434     unfolding adjoint_def choice_iff
```
```   435     by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
```
```   436 qed
```
```   437
```
```   438 lemma adjoint_clauses:
```
```   439   fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   440   assumes lf: "linear f"
```
```   441   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```   442     and "adjoint f y \<bullet> x = y \<bullet> f x"
```
```   443   by (simp_all add: adjoint_works[OF lf] inner_commute)
```
```   444
```
```   445 lemma adjoint_linear:
```
```   446   fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   447   assumes lf: "linear f"
```
```   448   shows "linear (adjoint f)"
```
```   449   by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
```
```   450     adjoint_clauses[OF lf] inner_distrib)
```
```   451
```
```   452 lemma adjoint_adjoint:
```
```   453   fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   454   assumes lf: "linear f"
```
```   455   shows "adjoint (adjoint f) = f"
```
```   456   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
```
```   457
```
```   458
```
```   459 subsection {* Interlude: Some properties of real sets *}
```
```   460
```
```   461 lemma seq_mono_lemma:
```
```   462   assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
```
```   463     and "\<forall>n \<ge> m. e n \<le> e m"
```
```   464   shows "\<forall>n \<ge> m. d n < e m"
```
```   465   using assms
```
```   466   apply auto
```
```   467   apply (erule_tac x="n" in allE)
```
```   468   apply (erule_tac x="n" in allE)
```
```   469   apply auto
```
```   470   done
```
```   471
```
```   472 lemma infinite_enumerate:
```
```   473   assumes fS: "infinite S"
```
```   474   shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
```
```   475   unfolding subseq_def
```
```   476   using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
```
```   477
```
```   478 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)"
```
```   479   apply auto
```
```   480   apply (rule_tac x="d/2" in exI)
```
```   481   apply auto
```
```   482   done
```
```   483
```
```   484 lemma triangle_lemma:
```
```   485   fixes x y z :: real
```
```   486   assumes x: "0 \<le> x"
```
```   487     and y: "0 \<le> y"
```
```   488     and z: "0 \<le> z"
```
```   489     and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
```
```   490   shows "x \<le> y + z"
```
```   491 proof -
```
```   492   have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 *y * z + z\<^sup>2"
```
```   493     using z y by (simp add: mult_nonneg_nonneg)
```
```   494   with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
```
```   495     by (simp add: power2_eq_square field_simps)
```
```   496   from y z have yz: "y + z \<ge> 0"
```
```   497     by arith
```
```   498   from power2_le_imp_le[OF th yz] show ?thesis .
```
```   499 qed
```
```   500
```
```   501
```
```   502 subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
```
```   503
```
```   504 definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
```
```   505   where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
```
```   506
```
```   507 lemma hull_same: "S s \<Longrightarrow> S hull s = s"
```
```   508   unfolding hull_def by auto
```
```   509
```
```   510 lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
```
```   511   unfolding hull_def Ball_def by auto
```
```   512
```
```   513 lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
```
```   514   using hull_same[of S s] hull_in[of S s] by metis
```
```   515
```
```   516 lemma hull_hull: "S hull (S hull s) = S hull s"
```
```   517   unfolding hull_def by blast
```
```   518
```
```   519 lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
```
```   520   unfolding hull_def by blast
```
```   521
```
```   522 lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
```
```   523   unfolding hull_def by blast
```
```   524
```
```   525 lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
```
```   526   unfolding hull_def by blast
```
```   527
```
```   528 lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
```
```   529   unfolding hull_def by blast
```
```   530
```
```   531 lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
```
```   532   unfolding hull_def by blast
```
```   533
```
```   534 lemma hull_UNIV: "S hull UNIV = UNIV"
```
```   535   unfolding hull_def by auto
```
```   536
```
```   537 lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
```
```   538   unfolding hull_def by auto
```
```   539
```
```   540 lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
```
```   541   using hull_minimal[of S "{x. P x}" Q]
```
```   542   by (auto simp add: subset_eq)
```
```   543
```
```   544 lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
```
```   545   by (metis hull_subset subset_eq)
```
```   546
```
```   547 lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
```
```   548   unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
```
```   549
```
```   550 lemma hull_union:
```
```   551   assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
```
```   552   shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
```
```   553   apply rule
```
```   554   apply (rule hull_mono)
```
```   555   unfolding Un_subset_iff
```
```   556   apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
```
```   557   apply (rule hull_minimal)
```
```   558   apply (metis hull_union_subset)
```
```   559   apply (metis hull_in T)
```
```   560   done
```
```   561
```
```   562 lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
```
```   563   unfolding hull_def by blast
```
```   564
```
```   565 lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> (S hull (insert a s) = S hull s)"
```
```   566   by (metis hull_redundant_eq)
```
```   567
```
```   568
```
```   569 subsection {* Archimedean properties and useful consequences *}
```
```   570
```
```   571 lemma real_arch_simple: "\<exists>n. x \<le> real (n::nat)"
```
```   572   unfolding real_of_nat_def by (rule ex_le_of_nat)
```
```   573
```
```   574 lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
```
```   575   using reals_Archimedean
```
```   576   apply (auto simp add: field_simps)
```
```   577   apply (subgoal_tac "inverse (real n) > 0")
```
```   578   apply arith
```
```   579   apply simp
```
```   580   done
```
```   581
```
```   582 lemma real_pow_lbound: "0 \<le> x \<Longrightarrow> 1 + real n * x \<le> (1 + x) ^ n"
```
```   583 proof (induct n)
```
```   584   case 0
```
```   585   then show ?case by simp
```
```   586 next
```
```   587   case (Suc n)
```
```   588   then have h: "1 + real n * x \<le> (1 + x) ^ n"
```
```   589     by simp
```
```   590   from h have p: "1 \<le> (1 + x) ^ n"
```
```   591     using Suc.prems by simp
```
```   592   from h have "1 + real n * x + x \<le> (1 + x) ^ n + x"
```
```   593     by simp
```
```   594   also have "\<dots> \<le> (1 + x) ^ Suc n"
```
```   595     apply (subst diff_le_0_iff_le[symmetric])
```
```   596     apply (simp add: field_simps)
```
```   597     using mult_left_mono[OF p Suc.prems]
```
```   598     apply simp
```
```   599     done
```
```   600   finally show ?case
```
```   601     by (simp add: real_of_nat_Suc field_simps)
```
```   602 qed
```
```   603
```
```   604 lemma real_arch_pow:
```
```   605   fixes x :: real
```
```   606   assumes x: "1 < x"
```
```   607   shows "\<exists>n. y < x^n"
```
```   608 proof -
```
```   609   from x have x0: "x - 1 > 0"
```
```   610     by arith
```
```   611   from reals_Archimedean3[OF x0, rule_format, of y]
```
```   612   obtain n :: nat where n: "y < real n * (x - 1)" by metis
```
```   613   from x0 have x00: "x- 1 \<ge> 0" by arith
```
```   614   from real_pow_lbound[OF x00, of n] n
```
```   615   have "y < x^n" by auto
```
```   616   then show ?thesis by metis
```
```   617 qed
```
```   618
```
```   619 lemma real_arch_pow2:
```
```   620   fixes x :: real
```
```   621   shows "\<exists>n. x < 2^ n"
```
```   622   using real_arch_pow[of 2 x] by simp
```
```   623
```
```   624 lemma real_arch_pow_inv:
```
```   625   fixes x y :: real
```
```   626   assumes y: "y > 0"
```
```   627     and x1: "x < 1"
```
```   628   shows "\<exists>n. x^n < y"
```
```   629 proof (cases "x > 0")
```
```   630   case True
```
```   631   with x1 have ix: "1 < 1/x" by (simp add: field_simps)
```
```   632   from real_arch_pow[OF ix, of "1/y"]
```
```   633   obtain n where n: "1/y < (1/x)^n" by blast
```
```   634   then show ?thesis using y `x > 0`
```
```   635     by (auto simp add: field_simps power_divide)
```
```   636 next
```
```   637   case False
```
```   638   with y x1 show ?thesis
```
```   639     apply auto
```
```   640     apply (rule exI[where x=1])
```
```   641     apply auto
```
```   642     done
```
```   643 qed
```
```   644
```
```   645 lemma forall_pos_mono:
```
```   646   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```   647     (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
```
```   648   by (metis real_arch_inv)
```
```   649
```
```   650 lemma forall_pos_mono_1:
```
```   651   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```   652     (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
```
```   653   apply (rule forall_pos_mono)
```
```   654   apply auto
```
```   655   apply (atomize)
```
```   656   apply (erule_tac x="n - 1" in allE)
```
```   657   apply auto
```
```   658   done
```
```   659
```
```   660 lemma real_archimedian_rdiv_eq_0:
```
```   661   assumes x0: "x \<ge> 0"
```
```   662     and c: "c \<ge> 0"
```
```   663     and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
```
```   664   shows "x = 0"
```
```   665 proof (rule ccontr)
```
```   666   assume "x \<noteq> 0"
```
```   667   with x0 have xp: "x > 0" by arith
```
```   668   from reals_Archimedean3[OF xp, rule_format, of c]
```
```   669   obtain n :: nat where n: "c < real n * x"
```
```   670     by blast
```
```   671   with xc[rule_format, of n] have "n = 0"
```
```   672     by arith
```
```   673   with n c show False
```
```   674     by simp
```
```   675 qed
```
```   676
```
```   677
```
```   678 subsection{* A bit of linear algebra. *}
```
```   679
```
```   680 definition (in real_vector) subspace :: "'a set \<Rightarrow> bool"
```
```   681   where "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x\<in> S. \<forall>y \<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in>S. c *\<^sub>R x \<in>S )"
```
```   682
```
```   683 definition (in real_vector) "span S = (subspace hull S)"
```
```   684 definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))"
```
```   685 abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
```
```   686
```
```   687 text {* Closure properties of subspaces. *}
```
```   688
```
```   689 lemma subspace_UNIV[simp]: "subspace UNIV"
```
```   690   by (simp add: subspace_def)
```
```   691
```
```   692 lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
```
```   693   by (metis subspace_def)
```
```   694
```
```   695 lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
```
```   696   by (metis subspace_def)
```
```   697
```
```   698 lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
```
```   699   by (metis subspace_def)
```
```   700
```
```   701 lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
```
```   702   by (metis scaleR_minus1_left subspace_mul)
```
```   703
```
```   704 lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
```
```   705   using subspace_add [of S x "- y"] by (simp add: subspace_neg)
```
```   706
```
```   707 lemma (in real_vector) subspace_setsum:
```
```   708   assumes sA: "subspace A"
```
```   709     and fB: "finite B"
```
```   710     and f: "\<forall>x\<in> B. f x \<in> A"
```
```   711   shows "setsum f B \<in> A"
```
```   712   using  fB f sA
```
```   713   by (induct rule: finite_induct[OF fB])
```
```   714     (simp add: subspace_def sA, auto simp add: sA subspace_add)
```
```   715
```
```   716 lemma subspace_linear_image:
```
```   717   assumes lf: "linear f"
```
```   718     and sS: "subspace S"
```
```   719   shows "subspace (f ` S)"
```
```   720   using lf sS linear_0[OF lf]
```
```   721   unfolding linear_iff subspace_def
```
```   722   apply (auto simp add: image_iff)
```
```   723   apply (rule_tac x="x + y" in bexI)
```
```   724   apply auto
```
```   725   apply (rule_tac x="c *\<^sub>R x" in bexI)
```
```   726   apply auto
```
```   727   done
```
```   728
```
```   729 lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
```
```   730   by (auto simp add: subspace_def linear_iff linear_0[of f])
```
```   731
```
```   732 lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
```
```   733   by (auto simp add: subspace_def linear_iff linear_0[of f])
```
```   734
```
```   735 lemma subspace_trivial: "subspace {0}"
```
```   736   by (simp add: subspace_def)
```
```   737
```
```   738 lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
```
```   739   by (simp add: subspace_def)
```
```   740
```
```   741 lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
```
```   742   unfolding subspace_def zero_prod_def by simp
```
```   743
```
```   744 text {* Properties of span. *}
```
```   745
```
```   746 lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
```
```   747   by (metis span_def hull_mono)
```
```   748
```
```   749 lemma (in real_vector) subspace_span: "subspace (span S)"
```
```   750   unfolding span_def
```
```   751   apply (rule hull_in)
```
```   752   apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
```
```   753   apply auto
```
```   754   done
```
```   755
```
```   756 lemma (in real_vector) span_clauses:
```
```   757   "a \<in> S \<Longrightarrow> a \<in> span S"
```
```   758   "0 \<in> span S"
```
```   759   "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
```
```   760   "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
```
```   761   by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
```
```   762
```
```   763 lemma span_unique:
```
```   764   "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
```
```   765   unfolding span_def by (rule hull_unique)
```
```   766
```
```   767 lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
```
```   768   unfolding span_def by (rule hull_minimal)
```
```   769
```
```   770 lemma (in real_vector) span_induct:
```
```   771   assumes x: "x \<in> span S"
```
```   772     and P: "subspace P"
```
```   773     and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
```
```   774   shows "x \<in> P"
```
```   775 proof -
```
```   776   from SP have SP': "S \<subseteq> P"
```
```   777     by (simp add: subset_eq)
```
```   778   from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
```
```   779   show "x \<in> P"
```
```   780     by (metis subset_eq)
```
```   781 qed
```
```   782
```
```   783 lemma span_empty[simp]: "span {} = {0}"
```
```   784   apply (simp add: span_def)
```
```   785   apply (rule hull_unique)
```
```   786   apply (auto simp add: subspace_def)
```
```   787   done
```
```   788
```
```   789 lemma (in real_vector) independent_empty[intro]: "independent {}"
```
```   790   by (simp add: dependent_def)
```
```   791
```
```   792 lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
```
```   793   unfolding dependent_def by auto
```
```   794
```
```   795 lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
```
```   796   apply (clarsimp simp add: dependent_def span_mono)
```
```   797   apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
```
```   798   apply force
```
```   799   apply (rule span_mono)
```
```   800   apply auto
```
```   801   done
```
```   802
```
```   803 lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
```
```   804   by (metis order_antisym span_def hull_minimal)
```
```   805
```
```   806 lemma (in real_vector) span_induct':
```
```   807   assumes SP: "\<forall>x \<in> S. P x"
```
```   808     and P: "subspace {x. P x}"
```
```   809   shows "\<forall>x \<in> span S. P x"
```
```   810   using span_induct SP P by blast
```
```   811
```
```   812 inductive_set (in real_vector) span_induct_alt_help for S:: "'a set"
```
```   813 where
```
```   814   span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
```
```   815 | span_induct_alt_help_S:
```
```   816     "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
```
```   817       (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
```
```   818
```
```   819 lemma span_induct_alt':
```
```   820   assumes h0: "h 0"
```
```   821     and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
```
```   822   shows "\<forall>x \<in> span S. h x"
```
```   823 proof -
```
```   824   {
```
```   825     fix x :: 'a
```
```   826     assume x: "x \<in> span_induct_alt_help S"
```
```   827     have "h x"
```
```   828       apply (rule span_induct_alt_help.induct[OF x])
```
```   829       apply (rule h0)
```
```   830       apply (rule hS)
```
```   831       apply assumption
```
```   832       apply assumption
```
```   833       done
```
```   834   }
```
```   835   note th0 = this
```
```   836   {
```
```   837     fix x
```
```   838     assume x: "x \<in> span S"
```
```   839     have "x \<in> span_induct_alt_help S"
```
```   840     proof (rule span_induct[where x=x and S=S])
```
```   841       show "x \<in> span S" by (rule x)
```
```   842     next
```
```   843       fix x
```
```   844       assume xS: "x \<in> S"
```
```   845       from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
```
```   846       show "x \<in> span_induct_alt_help S"
```
```   847         by simp
```
```   848     next
```
```   849       have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
```
```   850       moreover
```
```   851       {
```
```   852         fix x y
```
```   853         assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
```
```   854         from h have "(x + y) \<in> span_induct_alt_help S"
```
```   855           apply (induct rule: span_induct_alt_help.induct)
```
```   856           apply simp
```
```   857           unfolding add_assoc
```
```   858           apply (rule span_induct_alt_help_S)
```
```   859           apply assumption
```
```   860           apply simp
```
```   861           done
```
```   862       }
```
```   863       moreover
```
```   864       {
```
```   865         fix c x
```
```   866         assume xt: "x \<in> span_induct_alt_help S"
```
```   867         then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
```
```   868           apply (induct rule: span_induct_alt_help.induct)
```
```   869           apply (simp add: span_induct_alt_help_0)
```
```   870           apply (simp add: scaleR_right_distrib)
```
```   871           apply (rule span_induct_alt_help_S)
```
```   872           apply assumption
```
```   873           apply simp
```
```   874           done }
```
```   875       ultimately show "subspace (span_induct_alt_help S)"
```
```   876         unfolding subspace_def Ball_def by blast
```
```   877     qed
```
```   878   }
```
```   879   with th0 show ?thesis by blast
```
```   880 qed
```
```   881
```
```   882 lemma span_induct_alt:
```
```   883   assumes h0: "h 0"
```
```   884     and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
```
```   885     and x: "x \<in> span S"
```
```   886   shows "h x"
```
```   887   using span_induct_alt'[of h S] h0 hS x by blast
```
```   888
```
```   889 text {* Individual closure properties. *}
```
```   890
```
```   891 lemma span_span: "span (span A) = span A"
```
```   892   unfolding span_def hull_hull ..
