src/HOL/Multivariate_Analysis/Linear_Algebra.thy
 author paulson Thu May 28 14:33:35 2015 +0100 (2015-05-28) changeset 60307 75e1aa7a450e parent 60303 00c06f1315d0 child 60420 884f54e01427 permissions -rw-r--r--
Convex hulls: theorems about interior, etc. And a few simple lemmas.
```     1 (*  Title:      HOL/Multivariate_Analysis/Linear_Algebra.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3 *)
```
```     4
```
```     5 section {* 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. \<bar>y - x\<bar> < d \<longrightarrow> \<bar>y * y - x * x\<bar> < e)"
```
```    31   using isCont_power[OF continuous_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
```
```    32   by (force simp add: power2_eq_square)
```
```    33
```
```    34 text{* Hence derive more interesting properties of the norm. *}
```
```    35
```
```    36 lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
```
```    37   by simp (* TODO: delete *)
```
```    38
```
```    39 lemma norm_triangle_sub:
```
```    40   fixes x y :: "'a::real_normed_vector"
```
```    41   shows "norm x \<le> norm y + norm (x - y)"
```
```    42   using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
```
```    43
```
```    44 lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
```
```    45   by (simp add: norm_eq_sqrt_inner)
```
```    46
```
```    47 lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
```
```    48   by (simp add: norm_eq_sqrt_inner)
```
```    49
```
```    50 lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
```
```    51   apply (subst order_eq_iff)
```
```    52   apply (auto simp: norm_le)
```
```    53   done
```
```    54
```
```    55 lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
```
```    56   by (simp add: norm_eq_sqrt_inner)
```
```    57
```
```    58 text{* Squaring equations and inequalities involving norms.  *}
```
```    59
```
```    60 lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
```
```    61   by (simp only: power2_norm_eq_inner) (* TODO: move? *)
```
```    62
```
```    63 lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
```
```    64   by (auto simp add: norm_eq_sqrt_inner)
```
```    65
```
```    66 lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2"
```
```    67   apply (simp add: dot_square_norm abs_le_square_iff[symmetric])
```
```    68   using norm_ge_zero[of x]
```
```    69   apply arith
```
```    70   done
```
```    71
```
```    72 lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2"
```
```    73   apply (simp add: dot_square_norm abs_le_square_iff[symmetric])
```
```    74   using norm_ge_zero[of x]
```
```    75   apply arith
```
```    76   done
```
```    77
```
```    78 lemma norm_lt_square: "norm x < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2"
```
```    79   by (metis not_le norm_ge_square)
```
```    80
```
```    81 lemma norm_gt_square: "norm x > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2"
```
```    82   by (metis norm_le_square not_less)
```
```    83
```
```    84 text{* Dot product in terms of the norm rather than conversely. *}
```
```    85
```
```    86 lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
```
```    87   inner_scaleR_left inner_scaleR_right
```
```    88
```
```    89 lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
```
```    90   unfolding power2_norm_eq_inner inner_simps inner_commute by auto
```
```    91
```
```    92 lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
```
```    93   unfolding power2_norm_eq_inner inner_simps inner_commute
```
```    94   by (auto simp add: algebra_simps)
```
```    95
```
```    96 text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
```
```    97
```
```    98 lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
```
```    99   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   100 proof
```
```   101   assume ?lhs
```
```   102   then show ?rhs by simp
```
```   103 next
```
```   104   assume ?rhs
```
```   105   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
```
```   106     by simp
```
```   107   then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
```
```   108     by (simp add: inner_diff inner_commute)
```
```   109   then have "(x - y) \<bullet> (x - y) = 0"
```
```   110     by (simp add: field_simps inner_diff inner_commute)
```
```   111   then show "x = y" by simp
```
```   112 qed
```
```   113
```
```   114 lemma norm_triangle_half_r:
```
```   115   "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
```
```   116   using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
```
```   117
```
```   118 lemma norm_triangle_half_l:
```
```   119   assumes "norm (x - y) < e / 2"
```
```   120     and "norm (x' - y) < e / 2"
```
```   121   shows "norm (x - x') < e"
```
```   122   using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
```
```   123   unfolding dist_norm[symmetric] .
```
```   124
```
```   125 lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
```
```   126   by (rule norm_triangle_ineq [THEN order_trans])
```
```   127
```
```   128 lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
```
```   129   by (rule norm_triangle_ineq [THEN le_less_trans])
```
```   130
```
```   131 lemma setsum_clauses:
```
```   132   shows "setsum f {} = 0"
```
```   133     and "finite S \<Longrightarrow> setsum f (insert x S) = (if x \<in> S then setsum f S else f x + setsum f S)"
```
```   134   by (auto simp add: insert_absorb)
```
```   135
```
```   136 lemma setsum_norm_le:
```
```   137   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   138   assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
```
```   139   shows "norm (setsum f S) \<le> setsum g S"
```
```   140   by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)
```
```   141
```
```   142 lemma setsum_norm_bound:
```
```   143   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   144   assumes K: "\<forall>x \<in> S. norm (f x) \<le> K"
```
```   145   shows "norm (setsum f S) \<le> of_nat (card S) * K"
```
```   146   using setsum_norm_le[OF K] setsum_constant[symmetric]
```
```   147   by simp
```
```   148
```
```   149 lemma setsum_group:
```
```   150   assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
```
```   151   shows "setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) T = setsum g S"
```
```   152   apply (subst setsum_image_gen[OF fS, of g f])
```
```   153   apply (rule setsum.mono_neutral_right[OF fT fST])
```
```   154   apply (auto intro: setsum.neutral)
```
```   155   done
```
```   156
```
```   157 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
```
```   158 proof
```
```   159   assume "\<forall>x. x \<bullet> y = x \<bullet> z"
```
```   160   then have "\<forall>x. x \<bullet> (y - z) = 0"
```
```   161     by (simp add: inner_diff)
```
```   162   then have "(y - z) \<bullet> (y - z) = 0" ..
```
```   163   then show "y = z" by simp
```
```   164 qed simp
```
```   165
```
```   166 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
```
```   167 proof
```
```   168   assume "\<forall>z. x \<bullet> z = y \<bullet> z"
```
```   169   then have "\<forall>z. (x - y) \<bullet> z = 0"
```
```   170     by (simp add: inner_diff)
```
```   171   then have "(x - y) \<bullet> (x - y) = 0" ..
```
```   172   then show "x = y" by simp
```
```   173 qed simp
```
```   174
```
```   175
```
```   176 subsection {* Orthogonality. *}
```
```   177
```
```   178 context real_inner
```
```   179 begin
```
```   180
```
```   181 definition "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
```
```   182
```
```   183 lemma orthogonal_clauses:
```
```   184   "orthogonal a 0"
```
```   185   "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
```
```   186   "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
```
```   187   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
```
```   188   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
```
```   189   "orthogonal 0 a"
```
```   190   "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
```
```   191   "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
```
```   192   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
```
```   193   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
```
```   194   unfolding orthogonal_def inner_add inner_diff by auto
```
```   195
```
```   196 end
```
```   197
```
```   198 lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
```
```   199   by (simp add: orthogonal_def inner_commute)
```
```   200
```
```   201
```
```   202 subsection {* Linear functions. *}
```
```   203
```
```   204 lemma linear_iff:
```
```   205   "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)"
```
```   206   (is "linear f \<longleftrightarrow> ?rhs")
```
```   207 proof
```
```   208   assume "linear f"
```
```   209   then interpret f: linear f .
```
```   210   show "?rhs" by (simp add: f.add f.scaleR)
```
```   211 next
```
```   212   assume "?rhs"
```
```   213   then show "linear f" by unfold_locales simp_all
```
```   214 qed
```
```   215
```
```   216 lemma linear_compose_cmul: "linear f \<Longrightarrow> linear (\<lambda>x. c *\<^sub>R f x)"
```
```   217   by (simp add: linear_iff algebra_simps)
```
```   218
```
```   219 lemma linear_compose_neg: "linear f \<Longrightarrow> linear (\<lambda>x. - f x)"
```
```   220   by (simp add: linear_iff)
```
```   221
```
```   222 lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x + g x)"
```
```   223   by (simp add: linear_iff algebra_simps)
```
```   224
```
```   225 lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x - g x)"
```
```   226   by (simp add: linear_iff algebra_simps)
```
```   227
```
```   228 lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> f)"
```
```   229   by (simp add: linear_iff)
```
```   230
```
```   231 lemma linear_id: "linear id"
```
```   232   by (simp add: linear_iff id_def)
```
```   233
```
```   234 lemma linear_zero: "linear (\<lambda>x. 0)"
```
```   235   by (simp add: linear_iff)
```
```   236
```
```   237 lemma linear_compose_setsum:
```
```   238   assumes lS: "\<forall>a \<in> S. linear (f a)"
```
```   239   shows "linear (\<lambda>x. setsum (\<lambda>a. f a x) S)"
```
```   240 proof (cases "finite S")
```
```   241   case True
```
```   242   then show ?thesis
```
```   243     using lS by induct (simp_all add: linear_zero linear_compose_add)
```
```   244 next
```
```   245   case False
```
```   246   then show ?thesis
```
```   247     by (simp add: linear_zero)
```
```   248 qed
```
```   249
```
```   250 lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
```
```   251   unfolding linear_iff
```
```   252   apply clarsimp
```
```   253   apply (erule allE[where x="0::'a"])
```
```   254   apply simp
```
```   255   done
```
```   256
```
```   257 lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```   258   by (simp add: linear_iff)
```
```   259
```
```   260 lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x"
```
```   261   using linear_cmul [where c="-1"] by simp
```
```   262
```
```   263 lemma linear_add: "linear f \<Longrightarrow> f (x + y) = f x + f y"
```
```   264   by (metis linear_iff)
```
```   265
```
```   266 lemma linear_sub: "linear f \<Longrightarrow> f (x - y) = f x - f y"
```
```   267   using linear_add [of f x "- y"] by (simp add: linear_neg)
```
```   268
```
```   269 lemma linear_setsum:
```
```   270   assumes f: "linear f"
```
```   271   shows "f (setsum g S) = setsum (f \<circ> g) S"
```
```   272 proof (cases "finite S")
```
```   273   case True
```
```   274   then show ?thesis
```
```   275     by induct (simp_all add: linear_0 [OF f] linear_add [OF f])
```
```   276 next
```
```   277   case False
```
```   278   then show ?thesis
```
```   279     by (simp add: linear_0 [OF f])
```
```   280 qed
```
```   281
```
```   282 lemma linear_setsum_mul:
```
```   283   assumes lin: "linear f"
```
```   284   shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
```
```   285   using linear_setsum[OF lin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
```
```   286   by simp
```
```   287
```
```   288 lemma linear_injective_0:
```
```   289   assumes lin: "linear f"
```
```   290   shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
```
```   291 proof -
```
```   292   have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)"
```
```   293     by (simp add: inj_on_def)
```
```   294   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)"
```
```   295     by simp
```
```   296   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
```
```   297     by (simp add: linear_sub[OF lin])
```
```   298   also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
```
```   299     by auto
```
```   300   finally show ?thesis .
```
```   301 qed
```
```   302
```
```   303
```
```   304 subsection {* Bilinear functions. *}
```
```   305
```
```   306 definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
```
```   307
```
```   308 lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
```
```   309   by (simp add: bilinear_def linear_iff)
```
```   310
```
```   311 lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
```
```   312   by (simp add: bilinear_def linear_iff)
```
```   313
```
```   314 lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
```
```   315   by (simp add: bilinear_def linear_iff)
```
```   316
```
```   317 lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
```
```   318   by (simp add: bilinear_def linear_iff)
```
```   319
```
```   320 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
```
```   321   by (drule bilinear_lmul [of _ "- 1"]) simp
```
```   322
```
```   323 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
```
```   324   by (drule bilinear_rmul [of _ _ "- 1"]) simp
```
```   325
```
```   326 lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
```
```   327   using add_left_imp_eq[of x y 0] by auto
```
```   328
```
```   329 lemma bilinear_lzero:
```
```   330   assumes "bilinear h"
```
```   331   shows "h 0 x = 0"
```
```   332   using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
```
```   333
```
```   334 lemma bilinear_rzero:
```
```   335   assumes "bilinear h"
```
```   336   shows "h x 0 = 0"
```
```   337   using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
```
```   338
```
```   339 lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
```
```   340   using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
```
```   341
```
```   342 lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
```
```   343   using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
```
```   344
```
```   345 lemma bilinear_setsum:
```
```   346   assumes bh: "bilinear h"
```
```   347     and fS: "finite S"
```
```   348     and fT: "finite T"
```
```   349   shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
```
```   350 proof -
```
```   351   have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
```
```   352     apply (rule linear_setsum[unfolded o_def])
```
```   353     using bh fS
```
```   354     apply (auto simp add: bilinear_def)
```
```   355     done
```
```   356   also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
```
```   357     apply (rule setsum.cong, simp)
```
```   358     apply (rule linear_setsum[unfolded o_def])
```
```   359     using bh fT
```
```   360     apply (auto simp add: bilinear_def)
```
```   361     done
```
```   362   finally show ?thesis
```
```   363     unfolding setsum.cartesian_product .
```
```   364 qed
```
```   365
```
```   366
```
```   367 subsection {* Adjoints. *}
```
```   368
```
```   369 definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
```
```   370
```
```   371 lemma adjoint_unique:
```
```   372   assumes "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   373   shows "adjoint f = g"
```
```   374   unfolding adjoint_def
```
```   375 proof (rule some_equality)
```
```   376   show "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   377     by (rule assms)
```
```   378 next
```
```   379   fix h
```
```   380   assume "\<forall>x y. inner (f x) y = inner x (h y)"
```
```   381   then have "\<forall>x y. inner x (g y) = inner x (h y)"
```
```   382     using assms by simp
```
```   383   then have "\<forall>x y. inner x (g y - h y) = 0"
```
```   384     by (simp add: inner_diff_right)
```
```   385   then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
```
```   386     by simp
```
```   387   then have "\<forall>y. h y = g y"
```
```   388     by simp
```
```   389   then show "h = g" by (simp add: ext)
```
```   390 qed
```
```   391
```
```   392 text {* TODO: The following lemmas about adjoints should hold for any
```
```   393 Hilbert space (i.e. complete inner product space).
