src/HOL/Multivariate_Analysis/Linear_Algebra.thy
author huffman
Tue Mar 18 09:39:07 2014 -0700 (2014-03-18)
changeset 56196 32b7eafc5a52
parent 56166 9a241bc276cd
child 56409 36489d77c484
permissions -rw-r--r--
remove unnecessary finiteness assumptions from lemmas about setsum
huffman@44133
     1
(*  Title:      HOL/Multivariate_Analysis/Linear_Algebra.thy
huffman@44133
     2
    Author:     Amine Chaieb, University of Cambridge
huffman@44133
     3
*)
huffman@44133
     4
huffman@44133
     5
header {* Elementary linear algebra on Euclidean spaces *}
huffman@44133
     6
huffman@44133
     7
theory Linear_Algebra
huffman@44133
     8
imports
huffman@44133
     9
  Euclidean_Space
huffman@44133
    10
  "~~/src/HOL/Library/Infinite_Set"
huffman@44133
    11
begin
huffman@44133
    12
huffman@44133
    13
lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
huffman@44133
    14
  by auto
huffman@44133
    15
huffman@44133
    16
notation inner (infix "\<bullet>" 70)
huffman@44133
    17
wenzelm@53716
    18
lemma square_bound_lemma:
wenzelm@53716
    19
  fixes x :: real
wenzelm@53716
    20
  shows "x < (1 + x) * (1 + x)"
wenzelm@49522
    21
proof -
wenzelm@53406
    22
  have "(x + 1/2)\<^sup>2 + 3/4 > 0"
wenzelm@53406
    23
    using zero_le_power2[of "x+1/2"] by arith
wenzelm@53406
    24
  then show ?thesis
wenzelm@53406
    25
    by (simp add: field_simps power2_eq_square)
huffman@44133
    26
qed
huffman@44133
    27
wenzelm@53406
    28
lemma square_continuous:
wenzelm@53406
    29
  fixes e :: real
wenzelm@53406
    30
  shows "e > 0 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. abs (y - x) < d \<longrightarrow> abs (y * y - x * x) < e)"
hoelzl@51478
    31
  using isCont_power[OF isCont_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
wenzelm@49522
    32
  apply (auto simp add: power2_eq_square)
huffman@44133
    33
  apply (rule_tac x="s" in exI)
huffman@44133
    34
  apply auto
huffman@44133
    35
  apply (erule_tac x=y in allE)
huffman@44133
    36
  apply auto
huffman@44133
    37
  done
huffman@44133
    38
huffman@44133
    39
text{* Hence derive more interesting properties of the norm. *}
huffman@44133
    40
wenzelm@53406
    41
lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
huffman@44666
    42
  by simp (* TODO: delete *)
huffman@44133
    43
huffman@44133
    44
lemma norm_triangle_sub:
huffman@44133
    45
  fixes x y :: "'a::real_normed_vector"
wenzelm@53406
    46
  shows "norm x \<le> norm y + norm (x - y)"
huffman@44133
    47
  using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
huffman@44133
    48
wenzelm@53406
    49
lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
huffman@44133
    50
  by (simp add: norm_eq_sqrt_inner)
huffman@44666
    51
wenzelm@53406
    52
lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
wenzelm@53406
    53
  by (simp add: norm_eq_sqrt_inner)
wenzelm@53406
    54
wenzelm@53406
    55
lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
wenzelm@49522
    56
  apply (subst order_eq_iff)
wenzelm@49522
    57
  apply (auto simp: norm_le)
wenzelm@49522
    58
  done
huffman@44666
    59
wenzelm@53406
    60
lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
huffman@44666
    61
  by (simp add: norm_eq_sqrt_inner)
huffman@44133
    62
huffman@44133
    63
text{* Squaring equations and inequalities involving norms.  *}
huffman@44133
    64
wenzelm@53077
    65
lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
huffman@44666
    66
  by (simp only: power2_norm_eq_inner) (* TODO: move? *)
huffman@44133
    67
wenzelm@53406
    68
lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
huffman@44133
    69
  by (auto simp add: norm_eq_sqrt_inner)
huffman@44133
    70
wenzelm@53077
    71
lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)\<^sup>2 \<le> y\<^sup>2"
huffman@44133
    72
proof
huffman@44133
    73
  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
wenzelm@53015
    74
  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
wenzelm@53015
    75
  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
huffman@44133
    76
next
wenzelm@53015
    77
  assume "x\<^sup>2 \<le> y\<^sup>2"
wenzelm@53015
    78
  then have "sqrt (x\<^sup>2) \<le> sqrt (y\<^sup>2)" by (rule real_sqrt_le_mono)
huffman@44133
    79
  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
huffman@44133
    80
qed
huffman@44133
    81
wenzelm@53406
    82
lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2"
huffman@44133
    83
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
huffman@44133
    84
  using norm_ge_zero[of x]
huffman@44133
    85
  apply arith
huffman@44133
    86
  done
huffman@44133
    87
wenzelm@53406
    88
lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2"
huffman@44133
    89
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
huffman@44133
    90
  using norm_ge_zero[of x]
huffman@44133
    91
  apply arith
huffman@44133
    92
  done
huffman@44133
    93
wenzelm@53716
    94
lemma norm_lt_square: "norm x < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2"
huffman@44133
    95
  by (metis not_le norm_ge_square)
wenzelm@53406
    96
wenzelm@53716
    97
lemma norm_gt_square: "norm x > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2"
huffman@44133
    98
  by (metis norm_le_square not_less)
huffman@44133
    99
huffman@44133
   100
text{* Dot product in terms of the norm rather than conversely. *}
huffman@44133
   101
wenzelm@53406
   102
lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
wenzelm@49522
   103
  inner_scaleR_left inner_scaleR_right
huffman@44133
   104
wenzelm@53077
   105
lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
wenzelm@53406
   106
  unfolding power2_norm_eq_inner inner_simps inner_commute by auto
huffman@44133
   107
wenzelm@53077
   108
lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
wenzelm@49525
   109
  unfolding power2_norm_eq_inner inner_simps inner_commute
wenzelm@49525
   110
  by (auto simp add: algebra_simps)
huffman@44133
   111
huffman@44133
   112
text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
huffman@44133
   113
wenzelm@53406
   114
lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
wenzelm@53406
   115
  (is "?lhs \<longleftrightarrow> ?rhs")
huffman@44133
   116
proof
wenzelm@49652
   117
  assume ?lhs
wenzelm@49652
   118
  then show ?rhs by simp
huffman@44133
   119
next
huffman@44133
   120
  assume ?rhs
wenzelm@53406
   121
  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
wenzelm@53406
   122
    by simp
wenzelm@53406
   123
  then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
wenzelm@53406
   124
    by (simp add: inner_diff inner_commute)
wenzelm@53406
   125
  then have "(x - y) \<bullet> (x - y) = 0"
wenzelm@53406
   126
    by (simp add: field_simps inner_diff inner_commute)
wenzelm@53406
   127
  then show "x = y" by simp
huffman@44133
   128
qed
huffman@44133
   129
huffman@44133
   130
lemma norm_triangle_half_r:
wenzelm@53406
   131
  "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
wenzelm@53406
   132
  using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
huffman@44133
   133
wenzelm@49522
   134
lemma norm_triangle_half_l:
wenzelm@53406
   135
  assumes "norm (x - y) < e / 2"
wenzelm@53842
   136
    and "norm (x' - y) < e / 2"
huffman@44133
   137
  shows "norm (x - x') < e"
wenzelm@53406
   138
  using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
wenzelm@53406
   139
  unfolding dist_norm[symmetric] .