```
```   893
```
```   894 lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
```
```   895   by (metis span_clauses(1))
```
```   896
```
```   897 lemma (in real_vector) span_0: "0 \<in> span S"
```
```   898   by (metis subspace_span subspace_0)
```
```   899
```
```   900 lemma span_inc: "S \<subseteq> span S"
```
```   901   by (metis subset_eq span_superset)
```
```   902
```
```   903 lemma (in real_vector) dependent_0:
```
```   904   assumes "0 \<in> A"
```
```   905   shows "dependent A"
```
```   906   unfolding dependent_def
```
```   907   apply (rule_tac x=0 in bexI)
```
```   908   using assms span_0
```
```   909   apply auto
```
```   910   done
```
```   911
```
```   912 lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
```
```   913   by (metis subspace_add subspace_span)
```
```   914
```
```   915 lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
```
```   916   by (metis subspace_span subspace_mul)
```
```   917
```
```   918 lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
```
```   919   by (metis subspace_neg subspace_span)
```
```   920
```
```   921 lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
```
```   922   by (metis subspace_span subspace_sub)
```
```   923
```
```   924 lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S"
```
```   925   by (rule subspace_setsum, rule subspace_span)
```
```   926
```
```   927 lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
```
```   928   by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
```
```   929
```
```   930 text {* Mapping under linear image. *}
```
```   931
```
```   932 lemma span_linear_image:
```
```   933   assumes lf: "linear f"
```
```   934   shows "span (f ` S) = f ` (span S)"
```
```   935 proof (rule span_unique)
```
```   936   show "f ` S \<subseteq> f ` span S"
```
```   937     by (intro image_mono span_inc)
```
```   938   show "subspace (f ` span S)"
```
```   939     using lf subspace_span by (rule subspace_linear_image)
```
```   940 next
```
```   941   fix T
```
```   942   assume "f ` S \<subseteq> T" and "subspace T"
```
```   943   then show "f ` span S \<subseteq> T"
```
```   944     unfolding image_subset_iff_subset_vimage
```
```   945     by (intro span_minimal subspace_linear_vimage lf)
```
```   946 qed
```
```   947
```
```   948 lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
```
```   949 proof (rule span_unique)
```
```   950   show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
```
```   951     by safe (force intro: span_clauses)+
```
```   952 next
```
```   953   have "linear (\<lambda>(a, b). a + b)"
```
```   954     by (simp add: linear_iff scaleR_add_right)
```
```   955   moreover have "subspace (span A \<times> span B)"
```
```   956     by (intro subspace_Times subspace_span)
```
```   957   ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
```
```   958     by (rule subspace_linear_image)
```
```   959 next
```
```   960   fix T
```
```   961   assume "A \<union> B \<subseteq> T" and "subspace T"
```
```   962   then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
```
```   963     by (auto intro!: subspace_add elim: span_induct)
```
```   964 qed
```
```   965
```
```   966 text {* The key breakdown property. *}
```
```   967
```
```   968 lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
```
```   969 proof (rule span_unique)
```
```   970   show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
```
```   971     by (fast intro: scaleR_one [symmetric])
```
```   972   show "subspace (range (\<lambda>k. k *\<^sub>R x))"
```
```   973     unfolding subspace_def
```
```   974     by (auto intro: scaleR_add_left [symmetric])
```
```   975 next
```
```   976   fix T
```
```   977   assume "{x} \<subseteq> T" and "subspace T"
```
```   978   then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
```
```   979     unfolding subspace_def by auto
```
```   980 qed
```
```   981
```
```   982 lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
```
```   983 proof -
```
```   984   have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
```
```   985     unfolding span_union span_singleton
```
```   986     apply safe
```
```   987     apply (rule_tac x=k in exI, simp)
```
```   988     apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
```
```   989     apply auto
```
```   990     done
```
```   991   then show ?thesis by simp
```
```   992 qed
```
```   993
```
```   994 lemma span_breakdown:
```
```   995   assumes bS: "b \<in> S"
```
```   996     and aS: "a \<in> span S"
```
```   997   shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
```
```   998   using assms span_insert [of b "S - {b}"]
```
```   999   by (simp add: insert_absorb)
```
```  1000
```
```  1001 lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
```
```  1002   by (simp add: span_insert)
```
```  1003
```
```  1004 text {* Hence some "reversal" results. *}
```
```  1005
```
```  1006 lemma in_span_insert:
```
```  1007   assumes a: "a \<in> span (insert b S)"
```
```  1008     and na: "a \<notin> span S"
```
```  1009   shows "b \<in> span (insert a S)"
```
```  1010 proof -
```
```  1011   from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
```
```  1012     unfolding span_insert by fast
```
```  1013   show ?thesis
```
```  1014   proof (cases "k = 0")
```
```  1015     case True
```
```  1016     with k have "a \<in> span S" by simp
```
```  1017     with na show ?thesis by simp
```
```  1018   next
```
```  1019     case False
```
```  1020     from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
```
```  1021       by (rule span_mul)
```
```  1022     then have "b - inverse k *\<^sub>R a \<in> span S"
```
```  1023       using `k \<noteq> 0` by (simp add: scaleR_diff_right)
```
```  1024     then show ?thesis
```
```  1025       unfolding span_insert by fast
```
```  1026   qed
```
```  1027 qed
```
```  1028
```
```  1029 lemma in_span_delete:
```
```  1030   assumes a: "a \<in> span S"
```
```  1031     and na: "a \<notin> span (S - {b})"
```
```  1032   shows "b \<in> span (insert a (S - {b}))"
```
```  1033   apply (rule in_span_insert)
```
```  1034   apply (rule set_rev_mp)
```
```  1035   apply (rule a)
```
```  1036   apply (rule span_mono)
```
```  1037   apply blast
```
```  1038   apply (rule na)
```
```  1039   done
```
```  1040
```
```  1041 text {* Transitivity property. *}
```
```  1042
```
```  1043 lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
```
```  1044   unfolding span_def by (rule hull_redundant)
```
```  1045
```
```  1046 lemma span_trans:
```
```  1047   assumes x: "x \<in> span S"
```
```  1048     and y: "y \<in> span (insert x S)"
```
```  1049   shows "y \<in> span S"
```
```  1050   using assms by (simp only: span_redundant)
```
```  1051
```
```  1052 lemma span_insert_0[simp]: "span (insert 0 S) = span S"
```
```  1053   by (simp only: span_redundant span_0)
```
```  1054
```
```  1055 text {* An explicit expansion is sometimes needed. *}
```
```  1056
```
```  1057 lemma span_explicit:
```
```  1058   "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
```
```  1059   (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
```
```  1060 proof -
```
```  1061   {
```
```  1062     fix x
```
```  1063     assume "?h x"
```
```  1064     then obtain S u where "finite S" and "S \<subseteq> P" and "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
```
```  1065       by blast
```
```  1066     then have "x \<in> span P"
```
```  1067       by (auto intro: span_setsum span_mul span_superset)
```
```  1068   }
```
```  1069   moreover
```
```  1070   have "\<forall>x \<in> span P. ?h x"
```
```  1071   proof (rule span_induct_alt')
```
```  1072     show "?h 0"
```
```  1073       by (rule exI[where x="{}"], simp)
```
```  1074   next
```
```  1075     fix c x y
```
```  1076     assume x: "x \<in> P"
```
```  1077     assume hy: "?h y"
```
```  1078     from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
```
```  1079       and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
```
```  1080     let ?S = "insert x S"
```
```  1081     let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
```
```  1082     from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
```
```  1083       by blast+
```
```  1084     have "?Q ?S ?u (c*\<^sub>R x + y)"
```
```  1085     proof cases
```
```  1086       assume xS: "x \<in> S"
```
```  1087       have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = (\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
```
```  1088         using xS by (simp add: setsum.remove [OF fS xS] insert_absorb)
```
```  1089       also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
```
```  1090         by (simp add: setsum.remove [OF fS xS] algebra_simps)
```
```  1091       also have "\<dots> = c*\<^sub>R x + y"
```
```  1092         by (simp add: add_commute u)
```
```  1093       finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
```
```  1094       then show ?thesis using th0 by blast
```
```  1095     next
```
```  1096       assume xS: "x \<notin> S"
```
```  1097       have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
```
```  1098         unfolding u[symmetric]
```
```  1099         apply (rule setsum_cong2)
```
```  1100         using xS
```
```  1101         apply auto
```
```  1102         done
```
```  1103       show ?thesis using fS xS th0
```
```  1104         by (simp add: th00 add_commute cong del: if_weak_cong)
```
```  1105     qed
```
```  1106     then show "?h (c*\<^sub>R x + y)"
```
```  1107       by fast
```
```  1108   qed
```
```  1109   ultimately show ?thesis by blast
```
```  1110 qed
```
```  1111
```
```  1112 lemma dependent_explicit:
```
```  1113   "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))"
```
```  1114   (is "?lhs = ?rhs")
```
```  1115 proof -
```
```  1116   {
```
```  1117     assume dP: "dependent P"
```
```  1118     then obtain a S u where aP: "a \<in> P" and fS: "finite S"
```
```  1119       and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
```
```  1120       unfolding dependent_def span_explicit by blast
```
```  1121     let ?S = "insert a S"
```
```  1122     let ?u = "\<lambda>y. if y = a then - 1 else u y"
```
```  1123     let ?v = a
```
```  1124     from aP SP have aS: "a \<notin> S"
```
```  1125       by blast
```
```  1126     from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
```
```  1127       by auto
```
```  1128     have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
```
```  1129       using fS aS
```
```  1130       apply simp
```
```  1131       apply (subst (2) ua[symmetric])
```
```  1132       apply (rule setsum_cong2)
```
```  1133       apply auto
```
```  1134       done
```
```  1135     with th0 have ?rhs by fast
```
```  1136   }
```
```  1137   moreover
```
```  1138   {
```
```  1139     fix S u v
```
```  1140     assume fS: "finite S"
```
```  1141       and SP: "S \<subseteq> P"
```
```  1142       and vS: "v \<in> S"
```
```  1143       and uv: "u v \<noteq> 0"
```
```  1144       and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
```
```  1145     let ?a = v
```
```  1146     let ?S = "S - {v}"
```
```  1147     let ?u = "\<lambda>i. (- u i) / u v"
```
```  1148     have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
```
```  1149       using fS SP vS by auto
```
```  1150     have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
```
```  1151       setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
```
```  1152       using fS vS uv by (simp add: setsum_diff1 divide_inverse field_simps)
```
```  1153     also have "\<dots> = ?a"
```
```  1154       unfolding scaleR_right.setsum [symmetric] u using uv by simp
```
```  1155     finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
```
```  1156     with th0 have ?lhs
```
```  1157       unfolding dependent_def span_explicit
```
```  1158       apply -
```
```  1159       apply (rule bexI[where x= "?a"])
```
```  1160       apply (simp_all del: scaleR_minus_left)
```
```  1161       apply (rule exI[where x= "?S"])
```
```  1162       apply (auto simp del: scaleR_minus_left)
```
```  1163       done
```
```  1164   }
```
```  1165   ultimately show ?thesis by blast
```
```  1166 qed
```
```  1167
```
```  1168
```
```  1169 lemma span_finite:
```
```  1170   assumes fS: "finite S"
```
```  1171   shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
```
```  1172   (is "_ = ?rhs")
```
```  1173 proof -
```
```  1174   {
```
```  1175     fix y
```
```  1176     assume y: "y \<in> span S"
```
```  1177     from y obtain S' u where fS': "finite S'"
```
```  1178       and SS': "S' \<subseteq> S"
```
```  1179       and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
```
```  1180       unfolding span_explicit by blast
```
```  1181     let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
```
```  1182     have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
```
```  1183       using SS' fS by (auto intro!: setsum_mono_zero_cong_right)
```