```
```   394 (see @{url "http://en.wikipedia.org/wiki/Hermitian_adjoint"})
```
```   395 *}
```
```   396
```
```   397 lemma adjoint_works:
```
```   398   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   399   assumes lf: "linear f"
```
```   400   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```   401 proof -
```
```   402   have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
```
```   403   proof (intro allI exI)
```
```   404     fix y :: "'m" and x
```
```   405     let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
```
```   406     have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
```
```   407       by (simp add: euclidean_representation)
```
```   408     also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
```
```   409       unfolding linear_setsum[OF lf]
```
```   410       by (simp add: linear_cmul[OF lf])
```
```   411     finally show "f x \<bullet> y = x \<bullet> ?w"
```
```   412       by (simp add: inner_setsum_left inner_setsum_right mult.commute)
```
```   413   qed
```
```   414   then show ?thesis
```
```   415     unfolding adjoint_def choice_iff
```
```   416     by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
```
```   417 qed
```
```   418
```
```   419 lemma adjoint_clauses:
```
```   420   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   421   assumes lf: "linear f"
```
```   422   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```   423     and "adjoint f y \<bullet> x = y \<bullet> f x"
```
```   424   by (simp_all add: adjoint_works[OF lf] inner_commute)
```
```   425
```
```   426 lemma adjoint_linear:
```
```   427   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   428   assumes lf: "linear f"
```
```   429   shows "linear (adjoint f)"
```
```   430   by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
```
```   431     adjoint_clauses[OF lf] inner_distrib)
```
```   432
```
```   433 lemma adjoint_adjoint:
```
```   434   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   435   assumes lf: "linear f"
```
```   436   shows "adjoint (adjoint f) = f"
```
```   437   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
```
```   438
```
```   439
```
```   440 subsection {* Interlude: Some properties of real sets *}
```
```   441
```
```   442 lemma seq_mono_lemma:
```
```   443   assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
```
```   444     and "\<forall>n \<ge> m. e n \<le> e m"
```
```   445   shows "\<forall>n \<ge> m. d n < e m"
```
```   446   using assms
```
```   447   apply auto
```
```   448   apply (erule_tac x="n" in allE)
```
```   449   apply (erule_tac x="n" in allE)
```
```   450   apply auto
```
```   451   done
```
```   452
```
```   453 lemma infinite_enumerate:
```
```   454   assumes fS: "infinite S"
```
```   455   shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
```
```   456   unfolding subseq_def
```
```   457   using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
```
```   458
```
```   459 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)"
```
```   460   apply auto
```
```   461   apply (rule_tac x="d/2" in exI)
```
```   462   apply auto
```
```   463   done
```
```   464
```
```   465 lemma triangle_lemma:
```
```   466   fixes x y z :: real
```
```   467   assumes x: "0 \<le> x"
```
```   468     and y: "0 \<le> y"
```
```   469     and z: "0 \<le> z"
```
```   470     and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
```
```   471   shows "x \<le> y + z"
```
```   472 proof -
```
```   473   have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
```
```   474     using z y by simp
```
```   475   with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
```
```   476     by (simp add: power2_eq_square field_simps)
```
```   477   from y z have yz: "y + z \<ge> 0"
```
```   478     by arith
```
```   479   from power2_le_imp_le[OF th yz] show ?thesis .
```
```   480 qed
```
```   481
```
```   482
```
```   483 subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
```
```   484
```
```   485 definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
```
```   486   where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
```
```   487
```
```   488 lemma hull_same: "S s \<Longrightarrow> S hull s = s"
```
```   489   unfolding hull_def by auto
```
```   490
```
```   491 lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
```
```   492   unfolding hull_def Ball_def by auto
```
```   493
```
```   494 lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
```
```   495   using hull_same[of S s] hull_in[of S s] by metis
```
```   496
```
```   497 lemma hull_hull: "S hull (S hull s) = S hull s"
```
```   498   unfolding hull_def by blast
```
```   499
```
```   500 lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
```
```   501   unfolding hull_def by blast
```
```   502
```
```   503 lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
```
```   504   unfolding hull_def by blast
```
```   505
```
```   506 lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
```
```   507   unfolding hull_def by blast
```
```   508
```
```   509 lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
```
```   510   unfolding hull_def by blast
```
```   511
```
```   512 lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
```
```   513   unfolding hull_def by blast
```
```   514
```
```   515 lemma hull_UNIV: "S hull UNIV = UNIV"
```
```   516   unfolding hull_def by auto
```
```   517
```
```   518 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)"
```
```   519   unfolding hull_def by auto
```
```   520
```
```   521 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"
```
```   522   using hull_minimal[of S "{x. P x}" Q]
```
```   523   by (auto simp add: subset_eq)
```
```   524
```
```   525 lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
```
```   526   by (metis hull_subset subset_eq)
```
```   527
```
```   528 lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
```
```   529   unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
```
```   530
```
```   531 lemma hull_union:
```
```   532   assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
```
```   533   shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
```
```   534   apply rule
```
```   535   apply (rule hull_mono)
```
```   536   unfolding Un_subset_iff
```
```   537   apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
```
```   538   apply (rule hull_minimal)
```
```   539   apply (metis hull_union_subset)
```
```   540   apply (metis hull_in T)
```
```   541   done
```
```   542
```
```   543 lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> S hull (insert a s) = S hull s"
```
```   544   unfolding hull_def by blast
```
```   545
```
```   546 lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> S hull (insert a s) = S hull s"
```
```   547   by (metis hull_redundant_eq)
```
```   548
```
```   549
```
```   550 subsection {* Archimedean properties and useful consequences *}
```
```   551
```
```   552 lemma real_arch_simple: "\<exists>n::nat. x \<le> real n"
```
```   553   unfolding real_of_nat_def by (rule ex_le_of_nat)
```
```   554
```
```   555 lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
```
```   556   using reals_Archimedean[of e] less_trans[of 0 "1 / real n" e for n::nat]
```
```   557   by (auto simp add: field_simps cong: conj_cong)
```
```   558
```
```   559 lemma real_pow_lbound: "0 \<le> x \<Longrightarrow> 1 + real n * x \<le> (1 + x) ^ n"
```
```   560 proof (induct n)
```
```   561   case 0
```
```   562   then show ?case by simp
```
```   563 next
```
```   564   case (Suc n)
```
```   565   then have h: "1 + real n * x \<le> (1 + x) ^ n"
```
```   566     by simp
```
```   567   from h have p: "1 \<le> (1 + x) ^ n"
```
```   568     using Suc.prems by simp
```
```   569   from h have "1 + real n * x + x \<le> (1 + x) ^ n + x"
```
```   570     by simp
```
```   571   also have "\<dots> \<le> (1 + x) ^ Suc n"
```
```   572     apply (subst diff_le_0_iff_le[symmetric])
```
```   573     using mult_left_mono[OF p Suc.prems]
```
```   574     apply (simp add: field_simps)
```
```   575     done
```
```   576   finally show ?case
```
```   577     by (simp add: real_of_nat_Suc field_simps)
```
```   578 qed
```
```   579
```
```   580 lemma real_arch_pow:
```
```   581   fixes x :: real
```
```   582   assumes x: "1 < x"
```
```   583   shows "\<exists>n. y < x^n"
```
```   584 proof -
```
```   585   from x have x0: "x - 1 > 0"
```
```   586     by arith
```
```   587   from reals_Archimedean3[OF x0, rule_format, of y]
```
```   588   obtain n :: nat where n: "y < real n * (x - 1)" by metis
```
```   589   from x0 have x00: "x- 1 \<ge> 0" by arith
```
```   590   from real_pow_lbound[OF x00, of n] n
```
```   591   have "y < x^n" by auto
```
```   592   then show ?thesis by metis
```
```   593 qed
```
```   594
```
```   595 lemma real_arch_pow2:
```
```   596   fixes x :: real
```
```   597   shows "\<exists>n. x < 2^ n"
```
```   598   using real_arch_pow[of 2 x] by simp
```
```   599
```
```   600 lemma real_arch_pow_inv:
```
```   601   fixes x y :: real
```
```   602   assumes y: "y > 0"
```
```   603     and x1: "x < 1"
```
```   604   shows "\<exists>n. x^n < y"
```
```   605 proof (cases "x > 0")
```
```   606   case True
```
```   607   with x1 have ix: "1 < 1/x" by (simp add: field_simps)
```
```   608   from real_arch_pow[OF ix, of "1/y"]
```
```   609   obtain n where n: "1/y < (1/x)^n" by blast
```
```   610   then show ?thesis using y `x > 0`
```
```   611     by (auto simp add: field_simps)
```
```   612 next
```
```   613   case False
```
```   614   with y x1 show ?thesis
```
```   615     apply auto
```
```   616     apply (rule exI[where x=1])
```
```   617     apply auto
```
```   618     done
```
```   619 qed
```
```   620
```
```   621 lemma forall_pos_mono:
```
```   622   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```   623     (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
```
```   624   by (metis real_arch_inv)
```
```   625
```
```   626 lemma forall_pos_mono_1:
```
```   627   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```   628     (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
```
```   629   apply (rule forall_pos_mono)
```
```   630   apply auto
```
```   631   apply (atomize)
```
```   632   apply (erule_tac x="n - 1" in allE)
```
```   633   apply auto
```
```   634   done
```
```   635
```
```   636 lemma real_archimedian_rdiv_eq_0:
```
```   637   assumes x0: "x \<ge> 0"
```
```   638     and c: "c \<ge> 0"
```
```   639     and xc: "\<forall>(m::nat) > 0. real m * x \<le> c"
```
```   640   shows "x = 0"
```
```   641 proof (rule ccontr)
```
```   642   assume "x \<noteq> 0"
```
```   643   with x0 have xp: "x > 0" by arith
```
```   644   from reals_Archimedean3[OF xp, rule_format, of c]
```
```   645   obtain n :: nat where n: "c < real n * x"
```
```   646     by blast
```
```   647   with xc[rule_format, of n] have "n = 0"
```
```   648     by arith
```
```   649   with n c show False
```
```   650     by simp
```
```   651 qed
```
```   652
```
```   653
```
```   654 subsection{* A bit of linear algebra. *}
```
```   655
```
```   656 definition (in real_vector) subspace :: "'a set \<Rightarrow> bool"
```
```   657   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)"
```
```   658
```
```   659 definition (in real_vector) "span S = (subspace hull S)"
```
```   660 definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))"
```
```   661 abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
```
```   662
```
```   663 text {* Closure properties of subspaces. *}
```
```   664
```
```   665 lemma subspace_UNIV[simp]: "subspace UNIV"
```
```   666   by (simp add: subspace_def)
```
```   667
```
```   668 lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
```
```   669   by (metis subspace_def)
```
```   670
```
```   671 lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
```
```   672   by (metis subspace_def)
```
```   673
```
```   674 lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
```
```   675   by (metis subspace_def)
```
```   676
```
```   677 lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
```
```   678   by (metis scaleR_minus1_left subspace_mul)
```
```   679
```
```   680 lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
```
```   681   using subspace_add [of S x "- y"] by (simp add: subspace_neg)
```
```   682
```
```   683 lemma (in real_vector) subspace_setsum:
```
```   684   assumes sA: "subspace A"
```
```   685     and f: "\<forall>x\<in>B. f x \<in> A"
```
```   686   shows "setsum f B \<in> A"
```
```   687 proof (cases "finite B")
```
```   688   case True
```
```   689   then show ?thesis
```
```   690     using f by induct (simp_all add: subspace_0 [OF sA] subspace_add [OF sA])
```
```   691 qed (simp add: subspace_0 [OF sA])
```
```   692
```
```   693 lemma subspace_linear_image:
```
```   694   assumes lf: "linear f"
```
```   695     and sS: "subspace S"
```
```   696   shows "subspace (f ` S)"
```
```   697   using lf sS linear_0[OF lf]
```
```   698   unfolding linear_iff subspace_def
```
```   699   apply (auto simp add: image_iff)
```
```   700   apply (rule_tac x="x + y" in bexI)
```
```   701   apply auto
```
```   702   apply (rule_tac x="c *\<^sub>R x" in bexI)
```
```   703   apply auto
```
```   704   done
```
```   705
```
```   706 lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
```
```   707   by (auto simp add: subspace_def linear_iff linear_0[of f])
```
```   708
```
```   709 lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
```
```   710   by (auto simp add: subspace_def linear_iff linear_0[of f])
```
```   711
```
```   712 lemma subspace_trivial: "subspace {0}"
```
```   713   by (simp add: subspace_def)
```
```   714
```
```   715 lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
```
```   716   by (simp add: subspace_def)
```
```   717
```
```   718 lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
```
```   719   unfolding subspace_def zero_prod_def by simp
```
```   720
```
```   721 text {* Properties of span. *}
```
```   722
```
```   723 lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
```
```   724   by (metis span_def hull_mono)
```
```   725
```
```   726 lemma (in real_vector) subspace_span: "subspace (span S)"
```
```   727   unfolding span_def
```
```   728   apply (rule hull_in)
```
```   729   apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
```
```   730   apply auto
```
```   731   done
```
```   732
```
```   733 lemma (in real_vector) span_clauses:
```
```   734   "a \<in> S \<Longrightarrow> a \<in> span S"
```
```   735   "0 \<in> span S"
```
```   736   "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
```
```   737   "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
```
```   738   by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
```
```   739
```
```   740 lemma span_unique:
```
```   741   "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
```
```   742   unfolding span_def by (rule hull_unique)
```
```   743
```
```   744 lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
```
```   745   unfolding span_def by (rule hull_minimal)
```
```   746
```
```   747 lemma (in real_vector) span_induct:
```
```   748   assumes x: "x \<in> span S"
```
```   749     and P: "subspace P"
```
```   750     and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
```
```   751   shows "x \<in> P"
```
```   752 proof -
```
```   753   from SP have SP': "S \<subseteq> P"
```
```   754     by (simp add: subset_eq)
```
```   755   from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
```
```   756   show "x \<in> P"
```
```   757     by (metis subset_eq)
```
```   758 qed
```
```   759
```
```   760 lemma span_empty[simp]: "span {} = {0}"
```
```   761   apply (simp add: span_def)
```
```   762   apply (rule hull_unique)
```
```   763   apply (auto simp add: subspace_def)
```
```   764   done
```
```   765
```
```   766 lemma (in real_vector) independent_empty[intro]: "independent {}"
```
```   767   by (simp add: dependent_def)
```
```   768
```
```   769 lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
```
```   770   unfolding dependent_def by auto
```
```   771
```
```   772 lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
```
```   773   apply (clarsimp simp add: dependent_def span_mono)
```
```   774   apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
```
```   775   apply force
```
```   776   apply (rule span_mono)
```
```   777   apply auto
```
```   778   done
```
```   779
```
```   780 lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
```
```   781   by (metis order_antisym span_def hull_minimal)
```
```   782
```
```   783 lemma (in real_vector) span_induct':
```
```   784   assumes SP: "\<forall>x \<in> S. P x"
```
```   785     and P: "subspace {x. P x}"
```
```   786   shows "\<forall>x \<in> span S. P x"
```
```   787   using span_induct SP P by blast
```
```   788
```
```   789 inductive_set (in real_vector) span_induct_alt_help for S :: "'a set"
```
```   790 where
```
```   791   span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
```
```   792 | span_induct_alt_help_S:
```
```   793     "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
```
```   794       (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
```
```   795
```
```   796 lemma span_induct_alt':
```
```   797   assumes h0: "h 0"
```
```   798     and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
```
```   799   shows "\<forall>x \<in> span S. h x"
```
```   800 proof -
```
```   801   {
```
```   802     fix x :: 'a
```
```   803     assume x: "x \<in> span_induct_alt_help S"
```
```   804     have "h x"
```
```   805       apply (rule span_induct_alt_help.induct[OF x])
```
```   806       apply (rule h0)
```
```   807       apply (rule hS)
```
```   808       apply assumption
```
```   809       apply assumption
```
```   810       done
```
```   811   }
```
```   812   note th0 = this
```
```   813   {
```
```   814     fix x
```
```   815     assume x: "x \<in> span S"
```
```   816     have "x \<in> span_induct_alt_help S"
```
```   817     proof (rule span_induct[where x=x and S=S])
```
```   818       show "x \<in> span S" by (rule x)
```
```   819     next
```
```   820       fix x
```
```   821       assume xS: "x \<in> S"
```
```   822       from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
```
```   823       show "x \<in> span_induct_alt_help S"
```
```   824         by simp
```
```   825     next
```
```   826       have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
```
```   827       moreover
```
```   828       {
```
```   829         fix x y
```
```   830         assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
```
```   831         from h have "(x + y) \<in> span_induct_alt_help S"
```
```   832           apply (induct rule: span_induct_alt_help.induct)
```
```   833           apply simp
```
```   834           unfolding add.assoc
```
```   835           apply (rule span_induct_alt_help_S)
```
```   836           apply assumption
```
```   837           apply simp
```
```   838           done
```
```   839       }
```
```   840       moreover
```
```   841       {
```
```   842         fix c x
```
```   843         assume xt: "x \<in> span_induct_alt_help S"
```
```   844         then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
```
```   845           apply (induct rule: span_induct_alt_help.induct)
```
```   846           apply (simp add: span_induct_alt_help_0)
```
```   847           apply (simp add: scaleR_right_distrib)
```
```   848           apply (rule span_induct_alt_help_S)
```
```   849           apply assumption
```
```   850           apply simp
```
```   851           done }
```
```   852       ultimately show "subspace (span_induct_alt_help S)"
```
```   853         unfolding subspace_def Ball_def by blast
```
```   854     qed
```
```   855   }
```
```   856   with th0 show ?thesis by blast
```
```   857 qed
```
```   858
```
```   859 lemma span_induct_alt:
```
```   860   assumes h0: "h 0"
```
```   861     and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
```
```   862     and x: "x \<in> span S"
```
```   863   shows "h x"
```
```   864   using span_induct_alt'[of h S] h0 hS x by blast
```
```   865
```
```   866 text {* Individual closure properties. *}
```
```   867
```
```   868 lemma span_span: "span (span A) = span A"
```
```   869   unfolding span_def hull_hull ..