wenzelm@53406
   140
wenzelm@53406
   141
lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
huffman@44666
   142
  by (rule norm_triangle_ineq [THEN order_trans])
huffman@44133
   143
wenzelm@53406
   144
lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
huffman@44666
   145
  by (rule norm_triangle_ineq [THEN le_less_trans])
huffman@44133
   146
huffman@44133
   147
lemma setsum_clauses:
huffman@44133
   148
  shows "setsum f {} = 0"
wenzelm@49525
   149
    and "finite S \<Longrightarrow> setsum f (insert x S) = (if x \<in> S then setsum f S else f x + setsum f S)"
huffman@44133
   150
  by (auto simp add: insert_absorb)
huffman@44133
   151
huffman@44133
   152
lemma setsum_norm_le:
huffman@44133
   153
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
huffman@44176
   154
  assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
huffman@44133
   155
  shows "norm (setsum f S) \<le> setsum g S"
wenzelm@49522
   156
  by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)
huffman@44133
   157
huffman@44133
   158
lemma setsum_norm_bound:
huffman@44133
   159
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
huffman@56196
   160
  assumes K: "\<forall>x \<in> S. norm (f x) \<le> K"
huffman@44133
   161
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
huffman@44176
   162
  using setsum_norm_le[OF K] setsum_constant[symmetric]
huffman@44133
   163
  by simp
huffman@44133
   164
huffman@44133
   165
lemma setsum_group:
huffman@44133
   166
  assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
huffman@53939
   167
  shows "setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) T = setsum g S"
huffman@44133
   168
  apply (subst setsum_image_gen[OF fS, of g f])
huffman@44133
   169
  apply (rule setsum_mono_zero_right[OF fT fST])
wenzelm@49522
   170
  apply (auto intro: setsum_0')
wenzelm@49522
   171
  done
huffman@44133
   172
huffman@44133
   173
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
huffman@44133
   174
proof
huffman@44133
   175
  assume "\<forall>x. x \<bullet> y = x \<bullet> z"
wenzelm@53406
   176
  then have "\<forall>x. x \<bullet> (y - z) = 0"
wenzelm@53406
   177
    by (simp add: inner_diff)
wenzelm@49522
   178
  then have "(y - z) \<bullet> (y - z) = 0" ..
wenzelm@49652
   179
  then show "y = z" by simp
huffman@44133
   180
qed simp
huffman@44133
   181
huffman@44133
   182
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
huffman@44133
   183
proof
huffman@44133
   184
  assume "\<forall>z. x \<bullet> z = y \<bullet> z"
wenzelm@53406
   185
  then have "\<forall>z. (x - y) \<bullet> z = 0"
wenzelm@53406
   186
    by (simp add: inner_diff)
wenzelm@49522
   187
  then have "(x - y) \<bullet> (x - y) = 0" ..
wenzelm@49652
   188
  then show "x = y" by simp
huffman@44133
   189
qed simp
huffman@44133
   190
wenzelm@49522
   191
wenzelm@49522
   192
subsection {* Orthogonality. *}
huffman@44133
   193
huffman@44133
   194
context real_inner
huffman@44133
   195
begin
huffman@44133
   196
wenzelm@53842
   197
definition "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
huffman@44133
   198
huffman@44133
   199
lemma orthogonal_clauses:
huffman@44133
   200
  "orthogonal a 0"
huffman@44133
   201
  "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
wenzelm@53842
   202
  "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
huffman@44133
   203
  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
huffman@44133
   204
  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
huffman@44133
   205
  "orthogonal 0 a"
huffman@44133
   206
  "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
wenzelm@53842
   207
  "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
huffman@44133
   208
  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
huffman@44133
   209
  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
huffman@44666
   210
  unfolding orthogonal_def inner_add inner_diff by auto
huffman@44666
   211
huffman@44133
   212
end
huffman@44133
   213
huffman@44133
   214
lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
huffman@44133
   215
  by (simp add: orthogonal_def inner_commute)
huffman@44133
   216
wenzelm@49522
   217
wenzelm@49522
   218
subsection {* Linear functions. *}
wenzelm@49522
   219
huffman@53600
   220
lemma linear_iff:
wenzelm@53716
   221
  "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)"
huffman@53600
   222
  (is "linear f \<longleftrightarrow> ?rhs")
huffman@53600
   223
proof
huffman@53600
   224
  assume "linear f" then interpret f: linear f .