```  1184     then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
```
```  1185     then have "y \<in> ?rhs" by auto
```
```  1186   }
```
```  1187   moreover
```
```  1188   {
```
```  1189     fix y u
```
```  1190     assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
```
```  1191     then have "y \<in> span S" using fS unfolding span_explicit by auto
```
```  1192   }
```
```  1193   ultimately show ?thesis by blast
```
```  1194 qed
```
```  1195
```
```  1196 text {* This is useful for building a basis step-by-step. *}
```
```  1197
```
```  1198 lemma independent_insert:
```
```  1199   "independent (insert a S) \<longleftrightarrow>
```
```  1200     (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
```
```  1201   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1202 proof (cases "a \<in> S")
```
```  1203   case True
```
```  1204   then show ?thesis
```
```  1205     using insert_absorb[OF True] by simp
```
```  1206 next
```
```  1207   case False
```
```  1208   show ?thesis
```
```  1209   proof
```
```  1210     assume i: ?lhs
```
```  1211     then show ?rhs
```
```  1212       using False
```
```  1213       apply simp
```
```  1214       apply (rule conjI)
```
```  1215       apply (rule independent_mono)
```
```  1216       apply assumption
```
```  1217       apply blast
```
```  1218       apply (simp add: dependent_def)
```
```  1219       done
```
```  1220   next
```
```  1221     assume i: ?rhs
```
```  1222     show ?lhs
```
```  1223       using i False
```
```  1224       apply (auto simp add: dependent_def)
```
```  1225       by (metis in_span_insert insert_Diff insert_Diff_if insert_iff)
```
```  1226   qed
```
```  1227 qed
```
```  1228
```
```  1229 text {* The degenerate case of the Exchange Lemma. *}
```
```  1230
```
```  1231 lemma spanning_subset_independent:
```
```  1232   assumes BA: "B \<subseteq> A"
```
```  1233     and iA: "independent A"
```
```  1234     and AsB: "A \<subseteq> span B"
```
```  1235   shows "A = B"
```
```  1236 proof
```
```  1237   show "B \<subseteq> A" by (rule BA)
```
```  1238
```
```  1239   from span_mono[OF BA] span_mono[OF AsB]
```
```  1240   have sAB: "span A = span B" unfolding span_span by blast
```
```  1241
```
```  1242   {
```
```  1243     fix x
```
```  1244     assume x: "x \<in> A"
```
```  1245     from iA have th0: "x \<notin> span (A - {x})"
```
```  1246       unfolding dependent_def using x by blast
```
```  1247     from x have xsA: "x \<in> span A"
```
```  1248       by (blast intro: span_superset)
```
```  1249     have "A - {x} \<subseteq> A" by blast
```
```  1250     then have th1: "span (A - {x}) \<subseteq> span A"
```
```  1251       by (metis span_mono)
```
```  1252     {
```
```  1253       assume xB: "x \<notin> B"
```
```  1254       from xB BA have "B \<subseteq> A - {x}"
```
```  1255         by blast
```
```  1256       then have "span B \<subseteq> span (A - {x})"
```
```  1257         by (metis span_mono)
```
```  1258       with th1 th0 sAB have "x \<notin> span A"
```
```  1259         by blast
```
```  1260       with x have False
```
```  1261         by (metis span_superset)
```
```  1262     }
```
```  1263     then have "x \<in> B" by blast
```
```  1264   }
```
```  1265   then show "A \<subseteq> B" by blast
```
```  1266 qed
```
```  1267
```
```  1268 text {* The general case of the Exchange Lemma, the key to what follows. *}
```
```  1269
```
```  1270 lemma exchange_lemma:
```
```  1271   assumes f:"finite t"
```
```  1272     and i: "independent s"
```
```  1273     and sp: "s \<subseteq> span t"
```
```  1274   shows "\<exists>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
```
```  1275   using f i sp
```
```  1276 proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
```
```  1277   case less
```
```  1278   note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
```
```  1279   let ?P = "\<lambda>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
```
```  1280   let ?ths = "\<exists>t'. ?P t'"
```
```  1281   {
```
```  1282     assume "s \<subseteq> t"
```
```  1283     then have ?ths
```
```  1284       by (metis ft Un_commute sp sup_ge1)
```
```  1285   }
```
```  1286   moreover
```
```  1287   {
```
```  1288     assume st: "t \<subseteq> s"
```
```  1289     from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
```
```  1290     have ?ths
```
```  1291       by (metis Un_absorb sp)
```
```  1292   }
```
```  1293   moreover
```
```  1294   {
```
```  1295     assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
```
```  1296     from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
```
```  1297       by blast
```
```  1298     from b have "t - {b} - s \<subset> t - s"
```
```  1299       by blast
```
```  1300     then have cardlt: "card (t - {b} - s) < card (t - s)"
```
```  1301       using ft by (auto intro: psubset_card_mono)
```
```  1302     from b ft have ct0: "card t \<noteq> 0"
```
```  1303       by auto
```
```  1304     have ?ths
```
```  1305     proof cases
```
```  1306       assume stb: "s \<subseteq> span (t - {b})"
```
```  1307       from ft have ftb: "finite (t - {b})"
```
```  1308         by auto
```
```  1309       from less(1)[OF cardlt ftb s stb]
```
```  1310       obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
```
```  1311         and fu: "finite u" by blast
```
```  1312       let ?w = "insert b u"
```
```  1313       have th0: "s \<subseteq> insert b u"
```
```  1314         using u by blast
```
```  1315       from u(3) b have "u \<subseteq> s \<union> t"
```
```  1316         by blast
```
```  1317       then have th1: "insert b u \<subseteq> s \<union> t"
```
```  1318         using u b by blast
```
```  1319       have bu: "b \<notin> u"
```
```  1320         using b u by blast
```
```  1321       from u(1) ft b have "card u = (card t - 1)"
```
```  1322         by auto
```
```  1323       then have th2: "card (insert b u) = card t"
```
```  1324         using card_insert_disjoint[OF fu bu] ct0 by auto
```
```  1325       from u(4) have "s \<subseteq> span u" .
```
```  1326       also have "\<dots> \<subseteq> span (insert b u)"
```
```  1327         by (rule span_mono) blast
```
```  1328       finally have th3: "s \<subseteq> span (insert b u)" .
```
```  1329       from th0 th1 th2 th3 fu have th: "?P ?w"
```
```  1330         by blast
```
```  1331       from th show ?thesis by blast
```
```  1332     next
```
```  1333       assume stb: "\<not> s \<subseteq> span (t - {b})"
```
```  1334       from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
```
```  1335         by blast
```
```  1336       have ab: "a \<noteq> b"
```
```  1337         using a b by blast
```
```  1338       have at: "a \<notin> t"
```
```  1339         using a ab span_superset[of a "t- {b}"] by auto
```
```  1340       have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
```
```  1341         using cardlt ft a b by auto
```
```  1342       have ft': "finite (insert a (t - {b}))"
```
```  1343         using ft by auto
```
```  1344       {
```
```  1345         fix x
```
```  1346         assume xs: "x \<in> s"
```
```  1347         have t: "t \<subseteq> insert b (insert a (t - {b}))"
```
```  1348           using b by auto
```
```  1349         from b(1) have "b \<in> span t"
```
```  1350           by (simp add: span_superset)
```
```  1351         have bs: "b \<in> span (insert a (t - {b}))"
```
```  1352           apply (rule in_span_delete)
```
```  1353           using a sp unfolding subset_eq
```
```  1354           apply auto
```
```  1355           done
```
```  1356         from xs sp have "x \<in> span t"
```
```  1357           by blast
```
```  1358         with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
```
```  1359         from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
```
```  1360       }
```
```  1361       then have sp': "s \<subseteq> span (insert a (t - {b}))"
```
```  1362         by blast
```
```  1363       from less(1)[OF mlt ft' s sp'] obtain u where u:
```
```  1364         "card u = card (insert a (t - {b}))"
```
```  1365         "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
```
```  1366         "s \<subseteq> span u" by blast
```
```  1367       from u a b ft at ct0 have "?P u"
```
```  1368         by auto
```
```  1369       then show ?thesis by blast
```
```  1370     qed
```
```  1371   }
```
```  1372   ultimately show ?ths by blast
```
```  1373 qed
```
```  1374
```
```  1375 text {* This implies corresponding size bounds. *}
```
```  1376
```
```  1377 lemma independent_span_bound:
```
```  1378   assumes f: "finite t"
```
```  1379     and i: "independent s"
```
```  1380     and sp: "s \<subseteq> span t"
```
```  1381   shows "finite s \<and> card s \<le> card t"
```
```  1382   by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
```
```  1383
```
```  1384 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
```
```  1385 proof -
```
```  1386   have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
```
```  1387     by auto
```
```  1388   show ?thesis unfolding eq
```
```  1389     apply (rule finite_imageI)
```
```  1390     apply (rule finite)
```
```  1391     done
```
```  1392 qed
```
```  1393
```
```  1394
```
```  1395 subsection {* Euclidean Spaces as Typeclass *}
```
```  1396
```
```  1397 lemma independent_Basis: "independent Basis"
```
```  1398   unfolding dependent_def
```
```  1399   apply (subst span_finite)
```
```  1400   apply simp
```
```  1401   apply clarify
```
```  1402   apply (drule_tac f="inner a" in arg_cong)
```
```  1403   apply (simp add: inner_Basis inner_setsum_right eq_commute)
```
```  1404   done
```
```  1405
```
```  1406 lemma span_Basis [simp]: "span Basis = UNIV"
```
```  1407   unfolding span_finite [OF finite_Basis]
```
```  1408   by (fast intro: euclidean_representation)
```
```  1409
```
```  1410 lemma in_span_Basis: "x \<in> span Basis"
```
```  1411   unfolding span_Basis ..
```
```  1412
```
```  1413 lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
```
```  1414   by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
```
```  1415
```
```  1416 lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
```
```  1417   by (metis Basis_le_norm order_trans)
```
```  1418
```
```  1419 lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
```
```  1420   by (metis Basis_le_norm le_less_trans)
```
```  1421
```
```  1422 lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
```
```  1423   apply (subst euclidean_representation[of x, symmetric])
```
```  1424   apply (rule order_trans[OF norm_setsum])
```
```  1425   apply (auto intro!: setsum_mono)
```
```  1426   done
```
```  1427
```
```  1428 lemma setsum_norm_allsubsets_bound:
```
```  1429   fixes f:: "'a \<Rightarrow> 'n::euclidean_space"
```
```  1430   assumes fP: "finite P"
```
```  1431     and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
```
```  1432   shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
```
```  1433 proof -
```
```  1434   have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
```
```  1435     by (rule setsum_mono) (rule norm_le_l1)
```
```  1436   also have "(\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>) = (\<Sum>b\<in>Basis. \<Sum>x\<in>P. \<bar>f x \<bullet> b\<bar>)"
```
```  1437     by (rule setsum_commute)
```
```  1438   also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
```
```  1439   proof (rule setsum_bounded)
```
```  1440     fix i :: 'n
```
```  1441     assume i: "i \<in> Basis"
```
```  1442     have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
```
```  1443       norm ((\<Sum>x\<in>P \<inter> - {x. f x \<bullet> i < 0}. f x) \<bullet> i) + norm ((\<Sum>x\<in>P \<inter> {x. f x \<bullet> i < 0}. f x) \<bullet> i)"
```
```  1444       by (simp add: abs_real_def setsum_cases[OF fP] setsum_negf norm_triangle_ineq4 inner_setsum_left del: real_norm_def)
```
```  1445     also have "\<dots> \<le> e + e"
```
```  1446       unfolding real_norm_def
```
```  1447       by (intro add_mono norm_bound_Basis_le i fPs) auto
```
```  1448     finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
```
```  1449   qed
```
```  1450   also have "\<dots> = 2 * real DIM('n) * e"
```
```  1451     by (simp add: real_of_nat_def)
```
```  1452   finally show ?thesis .
```
```  1453 qed
```
```  1454
```
```  1455
```
```  1456 subsection {* Linearity and Bilinearity continued *}
```
```  1457
```
```  1458 lemma linear_bounded:
```
```  1459   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1460   assumes lf: "linear f"
```
```  1461   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```  1462 proof
```
```  1463   let ?B = "\<Sum>b\<in>Basis. norm (f b)"
```
```  1464   show "\<forall>x. norm (f x) \<le> ?B * norm x"
```
```  1465   proof
```
```  1466     fix x :: 'a
```
```  1467     let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
```
```  1468     have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
```
```  1469       unfolding euclidean_representation ..