```
```   870
```
```   871 lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
```
```   872   by (metis span_clauses(1))
```
```   873
```
```   874 lemma (in real_vector) span_0: "0 \<in> span S"
```
```   875   by (metis subspace_span subspace_0)
```
```   876
```
```   877 lemma span_inc: "S \<subseteq> span S"
```
```   878   by (metis subset_eq span_superset)
```
```   879
```
```   880 lemma (in real_vector) dependent_0:
```
```   881   assumes "0 \<in> A"
```
```   882   shows "dependent A"
```
```   883   unfolding dependent_def
```
```   884   using assms span_0
```
```   885   by auto
```
```   886
```
```   887 lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
```
```   888   by (metis subspace_add subspace_span)
```
```   889
```
```   890 lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
```
```   891   by (metis subspace_span subspace_mul)
```
```   892
```
```   893 lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
```
```   894   by (metis subspace_neg subspace_span)
```
```   895
```
```   896 lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
```
```   897   by (metis subspace_span subspace_sub)
```
```   898
```
```   899 lemma (in real_vector) span_setsum: "\<forall>x\<in>A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S"
```
```   900   by (rule subspace_setsum [OF subspace_span])
```
```   901
```
```   902 lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
```
```   903   by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
```
```   904
```
```   905 text {* Mapping under linear image. *}
```
```   906
```
```   907 lemma span_linear_image:
```
```   908   assumes lf: "linear f"
```
```   909   shows "span (f ` S) = f ` span S"
```
```   910 proof (rule span_unique)
```
```   911   show "f ` S \<subseteq> f ` span S"
```
```   912     by (intro image_mono span_inc)
```
```   913   show "subspace (f ` span S)"
```
```   914     using lf subspace_span by (rule subspace_linear_image)
```
```   915 next
```
```   916   fix T
```
```   917   assume "f ` S \<subseteq> T" and "subspace T"
```
```   918   then show "f ` span S \<subseteq> T"
```
```   919     unfolding image_subset_iff_subset_vimage
```
```   920     by (intro span_minimal subspace_linear_vimage lf)
```
```   921 qed
```
```   922
```
```   923 lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
```
```   924 proof (rule span_unique)
```
```   925   show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
```
```   926     by safe (force intro: span_clauses)+
```
```   927 next
```
```   928   have "linear (\<lambda>(a, b). a + b)"
```
```   929     by (simp add: linear_iff scaleR_add_right)
```
```   930   moreover have "subspace (span A \<times> span B)"
```
```   931     by (intro subspace_Times subspace_span)
```
```   932   ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
```
```   933     by (rule subspace_linear_image)
```
```   934 next
```
```   935   fix T
```
```   936   assume "A \<union> B \<subseteq> T" and "subspace T"
```
```   937   then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
```
```   938     by (auto intro!: subspace_add elim: span_induct)
```
```   939 qed
```
```   940
```
```   941 text {* The key breakdown property. *}
```
```   942
```
```   943 lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
```
```   944 proof (rule span_unique)
```
```   945   show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
```
```   946     by (fast intro: scaleR_one [symmetric])
```
```   947   show "subspace (range (\<lambda>k. k *\<^sub>R x))"
```
```   948     unfolding subspace_def
```
```   949     by (auto intro: scaleR_add_left [symmetric])
```
```   950 next
```
```   951   fix T
```
```   952   assume "{x} \<subseteq> T" and "subspace T"
```
```   953   then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
```
```   954     unfolding subspace_def by auto
```
```   955 qed
```
```   956
```
```   957 lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
```
```   958 proof -
```
```   959   have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
```
```   960     unfolding span_union span_singleton
```
```   961     apply safe
```
```   962     apply (rule_tac x=k in exI, simp)
```
```   963     apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
```
```   964     apply auto
```
```   965     done
```
```   966   then show ?thesis by simp
```
```   967 qed
```
```   968
```
```   969 lemma span_breakdown:
```
```   970   assumes bS: "b \<in> S"
```
```   971     and aS: "a \<in> span S"
```
```   972   shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
```
```   973   using assms span_insert [of b "S - {b}"]
```
```   974   by (simp add: insert_absorb)
```
```   975
```
```   976 lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
```
```   977   by (simp add: span_insert)
```
```   978
```
```   979 text {* Hence some "reversal" results. *}
```
```   980
```
```   981 lemma in_span_insert:
```
```   982   assumes a: "a \<in> span (insert b S)"
```
```   983     and na: "a \<notin> span S"
```
```   984   shows "b \<in> span (insert a S)"
```
```   985 proof -
```
```   986   from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
```
```   987     unfolding span_insert by fast
```
```   988   show ?thesis
```
```   989   proof (cases "k = 0")
```
```   990     case True
```
```   991     with k have "a \<in> span S" by simp
```
```   992     with na show ?thesis by simp
```
```   993   next
```
```   994     case False
```
```   995     from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
```
```   996       by (rule span_mul)
```
```   997     then have "b - inverse k *\<^sub>R a \<in> span S"
```
```   998       using `k \<noteq> 0` by (simp add: scaleR_diff_right)
```
```   999     then show ?thesis
```
```  1000       unfolding span_insert by fast
```
```  1001   qed
```
```  1002 qed
```
```  1003
```
```  1004 lemma in_span_delete:
```
```  1005   assumes a: "a \<in> span S"
```
```  1006     and na: "a \<notin> span (S - {b})"
```
```  1007   shows "b \<in> span (insert a (S - {b}))"
```
```  1008   apply (rule in_span_insert)
```
```  1009   apply (rule set_rev_mp)
```
```  1010   apply (rule a)
```
```  1011   apply (rule span_mono)
```
```  1012   apply blast
```
```  1013   apply (rule na)
```
```  1014   done
```
```  1015
```
```  1016 text {* Transitivity property. *}
```
```  1017
```
```  1018 lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
```
```  1019   unfolding span_def by (rule hull_redundant)
```
```  1020
```
```  1021 lemma span_trans:
```
```  1022   assumes x: "x \<in> span S"
```
```  1023     and y: "y \<in> span (insert x S)"
```
```  1024   shows "y \<in> span S"
```
```  1025   using assms by (simp only: span_redundant)
```
```  1026
```
```  1027 lemma span_insert_0[simp]: "span (insert 0 S) = span S"
```
```  1028   by (simp only: span_redundant span_0)
```
```  1029
```
```  1030 text {* An explicit expansion is sometimes needed. *}
```
```  1031
```
```  1032 lemma span_explicit:
```
```  1033   "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
```
```  1034   (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
```
```  1035 proof -
```
```  1036   {
```
```  1037     fix x
```
```  1038     assume "?h x"
```
```  1039     then obtain S u where "finite S" and "S \<subseteq> P" and "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
```
```  1040       by blast
```
```  1041     then have "x \<in> span P"
```
```  1042       by (auto intro: span_setsum span_mul span_superset)
```
```  1043   }
```
```  1044   moreover
```
```  1045   have "\<forall>x \<in> span P. ?h x"
```
```  1046   proof (rule span_induct_alt')
```
```  1047     show "?h 0"
```
```  1048       by (rule exI[where x="{}"], simp)
```
```  1049   next
```
```  1050     fix c x y
```
```  1051     assume x: "x \<in> P"
```
```  1052     assume hy: "?h y"
```
```  1053     from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
```
```  1054       and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
```
```  1055     let ?S = "insert x S"
```
```  1056     let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
```
```  1057     from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
```
```  1058       by blast+
```
```  1059     have "?Q ?S ?u (c*\<^sub>R x + y)"
```
```  1060     proof cases
```
```  1061       assume xS: "x \<in> S"
```
```  1062       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"
```
```  1063         using xS by (simp add: setsum.remove [OF fS xS] insert_absorb)
```
```  1064       also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
```
```  1065         by (simp add: setsum.remove [OF fS xS] algebra_simps)
```
```  1066       also have "\<dots> = c*\<^sub>R x + y"
```
```  1067         by (simp add: add.commute u)
```
```  1068       finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
```
```  1069       then show ?thesis using th0 by blast
```
```  1070     next
```
```  1071       assume xS: "x \<notin> S"
```
```  1072       have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
```
```  1073         unfolding u[symmetric]
```
```  1074         apply (rule setsum.cong)
```
```  1075         using xS
```
```  1076         apply auto
```
```  1077         done
```
```  1078       show ?thesis using fS xS th0
```
```  1079         by (simp add: th00 add.commute cong del: if_weak_cong)
```
```  1080     qed
```
```  1081     then show "?h (c*\<^sub>R x + y)"
```
```  1082       by fast
```
```  1083   qed
```
```  1084   ultimately show ?thesis by blast
```
```  1085 qed
```
```  1086
```
```  1087 lemma dependent_explicit:
```
```  1088   "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))"
```
```  1089   (is "?lhs = ?rhs")
```
```  1090 proof -
```
```  1091   {
```
```  1092     assume dP: "dependent P"
```
```  1093     then obtain a S u where aP: "a \<in> P" and fS: "finite S"
```
```  1094       and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
```
```  1095       unfolding dependent_def span_explicit by blast
```
```  1096     let ?S = "insert a S"
```
```  1097     let ?u = "\<lambda>y. if y = a then - 1 else u y"
```
```  1098     let ?v = a
```
```  1099     from aP SP have aS: "a \<notin> S"
```
```  1100       by blast
```
```  1101     from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
```
```  1102       by auto
```
```  1103     have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
```
```  1104       using fS aS
```
```  1105       apply simp
```
```  1106       apply (subst (2) ua[symmetric])
```
```  1107       apply (rule setsum.cong)
```
```  1108       apply auto
```
```  1109       done
```
```  1110     with th0 have ?rhs by fast
```
```  1111   }
```
```  1112   moreover
```
```  1113   {
```
```  1114     fix S u v
```
```  1115     assume fS: "finite S"
```
```  1116       and SP: "S \<subseteq> P"
```
```  1117       and vS: "v \<in> S"
```
```  1118       and uv: "u v \<noteq> 0"
```
```  1119       and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
```
```  1120     let ?a = v
```
```  1121     let ?S = "S - {v}"
```
```  1122     let ?u = "\<lambda>i. (- u i) / u v"
```
```  1123     have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
```
```  1124       using fS SP vS by auto
```
```  1125     have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
```
```  1126       setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
```
```  1127       using fS vS uv by (simp add: setsum_diff1 field_simps)
```
```  1128     also have "\<dots> = ?a"
```
```  1129       unfolding scaleR_right.setsum [symmetric] u using uv by simp
```
```  1130     finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
```
```  1131     with th0 have ?lhs
```
```  1132       unfolding dependent_def span_explicit
```
```  1133       apply -
```
```  1134       apply (rule bexI[where x= "?a"])
```
```  1135       apply (simp_all del: scaleR_minus_left)
```
```  1136       apply (rule exI[where x= "?S"])
```
```  1137       apply (auto simp del: scaleR_minus_left)
```
```  1138       done
```
```  1139   }
```
```  1140   ultimately show ?thesis by blast
```
```  1141 qed
```
```  1142
```
```  1143
```
```  1144 lemma span_finite:
```
```  1145   assumes fS: "finite S"
```
```  1146   shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
```
```  1147   (is "_ = ?rhs")
```
```  1148 proof -
```
```  1149   {
```
```  1150     fix y
```
```  1151     assume y: "y \<in> span S"
```
```  1152     from y obtain S' u where fS': "finite S'"
```
```  1153       and SS': "S' \<subseteq> S"
```
```  1154       and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
```
```  1155       unfolding span_explicit by blast
```
```  1156     let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
```
```  1157     have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
```
```  1158       using SS' fS by (auto intro!: setsum.mono_neutral_cong_right)
```
```  1159     then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
```
```  1160     then have "y \<in> ?rhs" by auto
```
```  1161   }
```
```  1162   moreover
```
```  1163   {
```
```  1164     fix y u
```
```  1165     assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
```
```  1166     then have "y \<in> span S" using fS unfolding span_explicit by auto
```
```  1167   }
```
```  1168   ultimately show ?thesis by blast
```
```  1169 qed
```
```  1170
```
```  1171 text {* This is useful for building a basis step-by-step. *}
```
```  1172
```
```  1173 lemma independent_insert:
```
```  1174   "independent (insert a S) \<longleftrightarrow>
```
```  1175     (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
```
```  1176   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1177 proof (cases "a \<in> S")
```
```  1178   case True
```
```  1179   then show ?thesis
```
```  1180     using insert_absorb[OF True] by simp
```
```  1181 next
```
```  1182   case False
```
```  1183   show ?thesis
```
```  1184   proof
```
```  1185     assume i: ?lhs
```
```  1186     then show ?rhs
```
```  1187       using False
```
```  1188       apply simp
```
```  1189       apply (rule conjI)
```
```  1190       apply (rule independent_mono)
```
```  1191       apply assumption
```
```  1192       apply blast
```
```  1193       apply (simp add: dependent_def)
```
```  1194       done
```
```  1195   next
```
```  1196     assume i: ?rhs
```
```  1197     show ?lhs
```
```  1198       using i False
```
```  1199       apply (auto simp add: dependent_def)
```
```  1200       by (metis in_span_insert insert_Diff insert_Diff_if insert_iff)
```
```  1201   qed
```
```  1202 qed
```
```  1203
```
```  1204 text {* The degenerate case of the Exchange Lemma. *}
```
```  1205
```
```  1206 lemma spanning_subset_independent:
```
```  1207   assumes BA: "B \<subseteq> A"
```
```  1208     and iA: "independent A"
```
```  1209     and AsB: "A \<subseteq> span B"
```
```  1210   shows "A = B"
```
```  1211 proof
```
```  1212   show "B \<subseteq> A" by (rule BA)
```
```  1213
```
```  1214   from span_mono[OF BA] span_mono[OF AsB]
```
```  1215   have sAB: "span A = span B" unfolding span_span by blast
```
```  1216
```
```  1217   {
```
```  1218     fix x
```
```  1219     assume x: "x \<in> A"
```
```  1220     from iA have th0: "x \<notin> span (A - {x})"
```
```  1221       unfolding dependent_def using x by blast
```
```  1222     from x have xsA: "x \<in> span A"
```
```  1223       by (blast intro: span_superset)
```
```  1224     have "A - {x} \<subseteq> A" by blast
```
```  1225     then have th1: "span (A - {x}) \<subseteq> span A"
```
```  1226       by (metis span_mono)
```
```  1227     {
```
```  1228       assume xB: "x \<notin> B"
```
```  1229       from xB BA have "B \<subseteq> A - {x}"
```
```  1230         by blast
```
```  1231       then have "span B \<subseteq> span (A - {x})"
```
```  1232         by (metis span_mono)
```
```  1233       with th1 th0 sAB have "x \<notin> span A"
```
```  1234         by blast
```
```  1235       with x have False
```
```  1236         by (metis span_superset)
```
```  1237     }
```
```  1238     then have "x \<in> B" by blast
```
```  1239   }
```
```  1240   then show "A \<subseteq> B" by blast
```
```  1241 qed
```
```  1242
```
```  1243 text {* The general case of the Exchange Lemma, the key to what follows. *}
```
```  1244
```
```  1245 lemma exchange_lemma:
```
```  1246   assumes f:"finite t"
```
```  1247     and i: "independent s"
```
```  1248     and sp: "s \<subseteq> span t"
```
```  1249   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'"
```
```  1250   using f i sp
```
```  1251 proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
```
```  1252   case less
```
```  1253   note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
```
```  1254   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'"
```
```  1255   let ?ths = "\<exists>t'. ?P t'"
```
```  1256   {
```
```  1257     assume "s \<subseteq> t"
```
```  1258     then have ?ths
```
```  1259       by (metis ft Un_commute sp sup_ge1)
```
```  1260   }
```
```  1261   moreover
```
```  1262   {
```
```  1263     assume st: "t \<subseteq> s"
```
```  1264     from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
```
```  1265     have ?ths
```
```  1266       by (metis Un_absorb sp)
```
```  1267   }
```
```  1268   moreover
```
```  1269   {
```
```  1270     assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
```
```  1271     from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
```
```  1272       by blast
```
```  1273     from b have "t - {b} - s \<subset> t - s"
```
```  1274       by blast
```
```  1275     then have cardlt: "card (t - {b} - s) < card (t - s)"
```
```  1276       using ft by (auto intro: psubset_card_mono)
```
```  1277     from b ft have ct0: "card t \<noteq> 0"
```
```  1278       by auto
```
```  1279     have ?ths
```
```  1280     proof cases
```
```  1281       assume stb: "s \<subseteq> span (t - {b})"
```
```  1282       from ft have ftb: "finite (t - {b})"
```
```  1283         by auto
```
```  1284       from less(1)[OF cardlt ftb s stb]
```
```  1285       obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
```
```  1286         and fu: "finite u" by blast
```
```  1287       let ?w = "insert b u"
```
```  1288       have th0: "s \<subseteq> insert b u"
```
```  1289         using u by blast
```
```  1290       from u(3) b have "u \<subseteq> s \<union> t"
```
```  1291         by blast
```
```  1292       then have th1: "insert b u \<subseteq> s \<union> t"
```
```  1293         using u b by blast
```
```  1294       have bu: "b \<notin> u"
```
```  1295         using b u by blast
```
```  1296       from u(1) ft b have "card u = (card t - 1)"
```
```  1297         by auto
```
```  1298       then have th2: "card (insert b u) = card t"
```
```  1299         using card_insert_disjoint[OF fu bu] ct0 by auto
```
```  1300       from u(4) have "s \<subseteq> span u" .