huffman@53600
   225
  show "?rhs" by (simp add: f.add f.scaleR)
huffman@53600
   226
next
huffman@53600
   227
  assume "?rhs" then show "linear f" by unfold_locales simp_all
huffman@53600
   228
qed
huffman@44133
   229
wenzelm@53406
   230
lemma linear_compose_cmul: "linear f \<Longrightarrow> linear (\<lambda>x. c *\<^sub>R f x)"
huffman@53600
   231
  by (simp add: linear_iff algebra_simps)
huffman@44133
   232
wenzelm@53406
   233
lemma linear_compose_neg: "linear f \<Longrightarrow> linear (\<lambda>x. - f x)"
huffman@53600
   234
  by (simp add: linear_iff)
huffman@44133
   235
wenzelm@53406
   236
lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x + g x)"
huffman@53600
   237
  by (simp add: linear_iff algebra_simps)
huffman@44133
   238
wenzelm@53406
   239
lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x - g x)"
huffman@53600
   240
  by (simp add: linear_iff algebra_simps)
huffman@44133
   241
wenzelm@53406
   242
lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> f)"
huffman@53600
   243
  by (simp add: linear_iff)
huffman@44133
   244
wenzelm@53406
   245
lemma linear_id: "linear id"
huffman@53600
   246
  by (simp add: linear_iff id_def)
wenzelm@53406
   247
wenzelm@53406
   248
lemma linear_zero: "linear (\<lambda>x. 0)"
huffman@53600
   249
  by (simp add: linear_iff)
huffman@44133
   250
huffman@44133
   251
lemma linear_compose_setsum:
huffman@56196
   252
  assumes lS: "\<forall>a \<in> S. linear (f a)"
wenzelm@53716
   253
  shows "linear (\<lambda>x. setsum (\<lambda>a. f a x) S)"
huffman@56196
   254
proof (cases "finite S")
huffman@56196
   255
  case True
huffman@56196
   256
  then show ?thesis
huffman@56196
   257
    using lS by induct (simp_all add: linear_zero linear_compose_add)
huffman@56196
   258
qed (simp add: linear_zero)
huffman@44133
   259
huffman@44133
   260
lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
huffman@53600
   261
  unfolding linear_iff
huffman@44133
   262
  apply clarsimp
huffman@44133
   263
  apply (erule allE[where x="0::'a"])
huffman@44133
   264
  apply simp
huffman@44133
   265
  done
huffman@44133
   266
wenzelm@53406
   267
lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
huffman@53600
   268
  by (simp add: linear_iff)
huffman@44133
   269
wenzelm@53406
   270
lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x"
huffman@44133
   271
  using linear_cmul [where c="-1"] by simp
huffman@44133
   272
wenzelm@53716
   273
lemma linear_add: "linear f \<Longrightarrow> f (x + y) = f x + f y"
huffman@53600
   274
  by (metis linear_iff)
huffman@44133
   275
wenzelm@53716
   276
lemma linear_sub: "linear f \<Longrightarrow> f (x - y) = f x - f y"
haftmann@54230
   277
  using linear_add [of f x "- y"] by (simp add: linear_neg)
huffman@44133
   278
huffman@44133
   279
lemma linear_setsum:
huffman@56196
   280
  assumes f: "linear f"
wenzelm@53406
   281
  shows "f (setsum g S) = setsum (f \<circ> g) S"
huffman@56196
   282
proof (cases "finite S")
huffman@56196
   283
  case True
huffman@56196
   284
  then show ?thesis
huffman@56196
   285
    by induct (simp_all add: linear_0 [OF f] linear_add [OF f])
huffman@56196
   286
qed (simp add: linear_0 [OF f])
huffman@44133
   287
huffman@44133
   288
lemma linear_setsum_mul:
wenzelm@53406
   289
  assumes lin: "linear f"
huffman@44133
   290
  shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
huffman@56196
   291
  using linear_setsum[OF lin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
wenzelm@49522
   292
  by simp
huffman@44133
   293
huffman@44133
   294
lemma linear_injective_0:
wenzelm@53406
   295
  assumes lin: "linear f"
huffman@44133
   296
  shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
wenzelm@49663
   297
proof -
wenzelm@53406
   298
  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)"
wenzelm@53406
   299
    by (simp add: inj_on_def)
wenzelm@53406
   300
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)"
wenzelm@53406
   301
    by simp
huffman@44133
   302
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
wenzelm@53406
   303
    by (simp add: linear_sub[OF lin])
wenzelm@53406
   304
  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
wenzelm@53406
   305
    by auto
huffman@44133
   306
  finally show ?thesis .
huffman@44133
   307
qed
huffman@44133
   308
wenzelm@49522
   309
wenzelm@49522
   310
subsection {* Bilinear functions. *}
huffman@44133
   311
wenzelm@53406
   312
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
wenzelm@53406
   313
wenzelm@53406
   314
lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
huffman@53600
   315
  by (simp add: bilinear_def linear_iff)
wenzelm@49663
   316
wenzelm@53406
   317
lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
huffman@53600
   318
  by (simp add: bilinear_def linear_iff)
huffman@44133
   319
wenzelm@53406
   320
lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
huffman@53600
   321
  by (simp add: bilinear_def linear_iff)
huffman@44133
   322
wenzelm@53406
   323
lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
huffman@53600
   324
  by (simp add: bilinear_def linear_iff)
huffman@44133
   325
wenzelm@53406
   326
lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
haftmann@54489
   327
  by (drule bilinear_lmul [of _ "- 1"]) simp
huffman@44133
   328
wenzelm@53406
   329
lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
haftmann@54489
   330
  by (drule bilinear_rmul [of _ _ "- 1"]) simp
huffman@44133
   331
wenzelm@53406
   332
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
huffman@44133
   333
  using add_imp_eq[of x y 0] by auto
huffman@44133
   334
wenzelm@53406
   335
lemma bilinear_lzero:
wenzelm@53406
   336
  assumes "bilinear h"
wenzelm@53406
   337
  shows "h 0 x = 0"
wenzelm@49663
   338
  using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
wenzelm@49663
   339
wenzelm@53406
   340
lemma bilinear_rzero:
wenzelm@53406
   341
  assumes "bilinear h"
wenzelm@53406
   342
  shows "h x 0 = 0"
wenzelm@49663
   343
  using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
huffman@44133
   344
wenzelm@53406
   345
lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
haftmann@54230
   346
  using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
huffman@44133
   347
wenzelm@53406
   348
lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
haftmann@54230
   349
  using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
huffman@44133
   350
huffman@44133
   351
lemma bilinear_setsum:
wenzelm@49663
   352
  assumes bh: "bilinear h"
wenzelm@49663
   353
    and fS: "finite S"
wenzelm@49663
   354
    and fT: "finite T"
huffman@44133
   355
  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
wenzelm@49522
   356
proof -
huffman@44133
   357
  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
huffman@44133
   358
    apply (rule linear_setsum[unfolded o_def])
wenzelm@53406
   359
    using bh fS
wenzelm@53406
   360
    apply (auto simp add: bilinear_def)
wenzelm@49522
   361
    done
huffman@44133
   362
  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
huffman@44133
   363
    apply (rule setsum_cong, simp)
huffman@44133
   364
    apply (rule linear_setsum[unfolded o_def])
wenzelm@49522
   365
    using bh fT
wenzelm@49522
   366
    apply (auto simp add: bilinear_def)
wenzelm@49522
   367
    done
wenzelm@53406
   368
  finally show ?thesis
wenzelm@53406
   369
    unfolding setsum_cartesian_product .