```
```  1470     also have "\<dots> = norm (setsum ?g Basis)"
```
```  1471       by (simp add: linear_setsum [OF lf] linear_cmul [OF lf])
```
```  1472     finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
```
```  1473     have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
```
```  1474     proof
```
```  1475       fix i :: 'a
```
```  1476       assume i: "i \<in> Basis"
```
```  1477       from Basis_le_norm[OF i, of x]
```
```  1478       show "norm (?g i) \<le> norm (f i) * norm x"
```
```  1479         unfolding norm_scaleR
```
```  1480         apply (subst mult_commute)
```
```  1481         apply (rule mult_mono)
```
```  1482         apply (auto simp add: field_simps)
```
```  1483         done
```
```  1484     qed
```
```  1485     from setsum_norm_le[of _ ?g, OF th]
```
```  1486     show "norm (f x) \<le> ?B * norm x"
```
```  1487       unfolding th0 setsum_left_distrib by metis
```
```  1488   qed
```
```  1489 qed
```
```  1490
```
```  1491 lemma linear_conv_bounded_linear:
```
```  1492   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1493   shows "linear f \<longleftrightarrow> bounded_linear f"
```
```  1494 proof
```
```  1495   assume "linear f"
```
```  1496   then interpret f: linear f .
```
```  1497   show "bounded_linear f"
```
```  1498   proof
```
```  1499     have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```  1500       using `linear f` by (rule linear_bounded)
```
```  1501     then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1502       by (simp add: mult_commute)
```
```  1503   qed
```
```  1504 next
```
```  1505   assume "bounded_linear f"
```
```  1506   then interpret f: bounded_linear f .
```
```  1507   show "linear f" ..
```
```  1508 qed
```
```  1509
```
```  1510 lemma linear_bounded_pos:
```
```  1511   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1512   assumes lf: "linear f"
```
```  1513   shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
```
```  1514 proof -
```
```  1515   have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
```
```  1516     using lf unfolding linear_conv_bounded_linear
```
```  1517     by (rule bounded_linear.pos_bounded)
```
```  1518   then show ?thesis
```
```  1519     by (simp only: mult_commute)
```
```  1520 qed
```
```  1521
```
```  1522 lemma bounded_linearI':
```
```  1523   fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1524   assumes "\<And>x y. f (x + y) = f x + f y"
```
```  1525     and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```  1526   shows "bounded_linear f"
```
```  1527   unfolding linear_conv_bounded_linear[symmetric]
```
```  1528   by (rule linearI[OF assms])
```
```  1529
```
```  1530 lemma bilinear_bounded:
```
```  1531   fixes h:: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
```
```  1532   assumes bh: "bilinear h"
```
```  1533   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```  1534 proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
```
```  1535   fix x :: 'm
```
```  1536   fix y :: 'n
```
```  1537   have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
```
```  1538     apply (subst euclidean_representation[where 'a='m])
```
```  1539     apply (subst euclidean_representation[where 'a='n])
```
```  1540     apply rule
```
```  1541     done
```
```  1542   also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
```
```  1543     unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
```
```  1544   finally have th: "norm (h x y) = \<dots>" .
```
```  1545   show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
```
```  1546     apply (auto simp add: setsum_left_distrib th setsum_cartesian_product)
```
```  1547     apply (rule setsum_norm_le)
```
```  1548     apply simp
```
```  1549     apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
```
```  1550       field_simps simp del: scaleR_scaleR)
```
```  1551     apply (rule mult_mono)
```
```  1552     apply (auto simp add: zero_le_mult_iff Basis_le_norm)
```
```  1553     apply (rule mult_mono)
```
```  1554     apply (auto simp add: zero_le_mult_iff Basis_le_norm)
```
```  1555     done
```
```  1556 qed
```
```  1557
```
```  1558 lemma bilinear_conv_bounded_bilinear:
```
```  1559   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```  1560   shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
```
```  1561 proof
```
```  1562   assume "bilinear h"
```
```  1563   show "bounded_bilinear h"
```
```  1564   proof
```
```  1565     fix x y z
```
```  1566     show "h (x + y) z = h x z + h y z"
```
```  1567       using `bilinear h` unfolding bilinear_def linear_iff by simp
```
```  1568   next
```
```  1569     fix x y z
```
```  1570     show "h x (y + z) = h x y + h x z"
```
```  1571       using `bilinear h` unfolding bilinear_def linear_iff by simp
```
```  1572   next
```
```  1573     fix r x y
```
```  1574     show "h (scaleR r x) y = scaleR r (h x y)"
```
```  1575       using `bilinear h` unfolding bilinear_def linear_iff
```
```  1576       by simp
```
```  1577   next
```
```  1578     fix r x y
```
```  1579     show "h x (scaleR r y) = scaleR r (h x y)"
```
```  1580       using `bilinear h` unfolding bilinear_def linear_iff
```
```  1581       by simp
```
```  1582   next
```
```  1583     have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```  1584       using `bilinear h` by (rule bilinear_bounded)
```
```  1585     then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
```
```  1586       by (simp add: mult_ac)
```
```  1587   qed
```
```  1588 next
```
```  1589   assume "bounded_bilinear h"
```
```  1590   then interpret h: bounded_bilinear h .
```
```  1591   show "bilinear h"
```
```  1592     unfolding bilinear_def linear_conv_bounded_linear
```
```  1593     using h.bounded_linear_left h.bounded_linear_right by simp
```
```  1594 qed
```
```  1595
```
```  1596 lemma bilinear_bounded_pos:
```
```  1597   fixes h:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```  1598   assumes bh: "bilinear h"
```
```  1599   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```  1600 proof -
```
```  1601   have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
```
```  1602     using bh [unfolded bilinear_conv_bounded_bilinear]
```
```  1603     by (rule bounded_bilinear.pos_bounded)
```
```  1604   then show ?thesis
```
```  1605     by (simp only: mult_ac)
```
```  1606 qed
```
```  1607
```
```  1608
```
```  1609 subsection {* We continue. *}
```
```  1610
```
```  1611 lemma independent_bound:
```
```  1612   fixes S :: "'a::euclidean_space set"
```
```  1613   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
```
```  1614   using independent_span_bound[OF finite_Basis, of S] by auto
```
```  1615
```
```  1616 lemma dependent_biggerset:
```
```  1617   "(finite (S::('a::euclidean_space) set) \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
```
```  1618   by (metis independent_bound not_less)
```
```  1619
```
```  1620 text {* Hence we can create a maximal independent subset. *}
```
```  1621
```
```  1622 lemma maximal_independent_subset_extend:
```
```  1623   fixes S :: "'a::euclidean_space set"
```
```  1624   assumes sv: "S \<subseteq> V"
```
```  1625     and iS: "independent S"
```
```  1626   shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
```
```  1627   using sv iS
```
```  1628 proof (induct "DIM('a) - card S" arbitrary: S rule: less_induct)
```
```  1629   case less
```
```  1630   note sv = `S \<subseteq> V` and i = `independent S`
```
```  1631   let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
```
```  1632   let ?ths = "\<exists>x. ?P x"
```
```  1633   let ?d = "DIM('a)"
```
```  1634   show ?ths
```
```  1635   proof (cases "V \<subseteq> span S")
```
```  1636     case True
```
```  1637     then show ?thesis
```
```  1638       using sv i by blast
```
```  1639   next
```
```  1640     case False
```
```  1641     then obtain a where a: "a \<in> V" "a \<notin> span S"
```
```  1642       by blast
```
```  1643     from a have aS: "a \<notin> S"
```
```  1644       by (auto simp add: span_superset)
```
```  1645     have th0: "insert a S \<subseteq> V"
```
```  1646       using a sv by blast
```
```  1647     from independent_insert[of a S]  i a
```
```  1648     have th1: "independent (insert a S)"
```
```  1649       by auto
```
```  1650     have mlt: "?d - card (insert a S) < ?d - card S"
```
```  1651       using aS a independent_bound[OF th1] by auto
```
```  1652
```
```  1653     from less(1)[OF mlt th0 th1]
```
```  1654     obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
```
```  1655       by blast
```
```  1656     from B have "?P B" by auto
```
```  1657     then show ?thesis by blast
```
```  1658   qed
```
```  1659 qed
```
```  1660
```
```  1661 lemma maximal_independent_subset:
```
```  1662   "\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
```
```  1663   by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"]
```
```  1664     empty_subsetI independent_empty)
```
```  1665
```
```  1666
```
```  1667 text {* Notion of dimension. *}
```
```  1668
```
```  1669 definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
```
```  1670
```
```  1671 lemma basis_exists:
```
```  1672   "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
```
```  1673   unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n)"]
```
```  1674   using maximal_independent_subset[of V] independent_bound
```
```  1675   by auto
```
```  1676
```
```  1677 text {* Consequences of independence or spanning for cardinality. *}
```
```  1678
```
```  1679 lemma independent_card_le_dim:
```
```  1680   fixes B :: "'a::euclidean_space set"
```
```  1681   assumes "B \<subseteq> V"
```
```  1682     and "independent B"
```
```  1683   shows "card B \<le> dim V"
```
```  1684 proof -
```
```  1685   from basis_exists[of V] `B \<subseteq> V`
```
```  1686   obtain B' where "independent B'"
```
```  1687     and "B \<subseteq> span B'"
```
```  1688     and "card B' = dim V"
```
```  1689     by blast
```
```  1690   with independent_span_bound[OF _ `independent B` `B \<subseteq> span B'`] independent_bound[of B']
```
```  1691   show ?thesis by auto
```
```  1692 qed
```
```  1693
```
```  1694 lemma span_card_ge_dim:
```
```  1695   fixes B :: "'a::euclidean_space set"
```
```  1696   shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
```
```  1697   by (metis basis_exists[of V] independent_span_bound subset_trans)
```
```  1698
```
```  1699 lemma basis_card_eq_dim:
```
```  1700   fixes V :: "'a::euclidean_space set"
```
```  1701   shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
```
```  1702   by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
```
```  1703
```
```  1704 lemma dim_unique:
```
```  1705   fixes B :: "'a::euclidean_space set"
```
```  1706   shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
```
```  1707   by (metis basis_card_eq_dim)
```
```  1708
```
```  1709 text {* More lemmas about dimension. *}
```
```  1710
```
```  1711 lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
```
```  1712   using independent_Basis
```
```  1713   by (intro dim_unique[of Basis]) auto
```
```  1714
```
```  1715 lemma dim_subset:
```
```  1716   fixes S :: "'a::euclidean_space set"
```
```  1717   shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
```
```  1718   using basis_exists[of T] basis_exists[of S]
```
```  1719   by (metis independent_card_le_dim subset_trans)
```
```  1720
```
```  1721 lemma dim_subset_UNIV:
```
```  1722   fixes S :: "'a::euclidean_space set"
```
```  1723   shows "dim S \<le> DIM('a)"
```
```  1724   by (metis dim_subset subset_UNIV dim_UNIV)
```
```  1725
```
```  1726 text {* Converses to those. *}
```
```  1727
```
```  1728 lemma card_ge_dim_independent:
```
```  1729   fixes B :: "'a::euclidean_space set"
```
```  1730   assumes BV: "B \<subseteq> V"
```
```  1731     and iB: "independent B"
```
```  1732     and dVB: "dim V \<le> card B"
```
```  1733   shows "V \<subseteq> span B"
```
```  1734 proof
```
```  1735   fix a
```
```  1736   assume aV: "a \<in> V"
```
```  1737   {
```
```  1738     assume aB: "a \<notin> span B"
```
```  1739     then have iaB: "independent (insert a B)"
```
```  1740       using iB aV BV by (simp add: independent_insert)
```
```  1741     from aV BV have th0: "insert a B \<subseteq> V"
```
```  1742       by blast
```
```  1743     from aB have "a \<notin>B"
```
```  1744       by (auto simp add: span_superset)
```
```  1745     with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
```
```  1746     have False by auto
```
```  1747   }
```
```  1748   then show "a \<in> span B" by blast
```
```  1749 qed
```
```  1750
```
```  1751 lemma card_le_dim_spanning:
```
```  1752   assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
```
```  1753     and VB: "V \<subseteq> span B"
```
```  1754     and fB: "finite B"
```
```  1755     and dVB: "dim V \<ge> card B"
```
```  1756   shows "independent B"
```
```  1757 proof -
```
```  1758   {
```
```  1759     fix a
```
```  1760     assume a: "a \<in> B" "a \<in> span (B - {a})"
```
```  1761     from a fB have c0: "card B \<noteq> 0"
```
```  1762       by auto
```
```  1763     from a fB have cb: "card (B - {a}) = card B - 1"
```
```  1764       by auto
```
```  1765     from BV a have th0: "B - {a} \<subseteq> V"
```
```  1766       by blast
```
```  1767     {
```
```  1768       fix x
```
```  1769       assume x: "x \<in> V"
```
```  1770       from a have eq: "insert a (B - {a}) = B"
```
```  1771         by blast
```
```  1772       from x VB have x': "x \<in> span B"
```
```  1773         by blast
```
```  1774       from span_trans[OF a(2), unfolded eq, OF x']
```
```  1775       have "x \<in> span (B - {a})" .
```
```  1776     }
```
```  1777     then have th1: "V \<subseteq> span (B - {a})"
```
```  1778       by blast
```
```  1779     have th2: "finite (B - {a})"
```
```  1780       using fB by auto
```
```  1781     from span_card_ge_dim[OF th0 th1 th2]
```
```  1782     have c: "dim V \<le> card (B - {a})" .