```
```  1301       also have "\<dots> \<subseteq> span (insert b u)"
```
```  1302         by (rule span_mono) blast
```
```  1303       finally have th3: "s \<subseteq> span (insert b u)" .
```
```  1304       from th0 th1 th2 th3 fu have th: "?P ?w"
```
```  1305         by blast
```
```  1306       from th show ?thesis by blast
```
```  1307     next
```
```  1308       assume stb: "\<not> s \<subseteq> span (t - {b})"
```
```  1309       from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
```
```  1310         by blast
```
```  1311       have ab: "a \<noteq> b"
```
```  1312         using a b by blast
```
```  1313       have at: "a \<notin> t"
```
```  1314         using a ab span_superset[of a "t- {b}"] by auto
```
```  1315       have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
```
```  1316         using cardlt ft a b by auto
```
```  1317       have ft': "finite (insert a (t - {b}))"
```
```  1318         using ft by auto
```
```  1319       {
```
```  1320         fix x
```
```  1321         assume xs: "x \<in> s"
```
```  1322         have t: "t \<subseteq> insert b (insert a (t - {b}))"
```
```  1323           using b by auto
```
```  1324         from b(1) have "b \<in> span t"
```
```  1325           by (simp add: span_superset)
```
```  1326         have bs: "b \<in> span (insert a (t - {b}))"
```
```  1327           apply (rule in_span_delete)
```
```  1328           using a sp unfolding subset_eq
```
```  1329           apply auto
```
```  1330           done
```
```  1331         from xs sp have "x \<in> span t"
```
```  1332           by blast
```
```  1333         with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
```
```  1334         from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
```
```  1335       }
```
```  1336       then have sp': "s \<subseteq> span (insert a (t - {b}))"
```
```  1337         by blast
```
```  1338       from less(1)[OF mlt ft' s sp'] obtain u where u:
```
```  1339         "card u = card (insert a (t - {b}))"
```
```  1340         "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
```
```  1341         "s \<subseteq> span u" by blast
```
```  1342       from u a b ft at ct0 have "?P u"
```
```  1343         by auto
```
```  1344       then show ?thesis by blast
```
```  1345     qed
```
```  1346   }
```
```  1347   ultimately show ?ths by blast
```
```  1348 qed
```
```  1349
```
```  1350 text {* This implies corresponding size bounds. *}
```
```  1351
```
```  1352 lemma independent_span_bound:
```
```  1353   assumes f: "finite t"
```
```  1354     and i: "independent s"
```
```  1355     and sp: "s \<subseteq> span t"
```
```  1356   shows "finite s \<and> card s \<le> card t"
```
```  1357   by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
```
```  1358
```
```  1359 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
```
```  1360 proof -
```
```  1361   have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
```
```  1362     by auto
```
```  1363   show ?thesis unfolding eq
```
```  1364     apply (rule finite_imageI)
```
```  1365     apply (rule finite)
```
```  1366     done
```
```  1367 qed
```
```  1368
```
```  1369
```
```  1370 subsection {* Euclidean Spaces as Typeclass *}
```
```  1371
```
```  1372 lemma independent_Basis: "independent Basis"
```
```  1373   unfolding dependent_def
```
```  1374   apply (subst span_finite)
```
```  1375   apply simp
```
```  1376   apply clarify
```
```  1377   apply (drule_tac f="inner a" in arg_cong)
```
```  1378   apply (simp add: inner_Basis inner_setsum_right eq_commute)
```
```  1379   done
```
```  1380
```
```  1381 lemma span_Basis [simp]: "span Basis = UNIV"
```
```  1382   unfolding span_finite [OF finite_Basis]
```
```  1383   by (fast intro: euclidean_representation)
```
```  1384
```
```  1385 lemma in_span_Basis: "x \<in> span Basis"
```
```  1386   unfolding span_Basis ..
```
```  1387
```
```  1388 lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
```
```  1389   by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
```
```  1390
```
```  1391 lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
```
```  1392   by (metis Basis_le_norm order_trans)
```
```  1393
```
```  1394 lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
```
```  1395   by (metis Basis_le_norm le_less_trans)
```
```  1396
```
```  1397 lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
```
```  1398   apply (subst euclidean_representation[of x, symmetric])
```
```  1399   apply (rule order_trans[OF norm_setsum])
```
```  1400   apply (auto intro!: setsum_mono)
```
```  1401   done
```
```  1402
```
```  1403 lemma setsum_norm_allsubsets_bound:
```
```  1404   fixes f :: "'a \<Rightarrow> 'n::euclidean_space"
```
```  1405   assumes fP: "finite P"
```
```  1406     and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
```
```  1407   shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
```
```  1408 proof -
```
```  1409   have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
```
```  1410     by (rule setsum_mono) (rule norm_le_l1)
```
```  1411   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>)"
```
```  1412     by (rule setsum.commute)
```
```  1413   also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
```
```  1414   proof (rule setsum_bounded)
```
```  1415     fix i :: 'n
```
```  1416     assume i: "i \<in> Basis"
```
```  1417     have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
```
```  1418       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)"
```
```  1419       by (simp add: abs_real_def setsum.If_cases[OF fP] setsum_negf norm_triangle_ineq4 inner_setsum_left
```
```  1420         del: real_norm_def)
```
```  1421     also have "\<dots> \<le> e + e"
```
```  1422       unfolding real_norm_def
```
```  1423       by (intro add_mono norm_bound_Basis_le i fPs) auto
```
```  1424     finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
```
```  1425   qed
```
```  1426   also have "\<dots> = 2 * real DIM('n) * e"
```
```  1427     by (simp add: real_of_nat_def)
```
```  1428   finally show ?thesis .
```
```  1429 qed
```
```  1430
```
```  1431
```
```  1432 subsection {* Linearity and Bilinearity continued *}
```
```  1433
```
```  1434 lemma linear_bounded:
```
```  1435   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1436   assumes lf: "linear f"
```
```  1437   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```  1438 proof
```
```  1439   let ?B = "\<Sum>b\<in>Basis. norm (f b)"
```
```  1440   show "\<forall>x. norm (f x) \<le> ?B * norm x"
```
```  1441   proof
```
```  1442     fix x :: 'a
```
```  1443     let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
```
```  1444     have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
```
```  1445       unfolding euclidean_representation ..
```
```  1446     also have "\<dots> = norm (setsum ?g Basis)"
```
```  1447       by (simp add: linear_setsum [OF lf] linear_cmul [OF lf])
```
```  1448     finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
```
```  1449     have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
```
```  1450     proof
```
```  1451       fix i :: 'a
```
```  1452       assume i: "i \<in> Basis"
```
```  1453       from Basis_le_norm[OF i, of x]
```
```  1454       show "norm (?g i) \<le> norm (f i) * norm x"
```
```  1455         unfolding norm_scaleR
```
```  1456         apply (subst mult.commute)
```
```  1457         apply (rule mult_mono)
```
```  1458         apply (auto simp add: field_simps)
```
```  1459         done
```
```  1460     qed
```
```  1461     from setsum_norm_le[of _ ?g, OF th]
```
```  1462     show "norm (f x) \<le> ?B * norm x"
```
```  1463       unfolding th0 setsum_left_distrib by metis
```
```  1464   qed
```
```  1465 qed
```
```  1466
```
```  1467 lemma linear_conv_bounded_linear:
```
```  1468   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1469   shows "linear f \<longleftrightarrow> bounded_linear f"
```
```  1470 proof
```
```  1471   assume "linear f"
```
```  1472   then interpret f: linear f .
```
```  1473   show "bounded_linear f"
```
```  1474   proof
```
```  1475     have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```  1476       using `linear f` by (rule linear_bounded)
```
```  1477     then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1478       by (simp add: mult.commute)
```
```  1479   qed
```
```  1480 next
```
```  1481   assume "bounded_linear f"
```
```  1482   then interpret f: bounded_linear f .
```
```  1483   show "linear f" ..
```
```  1484 qed
```
```  1485
```
```  1486 lemma linear_bounded_pos:
```
```  1487   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1488   assumes lf: "linear f"
```
```  1489   shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
```
```  1490 proof -
```
```  1491   have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
```
```  1492     using lf unfolding linear_conv_bounded_linear
```
```  1493     by (rule bounded_linear.pos_bounded)
```
```  1494   then show ?thesis
```
```  1495     by (simp only: mult.commute)
```
```  1496 qed
```
```  1497
```
```  1498 lemma bounded_linearI':
```
```  1499   fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1500   assumes "\<And>x y. f (x + y) = f x + f y"
```
```  1501     and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```  1502   shows "bounded_linear f"
```
```  1503   unfolding linear_conv_bounded_linear[symmetric]
```
```  1504   by (rule linearI[OF assms])
```
```  1505
```
```  1506 lemma bilinear_bounded:
```
```  1507   fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
```
```  1508   assumes bh: "bilinear h"
```
```  1509   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```  1510 proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
```
```  1511   fix x :: 'm
```
```  1512   fix y :: 'n
```
```  1513   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))"
```
```  1514     apply (subst euclidean_representation[where 'a='m])
```
```  1515     apply (subst euclidean_representation[where 'a='n])
```
```  1516     apply rule
```
```  1517     done
```
```  1518   also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
```
```  1519     unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
```
```  1520   finally have th: "norm (h x y) = \<dots>" .
```
```  1521   show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
```
```  1522     apply (auto simp add: setsum_left_distrib th setsum.cartesian_product)
```
```  1523     apply (rule setsum_norm_le)
```
```  1524     apply simp
```
```  1525     apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
```
```  1526       field_simps simp del: scaleR_scaleR)
```
```  1527     apply (rule mult_mono)
```
```  1528     apply (auto simp add: zero_le_mult_iff Basis_le_norm)
```
```  1529     apply (rule mult_mono)
```
```  1530     apply (auto simp add: zero_le_mult_iff Basis_le_norm)
```
```  1531     done
```
```  1532 qed
```
```  1533
```
```  1534 lemma bilinear_conv_bounded_bilinear:
```
```  1535   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```  1536   shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
```
```  1537 proof
```
```  1538   assume "bilinear h"
```
```  1539   show "bounded_bilinear h"
```
```  1540   proof
```
```  1541     fix x y z
```
```  1542     show "h (x + y) z = h x z + h y z"
```
```  1543       using `bilinear h` unfolding bilinear_def linear_iff by simp
```
```  1544   next
```
```  1545     fix x y z
```
```  1546     show "h x (y + z) = h x y + h x z"
```
```  1547       using `bilinear h` unfolding bilinear_def linear_iff by simp
```
```  1548   next
```
```  1549     fix r x y
```
```  1550     show "h (scaleR r x) y = scaleR r (h x y)"
```
```  1551       using `bilinear h` unfolding bilinear_def linear_iff
```
```  1552       by simp
```
```  1553   next
```
```  1554     fix r x y
```
```  1555     show "h x (scaleR r y) = scaleR r (h x y)"
```
```  1556       using `bilinear h` unfolding bilinear_def linear_iff
```
```  1557       by simp
```
```  1558   next
```
```  1559     have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```  1560       using `bilinear h` by (rule bilinear_bounded)
```
```  1561     then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
```
```  1562       by (simp add: ac_simps)
```
```  1563   qed
```
```  1564 next
```
```  1565   assume "bounded_bilinear h"
```
```  1566   then interpret h: bounded_bilinear h .