huffman@44133
   370
qed
huffman@44133
   371
wenzelm@49522
   372
wenzelm@49522
   373
subsection {* Adjoints. *}
huffman@44133
   374
huffman@44133
   375
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
huffman@44133
   376
huffman@44133
   377
lemma adjoint_unique:
huffman@44133
   378
  assumes "\<forall>x y. inner (f x) y = inner x (g y)"
huffman@44133
   379
  shows "adjoint f = g"
wenzelm@49522
   380
  unfolding adjoint_def
huffman@44133
   381
proof (rule some_equality)
wenzelm@53406
   382
  show "\<forall>x y. inner (f x) y = inner x (g y)"
wenzelm@53406
   383
    by (rule assms)
huffman@44133
   384
next
wenzelm@53406
   385
  fix h
wenzelm@53406
   386
  assume "\<forall>x y. inner (f x) y = inner x (h y)"
wenzelm@53406
   387
  then have "\<forall>x y. inner x (g y) = inner x (h y)"
wenzelm@53406
   388
    using assms by simp
wenzelm@53406
   389
  then have "\<forall>x y. inner x (g y - h y) = 0"
wenzelm@53406
   390
    by (simp add: inner_diff_right)
wenzelm@53406
   391
  then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
wenzelm@53406
   392
    by simp
wenzelm@53406
   393
  then have "\<forall>y. h y = g y"
wenzelm@53406
   394
    by simp
wenzelm@49652
   395
  then show "h = g" by (simp add: ext)
huffman@44133
   396
qed
huffman@44133
   397
hoelzl@50526
   398
text {* TODO: The following lemmas about adjoints should hold for any
hoelzl@50526
   399
Hilbert space (i.e. complete inner product space).
wenzelm@54703
   400
(see @{url "http://en.wikipedia.org/wiki/Hermitian_adjoint"})
hoelzl@50526
   401
*}
hoelzl@50526
   402
hoelzl@50526
   403
lemma adjoint_works:
hoelzl@50526
   404
  fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@50526
   405
  assumes lf: "linear f"
hoelzl@50526
   406
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@50526
   407
proof -
hoelzl@50526
   408
  have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
hoelzl@50526
   409
  proof (intro allI exI)
hoelzl@50526
   410
    fix y :: "'m" and x
hoelzl@50526
   411
    let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
hoelzl@50526
   412
    have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
hoelzl@50526
   413
      by (simp add: euclidean_representation)
hoelzl@50526
   414
    also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
huffman@56196
   415
      unfolding linear_setsum[OF lf]
hoelzl@50526
   416
      by (simp add: linear_cmul[OF lf])
hoelzl@50526
   417
    finally show "f x \<bullet> y = x \<bullet> ?w"
wenzelm@53406
   418
      by (simp add: inner_setsum_left inner_setsum_right mult_commute)
hoelzl@50526
   419
  qed
hoelzl@50526
   420
  then show ?thesis
hoelzl@50526
   421
    unfolding adjoint_def choice_iff
hoelzl@50526
   422
    by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
hoelzl@50526
   423
qed
hoelzl@50526
   424
hoelzl@50526
   425
lemma adjoint_clauses:
hoelzl@50526
   426
  fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@50526
   427
  assumes lf: "linear f"
hoelzl@50526
   428
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@50526
   429
    and "adjoint f y \<bullet> x = y \<bullet> f x"
hoelzl@50526
   430
  by (simp_all add: adjoint_works[OF lf] inner_commute)
hoelzl@50526
   431
hoelzl@50526
   432
lemma adjoint_linear:
hoelzl@50526
   433
  fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@50526
   434
  assumes lf: "linear f"
hoelzl@50526
   435
  shows "linear (adjoint f)"
huffman@53600
   436
  by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
huffman@53939
   437
    adjoint_clauses[OF lf] inner_distrib)
hoelzl@50526
   438
hoelzl@50526
   439
lemma adjoint_adjoint:
hoelzl@50526
   440
  fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@50526
   441
  assumes lf: "linear f"
hoelzl@50526
   442
  shows "adjoint (adjoint f) = f"
hoelzl@50526
   443
  by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
hoelzl@50526
   444
wenzelm@53406
   445
wenzelm@49522
   446
subsection {* Interlude: Some properties of real sets *}
huffman@44133
   447
wenzelm@53406
   448
lemma seq_mono_lemma:
wenzelm@53406
   449
  assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
wenzelm@53406
   450
    and "\<forall>n \<ge> m. e n \<le> e m"
huffman@44133
   451
  shows "\<forall>n \<ge> m. d n < e m"
wenzelm@53406
   452
  using assms
wenzelm@53406
   453
  apply auto
huffman@44133
   454
  apply (erule_tac x="n" in allE)
huffman@44133
   455
  apply (erule_tac x="n" in allE)
huffman@44133
   456
  apply auto
huffman@44133
   457
  done
huffman@44133
   458
wenzelm@53406
   459
lemma infinite_enumerate:
wenzelm@53406
   460
  assumes fS: "infinite S"
huffman@44133
   461
  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
wenzelm@49525
   462
  unfolding subseq_def
wenzelm@49525
   463
  using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
huffman@44133
   464
huffman@44133
   465
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)"
wenzelm@49522
   466
  apply auto
wenzelm@49522
   467
  apply (rule_tac x="d/2" in exI)
wenzelm@49522
   468
  apply auto
wenzelm@49522
   469
  done
huffman@44133
   470
huffman@44133
   471
lemma triangle_lemma:
wenzelm@53406
   472
  fixes x y z :: real
wenzelm@53406
   473
  assumes x: "0 \<le> x"
wenzelm@53406
   474
    and y: "0 \<le> y"
wenzelm@53406
   475
    and z: "0 \<le> z"
wenzelm@53406
   476
    and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
wenzelm@53406
   477
  shows "x \<le> y + z"
wenzelm@49522
   478
proof -
wenzelm@53406
   479
  have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 *y * z + z\<^sup>2"
wenzelm@53406
   480
    using z y by (simp add: mult_nonneg_nonneg)
wenzelm@53406
   481
  with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
wenzelm@53406
   482
    by (simp add: power2_eq_square field_simps)
wenzelm@53406
   483
  from y z have yz: "y + z \<ge> 0"
wenzelm@53406
   484
    by arith
huffman@44133
   485
  from power2_le_imp_le[OF th yz] show ?thesis .