```
```  1783     from c c0 dVB cb have False by simp
```
```  1784   }
```
```  1785   then show ?thesis
```
```  1786     unfolding dependent_def by blast
```
```  1787 qed
```
```  1788
```
```  1789 lemma card_eq_dim:
```
```  1790   fixes B :: "'a::euclidean_space set"
```
```  1791   shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
```
```  1792   by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
```
```  1793
```
```  1794 text {* More general size bound lemmas. *}
```
```  1795
```
```  1796 lemma independent_bound_general:
```
```  1797   fixes S :: "'a::euclidean_space set"
```
```  1798   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
```
```  1799   by (metis independent_card_le_dim independent_bound subset_refl)
```
```  1800
```
```  1801 lemma dependent_biggerset_general:
```
```  1802   fixes S :: "'a::euclidean_space set"
```
```  1803   shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
```
```  1804   using independent_bound_general[of S] by (metis linorder_not_le)
```
```  1805
```
```  1806 lemma dim_span:
```
```  1807   fixes S :: "'a::euclidean_space set"
```
```  1808   shows "dim (span S) = dim S"
```
```  1809 proof -
```
```  1810   have th0: "dim S \<le> dim (span S)"
```
```  1811     by (auto simp add: subset_eq intro: dim_subset span_superset)
```
```  1812   from basis_exists[of S]
```
```  1813   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
```
```  1814     by blast
```
```  1815   from B have fB: "finite B" "card B = dim S"
```
```  1816     using independent_bound by blast+
```
```  1817   have bSS: "B \<subseteq> span S"
```
```  1818     using B(1) by (metis subset_eq span_inc)
```
```  1819   have sssB: "span S \<subseteq> span B"
```
```  1820     using span_mono[OF B(3)] by (simp add: span_span)
```
```  1821   from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
```
```  1822     using fB(2) by arith
```
```  1823 qed
```
```  1824
```
```  1825 lemma subset_le_dim:
```
```  1826   fixes S :: "'a::euclidean_space set"
```
```  1827   shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
```
```  1828   by (metis dim_span dim_subset)
```
```  1829
```
```  1830 lemma span_eq_dim:
```
```  1831   fixes S:: "'a::euclidean_space set"
```
```  1832   shows "span S = span T \<Longrightarrow> dim S = dim T"
```
```  1833   by (metis dim_span)
```
```  1834
```
```  1835 lemma spans_image:
```
```  1836   assumes lf: "linear f"
```
```  1837     and VB: "V \<subseteq> span B"
```
```  1838   shows "f ` V \<subseteq> span (f ` B)"
```
```  1839   unfolding span_linear_image[OF lf] by (metis VB image_mono)
```
```  1840
```
```  1841 lemma dim_image_le:
```
```  1842   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1843   assumes lf: "linear f"
```
```  1844   shows "dim (f ` S) \<le> dim (S)"
```
```  1845 proof -
```
```  1846   from basis_exists[of S] obtain B where
```
```  1847     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
```
```  1848   from B have fB: "finite B" "card B = dim S"
```
```  1849     using independent_bound by blast+
```
```  1850   have "dim (f ` S) \<le> card (f ` B)"
```
```  1851     apply (rule span_card_ge_dim)
```
```  1852     using lf B fB
```
```  1853     apply (auto simp add: span_linear_image spans_image subset_image_iff)
```
```  1854     done
```
```  1855   also have "\<dots> \<le> dim S"
```
```  1856     using card_image_le[OF fB(1)] fB by simp
```
```  1857   finally show ?thesis .
```
```  1858 qed
```
```  1859
```
```  1860 text {* Relation between bases and injectivity/surjectivity of map. *}
```
```  1861
```
```  1862 lemma spanning_surjective_image:
```
```  1863   assumes us: "UNIV \<subseteq> span S"
```
```  1864     and lf: "linear f"
```
```  1865     and sf: "surj f"
```
```  1866   shows "UNIV \<subseteq> span (f ` S)"
```
```  1867 proof -
```
```  1868   have "UNIV \<subseteq> f ` UNIV"
```
```  1869     using sf by (auto simp add: surj_def)
```
```  1870   also have " \<dots> \<subseteq> span (f ` S)"
```
```  1871     using spans_image[OF lf us] .
```
```  1872   finally show ?thesis .
```
```  1873 qed
```
```  1874
```
```  1875 lemma independent_injective_image:
```
```  1876   assumes iS: "independent S"
```
```  1877     and lf: "linear f"
```
```  1878     and fi: "inj f"
```
```  1879   shows "independent (f ` S)"
```
```  1880 proof -
```
```  1881   {
```
```  1882     fix a
```
```  1883     assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
```
```  1884     have eq: "f ` S - {f a} = f ` (S - {a})"
```
```  1885       using fi by (auto simp add: inj_on_def)
```
```  1886     from a have "f a \<in> f ` span (S - {a})"
```
```  1887       unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
```
```  1888     then have "a \<in> span (S - {a})"
```
```  1889       using fi by (auto simp add: inj_on_def)
```
```  1890     with a(1) iS have False
```
```  1891       by (simp add: dependent_def)
```
```  1892   }
```
```  1893   then show ?thesis
```
```  1894     unfolding dependent_def by blast
```
```  1895 qed
```
```  1896
```
```  1897 text {* Picking an orthogonal replacement for a spanning set. *}
```
```  1898
```
```  1899 (* FIXME : Move to some general theory ?*)
```
```  1900 definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
```
```  1901
```
```  1902 lemma vector_sub_project_orthogonal:
```
```  1903   fixes b x :: "'a::euclidean_space"
```
```  1904   shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
```
```  1905   unfolding inner_simps by auto
```
```  1906
```
```  1907 lemma pairwise_orthogonal_insert:
```
```  1908   assumes "pairwise orthogonal S"
```
```  1909     and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
```
```  1910   shows "pairwise orthogonal (insert x S)"
```
```  1911   using assms unfolding pairwise_def
```
```  1912   by (auto simp add: orthogonal_commute)
```
```  1913
```
```  1914 lemma basis_orthogonal:
```
```  1915   fixes B :: "'a::real_inner set"
```
```  1916   assumes fB: "finite B"
```
```  1917   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
```
```  1918   (is " \<exists>C. ?P B C")
```
```  1919   using fB
```
```  1920 proof (induct rule: finite_induct)
```
```  1921   case empty
```
```  1922   then show ?case
```
```  1923     apply (rule exI[where x="{}"])
```
```  1924     apply (auto simp add: pairwise_def)
```
```  1925     done
```
```  1926 next
```
```  1927   case (insert a B)
```
```  1928   note fB = `finite B` and aB = `a \<notin> B`
```
```  1929   from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
```
```  1930   obtain C where C: "finite C" "card C \<le> card B"
```
```  1931     "span C = span B" "pairwise orthogonal C" by blast
```
```  1932   let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
```
```  1933   let ?C = "insert ?a C"
```
```  1934   from C(1) have fC: "finite ?C"
```
```  1935     by simp
```
```  1936   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
```
```  1937     by (simp add: card_insert_if)
```
```  1938   {
```
```  1939     fix x k
```
```  1940     have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
```
```  1941       by (simp add: field_simps)
```
```  1942     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"
```
```  1943       apply (simp only: scaleR_right_diff_distrib th0)
```
```  1944       apply (rule span_add_eq)
```
```  1945       apply (rule span_mul)
```
```  1946       apply (rule span_setsum[OF C(1)])
```
```  1947       apply clarify
```
```  1948       apply (rule span_mul)
```
```  1949       apply (rule span_superset)
```
```  1950       apply assumption
```
```  1951       done
```
```  1952   }
```
```  1953   then have SC: "span ?C = span (insert a B)"
```
```  1954     unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
```
```  1955   {
```
```  1956     fix y
```
```  1957     assume yC: "y \<in> C"
```
```  1958     then have Cy: "C = insert y (C - {y})"
```
```  1959       by blast
```
```  1960     have fth: "finite (C - {y})"
```
```  1961       using C by simp
```
```  1962     have "orthogonal ?a y"
```
```  1963       unfolding orthogonal_def
```
```  1964       unfolding inner_diff inner_setsum_left right_minus_eq
```
```  1965       unfolding setsum_diff1' [OF `finite C` `y \<in> C`]
```
```  1966       apply (clarsimp simp add: inner_commute[of y a])
```
```  1967       apply (rule setsum_0')
```
```  1968       apply clarsimp
```
```  1969       apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
```
```  1970       using `y \<in> C` by auto
```
```  1971   }
```
```  1972   with `pairwise orthogonal C` have CPO: "pairwise orthogonal ?C"
```
```  1973     by (rule pairwise_orthogonal_insert)
```
```  1974   from fC cC SC CPO have "?P (insert a B) ?C"
```
```  1975     by blast
```
```  1976   then show ?case by blast
```
```  1977 qed
```
```  1978
```
```  1979 lemma orthogonal_basis_exists:
```
```  1980   fixes V :: "('a::euclidean_space) set"
```
```  1981   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
```
```  1982 proof -
```
```  1983   from basis_exists[of V] obtain B where
```
```  1984     B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
```
```  1985     by blast
```
```  1986   from B have fB: "finite B" "card B = dim V"
```
```  1987     using independent_bound by auto
```
```  1988   from basis_orthogonal[OF fB(1)] obtain C where
```
```  1989     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
```
```  1990     by blast
```
```  1991   from C B have CSV: "C \<subseteq> span V"
```
```  1992     by (metis span_inc span_mono subset_trans)
```
```  1993   from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
```
```  1994     by (simp add: span_span)
```
```  1995   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
```
```  1996   have iC: "independent C"
```
```  1997     by (simp add: dim_span)
```
```  1998   from C fB have "card C \<le> dim V"
```
```  1999     by simp
```
```  2000   moreover have "dim V \<le> card C"
```
```  2001     using span_card_ge_dim[OF CSV SVC C(1)]
```
```  2002     by (simp add: dim_span)
```
```  2003   ultimately have CdV: "card C = dim V"
```
```  2004     using C(1) by simp
```
```  2005   from C B CSV CdV iC show ?thesis
```
```  2006     by auto
```
```  2007 qed
```
```  2008
```
```  2009 lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
```
```  2010   using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
```
```  2011   by (auto simp add: span_span)
```
```  2012
```
```  2013 text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
```
```  2014
```
```  2015 lemma span_not_univ_orthogonal:
```
```  2016   fixes S :: "'a::euclidean_space set"
```
```  2017   assumes sU: "span S \<noteq> UNIV"
```
```  2018   shows "\<exists>(a::'a). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
```
```  2019 proof -
```
```  2020   from sU obtain a where a: "a \<notin> span S"
```
```  2021     by blast
```
```  2022   from orthogonal_basis_exists obtain B where
```
```  2023     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
```
```  2024     by blast
```
```  2025   from B have fB: "finite B" "card B = dim S"
```
```  2026     using independent_bound by auto
```
```  2027   from span_mono[OF B(2)] span_mono[OF B(3)]
```
```  2028   have sSB: "span S = span B"
```
```  2029     by (simp add: span_span)
```
```  2030   let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
```
```  2031   have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
```
```  2032     unfolding sSB
```
```  2033     apply (rule span_setsum[OF fB(1)])
```
```  2034     apply clarsimp
```
```  2035     apply (rule span_mul)
```
```  2036     apply (rule span_superset)
```
```  2037     apply assumption
```
```  2038     done
```
```  2039   with a have a0:"?a  \<noteq> 0"
```
```  2040     by auto
```
```  2041   have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
```
```  2042   proof (rule span_induct')
```
```  2043     show "subspace {x. ?a \<bullet> x = 0}"
```
```  2044       by (auto simp add: subspace_def inner_add)
```
```  2045   next
```
```  2046     {
```
```  2047       fix x
```
```  2048       assume x: "x \<in> B"
```
```  2049       from x have B': "B = insert x (B - {x})"
```
```  2050         by blast
```
```  2051       have fth: "finite (B - {x})"
```
```  2052         using fB by simp
```
```  2053       have "?a \<bullet> x = 0"
```
```  2054         apply (subst B')
```
```  2055         using fB fth
```
```  2056         unfolding setsum_clauses(2)[OF fth]
```
```  2057         apply simp unfolding inner_simps
```
```  2058         apply (clarsimp simp add: inner_add inner_setsum_left)
```
```  2059         apply (rule setsum_0', rule ballI)
```
```  2060         unfolding inner_commute
```
```  2061         apply (auto simp add: x field_simps
```
```  2062           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
```
```  2063         done
```
```  2064     }
```
```  2065     then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
```
```  2066       by blast
```
```  2067   qed
```
```  2068   with a0 show ?thesis
```
```  2069     unfolding sSB by (auto intro: exI[where x="?a"])
```
```  2070 qed
```
```  2071
```
```  2072 lemma span_not_univ_subset_hyperplane:
```
```  2073   fixes S :: "'a::euclidean_space set"
```
```  2074   assumes SU: "span S \<noteq> UNIV"
```
```  2075   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```  2076   using span_not_univ_orthogonal[OF SU] by auto
```
```  2077
```
```  2078 lemma lowdim_subset_hyperplane:
```
```  2079   fixes S :: "'a::euclidean_space set"
```
```  2080   assumes d: "dim S < DIM('a)"
```
```  2081   shows "\<exists>(a::'a). a  \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```  2082 proof -
```
```  2083   {
```
```  2084     assume "span S = UNIV"
```
```  2085     then have "dim (span S) = dim (UNIV :: ('a) set)"
```
```  2086       by simp
```
```  2087     then have "dim S = DIM('a)"
```
```  2088       by (simp add: dim_span dim_UNIV)
```
```  2089     with d have False by arith
```
```  2090   }
```
```  2091   then have th: "span S \<noteq> UNIV"
```
```  2092     by blast
```
```  2093   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
```
```  2094 qed
```
```  2095
```
```  2096 text {* We can extend a linear basis-basis injection to the whole set. *}
```
```  2097
```
```  2098 lemma linear_indep_image_lemma:
```
```  2099   assumes lf: "linear f"
```
```  2100     and fB: "finite B"
```
```  2101     and ifB: "independent (f ` B)"
```
```  2102     and fi: "inj_on f B"
```
```  2103     and xsB: "x \<in> span B"
```
```  2104     and fx: "f x = 0"
```
```  2105   shows "x = 0"
```
```  2106   using fB ifB fi xsB fx
```
```  2107 proof (induct arbitrary: x rule: finite_induct[OF fB])
```
```  2108   case 1
```
```  2109   then show ?case by auto
```
```  2110 next
```
```  2111   case (2 a b x)
```
```  2112   have fb: "finite b" using "2.prems" by simp
```
```  2113   have th0: "f ` b \<subseteq> f ` (insert a b)"
```
```  2114     apply (rule image_mono)
```
```  2115     apply blast
```
```  2116     done
```
```  2117   from independent_mono[ OF "2.prems"(2) th0]
```
```  2118   have ifb: "independent (f ` b)"  .