```
```  1567   show "bilinear h"
```
```  1568     unfolding bilinear_def linear_conv_bounded_linear
```
```  1569     using h.bounded_linear_left h.bounded_linear_right by simp
```
```  1570 qed
```
```  1571
```
```  1572 lemma bilinear_bounded_pos:
```
```  1573   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```  1574   assumes bh: "bilinear h"
```
```  1575   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```  1576 proof -
```
```  1577   have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
```
```  1578     using bh [unfolded bilinear_conv_bounded_bilinear]
```
```  1579     by (rule bounded_bilinear.pos_bounded)
```
```  1580   then show ?thesis
```
```  1581     by (simp only: ac_simps)
```
```  1582 qed
```
```  1583
```
```  1584
```
```  1585 subsection {* We continue. *}
```
```  1586
```
```  1587 lemma independent_bound:
```
```  1588   fixes S :: "'a::euclidean_space set"
```
```  1589   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
```
```  1590   using independent_span_bound[OF finite_Basis, of S] by auto
```
```  1591
```
```  1592 corollary
```
```  1593   fixes S :: "'a::euclidean_space set"
```
```  1594   assumes "independent S"
```
```  1595   shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)"
```
```  1596 using assms independent_bound by auto
```
```  1597
```
```  1598 lemma dependent_biggerset:
```
```  1599   fixes S :: "'a::euclidean_space set"
```
```  1600   shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
```
```  1601   by (metis independent_bound not_less)
```
```  1602
```
```  1603 text {* Hence we can create a maximal independent subset. *}
```
```  1604
```
```  1605 lemma maximal_independent_subset_extend:
```
```  1606   fixes S :: "'a::euclidean_space set"
```
```  1607   assumes sv: "S \<subseteq> V"
```
```  1608     and iS: "independent S"
```
```  1609   shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
```
```  1610   using sv iS
```
```  1611 proof (induct "DIM('a) - card S" arbitrary: S rule: less_induct)
```
```  1612   case less
```
```  1613   note sv = `S \<subseteq> V` and i = `independent S`
```
```  1614   let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
```
```  1615   let ?ths = "\<exists>x. ?P x"
```
```  1616   let ?d = "DIM('a)"
```
```  1617   show ?ths
```
```  1618   proof (cases "V \<subseteq> span S")
```
```  1619     case True
```
```  1620     then show ?thesis
```
```  1621       using sv i by blast
```
```  1622   next
```
```  1623     case False
```
```  1624     then obtain a where a: "a \<in> V" "a \<notin> span S"
```
```  1625       by blast
```
```  1626     from a have aS: "a \<notin> S"
```
```  1627       by (auto simp add: span_superset)
```
```  1628     have th0: "insert a S \<subseteq> V"
```
```  1629       using a sv by blast
```
```  1630     from independent_insert[of a S]  i a
```
```  1631     have th1: "independent (insert a S)"
```
```  1632       by auto
```
```  1633     have mlt: "?d - card (insert a S) < ?d - card S"
```
```  1634       using aS a independent_bound[OF th1] by auto
```
```  1635
```
```  1636     from less(1)[OF mlt th0 th1]
```
```  1637     obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
```
```  1638       by blast
```
```  1639     from B have "?P B" by auto
```
```  1640     then show ?thesis by blast
```
```  1641   qed
```
```  1642 qed
```
```  1643
```
```  1644 lemma maximal_independent_subset:
```
```  1645   "\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
```
```  1646   by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"]
```
```  1647     empty_subsetI independent_empty)
```
```  1648
```
```  1649
```
```  1650 text {* Notion of dimension. *}
```
```  1651
```
```  1652 definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
```
```  1653
```
```  1654 lemma basis_exists:
```
```  1655   "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
```
```  1656   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)"]
```
```  1657   using maximal_independent_subset[of V] independent_bound
```
```  1658   by auto
```
```  1659
```
```  1660 corollary dim_le_card:
```
```  1661   fixes s :: "'a::euclidean_space set"
```
```  1662   shows "finite s \<Longrightarrow> dim s \<le> card s"
```
```  1663 by (metis basis_exists card_mono)
```
```  1664
```
```  1665 text {* Consequences of independence or spanning for cardinality. *}
```
```  1666
```
```  1667 lemma independent_card_le_dim:
```
```  1668   fixes B :: "'a::euclidean_space set"
```
```  1669   assumes "B \<subseteq> V"
```
```  1670     and "independent B"
```
```  1671   shows "card B \<le> dim V"
```
```  1672 proof -
```
```  1673   from basis_exists[of V] `B \<subseteq> V`
```
```  1674   obtain B' where "independent B'"
```
```  1675     and "B \<subseteq> span B'"
```
```  1676     and "card B' = dim V"
```
```  1677     by blast
```
```  1678   with independent_span_bound[OF _ `independent B` `B \<subseteq> span B'`] independent_bound[of B']
```
```  1679   show ?thesis by auto
```
```  1680 qed
```
```  1681
```
```  1682 lemma span_card_ge_dim:
```
```  1683   fixes B :: "'a::euclidean_space set"
```
```  1684   shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
```
```  1685   by (metis basis_exists[of V] independent_span_bound subset_trans)
```
```  1686
```
```  1687 lemma basis_card_eq_dim:
```
```  1688   fixes V :: "'a::euclidean_space set"
```
```  1689   shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
```
```  1690   by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
```
```  1691
```
```  1692 lemma dim_unique:
```
```  1693   fixes B :: "'a::euclidean_space set"
```
```  1694   shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
```
```  1695   by (metis basis_card_eq_dim)
```
```  1696
```
```  1697 text {* More lemmas about dimension. *}
```
```  1698
```
```  1699 lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
```
```  1700   using independent_Basis
```
```  1701   by (intro dim_unique[of Basis]) auto
```
```  1702
```
```  1703 lemma dim_subset:
```
```  1704   fixes S :: "'a::euclidean_space set"
```
```  1705   shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
```
```  1706   using basis_exists[of T] basis_exists[of S]
```
```  1707   by (metis independent_card_le_dim subset_trans)
```
```  1708
```
```  1709 lemma dim_subset_UNIV:
```
```  1710   fixes S :: "'a::euclidean_space set"
```
```  1711   shows "dim S \<le> DIM('a)"
```
```  1712   by (metis dim_subset subset_UNIV dim_UNIV)
```
```  1713
```
```  1714 text {* Converses to those. *}
```
```  1715
```
```  1716 lemma card_ge_dim_independent:
```
```  1717   fixes B :: "'a::euclidean_space set"
```
```  1718   assumes BV: "B \<subseteq> V"
```
```  1719     and iB: "independent B"
```
```  1720     and dVB: "dim V \<le> card B"
```
```  1721   shows "V \<subseteq> span B"
```
```  1722 proof
```
```  1723   fix a
```
```  1724   assume aV: "a \<in> V"
```
```  1725   {
```
```  1726     assume aB: "a \<notin> span B"
```
```  1727     then have iaB: "independent (insert a B)"
```
```  1728       using iB aV BV by (simp add: independent_insert)
```
```  1729     from aV BV have th0: "insert a B \<subseteq> V"
```
```  1730       by blast
```
```  1731     from aB have "a \<notin>B"
```
```  1732       by (auto simp add: span_superset)
```
```  1733     with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
```
```  1734     have False by auto
```
```  1735   }
```
```  1736   then show "a \<in> span B" by blast
```
```  1737 qed
```
```  1738
```
```  1739 lemma card_le_dim_spanning:
```
```  1740   assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
```
```  1741     and VB: "V \<subseteq> span B"
```
```  1742     and fB: "finite B"
```
```  1743     and dVB: "dim V \<ge> card B"
```
```  1744   shows "independent B"
```
```  1745 proof -
```
```  1746   {
```
```  1747     fix a
```
```  1748     assume a: "a \<in> B" "a \<in> span (B - {a})"
```
```  1749     from a fB have c0: "card B \<noteq> 0"
```
```  1750       by auto
```
```  1751     from a fB have cb: "card (B - {a}) = card B - 1"
```
```  1752       by auto
```
```  1753     from BV a have th0: "B - {a} \<subseteq> V"
```
```  1754       by blast
```
```  1755     {
```
```  1756       fix x
```
```  1757       assume x: "x \<in> V"
```
```  1758       from a have eq: "insert a (B - {a}) = B"
```
```  1759         by blast
```
```  1760       from x VB have x': "x \<in> span B"
```
```  1761         by blast
```
```  1762       from span_trans[OF a(2), unfolded eq, OF x']
```
```  1763       have "x \<in> span (B - {a})" .
```
```  1764     }
```
```  1765     then have th1: "V \<subseteq> span (B - {a})"
```
```  1766       by blast
```
```  1767     have th2: "finite (B - {a})"
```
```  1768       using fB by auto
```
```  1769     from span_card_ge_dim[OF th0 th1 th2]
```
```  1770     have c: "dim V \<le> card (B - {a})" .
```
```  1771     from c c0 dVB cb have False by simp
```
```  1772   }
```
```  1773   then show ?thesis
```
```  1774     unfolding dependent_def by blast
```
```  1775 qed
```
```  1776
```
```  1777 lemma card_eq_dim:
```
```  1778   fixes B :: "'a::euclidean_space set"
```
```  1779   shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
```
```  1780   by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
```
```  1781
```
```  1782 text {* More general size bound lemmas. *}
```
```  1783
```
```  1784 lemma independent_bound_general:
```
```  1785   fixes S :: "'a::euclidean_space set"
```
```  1786   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
```
```  1787   by (metis independent_card_le_dim independent_bound subset_refl)
```
```  1788
```
```  1789 lemma dependent_biggerset_general:
```
```  1790   fixes S :: "'a::euclidean_space set"
```
```  1791   shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
```
```  1792   using independent_bound_general[of S] by (metis linorder_not_le)
```
```  1793
```
```  1794 lemma dim_span [simp]:
```
```  1795   fixes S :: "'a::euclidean_space set"
```
```  1796   shows "dim (span S) = dim S"
```
```  1797 proof -
```
```  1798   have th0: "dim S \<le> dim (span S)"
```
```  1799     by (auto simp add: subset_eq intro: dim_subset span_superset)
```
```  1800   from basis_exists[of S]
```
```  1801   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
```
```  1802     by blast
```
```  1803   from B have fB: "finite B" "card B = dim S"
```
```  1804     using independent_bound by blast+
```
```  1805   have bSS: "B \<subseteq> span S"
```
```  1806     using B(1) by (metis subset_eq span_inc)
```
```  1807   have sssB: "span S \<subseteq> span B"
```
```  1808     using span_mono[OF B(3)] by (simp add: span_span)
```
```  1809   from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
```
```  1810     using fB(2) by arith
```
```  1811 qed
```
```  1812
```
```  1813 lemma subset_le_dim:
```
```  1814   fixes S :: "'a::euclidean_space set"
```
```  1815   shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
```
```  1816   by (metis dim_span dim_subset)
```
```  1817
```
```  1818 lemma span_eq_dim:
```
```  1819   fixes S :: "'a::euclidean_space set"
```
```  1820   shows "span S = span T \<Longrightarrow> dim S = dim T"
```
```  1821   by (metis dim_span)
```
```  1822
```
```  1823 lemma spans_image:
```
```  1824   assumes lf: "linear f"
```
```  1825     and VB: "V \<subseteq> span B"
```
```  1826   shows "f ` V \<subseteq> span (f ` B)"
```
```  1827   unfolding span_linear_image[OF lf] by (metis VB image_mono)
```
```  1828
```
```  1829 lemma dim_image_le:
```
```  1830   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1831   assumes lf: "linear f"
```
```  1832   shows "dim (f ` S) \<le> dim (S)"
```
```  1833 proof -
```
```  1834   from basis_exists[of S] obtain B where
```
```  1835     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
```
```  1836   from B have fB: "finite B" "card B = dim S"
```
```  1837     using independent_bound by blast+
```
```  1838   have "dim (f ` S) \<le> card (f ` B)"
```
```  1839     apply (rule span_card_ge_dim)
```
```  1840     using lf B fB
```
```  1841     apply (auto simp add: span_linear_image spans_image subset_image_iff)
```
```  1842     done
```
```  1843   also have "\<dots> \<le> dim S"
```
```  1844     using card_image_le[OF fB(1)] fB by simp
```
```  1845   finally show ?thesis .
```
```  1846 qed
```
```  1847
```
```  1848 text {* Relation between bases and injectivity/surjectivity of map. *}
```
```  1849
```
```  1850 lemma spanning_surjective_image:
```
```  1851   assumes us: "UNIV \<subseteq> span S"
```
```  1852     and lf: "linear f"
```
```  1853     and sf: "surj f"
```
```  1854   shows "UNIV \<subseteq> span (f ` S)"
```
```  1855 proof -
```
```  1856   have "UNIV \<subseteq> f ` UNIV"
```
```  1857     using sf by (auto simp add: surj_def)
```
```  1858   also have " \<dots> \<subseteq> span (f ` S)"
```
```  1859     using spans_image[OF lf us] .
```
```  1860   finally show ?thesis .