huffman@44133
   486
qed
huffman@44133
   487
wenzelm@49522
   488
huffman@44133
   489
subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
huffman@44133
   490
wenzelm@53406
   491
definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
wenzelm@53406
   492
  where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
huffman@44170
   493
huffman@44170
   494
lemma hull_same: "S s \<Longrightarrow> S hull s = s"
huffman@44133
   495
  unfolding hull_def by auto
huffman@44133
   496
wenzelm@53406
   497
lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
wenzelm@49522
   498
  unfolding hull_def Ball_def by auto
huffman@44170
   499
wenzelm@53406
   500
lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
wenzelm@49522
   501
  using hull_same[of S s] hull_in[of S s] by metis
huffman@44133
   502
huffman@44133
   503
lemma hull_hull: "S hull (S hull s) = S hull s"
huffman@44133
   504
  unfolding hull_def by blast
huffman@44133
   505
huffman@44133
   506
lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
huffman@44133
   507
  unfolding hull_def by blast
huffman@44133
   508
wenzelm@53406
   509
lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
huffman@44133
   510
  unfolding hull_def by blast
huffman@44133
   511
wenzelm@53406
   512
lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
huffman@44133
   513
  unfolding hull_def by blast
huffman@44133
   514
wenzelm@53406
   515
lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
huffman@44133
   516
  unfolding hull_def by blast
huffman@44133
   517
wenzelm@53406
   518
lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
huffman@44133
   519
  unfolding hull_def by blast
huffman@44133
   520
huffman@53596
   521
lemma hull_UNIV: "S hull UNIV = UNIV"
huffman@53596
   522
  unfolding hull_def by auto
huffman@53596
   523
wenzelm@53406
   524
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)"
wenzelm@49652
   525
  unfolding hull_def by auto
huffman@44133
   526
huffman@44133
   527
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"
huffman@44133
   528
  using hull_minimal[of S "{x. P x}" Q]
huffman@44170
   529
  by (auto simp add: subset_eq)
huffman@44133
   530
wenzelm@49522
   531
lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
wenzelm@49522
   532
  by (metis hull_subset subset_eq)
huffman@44133
   533
huffman@44133
   534
lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
wenzelm@49522
   535
  unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
wenzelm@49522
   536
wenzelm@49522
   537
lemma hull_union:
wenzelm@53406
   538
  assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
huffman@44133
   539
  shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
wenzelm@49522
   540
  apply rule
wenzelm@49522
   541
  apply (rule hull_mono)
wenzelm@49522
   542
  unfolding Un_subset_iff
wenzelm@49522
   543
  apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
wenzelm@49522
   544
  apply (rule hull_minimal)
wenzelm@49522
   545
  apply (metis hull_union_subset)
wenzelm@49522
   546
  apply (metis hull_in T)
wenzelm@49522
   547
  done
huffman@44133
   548
huffman@44133
   549
lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
huffman@44133
   550
  unfolding hull_def by blast
huffman@44133
   551
wenzelm@53406
   552
lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> (S hull (insert a s) = S hull s)"
wenzelm@49522
   553
  by (metis hull_redundant_eq)
wenzelm@49522
   554
huffman@44133
   555
huffman@44666
   556
subsection {* Archimedean properties and useful consequences *}
huffman@44133
   557
wenzelm@53406
   558
lemma real_arch_simple: "\<exists>n. x \<le> real (n::nat)"
huffman@44666
   559
  unfolding real_of_nat_def by (rule ex_le_of_nat)
huffman@44133
   560
huffman@44133
   561
lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
huffman@44133
   562
  using reals_Archimedean
huffman@44133
   563
  apply (auto simp add: field_simps)
huffman@44133
   564
  apply (subgoal_tac "inverse (real n) > 0")
huffman@44133
   565
  apply arith
huffman@44133
   566
  apply simp
huffman@44133
   567
  done
huffman@44133
   568
wenzelm@53406
   569
lemma real_pow_lbound: "0 \<le> x \<Longrightarrow> 1 + real n * x \<le> (1 + x) ^ n"
wenzelm@49522
   570
proof (induct n)
wenzelm@49522
   571
  case 0
wenzelm@49522
   572
  then show ?case by simp
huffman@44133
   573
next
huffman@44133
   574
  case (Suc n)
wenzelm@53406
   575
  then have h: "1 + real n * x \<le> (1 + x) ^ n"
wenzelm@53406
   576
    by simp
wenzelm@53406
   577
  from h have p: "1 \<le> (1 + x) ^ n"
wenzelm@53406
   578
    using Suc.prems by simp
wenzelm@53406
   579
  from h have "1 + real n * x + x \<le> (1 + x) ^ n + x"
wenzelm@53406
   580
    by simp
wenzelm@53406
   581
  also have "\<dots> \<le> (1 + x) ^ Suc n"
wenzelm@53406
   582
    apply (subst diff_le_0_iff_le[symmetric])
huffman@44133
   583
    apply (simp add: field_simps)
wenzelm@53406
   584
    using mult_left_mono[OF p Suc.prems]
wenzelm@53406
   585
    apply simp
wenzelm@49522
   586
    done
wenzelm@53406
   587
  finally show ?case
wenzelm@53406
   588
    by (simp add: real_of_nat_Suc field_simps)
huffman@44133
   589
qed
huffman@44133
   590
wenzelm@53406
   591
lemma real_arch_pow:
wenzelm@53406
   592
  fixes x :: real
wenzelm@53406
   593
  assumes x: "1 < x"
wenzelm@53406
   594
  shows "\<exists>n. y < x^n"
wenzelm@49522
   595
proof -
wenzelm@53406
   596
  from x have x0: "x - 1 > 0"
wenzelm@53406
   597
    by arith
huffman@44666
   598
  from reals_Archimedean3[OF x0, rule_format, of y]
wenzelm@53406
   599
  obtain n :: nat where n: "y < real n * (x - 1)" by metis
huffman@44133
   600
  from x0 have x00: "x- 1 \<ge> 0" by arith
huffman@44133
   601
  from real_pow_lbound[OF x00, of n] n
huffman@44133
   602
  have "y < x^n" by auto
huffman@44133
   603
  then show ?thesis by metis
huffman@44133
   604
qed
huffman@44133
   605
wenzelm@53406
   606
lemma real_arch_pow2:
wenzelm@53406
   607
  fixes x :: real
wenzelm@53406
   608
  shows "\<exists>n. x < 2^ n"
huffman@44133
   609
  using real_arch_pow[of 2 x] by simp
huffman@44133
   610
wenzelm@49522
   611
lemma real_arch_pow_inv:
wenzelm@53406
   612
  fixes x y :: real
wenzelm@53406
   613
  assumes y: "y > 0"
wenzelm@53406
   614
    and x1: "x < 1"
huffman@44133
   615
  shows "\<exists>n. x^n < y"
wenzelm@53406
   616
proof (cases "x > 0")
wenzelm@53406
   617
  case True
wenzelm@53406
   618
  with x1 have ix: "1 < 1/x" by (simp add: field_simps)