```
```  2119   have fib: "inj_on f b"
```
```  2120     apply (rule subset_inj_on [OF "2.prems"(3)])
```
```  2121     apply blast
```
```  2122     done
```
```  2123   from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
```
```  2124   obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
```
```  2125     by blast
```
```  2126   have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
```
```  2127     unfolding span_linear_image[OF lf]
```
```  2128     apply (rule imageI)
```
```  2129     using k span_mono[of "b - {a}" b]
```
```  2130     apply blast
```
```  2131     done
```
```  2132   then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
```
```  2133     by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
```
```  2134   then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
```
```  2135     using "2.prems"(5) by simp
```
```  2136   have xsb: "x \<in> span b"
```
```  2137   proof (cases "k = 0")
```
```  2138     case True
```
```  2139     with k have "x \<in> span (b - {a})" by simp
```
```  2140     then show ?thesis using span_mono[of "b - {a}" b]
```
```  2141       by blast
```
```  2142   next
```
```  2143     case False
```
```  2144     with span_mul[OF th, of "- 1/ k"]
```
```  2145     have th1: "f a \<in> span (f ` b)"
```
```  2146       by auto
```
```  2147     from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
```
```  2148     have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
```
```  2149     from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
```
```  2150     have "f a \<notin> span (f ` b)" using tha
```
```  2151       using "2.hyps"(2)
```
```  2152       "2.prems"(3) by auto
```
```  2153     with th1 have False by blast
```
```  2154     then show ?thesis by blast
```
```  2155   qed
```
```  2156   from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
```
```  2157 qed
```
```  2158
```
```  2159 text {* We can extend a linear mapping from basis. *}
```
```  2160
```
```  2161 lemma linear_independent_extend_lemma:
```
```  2162   fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
```
```  2163   assumes fi: "finite B"
```
```  2164     and ib: "independent B"
```
```  2165   shows "\<exists>g.
```
```  2166     (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y) \<and>
```
```  2167     (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
```
```  2168     (\<forall>x\<in> B. g x = f x)"
```
```  2169   using ib fi
```
```  2170 proof (induct rule: finite_induct[OF fi])
```
```  2171   case 1
```
```  2172   then show ?case by auto
```
```  2173 next
```
```  2174   case (2 a b)
```
```  2175   from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
```
```  2176     by (simp_all add: independent_insert)
```
```  2177   from "2.hyps"(3)[OF ibf] obtain g where
```
```  2178     g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
```
```  2179     "\<forall>x\<in>span b. \<forall>c. g (c *\<^sub>R x) = c *\<^sub>R g x" "\<forall>x\<in>b. g x = f x" by blast
```
```  2180   let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
```
```  2181   {
```
```  2182     fix z
```
```  2183     assume z: "z \<in> span (insert a b)"
```
```  2184     have th0: "z - ?h z *\<^sub>R a \<in> span b"
```
```  2185       apply (rule someI_ex)
```
```  2186       unfolding span_breakdown_eq[symmetric]
```
```  2187       apply (rule z)
```
```  2188       done
```
```  2189     {
```
```  2190       fix k
```
```  2191       assume k: "z - k *\<^sub>R a \<in> span b"
```
```  2192       have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
```
```  2193         by (simp add: field_simps scaleR_left_distrib [symmetric])
```
```  2194       from span_sub[OF th0 k] have khz: "(k - ?h z) *\<^sub>R a \<in> span b"
```
```  2195         by (simp add: eq)
```
```  2196       {
```
```  2197         assume "k \<noteq> ?h z"
```
```  2198         then have k0: "k - ?h z \<noteq> 0" by simp
```
```  2199         from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
```
```  2200         have "a \<in> span b" by simp
```
```  2201         with "2.prems"(1) "2.hyps"(2) have False
```
```  2202           by (auto simp add: dependent_def)
```
```  2203       }
```
```  2204       then have "k = ?h z" by blast
```
```  2205     }
```
```  2206     with th0 have "z - ?h z *\<^sub>R a \<in> span b \<and> (\<forall>k. z - k *\<^sub>R a \<in> span b \<longrightarrow> k = ?h z)"
```
```  2207       by blast
```
```  2208   }
```
```  2209   note h = this
```
```  2210   let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)"
```
```  2211   {
```
```  2212     fix x y
```
```  2213     assume x: "x \<in> span (insert a b)"
```
```  2214       and y: "y \<in> span (insert a b)"
```
```  2215     have tha: "\<And>(x::'a) y a k l. (x + y) - (k + l) *\<^sub>R a = (x - k *\<^sub>R a) + (y - l *\<^sub>R a)"
```
```  2216       by (simp add: algebra_simps)
```
```  2217     have addh: "?h (x + y) = ?h x + ?h y"
```
```  2218       apply (rule conjunct2[OF h, rule_format, symmetric])
```
```  2219       apply (rule span_add[OF x y])
```
```  2220       unfolding tha
```
```  2221       apply (metis span_add x y conjunct1[OF h, rule_format])
```
```  2222       done
```
```  2223     have "?g (x + y) = ?g x + ?g y"
```
```  2224       unfolding addh tha
```
```  2225       g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
```
```  2226       by (simp add: scaleR_left_distrib)}
```
```  2227   moreover
```
```  2228   {
```
```  2229     fix x :: "'a"
```
```  2230     fix c :: real
```
```  2231     assume x: "x \<in> span (insert a b)"
```
```  2232     have tha: "\<And>(x::'a) c k a. c *\<^sub>R x - (c * k) *\<^sub>R a = c *\<^sub>R (x - k *\<^sub>R a)"
```
```  2233       by (simp add: algebra_simps)
```
```  2234     have hc: "?h (c *\<^sub>R x) = c * ?h x"
```
```  2235       apply (rule conjunct2[OF h, rule_format, symmetric])
```
```  2236       apply (metis span_mul x)
```
```  2237       apply (metis tha span_mul x conjunct1[OF h])
```
```  2238       done
```
```  2239     have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
```
```  2240       unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
```
```  2241       by (simp add: algebra_simps)
```
```  2242   }
```
```  2243   moreover
```
```  2244   {
```
```  2245     fix x
```
```  2246     assume x: "x \<in> insert a b"
```
```  2247     {
```
```  2248       assume xa: "x = a"
```
```  2249       have ha1: "1 = ?h a"
```
```  2250         apply (rule conjunct2[OF h, rule_format])
```
```  2251         apply (metis span_superset insertI1)
```
```  2252         using conjunct1[OF h, OF span_superset, OF insertI1]
```
```  2253         apply (auto simp add: span_0)
```
```  2254         done
```
```  2255       from xa ha1[symmetric] have "?g x = f x"
```
```  2256         apply simp
```
```  2257         using g(2)[rule_format, OF span_0, of 0]
```
```  2258         apply simp
```
```  2259         done
```
```  2260     }
```
```  2261     moreover
```
```  2262     {
```
```  2263       assume xb: "x \<in> b"
```
```  2264       have h0: "0 = ?h x"
```
```  2265         apply (rule conjunct2[OF h, rule_format])
```
```  2266         apply (metis  span_superset x)
```
```  2267         apply simp
```
```  2268         apply (metis span_superset xb)
```
```  2269         done
```
```  2270       have "?g x = f x"
```
```  2271         by (simp add: h0[symmetric] g(3)[rule_format, OF xb])
```
```  2272     }
```
```  2273     ultimately have "?g x = f x"
```
```  2274       using x by blast
```
```  2275   }
```
```  2276   ultimately show ?case
```
```  2277     apply -
```
```  2278     apply (rule exI[where x="?g"])
```
```  2279     apply blast
```
```  2280     done
```
```  2281 qed
```
```  2282
```
```  2283 lemma linear_independent_extend:
```
```  2284   fixes B :: "'a::euclidean_space set"
```
```  2285   assumes iB: "independent B"
```
```  2286   shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
```
```  2287 proof -
```
```  2288   from maximal_independent_subset_extend[of B UNIV] iB
```
```  2289   obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C"
```
```  2290     by auto
```
```  2291
```
```  2292   from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
```
```  2293   obtain g where g:
```
```  2294     "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) \<and>
```
```  2295      (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
```
```  2296      (\<forall>x\<in> C. g x = f x)" by blast
```
```  2297   from g show ?thesis
```
```  2298     unfolding linear_iff
```
```  2299     using C
```
```  2300     apply clarsimp
```
```  2301     apply blast
```
```  2302     done
```
```  2303 qed
```
```  2304
```
```  2305 text {* Can construct an isomorphism between spaces of same dimension. *}
```
```  2306
```
```  2307 lemma subspace_isomorphism:
```
```  2308   fixes S :: "'a::euclidean_space set"
```
```  2309     and T :: "'b::euclidean_space set"
```
```  2310   assumes s: "subspace S"
```
```  2311     and t: "subspace T"
```
```  2312     and d: "dim S = dim T"
```
```  2313   shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
```
```  2314 proof -
```
```  2315   from basis_exists[of S] independent_bound
```
```  2316   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
```
```  2317     by blast
```
```  2318   from basis_exists[of T] independent_bound
```
```  2319   obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C"
```
```  2320     by blast
```
```  2321   from B(4) C(4) card_le_inj[of B C] d
```
```  2322   obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C`
```
```  2323     by auto
```
```  2324   from linear_independent_extend[OF B(2)]
```
```  2325   obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x"
```
```  2326     by blast
```
```  2327   from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
```
```  2328     by simp
```
```  2329   with B(4) C(4) have ceq: "card (f ` B) = card C"
```
```  2330     using d by simp
```
```  2331   have "g ` B = f ` B"
```
```  2332     using g(2) by (auto simp add: image_iff)
```
```  2333   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
```
```  2334   finally have gBC: "g ` B = C" .
```
```  2335   have gi: "inj_on g B"
```
```  2336     using f(2) g(2) by (auto simp add: inj_on_def)
```
```  2337   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
```
```  2338   {
```
```  2339     fix x y
```
```  2340     assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
```
```  2341     from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
```
```  2342       by blast+
```
```  2343     from gxy have th0: "g (x - y) = 0"
```
```  2344       by (simp add: linear_sub[OF g(1)])
```
```  2345     have th1: "x - y \<in> span B"
```
```  2346       using x' y' by (metis span_sub)
```
```  2347     have "x = y"
```
```  2348       using g0[OF th1 th0] by simp
```
```  2349   }
```
```  2350   then have giS: "inj_on g S"
```
```  2351     unfolding inj_on_def by blast
```
```  2352   from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)"
```
```  2353     by (simp add: span_linear_image[OF g(1)])
```
```  2354   also have "\<dots> = span C" unfolding gBC ..
```
```  2355   also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
```
```  2356   finally have gS: "g ` S = T" .