```
```  1861 qed
```
```  1862
```
```  1863 lemma independent_injective_image:
```
```  1864   assumes iS: "independent S"
```
```  1865     and lf: "linear f"
```
```  1866     and fi: "inj f"
```
```  1867   shows "independent (f ` S)"
```
```  1868 proof -
```
```  1869   {
```
```  1870     fix a
```
```  1871     assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
```
```  1872     have eq: "f ` S - {f a} = f ` (S - {a})"
```
```  1873       using fi by (auto simp add: inj_on_def)
```
```  1874     from a have "f a \<in> f ` span (S - {a})"
```
```  1875       unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
```
```  1876     then have "a \<in> span (S - {a})"
```
```  1877       using fi by (auto simp add: inj_on_def)
```
```  1878     with a(1) iS have False
```
```  1879       by (simp add: dependent_def)
```
```  1880   }
```
```  1881   then show ?thesis
```
```  1882     unfolding dependent_def by blast
```
```  1883 qed
```
```  1884
```
```  1885 text {* Picking an orthogonal replacement for a spanning set. *}
```
```  1886
```
```  1887 (* FIXME : Move to some general theory ?*)
```
```  1888 definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
```
```  1889
```
```  1890 lemma vector_sub_project_orthogonal:
```
```  1891   fixes b x :: "'a::euclidean_space"
```
```  1892   shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
```
```  1893   unfolding inner_simps by auto
```
```  1894
```
```  1895 lemma pairwise_orthogonal_insert:
```
```  1896   assumes "pairwise orthogonal S"
```
```  1897     and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
```
```  1898   shows "pairwise orthogonal (insert x S)"
```
```  1899   using assms unfolding pairwise_def
```
```  1900   by (auto simp add: orthogonal_commute)
```
```  1901
```
```  1902 lemma basis_orthogonal:
```
```  1903   fixes B :: "'a::real_inner set"
```
```  1904   assumes fB: "finite B"
```
```  1905   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
```
```  1906   (is " \<exists>C. ?P B C")
```
```  1907   using fB
```
```  1908 proof (induct rule: finite_induct)
```
```  1909   case empty
```
```  1910   then show ?case
```
```  1911     apply (rule exI[where x="{}"])
```
```  1912     apply (auto simp add: pairwise_def)
```
```  1913     done
```
```  1914 next
```
```  1915   case (insert a B)
```
```  1916   note fB = `finite B` and aB = `a \<notin> B`
```
```  1917   from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
```
```  1918   obtain C where C: "finite C" "card C \<le> card B"
```
```  1919     "span C = span B" "pairwise orthogonal C" by blast
```
```  1920   let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
```
```  1921   let ?C = "insert ?a C"
```
```  1922   from C(1) have fC: "finite ?C"
```
```  1923     by simp
```
```  1924   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
```
```  1925     by (simp add: card_insert_if)
```
```  1926   {
```
```  1927     fix x k
```
```  1928     have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
```
```  1929       by (simp add: field_simps)
```
```  1930     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"
```
```  1931       apply (simp only: scaleR_right_diff_distrib th0)
```
```  1932       apply (rule span_add_eq)
```
```  1933       apply (rule span_mul)
```
```  1934       apply (rule span_setsum)
```
```  1935       apply clarify
```
```  1936       apply (rule span_mul)
```
```  1937       apply (rule span_superset)
```
```  1938       apply assumption
```
```  1939       done
```
```  1940   }
```
```  1941   then have SC: "span ?C = span (insert a B)"
```
```  1942     unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
```
```  1943   {
```
```  1944     fix y
```
```  1945     assume yC: "y \<in> C"
```
```  1946     then have Cy: "C = insert y (C - {y})"
```
```  1947       by blast
```
```  1948     have fth: "finite (C - {y})"
```
```  1949       using C by simp
```
```  1950     have "orthogonal ?a y"
```
```  1951       unfolding orthogonal_def
```
```  1952       unfolding inner_diff inner_setsum_left right_minus_eq
```
```  1953       unfolding setsum.remove [OF `finite C` `y \<in> C`]
```
```  1954       apply (clarsimp simp add: inner_commute[of y a])
```
```  1955       apply (rule setsum.neutral)
```
```  1956       apply clarsimp
```
```  1957       apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
```
```  1958       using `y \<in> C` by auto
```
```  1959   }
```
```  1960   with `pairwise orthogonal C` have CPO: "pairwise orthogonal ?C"
```
```  1961     by (rule pairwise_orthogonal_insert)
```
```  1962   from fC cC SC CPO have "?P (insert a B) ?C"
```
```  1963     by blast
```
```  1964   then show ?case by blast
```
```  1965 qed
```
```  1966
```
```  1967 lemma orthogonal_basis_exists:
```
```  1968   fixes V :: "('a::euclidean_space) set"
```
```  1969   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
```
```  1970 proof -
```
```  1971   from basis_exists[of V] obtain B where
```
```  1972     B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
```
```  1973     by blast
```
```  1974   from B have fB: "finite B" "card B = dim V"
```
```  1975     using independent_bound by auto
```
```  1976   from basis_orthogonal[OF fB(1)] obtain C where
```
```  1977     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
```
```  1978     by blast
```
```  1979   from C B have CSV: "C \<subseteq> span V"
```
```  1980     by (metis span_inc span_mono subset_trans)
```
```  1981   from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
```
```  1982     by (simp add: span_span)
```
```  1983   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
```
```  1984   have iC: "independent C"
```
```  1985     by (simp add: dim_span)
```
```  1986   from C fB have "card C \<le> dim V"
```
```  1987     by simp
```
```  1988   moreover have "dim V \<le> card C"
```
```  1989     using span_card_ge_dim[OF CSV SVC C(1)]
```
```  1990     by (simp add: dim_span)
```
```  1991   ultimately have CdV: "card C = dim V"
```
```  1992     using C(1) by simp
```
```  1993   from C B CSV CdV iC show ?thesis
```
```  1994     by auto
```
```  1995 qed
```
```  1996
```
```  1997 lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
```
```  1998   using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
```
```  1999   by (auto simp add: span_span)
```
```  2000
```
```  2001 text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
```
```  2002
```
```  2003 lemma span_not_univ_orthogonal:
```
```  2004   fixes S :: "'a::euclidean_space set"
```
```  2005   assumes sU: "span S \<noteq> UNIV"
```
```  2006   shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
```
```  2007 proof -
```
```  2008   from sU obtain a where a: "a \<notin> span S"
```
```  2009     by blast
```
```  2010   from orthogonal_basis_exists obtain B where
```
```  2011     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
```
```  2012     by blast
```
```  2013   from B have fB: "finite B" "card B = dim S"
```
```  2014     using independent_bound by auto
```
```  2015   from span_mono[OF B(2)] span_mono[OF B(3)]
```
```  2016   have sSB: "span S = span B"
```
```  2017     by (simp add: span_span)
```
```  2018   let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
```
```  2019   have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
```
```  2020     unfolding sSB
```
```  2021     apply (rule span_setsum)
```
```  2022     apply clarsimp
```
```  2023     apply (rule span_mul)
```
```  2024     apply (rule span_superset)
```
```  2025     apply assumption
```
```  2026     done
```
```  2027   with a have a0:"?a  \<noteq> 0"
```
```  2028     by auto
```
```  2029   have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
```
```  2030   proof (rule span_induct')
```
```  2031     show "subspace {x. ?a \<bullet> x = 0}"
```
```  2032       by (auto simp add: subspace_def inner_add)
```
```  2033   next
```
```  2034     {
```
```  2035       fix x
```
```  2036       assume x: "x \<in> B"
```
```  2037       from x have B': "B = insert x (B - {x})"
```
```  2038         by blast
```
```  2039       have fth: "finite (B - {x})"
```
```  2040         using fB by simp
```
```  2041       have "?a \<bullet> x = 0"
```
```  2042         apply (subst B')
```
```  2043         using fB fth
```
```  2044         unfolding setsum_clauses(2)[OF fth]
```
```  2045         apply simp unfolding inner_simps
```
```  2046         apply (clarsimp simp add: inner_add inner_setsum_left)
```
```  2047         apply (rule setsum.neutral, rule ballI)
```
```  2048         unfolding inner_commute
```
```  2049         apply (auto simp add: x field_simps
```
```  2050           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
```
```  2051         done
```
```  2052     }
```
```  2053     then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
```
```  2054       by blast
```
```  2055   qed
```
```  2056   with a0 show ?thesis
```
```  2057     unfolding sSB by (auto intro: exI[where x="?a"])
```
```  2058 qed
```
```  2059
```
```  2060 lemma span_not_univ_subset_hyperplane:
```
```  2061   fixes S :: "'a::euclidean_space set"
```
```  2062   assumes SU: "span S \<noteq> UNIV"
```
```  2063   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```  2064   using span_not_univ_orthogonal[OF SU] by auto
```
```  2065
```
```  2066 lemma lowdim_subset_hyperplane:
```
```  2067   fixes S :: "'a::euclidean_space set"
```
```  2068   assumes d: "dim S < DIM('a)"
```
```  2069   shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```  2070 proof -
```
```  2071   {
```
```  2072     assume "span S = UNIV"
```
```  2073     then have "dim (span S) = dim (UNIV :: ('a) set)"
```
```  2074       by simp
```
```  2075     then have "dim S = DIM('a)"
```
```  2076       by (simp add: dim_span dim_UNIV)
```
```  2077     with d have False by arith
```
```  2078   }
```
```  2079   then have th: "span S \<noteq> UNIV"
```
```  2080     by blast
```
```  2081   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
```
```  2082 qed
```
```  2083
```
```  2084 text {* We can extend a linear basis-basis injection to the whole set. *}
```
```  2085
```
```  2086 lemma linear_indep_image_lemma:
```
```  2087   assumes lf: "linear f"
```
```  2088     and fB: "finite B"
```
```  2089     and ifB: "independent (f ` B)"
```
```  2090     and fi: "inj_on f B"
```
```  2091     and xsB: "x \<in> span B"
```
```  2092     and fx: "f x = 0"
```
```  2093   shows "x = 0"
```
```  2094   using fB ifB fi xsB fx
```
```  2095 proof (induct arbitrary: x rule: finite_induct[OF fB])
```
```  2096   case 1
```
```  2097   then show ?case by auto
```
```  2098 next
```
```  2099   case (2 a b x)
```
```  2100   have fb: "finite b" using "2.prems" by simp
```
```  2101   have th0: "f ` b \<subseteq> f ` (insert a b)"
```
```  2102     apply (rule image_mono)
```
```  2103     apply blast
```
```  2104     done
```
```  2105   from independent_mono[ OF "2.prems"(2) th0]
```
```  2106   have ifb: "independent (f ` b)"  .
```
```  2107   have fib: "inj_on f b"
```
```  2108     apply (rule subset_inj_on [OF "2.prems"(3)])
```
```  2109     apply blast
```
```  2110     done
```
```  2111   from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
```
```  2112   obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
```
```  2113     by blast
```
```  2114   have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
```
```  2115     unfolding span_linear_image[OF lf]
```
```  2116     apply (rule imageI)
```
```  2117     using k span_mono[of "b - {a}" b]
```
```  2118     apply blast
```
```  2119     done
```
```  2120   then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
```
```  2121     by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
```
```  2122   then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
```
```  2123     using "2.prems"(5) by simp
```
```  2124   have xsb: "x \<in> span b"
```
```  2125   proof (cases "k = 0")
```
```  2126     case True
```
```  2127     with k have "x \<in> span (b - {a})" by simp
```
```  2128     then show ?thesis using span_mono[of "b - {a}" b]
```
```  2129       by blast
```
```  2130   next
```
```  2131     case False
```
```  2132     with span_mul[OF th, of "- 1/ k"]
```
```  2133     have th1: "f a \<in> span (f ` b)"
```
```  2134       by auto
```
```  2135     from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
```
```  2136     have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
```
```  2137     from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
```
```  2138     have "f a \<notin> span (f ` b)" using tha
```
```  2139       using "2.hyps"(2)
```
```  2140       "2.prems"(3) by auto
```
```  2141     with th1 have False by blast
```
```  2142     then show ?thesis by blast
```
```  2143   qed
```
```  2144   from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
```
```  2145 qed
```
```  2146
```
```  2147 text {* We can extend a linear mapping from basis. *}
```
```  2148
```
```  2149 lemma linear_independent_extend_lemma:
```
```  2150   fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
```
```  2151   assumes fi: "finite B"
```
```  2152     and ib: "independent B"
```
```  2153   shows "\<exists>g.
```
```  2154     (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y) \<and>
```
```  2155     (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
```
```  2156     (\<forall>x\<in> B. g x = f x)"
```
```  2157   using ib fi
```
```  2158 proof (induct rule: finite_induct[OF fi])
```
```  2159   case 1
```
```  2160   then show ?case by auto
```
```  2161 next
```
```  2162   case (2 a b)
```
```  2163   from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
```
```  2164     by (simp_all add: independent_insert)
```
```  2165   from "2.hyps"(3)[OF ibf] obtain g where
```
```  2166     g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
```
```  2167     "\<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
```
```  2168   let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
```
```  2169   {
```
```  2170     fix z
```
```  2171     assume z: "z \<in> span (insert a b)"
```
```  2172     have th0: "z - ?h z *\<^sub>R a \<in> span b"
```
```  2173       apply (rule someI_ex)
```
```  2174       unfolding span_breakdown_eq[symmetric]
```
```  2175       apply (rule z)
```
```  2176       done
```
```  2177     {
```
```  2178       fix k
```
```  2179       assume k: "z - k *\<^sub>R a \<in> span b"
```
```  2180       have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
```
```  2181         by (simp add: field_simps scaleR_left_distrib [symmetric])
```
```  2182       from span_sub[OF th0 k] have khz: "(k - ?h z) *\<^sub>R a \<in> span b"
```
```  2183         by (simp add: eq)
```
```  2184       {
```
```  2185         assume "k \<noteq> ?h z"
```
```  2186         then have k0: "k - ?h z \<noteq> 0" by simp
```
```  2187         from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
```
```  2188         have "a \<in> span b" by simp
```
```  2189         with "2.prems"(1) "2.hyps"(2) have False
```
```  2190           by (auto simp add: dependent_def)
```
```  2191       }
```
```  2192       then have "k = ?h z" by blast
```
```  2193     }
```
```  2194     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)"
```
```  2195       by blast
```
```  2196   }
```
```  2197   note h = this
```
```  2198   let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)"
```
```  2199   {
```
```  2200     fix x y
```
```  2201     assume x: "x \<in> span (insert a b)"
```
```  2202       and y: "y \<in> span (insert a b)"
```
```  2203     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)"
```
```  2204       by (simp add: algebra_simps)
```
```  2205     have addh: "?h (x + y) = ?h x + ?h y"
```
```  2206       apply (rule conjunct2[OF h, rule_format, symmetric])
```
```  2207       apply (rule span_add[OF x y])
```
```  2208       unfolding tha
```
```  2209       apply (metis span_add x y conjunct1[OF h, rule_format])
```
```  2210       done
```
```  2211     have "?g (x + y) = ?g x + ?g y"
```
```  2212       unfolding addh tha
```
```  2213       g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
```
```  2214       by (simp add: scaleR_left_distrib)}
```
```  2215   moreover
```
```  2216   {
```
```  2217     fix x :: "'a"
```
```  2218     fix c :: real
```
```  2219     assume x: "x \<in> span (insert a b)"
```
```  2220     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)"
```
```  2221       by (simp add: algebra_simps)
```
```  2222     have hc: "?h (c *\<^sub>R x) = c * ?h x"
```
```  2223       apply (rule conjunct2[OF h, rule_format, symmetric])
```
```  2224       apply (metis span_mul x)
```
```  2225       apply (metis tha span_mul x conjunct1[OF h])
```
```  2226       done
```
```  2227     have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
```
```  2228       unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
```
```  2229       by (simp add: algebra_simps)
```
```  2230   }
```
```  2231   moreover
```
```  2232   {
```
```  2233     fix x
```
```  2234     assume x: "x \<in> insert a b"
```
```  2235     {
```
```  2236       assume xa: "x = a"
```
```  2237       have ha1: "1 = ?h a"
```
```  2238         apply (rule conjunct2[OF h, rule_format])
```
```  2239         apply (metis span_superset insertI1)
```
```  2240         using conjunct1[OF h, OF span_superset, OF insertI1]
```
```  2241         apply (auto simp add: span_0)
```
```  2242         done
```
```  2243       from xa ha1[symmetric] have "?g x = f x"
```
```  2244         apply simp
```
```  2245         using g(2)[rule_format, OF span_0, of 0]
```
```  2246         apply simp
```
```  2247         done
```
```  2248     }
```
```  2249     moreover
```
```  2250     {
```
```  2251       assume xb: "x \<in> b"
```
```  2252       have h0: "0 = ?h x"
```
```  2253         apply (rule conjunct2[OF h, rule_format])
```
```  2254         apply (metis  span_superset x)
```
```  2255         apply simp
```
```  2256         apply (metis span_superset xb)
```
```  2257         done
```
```  2258       have "?g x = f x"
```
```  2259         by (simp add: h0[symmetric] g(3)[rule_format, OF xb])
```
```  2260     }
```
```  2261     ultimately have "?g x = f x"
```
```  2262       using x by blast
```
```  2263   }
```
```  2264   ultimately show ?case
```
```  2265     apply -
```
```  2266     apply (rule exI[where x="?g"])
```
```  2267     apply blast
```
```  2268     done
```
```  2269 qed
```
```  2270
```
```  2271 lemma linear_independent_extend:
```
```  2272   fixes B :: "'a::euclidean_space set"
```
```  2273   assumes iB: "independent B"
```
```  2274   shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
```
```  2275 proof -
```
```  2276   from maximal_independent_subset_extend[of B UNIV] iB
```
```  2277   obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C"
```
```  2278     by auto
```
```  2279
```
```  2280   from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
```
```  2281   obtain g where g:
```
```  2282     "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) \<and>
```
```  2283      (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
```
```  2284      (\<forall>x\<in> C. g x = f x)" by blast
```
```  2285   from g show ?thesis
```
```  2286     unfolding linear_iff
```
```  2287     using C
```
```  2288     apply clarsimp
```
```  2289     apply blast
```
```  2290     done
```
```  2291 qed
```
```  2292
```
```  2293 text {* Can construct an isomorphism between spaces of same dimension. *}
```
```  2294
```
```  2295 lemma subspace_isomorphism:
```
```  2296   fixes S :: "'a::euclidean_space set"
```
```  2297     and T :: "'b::euclidean_space set"
```
```  2298   assumes s: "subspace S"
```
```  2299     and t: "subspace T"
```
```  2300     and d: "dim S = dim T"
```
```  2301   shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
```
```  2302 proof -
```
```  2303   from basis_exists[of S] independent_bound
```
```  2304   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
```
```  2305     by blast
```
```  2306   from basis_exists[of T] independent_bound
```
```  2307   obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C"
```
```  2308     by blast
```
```  2309   from B(4) C(4) card_le_inj[of B C] d
```
```  2310   obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C`
```
```  2311     by auto
```
```  2312   from linear_independent_extend[OF B(2)]
```
```  2313   obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x"
```
```  2314     by blast
```
```  2315   from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
```
```  2316     by simp
```
```  2317   with B(4) C(4) have ceq: "card (f ` B) = card C"
```
```  2318     using d by simp
```
```  2319   have "g ` B = f ` B"
```
```  2320     using g(2) by (auto simp add: image_iff)
```
```  2321   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
```
```  2322   finally have gBC: "g ` B = C" .