wenzelm@53406
   619
  from real_arch_pow[OF ix, of "1/y"]
wenzelm@53406
   620
  obtain n where n: "1/y < (1/x)^n" by blast
wenzelm@53406
   621
  then show ?thesis using y `x > 0`
wenzelm@53406
   622
    by (auto simp add: field_simps power_divide)
wenzelm@53406
   623
next
wenzelm@53406
   624
  case False
wenzelm@53406
   625
  with y x1 show ?thesis
wenzelm@53406
   626
    apply auto
wenzelm@53406
   627
    apply (rule exI[where x=1])
wenzelm@53406
   628
    apply auto
wenzelm@53406
   629
    done
huffman@44133
   630
qed
huffman@44133
   631
wenzelm@49522
   632
lemma forall_pos_mono:
wenzelm@53406
   633
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
wenzelm@53406
   634
    (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
huffman@44133
   635
  by (metis real_arch_inv)
huffman@44133
   636
wenzelm@49522
   637
lemma forall_pos_mono_1:
wenzelm@53406
   638
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
wenzelm@53716
   639
    (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
huffman@44133
   640
  apply (rule forall_pos_mono)
huffman@44133
   641
  apply auto
huffman@44133
   642
  apply (atomize)
huffman@44133
   643
  apply (erule_tac x="n - 1" in allE)
huffman@44133
   644
  apply auto
huffman@44133
   645
  done
huffman@44133
   646
wenzelm@49522
   647
lemma real_archimedian_rdiv_eq_0:
wenzelm@53406
   648
  assumes x0: "x \<ge> 0"
wenzelm@53406
   649
    and c: "c \<ge> 0"
wenzelm@53406
   650
    and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
huffman@44133
   651
  shows "x = 0"
wenzelm@53406
   652
proof (rule ccontr)
wenzelm@53406
   653
  assume "x \<noteq> 0"
wenzelm@53406
   654
  with x0 have xp: "x > 0" by arith
wenzelm@53406
   655
  from reals_Archimedean3[OF xp, rule_format, of c]
wenzelm@53406
   656
  obtain n :: nat where n: "c < real n * x"
wenzelm@53406
   657
    by blast
wenzelm@53406
   658
  with xc[rule_format, of n] have "n = 0"
wenzelm@53406
   659
    by arith
wenzelm@53406
   660
  with n c show False
wenzelm@53406
   661
    by simp
huffman@44133
   662
qed
huffman@44133
   663
wenzelm@49522
   664
huffman@44133
   665
subsection{* A bit of linear algebra. *}
huffman@44133
   666
wenzelm@49522
   667
definition (in real_vector) subspace :: "'a set \<Rightarrow> bool"
wenzelm@49522
   668
  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 )"
huffman@44133
   669
huffman@44133
   670
definition (in real_vector) "span S = (subspace hull S)"
wenzelm@53716
   671
definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))"
wenzelm@53406
   672
abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
huffman@44133
   673
huffman@44133
   674
text {* Closure properties of subspaces. *}
huffman@44133
   675
wenzelm@53406
   676
lemma subspace_UNIV[simp]: "subspace UNIV"
wenzelm@53406
   677
  by (simp add: subspace_def)
wenzelm@53406
   678
wenzelm@53406
   679
lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
wenzelm@53406
   680
  by (metis subspace_def)
wenzelm@53406
   681
wenzelm@53406
   682
lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
huffman@44133
   683
  by (metis subspace_def)
huffman@44133
   684
huffman@44133
   685
lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
huffman@44133
   686
  by (metis subspace_def)
huffman@44133
   687
huffman@44133
   688
lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
huffman@44133
   689
  by (metis scaleR_minus1_left subspace_mul)
huffman@44133
   690
huffman@44133
   691
lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
haftmann@54230
   692
  using subspace_add [of S x "- y"] by (simp add: subspace_neg)
huffman@44133
   693
huffman@44133
   694
lemma (in real_vector) subspace_setsum:
wenzelm@53406
   695
  assumes sA: "subspace A"
huffman@56196
   696
    and f: "\<forall>x\<in>B. f x \<in> A"
huffman@44133
   697
  shows "setsum f B \<in> A"
huffman@56196
   698
proof (cases "finite B")
huffman@56196
   699
  case True
huffman@56196
   700
  then show ?thesis
huffman@56196
   701
    using f by induct (simp_all add: subspace_0 [OF sA] subspace_add [OF sA])
huffman@56196
   702
qed (simp add: subspace_0 [OF sA])
huffman@44133
   703
huffman@44133
   704
lemma subspace_linear_image:
wenzelm@53406
   705
  assumes lf: "linear f"
wenzelm@53406
   706
    and sS: "subspace S"
wenzelm@53406
   707
  shows "subspace (f ` S)"
huffman@44133
   708
  using lf sS linear_0[OF lf]
huffman@53600
   709
  unfolding linear_iff subspace_def
huffman@44133
   710
  apply (auto simp add: image_iff)
wenzelm@53406
   711
  apply (rule_tac x="x + y" in bexI)
wenzelm@53406
   712
  apply auto
wenzelm@53406
   713
  apply (rule_tac x="c *\<^sub>R x" in bexI)
wenzelm@53406
   714
  apply auto
huffman@44133
   715
  done
huffman@44133
   716
huffman@44521
   717
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
huffman@53600
   718
  by (auto simp add: subspace_def linear_iff linear_0[of f])
huffman@44521
   719
wenzelm@53406
   720
lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
huffman@53600
   721
  by (auto simp add: subspace_def linear_iff linear_0[of f])
huffman@44133
   722
huffman@44133
   723
lemma subspace_trivial: "subspace {0}"
huffman@44133
   724
  by (simp add: subspace_def)
huffman@44133
   725
wenzelm@53406
   726
lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
huffman@44133
   727
  by (simp add: subspace_def)
huffman@44133
   728
wenzelm@53406
   729
lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
huffman@44521
   730
  unfolding subspace_def zero_prod_def by simp
huffman@44521
   731
huffman@44521
   732
text {* Properties of span. *}
huffman@44521
   733
wenzelm@53406
   734
lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
huffman@44133
   735
  by (metis span_def hull_mono)
huffman@44133
   736
wenzelm@53406
   737
lemma (in real_vector) subspace_span: "subspace (span S)"
huffman@44133
   738
  unfolding span_def
huffman@44170
   739
  apply (rule hull_in)
huffman@44133
   740
  apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
huffman@44133
   741
  apply auto
huffman@44133
   742
  done
huffman@44133
   743
huffman@44133
   744
lemma (in real_vector) span_clauses:
wenzelm@53406
   745
  "a \<in> S \<Longrightarrow> a \<in> span S"
huffman@44133
   746
  "0 \<in> span S"
wenzelm@53406
   747
  "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
huffman@44133
   748
  "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
wenzelm@53406
   749
  by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