```
```  2357   from g(1) gS giS show ?thesis
```
```  2358     by blast
```
```  2359 qed
```
```  2360
```
```  2361 text {* Linear functions are equal on a subspace if they are on a spanning set. *}
```
```  2362
```
```  2363 lemma subspace_kernel:
```
```  2364   assumes lf: "linear f"
```
```  2365   shows "subspace {x. f x = 0}"
```
```  2366   apply (simp add: subspace_def)
```
```  2367   apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
```
```  2368   done
```
```  2369
```
```  2370 lemma linear_eq_0_span:
```
```  2371   assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
```
```  2372   shows "\<forall>x \<in> span B. f x = 0"
```
```  2373   using f0 subspace_kernel[OF lf]
```
```  2374   by (rule span_induct')
```
```  2375
```
```  2376 lemma linear_eq_0:
```
```  2377   assumes lf: "linear f"
```
```  2378     and SB: "S \<subseteq> span B"
```
```  2379     and f0: "\<forall>x\<in>B. f x = 0"
```
```  2380   shows "\<forall>x \<in> S. f x = 0"
```
```  2381   by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
```
```  2382
```
```  2383 lemma linear_eq:
```
```  2384   assumes lf: "linear f"
```
```  2385     and lg: "linear g"
```
```  2386     and S: "S \<subseteq> span B"
```
```  2387     and fg: "\<forall> x\<in> B. f x = g x"
```
```  2388   shows "\<forall>x\<in> S. f x = g x"
```
```  2389 proof -
```
```  2390   let ?h = "\<lambda>x. f x - g x"
```
```  2391   from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
```
```  2392   from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
```
```  2393   show ?thesis by simp
```
```  2394 qed
```
```  2395
```
```  2396 lemma linear_eq_stdbasis:
```
```  2397   assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> _)"
```
```  2398     and lg: "linear g"
```
```  2399     and fg: "\<forall>b\<in>Basis. f b = g b"
```
```  2400   shows "f = g"
```
```  2401   using linear_eq[OF lf lg, of _ Basis] fg by auto
```
```  2402
```
```  2403 text {* Similar results for bilinear functions. *}
```
```  2404
```
```  2405 lemma bilinear_eq:
```
```  2406   assumes bf: "bilinear f"
```
```  2407     and bg: "bilinear g"
```
```  2408     and SB: "S \<subseteq> span B"
```
```  2409     and TC: "T \<subseteq> span C"
```
```  2410     and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
```
```  2411   shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
```
```  2412 proof -
```
```  2413   let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
```
```  2414   from bf bg have sp: "subspace ?P"
```
```  2415     unfolding bilinear_def linear_iff subspace_def bf bg
```
```  2416     by (auto simp add: span_0 bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def
```
```  2417       intro: bilinear_ladd[OF bf])
```
```  2418
```
```  2419   have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
```
```  2420     apply (rule span_induct' [OF _ sp])
```
```  2421     apply (rule ballI)
```
```  2422     apply (rule span_induct')
```
```  2423     apply (simp add: fg)
```
```  2424     apply (auto simp add: subspace_def)
```
```  2425     using bf bg unfolding bilinear_def linear_iff
```
```  2426     apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
```
```  2427       intro: bilinear_ladd[OF bf])
```
```  2428     done
```
```  2429   then show ?thesis
```
```  2430     using SB TC by auto
```
```  2431 qed
```
```  2432
```
```  2433 lemma bilinear_eq_stdbasis:
```
```  2434   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
```
```  2435   assumes bf: "bilinear f"
```
```  2436     and bg: "bilinear g"
```
```  2437     and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
```
```  2438   shows "f = g"
```
```  2439   using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast
```
```  2440
```
```  2441 text {* Detailed theorems about left and right invertibility in general case. *}
```
```  2442
```
```  2443 lemma linear_injective_left_inverse:
```
```  2444   fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  2445   assumes lf: "linear f" and fi: "inj f"
```
```  2446   shows "\<exists>g. linear g \<and> g o f = id"
```
```  2447 proof -
```
```  2448   from linear_independent_extend[OF independent_injective_image, OF independent_Basis, OF lf fi]
```
```  2449   obtain h:: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x"
```
```  2450     by blast
```
```  2451   from h(2) have th: "\<forall>i\<in>Basis. (h \<circ> f) i = id i"
```
```  2452     using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
```
```  2453     by auto
```
```  2454   from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
```
```  2455   have "h o f = id" .
```
```  2456   then show ?thesis
```
```  2457     using h(1) by blast
```
```  2458 qed
```
```  2459
```
```  2460 lemma linear_surjective_right_inverse:
```
```  2461   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  2462   assumes lf: "linear f"
```
```  2463     and sf: "surj f"
```
```  2464   shows "\<exists>g. linear g \<and> f o g = id"
```
```  2465 proof -
```
```  2466   from linear_independent_extend[OF independent_Basis[where 'a='b],of "inv f"]
```
```  2467   obtain h:: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x\<in>Basis. h x = inv f x"
```
```  2468     by blast
```
```  2469   from h(2) have th: "\<forall>i\<in>Basis. (f o h) i = id i"
```
```  2470     using sf by (auto simp add: surj_iff_all)
```
```  2471   from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
```
```  2472   have "f o h = id" .
```
```  2473   then show ?thesis
```
```  2474     using h(1) by blast
```
```  2475 qed
```
```  2476
```
```  2477 text {* An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective. *}
```
```  2478
```
```  2479 lemma linear_injective_imp_surjective:
```
```  2480   fixes f::"'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2481   assumes lf: "linear f"
```
```  2482     and fi: "inj f"
```
```  2483   shows "surj f"
```
```  2484 proof -
```
```  2485   let ?U = "UNIV :: 'a set"
```
```  2486   from basis_exists[of ?U] obtain B
```
```  2487     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
```
```  2488     by blast
```
```  2489   from B(4) have d: "dim ?U = card B"
```
```  2490     by simp
```
```  2491   have th: "?U \<subseteq> span (f ` B)"
```
```  2492     apply (rule card_ge_dim_independent)
```
```  2493     apply blast
```
```  2494     apply (rule independent_injective_image[OF B(2) lf fi])
```
```  2495     apply (rule order_eq_refl)
```
```  2496     apply (rule sym)
```
```  2497     unfolding d
```
```  2498     apply (rule card_image)
```
```  2499     apply (rule subset_inj_on[OF fi])
```
```  2500     apply blast
```
```  2501     done
```
```  2502   from th show ?thesis
```
```  2503     unfolding span_linear_image[OF lf] surj_def
```
```  2504     using B(3) by blast
```
```  2505 qed
```
```  2506
```
```  2507 text {* And vice versa. *}
```
```  2508
```
```  2509 lemma surjective_iff_injective_gen:
```
```  2510   assumes fS: "finite S"
```
```  2511     and fT: "finite T"
```
```  2512     and c: "card S = card T"
```
```  2513     and ST: "f ` S \<subseteq> T"
```
```  2514   shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S"
```
```  2515   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2516 proof
```
```  2517   assume h: "?lhs"
```
```  2518   {
```
```  2519     fix x y
```
```  2520     assume x: "x \<in> S"
```
```  2521     assume y: "y \<in> S"
```
```  2522     assume f: "f x = f y"
```
```  2523     from x fS have S0: "card S \<noteq> 0"
```
```  2524       by auto
```
```  2525     have "x = y"
```
```  2526     proof (rule ccontr)
```
```  2527       assume xy: "\<not> ?thesis"
```
```  2528       have th: "card S \<le> card (f ` (S - {y}))"
```
```  2529         unfolding c
```
```  2530         apply (rule card_mono)
```
```  2531         apply (rule finite_imageI)
```
```  2532         using fS apply simp
```
```  2533         using h xy x y f unfolding subset_eq image_iff
```
```  2534         apply auto
```
```  2535         apply (case_tac "xa = f x")
```
```  2536         apply (rule bexI[where x=x])
```
```  2537         apply auto
```
```  2538         done
```
```  2539       also have " \<dots> \<le> card (S - {y})"
```
```  2540         apply (rule card_image_le)
```
```  2541         using fS by simp
```
```  2542       also have "\<dots> \<le> card S - 1" using y fS by simp
```
```  2543       finally show False using S0 by arith
```
```  2544     qed
```
```  2545   }
```
```  2546   then show ?rhs
```
```  2547     unfolding inj_on_def by blast
```
```  2548 next
```
```  2549   assume h: ?rhs
```
```  2550   have "f ` S = T"
```
```  2551     apply (rule card_subset_eq[OF fT ST])
```
```  2552     unfolding card_image[OF h]
```
```  2553     apply (rule c)
```
```  2554     done
```
```  2555   then show ?lhs by blast
```
```  2556 qed
```
```  2557
```
```  2558 lemma linear_surjective_imp_injective:
```
```  2559   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2560   assumes lf: "linear f"
```
```  2561     and sf: "surj f"
```
```  2562   shows "inj f"
```
```  2563 proof -
```
```  2564   let ?U = "UNIV :: 'a set"
```
```  2565   from basis_exists[of ?U] obtain B
```
```  2566     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
```
```  2567     by blast
```
```  2568   {
```
```  2569     fix x
```
```  2570     assume x: "x \<in> span B"
```
```  2571     assume fx: "f x = 0"
```
```  2572     from B(2) have fB: "finite B"
```
```  2573       using independent_bound by auto
```
```  2574     have fBi: "independent (f ` B)"
```
```  2575       apply (rule card_le_dim_spanning[of "f ` B" ?U])
```
```  2576       apply blast
```
```  2577       using sf B(3)
```
```  2578       unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
```
```  2579       apply blast
```
```  2580       using fB apply blast
```
```  2581       unfolding d[symmetric]
```
```  2582       apply (rule card_image_le)
```
```  2583       apply (rule fB)
```
```  2584       done
```
```  2585     have th0: "dim ?U \<le> card (f ` B)"
```
```  2586       apply (rule span_card_ge_dim)
```
```  2587       apply blast
```
```  2588       unfolding span_linear_image[OF lf]
```
```  2589       apply (rule subset_trans[where B = "f ` UNIV"])
```
```  2590       using sf unfolding surj_def
```
```  2591       apply blast
```
```  2592       apply (rule image_mono)
```
```  2593       apply (rule B(3))
```
```  2594       apply (metis finite_imageI fB)
```
```  2595       done
```
```  2596     moreover have "card (f ` B) \<le> card B"
```
```  2597       by (rule card_image_le, rule fB)
```
```  2598     ultimately have th1: "card B = card (f ` B)"
```
```  2599       unfolding d by arith
```
```  2600     have fiB: "inj_on f B"
```
```  2601       unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric]
```
```  2602       by blast
```
```  2603     from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
```
```  2604     have "x = 0" by blast
```
```  2605   }
```
```  2606   then show ?thesis
```
```  2607     unfolding linear_injective_0[OF lf]
```
```  2608     using B(3)
```
```  2609     by blast
```
```  2610 qed
```
```  2611
```
```  2612 text {* Hence either is enough for isomorphism. *}
```
```  2613
```
```  2614 lemma left_right_inverse_eq:
```
```  2615   assumes fg: "f \<circ> g = id"
```
```  2616     and gh: "g \<circ> h = id"
```
```  2617   shows "f = h"
```
```  2618 proof -
```
```  2619   have "f = f \<circ> (g \<circ> h)"
```
```  2620     unfolding gh by simp
```
```  2621   also have "\<dots> = (f \<circ> g) \<circ> h"
```
```  2622     by (simp add: o_assoc)
```
```  2623   finally show "f = h"
```
```  2624     unfolding fg by simp
```
```  2625 qed
```
```  2626
```
```  2627 lemma isomorphism_expand:
```
```  2628   "f \<circ> g = id \<and> g \<circ> f = id \<longleftrightarrow> (\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x)"
```
```  2629   by (simp add: fun_eq_iff o_def id_def)
```
```  2630
```
```  2631 lemma linear_injective_isomorphism:
```
```  2632   fixes f::"'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2633   assumes lf: "linear f"
```
```  2634     and fi: "inj f"
```
```  2635   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
```
```  2636   unfolding isomorphism_expand[symmetric]
```
```  2637   using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]]
```
```  2638     linear_injective_left_inverse[OF lf fi]
```
```  2639   by (metis left_right_inverse_eq)
```
```  2640
```
```  2641 lemma linear_surjective_isomorphism:
```
```  2642   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2643   assumes lf: "linear f"
```
```  2644     and sf: "surj f"
```
```  2645   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
```
```  2646   unfolding isomorphism_expand[symmetric]
```
```  2647   using linear_surjective_right_inverse[OF lf sf]
```
```  2648     linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
```
```  2649   by (metis left_right_inverse_eq)
```
```  2650
```
```  2651 text {* Left and right inverses are the same for
```
```  2652   @{typ "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"}. *}
```
```  2653
```
```  2654 lemma linear_inverse_left:
```
```  2655   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2656   assumes lf: "linear f"
```
```  2657     and lf': "linear f'"
```
```  2658   shows "f \<circ> f' = id \<longleftrightarrow> f' \<circ> f = id"
```
```  2659 proof -
```
```  2660   {
```
```  2661     fix f f':: "'a \<Rightarrow> 'a"
```
```  2662     assume lf: "linear f" "linear f'"
```
```  2663     assume f: "f \<circ> f' = id"
```
```  2664     from f have sf: "surj f"
```
```  2665       apply (auto simp add: o_def id_def surj_def)
```
```  2666       apply metis
```
```  2667       done
```
```  2668     from linear_surjective_isomorphism[OF lf(1) sf] lf f
```
```  2669     have "f' \<circ> f = id"
```
```  2670       unfolding fun_eq_iff o_def id_def by metis
```
```  2671   }
```
```  2672   then show ?thesis
```
```  2673     using lf lf' by metis
```
```  2674 qed
```
```  2675
```
```  2676 text {* Moreover, a one-sided inverse is automatically linear. *}
```
```  2677
```
```  2678 lemma left_inverse_linear:
```
```  2679   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2680   assumes lf: "linear f"
```
```  2681     and gf: "g \<circ> f = id"
```
```  2682   shows "linear g"
```
```  2683 proof -
```
```  2684   from gf have fi: "inj f"
```
```  2685     apply (auto simp add: inj_on_def o_def id_def fun_eq_iff)
```
```  2686     apply metis
```
```  2687     done
```
```  2688   from linear_injective_isomorphism[OF lf fi]
```
```  2689   obtain h :: "'a \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x"
```
```  2690     by blast
```
```  2691   have "h = g"
```
```  2692     apply (rule ext) using gf h(2,3)
```
```  2693     apply (simp add: o_def id_def fun_eq_iff)
```
```  2694     apply metis
```
```  2695     done
```
```  2696   with h(1) show ?thesis by blast
```
```  2697 qed
```
```  2698
```
```  2699
```
```  2700 subsection {* Infinity norm *}
```
```  2701
```
```  2702 definition "infnorm (x::'a::euclidean_space) = Sup {abs (x \<bullet> b) |b. b \<in> Basis}"
```
```  2703
```
```  2704 lemma infnorm_set_image:
```
```  2705   fixes x :: "'a::euclidean_space"
```
```  2706   shows "{abs (x \<bullet> i) |i. i \<in> Basis} = (\<lambda>i. abs (x \<bullet> i)) ` Basis"
```
```  2707   by blast
```
```  2708
```
```  2709 lemma infnorm_Max:
```
```  2710   fixes x :: "'a::euclidean_space"
```
```  2711   shows "infnorm x = Max ((\<lambda>i. abs (x \<bullet> i)) ` Basis)"
```
```  2712   by (simp add: infnorm_def infnorm_set_image cSup_eq_Max del: Sup_image_eq)
```
```  2713
```
```  2714 lemma infnorm_set_lemma:
```
```  2715   fixes x :: "'a::euclidean_space"
```
```  2716   shows "finite {abs (x \<bullet> i) |i. i \<in> Basis}"
```
```  2717     and "{abs (x \<bullet> i) |i. i \<in> Basis} \<noteq> {}"
```
```  2718   unfolding infnorm_set_image
```
```  2719   by auto
```
```  2720
```
```  2721 lemma infnorm_pos_le:
```
```  2722   fixes x :: "'a::euclidean_space"
```
```  2723   shows "0 \<le> infnorm x"
```
```  2724   by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
```
```  2725
```
```  2726 lemma infnorm_triangle:
```
```  2727   fixes x :: "'a::euclidean_space"
```
```  2728   shows "infnorm (x + y) \<le> infnorm x + infnorm y"
```
```  2729 proof -
```
```  2730   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"
```
```  2731     by simp
```
```  2732   show ?thesis
```
```  2733     by (auto simp: infnorm_Max inner_add_left intro!: *)
```
```  2734 qed
```
```  2735
```
```  2736 lemma infnorm_eq_0:
```
```  2737   fixes x :: "'a::euclidean_space"
```
```  2738   shows "infnorm x = 0 \<longleftrightarrow> x = 0"
```
```  2739 proof -
```
```  2740   have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
```
```  2741     unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
```
```  2742   then show ?thesis
```
```  2743     using infnorm_pos_le[of x] by simp
```
```  2744 qed
```
```  2745
```
```  2746 lemma infnorm_0: "infnorm 0 = 0"
```
```  2747   by (simp add: infnorm_eq_0)
```
```  2748
```
```  2749 lemma infnorm_neg: "infnorm (- x) = infnorm x"
```
```  2750   unfolding infnorm_def
```
```  2751   apply (rule cong[of "Sup" "Sup"])
```
```  2752   apply blast
```
```  2753   apply auto
```
```  2754   done
```
```  2755
```
```  2756 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
```
```  2757 proof -
```
```  2758   have "y - x = - (x - y)" by simp
```
```  2759   then show ?thesis
```
```  2760     by (metis infnorm_neg)
```
```  2761 qed
```
```  2762
```
```  2763 lemma real_abs_sub_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
```
```  2764 proof -
```
```  2765   have th: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny <= n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
```
```  2766     by arith
```
```  2767   from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
```
```  2768   have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
```
```  2769     "infnorm y \<le> infnorm (x - y) + infnorm x"
```
```  2770     by (simp_all add: field_simps infnorm_neg)
```
```  2771   from th[OF ths] show ?thesis .