```
```  2323   have gi: "inj_on g B"
```
```  2324     using f(2) g(2) by (auto simp add: inj_on_def)
```
```  2325   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
```
```  2326   {
```
```  2327     fix x y
```
```  2328     assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
```
```  2329     from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
```
```  2330       by blast+
```
```  2331     from gxy have th0: "g (x - y) = 0"
```
```  2332       by (simp add: linear_sub[OF g(1)])
```
```  2333     have th1: "x - y \<in> span B"
```
```  2334       using x' y' by (metis span_sub)
```
```  2335     have "x = y"
```
```  2336       using g0[OF th1 th0] by simp
```
```  2337   }
```
```  2338   then have giS: "inj_on g S"
```
```  2339     unfolding inj_on_def by blast
```
```  2340   from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)"
```
```  2341     by (simp add: span_linear_image[OF g(1)])
```
```  2342   also have "\<dots> = span C" unfolding gBC ..
```
```  2343   also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
```
```  2344   finally have gS: "g ` S = T" .
```
```  2345   from g(1) gS giS show ?thesis
```
```  2346     by blast
```
```  2347 qed
```
```  2348
```
```  2349 text {* Linear functions are equal on a subspace if they are on a spanning set. *}
```
```  2350
```
```  2351 lemma subspace_kernel:
```
```  2352   assumes lf: "linear f"
```
```  2353   shows "subspace {x. f x = 0}"
```
```  2354   apply (simp add: subspace_def)
```
```  2355   apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
```
```  2356   done
```
```  2357
```
```  2358 lemma linear_eq_0_span:
```
```  2359   assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
```
```  2360   shows "\<forall>x \<in> span B. f x = 0"
```
```  2361   using f0 subspace_kernel[OF lf]
```
```  2362   by (rule span_induct')
```
```  2363
```
```  2364 lemma linear_eq_0:
```
```  2365   assumes lf: "linear f"
```
```  2366     and SB: "S \<subseteq> span B"
```
```  2367     and f0: "\<forall>x\<in>B. f x = 0"
```
```  2368   shows "\<forall>x \<in> S. f x = 0"
```
```  2369   by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
```
```  2370
```
```  2371 lemma linear_eq:
```
```  2372   assumes lf: "linear f"
```
```  2373     and lg: "linear g"
```
```  2374     and S: "S \<subseteq> span B"
```
```  2375     and fg: "\<forall> x\<in> B. f x = g x"
```
```  2376   shows "\<forall>x\<in> S. f x = g x"
```
```  2377 proof -
```
```  2378   let ?h = "\<lambda>x. f x - g x"
```
```  2379   from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
```
```  2380   from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
```
```  2381   show ?thesis by simp
```
```  2382 qed
```
```  2383
```
```  2384 lemma linear_eq_stdbasis:
```
```  2385   fixes f :: "'a::euclidean_space \<Rightarrow> _"
```
```  2386   assumes lf: "linear f"
```
```  2387     and lg: "linear g"
```
```  2388     and fg: "\<forall>b\<in>Basis. f b = g b"
```
```  2389   shows "f = g"
```
```  2390   using linear_eq[OF lf lg, of _ Basis] fg by auto
```
```  2391
```
```  2392 text {* Similar results for bilinear functions. *}
```
```  2393
```
```  2394 lemma bilinear_eq:
```
```  2395   assumes bf: "bilinear f"
```
```  2396     and bg: "bilinear g"
```
```  2397     and SB: "S \<subseteq> span B"
```
```  2398     and TC: "T \<subseteq> span C"
```
```  2399     and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
```
```  2400   shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
```
```  2401 proof -
```
```  2402   let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
```
```  2403   from bf bg have sp: "subspace ?P"
```
```  2404     unfolding bilinear_def linear_iff subspace_def bf bg
```
```  2405     by (auto simp add: span_0 bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def
```
```  2406       intro: bilinear_ladd[OF bf])
```
```  2407
```
```  2408   have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
```
```  2409     apply (rule span_induct' [OF _ sp])
```
```  2410     apply (rule ballI)
```
```  2411     apply (rule span_induct')
```
```  2412     apply (simp add: fg)
```
```  2413     apply (auto simp add: subspace_def)
```
```  2414     using bf bg unfolding bilinear_def linear_iff
```
```  2415     apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
```
```  2416       intro: bilinear_ladd[OF bf])
```
```  2417     done
```
```  2418   then show ?thesis
```
```  2419     using SB TC by auto
```
```  2420 qed
```
```  2421
```
```  2422 lemma bilinear_eq_stdbasis:
```
```  2423   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
```
```  2424   assumes bf: "bilinear f"
```
```  2425     and bg: "bilinear g"
```
```  2426     and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
```
```  2427   shows "f = g"
```
```  2428   using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast
```
```  2429
```
```  2430 text {* Detailed theorems about left and right invertibility in general case. *}
```
```  2431
```
```  2432 lemma linear_injective_left_inverse:
```
```  2433   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  2434   assumes lf: "linear f"
```
```  2435     and fi: "inj f"
```
```  2436   shows "\<exists>g. linear g \<and> g \<circ> f = id"
```
```  2437 proof -
```
```  2438   from linear_independent_extend[OF independent_injective_image, OF independent_Basis, OF lf fi]
```
```  2439   obtain h :: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x"
```
```  2440     by blast
```
```  2441   from h(2) have th: "\<forall>i\<in>Basis. (h \<circ> f) i = id i"
```
```  2442     using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
```
```  2443     by auto
```
```  2444   from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
```
```  2445   have "h \<circ> f = id" .
```
```  2446   then show ?thesis
```
```  2447     using h(1) by blast
```
```  2448 qed
```
```  2449
```
```  2450 lemma linear_surjective_right_inverse:
```
```  2451   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  2452   assumes lf: "linear f"
```
```  2453     and sf: "surj f"
```
```  2454   shows "\<exists>g. linear g \<and> f \<circ> g = id"
```
```  2455 proof -
```
```  2456   from linear_independent_extend[OF independent_Basis[where 'a='b],of "inv f"]
```
```  2457   obtain h :: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x\<in>Basis. h x = inv f x"
```
```  2458     by blast
```
```  2459   from h(2) have th: "\<forall>i\<in>Basis. (f \<circ> h) i = id i"
```
```  2460     using sf by (auto simp add: surj_iff_all)
```
```  2461   from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
```
```  2462   have "f \<circ> h = id" .
```
```  2463   then show ?thesis
```
```  2464     using h(1) by blast
```
```  2465 qed
```
```  2466
```
```  2467 text {* An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective. *}
```
```  2468
```
```  2469 lemma linear_injective_imp_surjective:
```
```  2470   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2471   assumes lf: "linear f"
```
```  2472     and fi: "inj f"
```
```  2473   shows "surj f"
```
```  2474 proof -
```
```  2475   let ?U = "UNIV :: 'a set"
```
```  2476   from basis_exists[of ?U] obtain B
```
```  2477     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
```
```  2478     by blast
```
```  2479   from B(4) have d: "dim ?U = card B"
```
```  2480     by simp
```
```  2481   have th: "?U \<subseteq> span (f ` B)"
```
```  2482     apply (rule card_ge_dim_independent)
```
```  2483     apply blast
```
```  2484     apply (rule independent_injective_image[OF B(2) lf fi])
```
```  2485     apply (rule order_eq_refl)
```
```  2486     apply (rule sym)
```
```  2487     unfolding d
```
```  2488     apply (rule card_image)
```
```  2489     apply (rule subset_inj_on[OF fi])
```
```  2490     apply blast
```
```  2491     done
```
```  2492   from th show ?thesis
```
```  2493     unfolding span_linear_image[OF lf] surj_def
```
```  2494     using B(3) by blast
```
```  2495 qed
```
```  2496
```
```  2497 text {* And vice versa. *}
```
```  2498
```
```  2499 lemma surjective_iff_injective_gen:
```
```  2500   assumes fS: "finite S"
```
```  2501     and fT: "finite T"
```
```  2502     and c: "card S = card T"
```
```  2503     and ST: "f ` S \<subseteq> T"
```
```  2504   shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S"
```
```  2505   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2506 proof
```
```  2507   assume h: "?lhs"
```
```  2508   {
```
```  2509     fix x y
```
```  2510     assume x: "x \<in> S"
```
```  2511     assume y: "y \<in> S"
```
```  2512     assume f: "f x = f y"
```
```  2513     from x fS have S0: "card S \<noteq> 0"
```
```  2514       by auto
```
```  2515     have "x = y"
```
```  2516     proof (rule ccontr)
```
```  2517       assume xy: "\<not> ?thesis"
```
```  2518       have th: "card S \<le> card (f ` (S - {y}))"
```
```  2519         unfolding c
```
```  2520         apply (rule card_mono)
```
```  2521         apply (rule finite_imageI)
```
```  2522         using fS apply simp
```
```  2523         using h xy x y f unfolding subset_eq image_iff
```
```  2524         apply auto
```
```  2525         apply (case_tac "xa = f x")
```
```  2526         apply (rule bexI[where x=x])
```
```  2527         apply auto
```
```  2528         done
```
```  2529       also have " \<dots> \<le> card (S - {y})"
```
```  2530         apply (rule card_image_le)
```
```  2531         using fS by simp
```
```  2532       also have "\<dots> \<le> card S - 1" using y fS by simp
```
```  2533       finally show False using S0 by arith
```
```  2534     qed
```
```  2535   }
```
```  2536   then show ?rhs
```
```  2537     unfolding inj_on_def by blast
```
```  2538 next
```
```  2539   assume h: ?rhs
```
```  2540   have "f ` S = T"
```
```  2541     apply (rule card_subset_eq[OF fT ST])
```
```  2542     unfolding card_image[OF h]
```
```  2543     apply (rule c)
```
```  2544     done
```
```  2545   then show ?lhs by blast
```
```  2546 qed
```
```  2547
```
```  2548 lemma linear_surjective_imp_injective:
```
```  2549   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2550   assumes lf: "linear f"
```
```  2551     and sf: "surj f"
```
```  2552   shows "inj f"
```
```  2553 proof -
```
```  2554   let ?U = "UNIV :: 'a set"
```
```  2555   from basis_exists[of ?U] obtain B
```
```  2556     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
```
```  2557     by blast
```
```  2558   {
```
```  2559     fix x
```
```  2560     assume x: "x \<in> span B"
```
```  2561     assume fx: "f x = 0"
```
```  2562     from B(2) have fB: "finite B"
```
```  2563       using independent_bound by auto
```
```  2564     have fBi: "independent (f ` B)"
```
```  2565       apply (rule card_le_dim_spanning[of "f ` B" ?U])
```
```  2566       apply blast
```
```  2567       using sf B(3)
```
```  2568       unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
```
```  2569       apply blast
```
```  2570       using fB apply blast
```
```  2571       unfolding d[symmetric]
```
```  2572       apply (rule card_image_le)
```
```  2573       apply (rule fB)
```
```  2574       done
```
```  2575     have th0: "dim ?U \<le> card (f ` B)"
```
```  2576       apply (rule span_card_ge_dim)
```
```  2577       apply blast
```
```  2578       unfolding span_linear_image[OF lf]
```
```  2579       apply (rule subset_trans[where B = "f ` UNIV"])
```
```  2580       using sf unfolding surj_def
```
```  2581       apply blast
```
```  2582       apply (rule image_mono)
```
```  2583       apply (rule B(3))
```
```  2584       apply (metis finite_imageI fB)
```
```  2585       done
```
```  2586     moreover have "card (f ` B) \<le> card B"
```
```  2587       by (rule card_image_le, rule fB)
```
```  2588     ultimately have th1: "card B = card (f ` B)"
```
```  2589       unfolding d by arith
```
```  2590     have fiB: "inj_on f B"
```
```  2591       unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric]
```
```  2592       by blast
```
```  2593     from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
```
```  2594     have "x = 0" by blast
```
```  2595   }
```
```  2596   then show ?thesis
```
```  2597     unfolding linear_injective_0[OF lf]
```
```  2598     using B(3)
```
```  2599     by blast
```
```  2600 qed
```
```  2601
```
```  2602 text {* Hence either is enough for isomorphism. *}
```
```  2603
```
```  2604 lemma left_right_inverse_eq:
```
```  2605   assumes fg: "f \<circ> g = id"
```
```  2606     and gh: "g \<circ> h = id"
```
```  2607   shows "f = h"
```
```  2608 proof -
```
```  2609   have "f = f \<circ> (g \<circ> h)"
```
```  2610     unfolding gh by simp
```
```  2611   also have "\<dots> = (f \<circ> g) \<circ> h"
```
```  2612     by (simp add: o_assoc)
```
```  2613   finally show "f = h"
```
```  2614     unfolding fg by simp
```
```  2615 qed
```
```  2616
```
```  2617 lemma isomorphism_expand:
```
```  2618   "f \<circ> g = id \<and> g \<circ> f = id \<longleftrightarrow> (\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x)"
```
```  2619   by (simp add: fun_eq_iff o_def id_def)
```
```  2620
```
```  2621 lemma linear_injective_isomorphism:
```
```  2622   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2623   assumes lf: "linear f"
```
```  2624     and fi: "inj f"
```
```  2625   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
```
```  2626   unfolding isomorphism_expand[symmetric]
```
```  2627   using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]]
```
```  2628     linear_injective_left_inverse[OF lf fi]
```
```  2629   by (metis left_right_inverse_eq)
```
```  2630
```
```  2631 lemma linear_surjective_isomorphism:
```
```  2632   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2633   assumes lf: "linear f"
```
```  2634     and sf: "surj 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 sf]
```
```  2638     linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
```
```  2639   by (metis left_right_inverse_eq)
```
```  2640
```
```  2641 text {* Left and right inverses are the same for
```
```  2642   @{typ "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"}. *}
```
```  2643
```
```  2644 lemma linear_inverse_left:
```
```  2645   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2646   assumes lf: "linear f"
```
```  2647     and lf': "linear f'"
```
```  2648   shows "f \<circ> f' = id \<longleftrightarrow> f' \<circ> f = id"
```
```  2649 proof -
```
```  2650   {
```
```  2651     fix f f':: "'a \<Rightarrow> 'a"
```
```  2652     assume lf: "linear f" "linear f'"
```
```  2653     assume f: "f \<circ> f' = id"
```
```  2654     from f have sf: "surj f"
```
```  2655       apply (auto simp add: o_def id_def surj_def)
```
```  2656       apply metis
```
```  2657       done
```
```  2658     from linear_surjective_isomorphism[OF lf(1) sf] lf f
```
```  2659     have "f' \<circ> f = id"
```
```  2660       unfolding fun_eq_iff o_def id_def by metis
```
```  2661   }
```
```  2662   then show ?thesis
```
```  2663     using lf lf' by metis
```
```  2664 qed
```
```  2665
```
```  2666 text {* Moreover, a one-sided inverse is automatically linear. *}
```
```  2667
```
```  2668 lemma left_inverse_linear:
```
```  2669   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
```
```  2670   assumes lf: "linear f"
```
```  2671     and gf: "g \<circ> f = id"
```
```  2672   shows "linear g"
```
```  2673 proof -
```
```  2674   from gf have fi: "inj f"
```
```  2675     apply (auto simp add: inj_on_def o_def id_def fun_eq_iff)
```
```  2676     apply metis
```
```  2677     done
```
```  2678   from linear_injective_isomorphism[OF lf fi]
```
```  2679   obtain h :: "'a \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x"
```
```  2680     by blast
```
```  2681   have "h = g"
```
```  2682     apply (rule ext) using gf h(2,3)
```
```  2683     apply (simp add: o_def id_def fun_eq_iff)
```
```  2684     apply metis
```
```  2685     done
```
```  2686   with h(1) show ?thesis by blast
```
```  2687 qed
```
```  2688
```
```  2689
```
```  2690 subsection {* Infinity norm *}
```
```  2691
```
```  2692 definition "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
```
```  2693
```
```  2694 lemma infnorm_set_image:
```
```  2695   fixes x :: "'a::euclidean_space"
```
```  2696   shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```  2697   by blast
```
```  2698
```
```  2699 lemma infnorm_Max:
```
```  2700   fixes x :: "'a::euclidean_space"
```
```  2701   shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)"
```
```  2702   by (simp add: infnorm_def infnorm_set_image cSup_eq_Max del: Sup_image_eq)
```
```  2703
```
```  2704 lemma infnorm_set_lemma:
```
```  2705   fixes x :: "'a::euclidean_space"
```
```  2706   shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
```
```  2707     and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}"
```
```  2708   unfolding infnorm_set_image
```
```  2709   by auto
```
```  2710
```
```  2711 lemma infnorm_pos_le:
```
```  2712   fixes x :: "'a::euclidean_space"
```
```  2713   shows "0 \<le> infnorm x"
```
```  2714   by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
```
```  2715
```
```  2716 lemma infnorm_triangle:
```
```  2717   fixes x :: "'a::euclidean_space"
```
```  2718   shows "infnorm (x + y) \<le> infnorm x + infnorm y"
```
```  2719 proof -
```
```  2720   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"
```
```  2721     by simp
```
```  2722   show ?thesis
```
```  2723     by (auto simp: infnorm_Max inner_add_left intro!: *)
```
```  2724 qed
```
```  2725
```
```  2726 lemma infnorm_eq_0:
```
```  2727   fixes x :: "'a::euclidean_space"
```
```  2728   shows "infnorm x = 0 \<longleftrightarrow> x = 0"
```
```  2729 proof -
```
```  2730   have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
```
```  2731     unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
```
```  2732   then show ?thesis
```
```  2733     using infnorm_pos_le[of x] by simp
```
```  2734 qed
```
```  2735
```
```  2736 lemma infnorm_0: "infnorm 0 = 0"
```
```  2737   by (simp add: infnorm_eq_0)
```
```  2738
```
```  2739 lemma infnorm_neg: "infnorm (- x) = infnorm x"
```
```  2740   unfolding infnorm_def
```
```  2741   apply (rule cong[of "Sup" "Sup"])
```
```  2742   apply blast
```
```  2743   apply auto
```
```  2744   done
```
```  2745
```
```  2746 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
```
```  2747 proof -
```
```  2748   have "y - x = - (x - y)" by simp
```
```  2749   then show ?thesis
```
```  2750     by (metis infnorm_neg)
```
```  2751 qed
```
```  2752
```
```  2753 lemma real_abs_sub_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
```
```  2754 proof -
```
```  2755   have th: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
```
```  2756     by arith
```
```  2757   from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
```
```  2758   have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
```
```  2759     "infnorm y \<le> infnorm (x - y) + infnorm x"
```
```  2760     by (simp_all add: field_simps infnorm_neg)
```
```  2761   from th[OF ths] show ?thesis .