huffman@44133
   750
huffman@44521
   751
lemma span_unique:
wenzelm@49522
   752
  "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
huffman@44521
   753
  unfolding span_def by (rule hull_unique)
huffman@44521
   754
huffman@44521
   755
lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
huffman@44521
   756
  unfolding span_def by (rule hull_minimal)
huffman@44521
   757
huffman@44521
   758
lemma (in real_vector) span_induct:
wenzelm@49522
   759
  assumes x: "x \<in> span S"
wenzelm@49522
   760
    and P: "subspace P"
wenzelm@53406
   761
    and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
huffman@44521
   762
  shows "x \<in> P"
wenzelm@49522
   763
proof -
wenzelm@53406
   764
  from SP have SP': "S \<subseteq> P"
wenzelm@53406
   765
    by (simp add: subset_eq)
huffman@44170
   766
  from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
wenzelm@53406
   767
  show "x \<in> P"
wenzelm@53406
   768
    by (metis subset_eq)
huffman@44133
   769
qed
huffman@44133
   770
huffman@44133
   771
lemma span_empty[simp]: "span {} = {0}"
huffman@44133
   772
  apply (simp add: span_def)
huffman@44133
   773
  apply (rule hull_unique)
huffman@44170
   774
  apply (auto simp add: subspace_def)
huffman@44133
   775
  done
huffman@44133
   776
huffman@44133
   777
lemma (in real_vector) independent_empty[intro]: "independent {}"
huffman@44133
   778
  by (simp add: dependent_def)
huffman@44133
   779
wenzelm@49522
   780
lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
huffman@44133
   781
  unfolding dependent_def by auto
huffman@44133
   782
wenzelm@53406
   783
lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
huffman@44133
   784
  apply (clarsimp simp add: dependent_def span_mono)
huffman@44133
   785
  apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
huffman@44133
   786
  apply force
huffman@44133
   787
  apply (rule span_mono)
huffman@44133
   788
  apply auto
huffman@44133
   789
  done
huffman@44133
   790
huffman@44133
   791
lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
huffman@44170
   792
  by (metis order_antisym span_def hull_minimal)
huffman@44133
   793
wenzelm@49711
   794
lemma (in real_vector) span_induct':
wenzelm@49711
   795
  assumes SP: "\<forall>x \<in> S. P x"
wenzelm@49711
   796
    and P: "subspace {x. P x}"
wenzelm@49711
   797
  shows "\<forall>x \<in> span S. P x"
huffman@44133
   798
  using span_induct SP P by blast
huffman@44133
   799
huffman@44170
   800
inductive_set (in real_vector) span_induct_alt_help for S:: "'a set"
wenzelm@53406
   801
where
huffman@44170
   802
  span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
wenzelm@49522
   803
| span_induct_alt_help_S:
wenzelm@53406
   804
    "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
wenzelm@53406
   805
      (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
huffman@44133
   806
huffman@44133
   807
lemma span_induct_alt':
wenzelm@53406
   808
  assumes h0: "h 0"
wenzelm@53406
   809
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@49522
   810
  shows "\<forall>x \<in> span S. h x"
wenzelm@49522
   811
proof -
wenzelm@53406
   812
  {
wenzelm@53406
   813
    fix x :: 'a
wenzelm@53406
   814
    assume x: "x \<in> span_induct_alt_help S"
huffman@44133
   815
    have "h x"
huffman@44133
   816
      apply (rule span_induct_alt_help.induct[OF x])
huffman@44133
   817
      apply (rule h0)
wenzelm@53406
   818
      apply (rule hS)
wenzelm@53406
   819
      apply assumption
wenzelm@53406
   820
      apply assumption
wenzelm@53406
   821
      done
wenzelm@53406
   822
  }
huffman@44133
   823
  note th0 = this
wenzelm@53406
   824
  {
wenzelm@53406
   825
    fix x
wenzelm@53406
   826
    assume x: "x \<in> span S"
huffman@44170
   827
    have "x \<in> span_induct_alt_help S"
wenzelm@49522
   828
    proof (rule span_induct[where x=x and S=S])
wenzelm@53406
   829
      show "x \<in> span S" by (rule x)
wenzelm@49522
   830
    next
wenzelm@53406
   831
      fix x
wenzelm@53406
   832
      assume xS: "x \<in> S"
wenzelm@53406
   833
      from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
wenzelm@53406
   834
      show "x \<in> span_induct_alt_help S"
wenzelm@53406
   835
        by simp
wenzelm@49522
   836
    next
wenzelm@49522
   837
      have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
wenzelm@49522
   838
      moreover
wenzelm@53406
   839
      {
wenzelm@53406
   840
        fix x y
wenzelm@49522
   841
        assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
wenzelm@49522
   842
        from h have "(x + y) \<in> span_induct_alt_help S"
wenzelm@49522
   843
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   844
          apply simp
wenzelm@49522
   845
          unfolding add_assoc
wenzelm@49522
   846
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   847
          apply assumption
wenzelm@49522
   848
          apply simp
wenzelm@53406
   849
          done
wenzelm@53406
   850
      }
wenzelm@49522
   851
      moreover
wenzelm@53406
   852
      {
wenzelm@53406
   853
        fix c x
wenzelm@49522
   854
        assume xt: "x \<in> span_induct_alt_help S"
wenzelm@49522
   855
        then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
wenzelm@49522
   856
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   857
          apply (simp add: span_induct_alt_help_0)
wenzelm@49522
   858
          apply (simp add: scaleR_right_distrib)
wenzelm@49522
   859
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   860
          apply assumption
wenzelm@49522
   861
          apply simp
wenzelm@49522
   862
          done }
wenzelm@53406
   863
      ultimately show "subspace (span_induct_alt_help S)"
wenzelm@49522
   864
        unfolding subspace_def Ball_def by blast
wenzelm@53406
   865
    qed
wenzelm@53406
   866
  }
huffman@44133
   867
  with th0 show ?thesis by blast
huffman@44133
   868
qed
huffman@44133
   869
huffman@44133
   870
lemma span_induct_alt:
wenzelm@53406
   871
  assumes h0: "h 0"
wenzelm@53406
   872
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@53406
   873
    and x: "x \<in> span S"
huffman@44133
   874
  shows "h x"
wenzelm@49522
   875
  using span_induct_alt'[of h S] h0 hS x by blast
huffman@44133
   876
huffman@44133
   877
text {* Individual closure properties. *}
huffman@44133
   878
huffman@44133
   879
lemma span_span: "span (span A) = span A"
huffman@44133
   880
  unfolding span_def hull_hull ..