```
```  2772 qed
```
```  2773
```
```  2774 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
```
```  2775   using infnorm_pos_le[of x] by arith
```
```  2776
```
```  2777 lemma Basis_le_infnorm:
```
```  2778   fixes x :: "'a::euclidean_space"
```
```  2779   shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
```
```  2780   by (simp add: infnorm_Max)
```
```  2781
```
```  2782 lemma infnorm_mul: "infnorm (a *\<^sub>R x) = abs a * infnorm x"
```
```  2783   unfolding infnorm_Max
```
```  2784 proof (safe intro!: Max_eqI)
```
```  2785   let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```  2786   {
```
```  2787     fix b :: 'a
```
```  2788     assume "b \<in> Basis"
```
```  2789     then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
```
```  2790       by (simp add: abs_mult mult_left_mono)
```
```  2791   next
```
```  2792     from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
```
```  2793       by (auto simp del: Max_in)
```
```  2794     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"
```
```  2795       by (intro image_eqI[where x=b]) (auto simp: abs_mult)
```
```  2796   }
```
```  2797 qed simp
```
```  2798
```
```  2799 lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
```
```  2800   unfolding infnorm_mul ..
```
```  2801
```
```  2802 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
```
```  2803   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
```
```  2804
```
```  2805 text {* Prove that it differs only up to a bound from Euclidean norm. *}
```
```  2806
```
```  2807 lemma infnorm_le_norm: "infnorm x \<le> norm x"
```
```  2808   by (simp add: Basis_le_norm infnorm_Max)
```
```  2809
```
```  2810 lemma (in euclidean_space) euclidean_inner: "inner x y = (\<Sum>b\<in>Basis. (x \<bullet> b) * (y \<bullet> b))"
```
```  2811   by (subst (1 2) euclidean_representation[symmetric])
```
```  2812     (simp add: inner_setsum_left inner_setsum_right setsum_cases inner_Basis ac_simps if_distrib)
```
```  2813
```
```  2814 lemma norm_le_infnorm:
```
```  2815   fixes x :: "'a::euclidean_space"
```
```  2816   shows "norm x \<le> sqrt DIM('a) * infnorm x"
```
```  2817 proof -
```
```  2818   let ?d = "DIM('a)"
```
```  2819   have "real ?d \<ge> 0"
```
```  2820     by simp
```
```  2821   then have d2: "(sqrt (real ?d))\<^sup>2 = real ?d"
```
```  2822     by (auto intro: real_sqrt_pow2)
```
```  2823   have th: "sqrt (real ?d) * infnorm x \<ge> 0"
```
```  2824     by (simp add: zero_le_mult_iff infnorm_pos_le)
```
```  2825   have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)\<^sup>2"
```
```  2826     unfolding power_mult_distrib d2
```
```  2827     unfolding real_of_nat_def
```
```  2828     apply (subst euclidean_inner)
```
```  2829     apply (subst power2_abs[symmetric])
```
```  2830     apply (rule order_trans[OF setsum_bounded[where K="\<bar>infnorm x\<bar>\<^sup>2"]])
```
```  2831     apply (auto simp add: power2_eq_square[symmetric])
```
```  2832     apply (subst power2_abs[symmetric])
```
```  2833     apply (rule power_mono)
```
```  2834     apply (auto simp: infnorm_Max)
```
```  2835     done
```
```  2836   from real_le_lsqrt[OF inner_ge_zero th th1]
```
```  2837   show ?thesis
```
```  2838     unfolding norm_eq_sqrt_inner id_def .
```
```  2839 qed
```
```  2840
```
```  2841 lemma tendsto_infnorm [tendsto_intros]:
```
```  2842   assumes "(f ---> a) F"
```
```  2843   shows "((\<lambda>x. infnorm (f x)) ---> infnorm a) F"
```
```  2844 proof (rule tendsto_compose [OF LIM_I assms])
```
```  2845   fix r :: real
```
```  2846   assume "r > 0"
```
```  2847   then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
```
```  2848     by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm)
```
```  2849 qed
```
```  2850
```
```  2851 text {* Equality in Cauchy-Schwarz and triangle inequalities. *}
```
```  2852
```
```  2853 lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  2854   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2855 proof -
```
```  2856   {
```
```  2857     assume h: "x = 0"
```
```  2858     then have ?thesis by simp
```
```  2859   }
```
```  2860   moreover
```
```  2861   {
```
```  2862     assume h: "y = 0"
```
```  2863     then have ?thesis by simp
```
```  2864   }
```
```  2865   moreover
```
```  2866   {
```
```  2867     assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
```
```  2868     from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
```
```  2869     have "?rhs \<longleftrightarrow>
```
```  2870       (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
```
```  2871         norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
```
```  2872       using x y
```
```  2873       unfolding inner_simps
```
```  2874       unfolding power2_norm_eq_inner[symmetric] power2_eq_square right_minus_eq
```
```  2875       apply (simp add: inner_commute)
```
```  2876       apply (simp add: field_simps)
```
```  2877       apply metis
```
```  2878       done
```
```  2879     also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
```
```  2880       by (simp add: field_simps inner_commute)
```
```  2881     also have "\<dots> \<longleftrightarrow> ?lhs" using x y
```
```  2882       apply simp
```
```  2883       apply metis
```
```  2884       done
```
```  2885     finally have ?thesis by blast
```
```  2886   }
```
```  2887   ultimately show ?thesis by blast
```
```  2888 qed
```
```  2889
```
```  2890 lemma norm_cauchy_schwarz_abs_eq:
```
```  2891   "abs (x \<bullet> y) = norm x * norm y \<longleftrightarrow>
```
```  2892     norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
```
```  2893   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2894 proof -
```
```  2895   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a"
```
```  2896     by arith
```
```  2897   have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
```
```  2898     by simp
```
```  2899   also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
```
```  2900     unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  2901     unfolding norm_minus_cancel norm_scaleR ..
```
```  2902   also have "\<dots> \<longleftrightarrow> ?lhs"
```
```  2903     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
```
```  2904     by auto
```
```  2905   finally show ?thesis ..
```
```  2906 qed
```
```  2907
```
```  2908 lemma norm_triangle_eq:
```
```  2909   fixes x y :: "'a::real_inner"
```
```  2910   shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  2911 proof -
```
```  2912   {
```
```  2913     assume x: "x = 0 \<or> y = 0"
```
```  2914     then have ?thesis
```
```  2915       by (cases "x = 0") simp_all
```
```  2916   }
```
```  2917   moreover
```
```  2918   {
```
```  2919     assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
```
```  2920     then have "norm x \<noteq> 0" "norm y \<noteq> 0"
```
```  2921       by simp_all
```
```  2922     then have n: "norm x > 0" "norm y > 0"
```
```  2923       using norm_ge_zero[of x] norm_ge_zero[of y] by arith+
```
```  2924     have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 \<Longrightarrow> a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2"
```
```  2925       by algebra
```
```  2926     have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
```
```  2927       apply (rule th)
```
```  2928       using n norm_ge_zero[of "x + y"]
```
```  2929       apply arith
```
```  2930       done
```
```  2931     also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  2932       unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  2933       unfolding power2_norm_eq_inner inner_simps
```
```  2934       by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
```
```  2935     finally have ?thesis .
```
```  2936   }
```
```  2937   ultimately show ?thesis by blast
```
```  2938 qed
```
```  2939
```
```  2940
```
```  2941 subsection {* Collinearity *}
```
```  2942
```
```  2943 definition collinear :: "'a::real_vector set \<Rightarrow> bool"
```
```  2944   where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
```
```  2945
```
```  2946 lemma collinear_empty: "collinear {}"
```
```  2947   by (simp add: collinear_def)
```
```  2948
```
```  2949 lemma collinear_sing: "collinear {x}"
```
```  2950   by (simp add: collinear_def)
```
```  2951
```
```  2952 lemma collinear_2: "collinear {x, y}"
```
```  2953   apply (simp add: collinear_def)
```
```  2954   apply (rule exI[where x="x - y"])
```
```  2955   apply auto
```
```  2956   apply (rule exI[where x=1], simp)
```
```  2957   apply (rule exI[where x="- 1"], simp)
```
```  2958   done
```
```  2959
```
```  2960 lemma collinear_lemma: "collinear {0,x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
```
```  2961   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2962 proof -
```
```  2963   {
```
```  2964     assume "x = 0 \<or> y = 0"
```
```  2965     then have ?thesis
```
```  2966       by (cases "x = 0") (simp_all add: collinear_2 insert_commute)
```
```  2967   }
```
```  2968   moreover
```
```  2969   {
```
```  2970     assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
```
```  2971     have ?thesis
```
```  2972     proof
```
```  2973       assume h: "?lhs"
```
```  2974       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"
```
```  2975         unfolding collinear_def by blast
```
```  2976       from u[rule_format, of x 0] u[rule_format, of y 0]
```
```  2977       obtain cx and cy where
```
```  2978         cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
```
```  2979         by auto
```
```  2980       from cx x have cx0: "cx \<noteq> 0" by auto
```
```  2981       from cy y have cy0: "cy \<noteq> 0" by auto
```
```  2982       let ?d = "cy / cx"
```
```  2983       from cx cy cx0 have "y = ?d *\<^sub>R x"
```
```  2984         by simp
```
```  2985       then show ?rhs using x y by blast
```
```  2986     next
```
```  2987       assume h: "?rhs"
```
```  2988       then obtain c where c: "y = c *\<^sub>R x"
```
```  2989         using x y by blast
```
```  2990       show ?lhs
```
```  2991         unfolding collinear_def c
```
```  2992         apply (rule exI[where x=x])
```
```  2993         apply auto
```
```  2994         apply (rule exI[where x="- 1"], simp)
```
```  2995         apply (rule exI[where x= "-c"], simp)
```
```  2996         apply (rule exI[where x=1], simp)
```
```  2997         apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
```
```  2998         apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
```
```  2999         done
```
```  3000     qed
```
```  3001   }
```
```  3002   ultimately show ?thesis by blast
```
```  3003 qed
```
```  3004
```
```  3005 lemma norm_cauchy_schwarz_equal: "abs (x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
```
```  3006   unfolding norm_cauchy_schwarz_abs_eq
```
```  3007   apply (cases "x=0", simp_all add: collinear_2)
```
```  3008   apply (cases "y=0", simp_all add: collinear_2 insert_commute)
```
```  3009   unfolding collinear_lemma
```
```  3010   apply simp
```
```  3011   apply (subgoal_tac "norm x \<noteq> 0")
```
```  3012   apply (subgoal_tac "norm y \<noteq> 0")
```
```  3013   apply (rule iffI)
```
```  3014   apply (cases "norm x *\<^sub>R y = norm y *\<^sub>R x")
```
```  3015   apply (rule exI[where x="(1/norm x) * norm y"])
```
```  3016   apply (drule sym)
```
```  3017   unfolding scaleR_scaleR[symmetric]
```
```  3018   apply (simp add: field_simps)
```
```  3019   apply (rule exI[where x="(1/norm x) * - norm y"])
```
```  3020   apply clarify
```
```  3021   apply (drule sym)
```
```  3022   unfolding scaleR_scaleR[symmetric]
```
```  3023   apply (simp add: field_simps)
```
```  3024   apply (erule exE)
```
```  3025   apply (erule ssubst)
```
```  3026   unfolding scaleR_scaleR
```
```  3027   unfolding norm_scaleR
```
```  3028   apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
```
```  3029   apply (auto simp add: field_simps)
```
```  3030   done
```
```  3031
```
```  3032 end
```
```  3033
```