```
```  2762 qed
```
```  2763
```
```  2764 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
```
```  2765   using infnorm_pos_le[of x] by arith
```
```  2766
```
```  2767 lemma Basis_le_infnorm:
```
```  2768   fixes x :: "'a::euclidean_space"
```
```  2769   shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
```
```  2770   by (simp add: infnorm_Max)
```
```  2771
```
```  2772 lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
```
```  2773   unfolding infnorm_Max
```
```  2774 proof (safe intro!: Max_eqI)
```
```  2775   let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```  2776   {
```
```  2777     fix b :: 'a
```
```  2778     assume "b \<in> Basis"
```
```  2779     then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
```
```  2780       by (simp add: abs_mult mult_left_mono)
```
```  2781   next
```
```  2782     from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
```
```  2783       by (auto simp del: Max_in)
```
```  2784     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"
```
```  2785       by (intro image_eqI[where x=b]) (auto simp: abs_mult)
```
```  2786   }
```
```  2787 qed simp
```
```  2788
```
```  2789 lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
```
```  2790   unfolding infnorm_mul ..
```
```  2791
```
```  2792 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
```
```  2793   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
```
```  2794
```
```  2795 text {* Prove that it differs only up to a bound from Euclidean norm. *}
```
```  2796
```
```  2797 lemma infnorm_le_norm: "infnorm x \<le> norm x"
```
```  2798   by (simp add: Basis_le_norm infnorm_Max)
```
```  2799
```
```  2800 lemma (in euclidean_space) euclidean_inner: "inner x y = (\<Sum>b\<in>Basis. (x \<bullet> b) * (y \<bullet> b))"
```
```  2801   by (subst (1 2) euclidean_representation [symmetric])
```
```  2802     (simp add: inner_setsum_right inner_Basis ac_simps)
```
```  2803
```
```  2804 lemma norm_le_infnorm:
```
```  2805   fixes x :: "'a::euclidean_space"
```
```  2806   shows "norm x \<le> sqrt DIM('a) * infnorm x"
```
```  2807 proof -
```
```  2808   let ?d = "DIM('a)"
```
```  2809   have "real ?d \<ge> 0"
```
```  2810     by simp
```
```  2811   then have d2: "(sqrt (real ?d))\<^sup>2 = real ?d"
```
```  2812     by (auto intro: real_sqrt_pow2)
```
```  2813   have th: "sqrt (real ?d) * infnorm x \<ge> 0"
```
```  2814     by (simp add: zero_le_mult_iff infnorm_pos_le)
```
```  2815   have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)\<^sup>2"
```
```  2816     unfolding power_mult_distrib d2
```
```  2817     unfolding real_of_nat_def
```
```  2818     apply (subst euclidean_inner)
```
```  2819     apply (subst power2_abs[symmetric])
```
```  2820     apply (rule order_trans[OF setsum_bounded[where K="\<bar>infnorm x\<bar>\<^sup>2"]])
```
```  2821     apply (auto simp add: power2_eq_square[symmetric])
```
```  2822     apply (subst power2_abs[symmetric])
```
```  2823     apply (rule power_mono)
```
```  2824     apply (auto simp: infnorm_Max)
```
```  2825     done
```
```  2826   from real_le_lsqrt[OF inner_ge_zero th th1]
```
```  2827   show ?thesis
```
```  2828     unfolding norm_eq_sqrt_inner id_def .
```
```  2829 qed
```
```  2830
```
```  2831 lemma tendsto_infnorm [tendsto_intros]:
```
```  2832   assumes "(f ---> a) F"
```
```  2833   shows "((\<lambda>x. infnorm (f x)) ---> infnorm a) F"
```
```  2834 proof (rule tendsto_compose [OF LIM_I assms])
```
```  2835   fix r :: real
```
```  2836   assume "r > 0"
```
```  2837   then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
```
```  2838     by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm)
```
```  2839 qed
```
```  2840
```
```  2841 text {* Equality in Cauchy-Schwarz and triangle inequalities. *}
```
```  2842
```
```  2843 lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  2844   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2845 proof -
```
```  2846   {
```
```  2847     assume h: "x = 0"
```
```  2848     then have ?thesis by simp
```
```  2849   }
```
```  2850   moreover
```
```  2851   {
```
```  2852     assume h: "y = 0"
```
```  2853     then have ?thesis by simp
```
```  2854   }
```
```  2855   moreover
```
```  2856   {
```
```  2857     assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
```
```  2858     from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
```
```  2859     have "?rhs \<longleftrightarrow>
```
```  2860       (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
```
```  2861         norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
```
```  2862       using x y
```
```  2863       unfolding inner_simps
```
```  2864       unfolding power2_norm_eq_inner[symmetric] power2_eq_square right_minus_eq
```
```  2865       apply (simp add: inner_commute)
```
```  2866       apply (simp add: field_simps)
```
```  2867       apply metis
```
```  2868       done
```
```  2869     also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
```
```  2870       by (simp add: field_simps inner_commute)
```
```  2871     also have "\<dots> \<longleftrightarrow> ?lhs" using x y
```
```  2872       apply simp
```
```  2873       apply metis
```
```  2874       done
```
```  2875     finally have ?thesis by blast
```
```  2876   }
```
```  2877   ultimately show ?thesis by blast
```
```  2878 qed
```
```  2879
```
```  2880 lemma norm_cauchy_schwarz_abs_eq:
```
```  2881   "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow>
```
```  2882     norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
```
```  2883   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2884 proof -
```
```  2885   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a"
```
```  2886     by arith
```
```  2887   have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
```
```  2888     by simp
```
```  2889   also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
```
```  2890     unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  2891     unfolding norm_minus_cancel norm_scaleR ..
```
```  2892   also have "\<dots> \<longleftrightarrow> ?lhs"
```
```  2893     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
```
```  2894     by auto
```
```  2895   finally show ?thesis ..
```
```  2896 qed
```
```  2897
```
```  2898 lemma norm_triangle_eq:
```
```  2899   fixes x y :: "'a::real_inner"
```
```  2900   shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  2901 proof -
```
```  2902   {
```
```  2903     assume x: "x = 0 \<or> y = 0"
```
```  2904     then have ?thesis
```
```  2905       by (cases "x = 0") simp_all
```
```  2906   }
```
```  2907   moreover
```
```  2908   {
```
```  2909     assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
```
```  2910     then have "norm x \<noteq> 0" "norm y \<noteq> 0"
```
```  2911       by simp_all
```
```  2912     then have n: "norm x > 0" "norm y > 0"
```
```  2913       using norm_ge_zero[of x] norm_ge_zero[of y] by arith+
```
```  2914     have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 \<Longrightarrow> a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2"
```
```  2915       by algebra
```
```  2916     have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
```
```  2917       apply (rule th)
```
```  2918       using n norm_ge_zero[of "x + y"]
```
```  2919       apply arith
```
```  2920       done
```
```  2921     also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  2922       unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  2923       unfolding power2_norm_eq_inner inner_simps
```
```  2924       by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
```
```  2925     finally have ?thesis .
```
```  2926   }
```
```  2927   ultimately show ?thesis by blast
```
```  2928 qed
```
```  2929
```
```  2930
```
```  2931 subsection {* Collinearity *}
```
```  2932
```
```  2933 definition collinear :: "'a::real_vector set \<Rightarrow> bool"
```
```  2934   where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
```
```  2935
```
```  2936 lemma collinear_empty: "collinear {}"
```
```  2937   by (simp add: collinear_def)
```
```  2938
```
```  2939 lemma collinear_sing: "collinear {x}"
```
```  2940   by (simp add: collinear_def)
```
```  2941
```
```  2942 lemma collinear_2: "collinear {x, y}"
```
```  2943   apply (simp add: collinear_def)
```
```  2944   apply (rule exI[where x="x - y"])
```
```  2945   apply auto
```
```  2946   apply (rule exI[where x=1], simp)
```
```  2947   apply (rule exI[where x="- 1"], simp)
```
```  2948   done
```
```  2949
```
```  2950 lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
```
```  2951   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2952 proof -
```
```  2953   {
```
```  2954     assume "x = 0 \<or> y = 0"
```
```  2955     then have ?thesis
```
```  2956       by (cases "x = 0") (simp_all add: collinear_2 insert_commute)
```
```  2957   }
```
```  2958   moreover
```
```  2959   {
```
```  2960     assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
```
```  2961     have ?thesis
```
```  2962     proof
```
```  2963       assume h: "?lhs"
```
```  2964       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"
```
```  2965         unfolding collinear_def by blast
```
```  2966       from u[rule_format, of x 0] u[rule_format, of y 0]
```
```  2967       obtain cx and cy where
```
```  2968         cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
```
```  2969         by auto
```
```  2970       from cx x have cx0: "cx \<noteq> 0" by auto
```
```  2971       from cy y have cy0: "cy \<noteq> 0" by auto
```
```  2972       let ?d = "cy / cx"
```
```  2973       from cx cy cx0 have "y = ?d *\<^sub>R x"
```
```  2974         by simp
```
```  2975       then show ?rhs using x y by blast
```
```  2976     next
```
```  2977       assume h: "?rhs"
```
```  2978       then obtain c where c: "y = c *\<^sub>R x"
```
```  2979         using x y by blast
```
```  2980       show ?lhs
```
```  2981         unfolding collinear_def c
```
```  2982         apply (rule exI[where x=x])
```
```  2983         apply auto
```
```  2984         apply (rule exI[where x="- 1"], simp)
```
```  2985         apply (rule exI[where x= "-c"], simp)
```
```  2986         apply (rule exI[where x=1], simp)
```
```  2987         apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
```
```  2988         apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
```
```  2989         done
```
```  2990     qed
```
```  2991   }
```
```  2992   ultimately show ?thesis by blast
```
```  2993 qed
```
```  2994
```
```  2995 lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
```
```  2996   unfolding norm_cauchy_schwarz_abs_eq
```
```  2997   apply (cases "x=0", simp_all add: collinear_2)
```
```  2998   apply (cases "y=0", simp_all add: collinear_2 insert_commute)
```
```  2999   unfolding collinear_lemma
```
```  3000   apply simp
```
```  3001   apply (subgoal_tac "norm x \<noteq> 0")
```
```  3002   apply (subgoal_tac "norm y \<noteq> 0")
```
```  3003   apply (rule iffI)
```
```  3004   apply (cases "norm x *\<^sub>R y = norm y *\<^sub>R x")
```
```  3005   apply (rule exI[where x="(1/norm x) * norm y"])
```
```  3006   apply (drule sym)
```
```  3007   unfolding scaleR_scaleR[symmetric]
```
```  3008   apply (simp add: field_simps)
```
```  3009   apply (rule exI[where x="(1/norm x) * - norm y"])
```
```  3010   apply clarify
```
```  3011   apply (drule sym)
```
```  3012   unfolding scaleR_scaleR[symmetric]
```
```  3013   apply (simp add: field_simps)
```
```  3014   apply (erule exE)
```
```  3015   apply (erule ssubst)
```
```  3016   unfolding scaleR_scaleR
```
```  3017   unfolding norm_scaleR
```
```  3018   apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
```
```  3019   apply (auto simp add: field_simps)
```
```  3020   done
```
```  3021
```
```  3022 end
```