huffman@44133
   881
wenzelm@53406
   882
lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
wenzelm@53406
   883
  by (metis span_clauses(1))
wenzelm@53406
   884
wenzelm@53406
   885
lemma (in real_vector) span_0: "0 \<in> span S"
wenzelm@53406
   886
  by (metis subspace_span subspace_0)
huffman@44133
   887
huffman@44133
   888
lemma span_inc: "S \<subseteq> span S"
huffman@44133
   889
  by (metis subset_eq span_superset)
huffman@44133
   890
wenzelm@53406
   891
lemma (in real_vector) dependent_0:
wenzelm@53406
   892
  assumes "0 \<in> A"
wenzelm@53406
   893
  shows "dependent A"
wenzelm@53406
   894
  unfolding dependent_def
wenzelm@53406
   895
  apply (rule_tac x=0 in bexI)
wenzelm@53406
   896
  using assms span_0
wenzelm@53406
   897
  apply auto
wenzelm@53406
   898
  done
wenzelm@53406
   899
wenzelm@53406
   900
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
huffman@44133
   901
  by (metis subspace_add subspace_span)
huffman@44133
   902
wenzelm@53406
   903
lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
huffman@44133
   904
  by (metis subspace_span subspace_mul)
huffman@44133
   905
wenzelm@53406
   906
lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
huffman@44133
   907
  by (metis subspace_neg subspace_span)
huffman@44133
   908
wenzelm@53406
   909
lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
huffman@44133
   910
  by (metis subspace_span subspace_sub)
huffman@44133
   911
huffman@56196
   912
lemma (in real_vector) span_setsum: "\<forall>x\<in>A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S"
huffman@56196
   913
  by (rule subspace_setsum [OF subspace_span])
huffman@44133
   914
huffman@44133
   915
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
lp15@55775
   916
  by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
huffman@44133
   917
huffman@44133
   918
text {* Mapping under linear image. *}
huffman@44133
   919
huffman@44521
   920
lemma span_linear_image:
huffman@44521
   921
  assumes lf: "linear f"
huffman@44133
   922
  shows "span (f ` S) = f ` (span S)"
huffman@44521
   923
proof (rule span_unique)
huffman@44521
   924
  show "f ` S \<subseteq> f ` span S"
huffman@44521
   925
    by (intro image_mono span_inc)
huffman@44521
   926
  show "subspace (f ` span S)"
huffman@44521
   927
    using lf subspace_span by (rule subspace_linear_image)
huffman@44521
   928
next
wenzelm@53406
   929
  fix T
wenzelm@53406
   930
  assume "f ` S \<subseteq> T" and "subspace T"
wenzelm@49522
   931
  then show "f ` span S \<subseteq> T"
huffman@44521
   932
    unfolding image_subset_iff_subset_vimage
huffman@44521
   933
    by (intro span_minimal subspace_linear_vimage lf)
huffman@44521
   934
qed
huffman@44521
   935
huffman@44521
   936
lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   937
proof (rule span_unique)
huffman@44521
   938
  show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   939
    by safe (force intro: span_clauses)+
huffman@44521
   940
next
huffman@44521
   941
  have "linear (\<lambda>(a, b). a + b)"
huffman@53600
   942
    by (simp add: linear_iff scaleR_add_right)
huffman@44521
   943
  moreover have "subspace (span A \<times> span B)"
huffman@44521
   944
    by (intro subspace_Times subspace_span)
huffman@44521
   945
  ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
huffman@44521
   946
    by (rule subspace_linear_image)
huffman@44521
   947
next
wenzelm@49711
   948
  fix T
wenzelm@49711
   949
  assume "A \<union> B \<subseteq> T" and "subspace T"
wenzelm@49522
   950
  then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
huffman@44521
   951
    by (auto intro!: subspace_add elim: span_induct)
huffman@44133
   952
qed
huffman@44133
   953
huffman@44133
   954
text {* The key breakdown property. *}
huffman@44133
   955
huffman@44521
   956
lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
huffman@44521
   957
proof (rule span_unique)
huffman@44521
   958
  show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
huffman@44521
   959
    by (fast intro: scaleR_one [symmetric])
huffman@44521
   960
  show "subspace (range (\<lambda>k. k *\<^sub>R x))"
huffman@44521
   961
    unfolding subspace_def
huffman@44521
   962
    by (auto intro: scaleR_add_left [symmetric])
wenzelm@53406
   963
next
wenzelm@53406
   964
  fix T
wenzelm@53406
   965
  assume "{x} \<subseteq> T" and "subspace T"
wenzelm@53406
   966
  then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
huffman@44521
   967
    unfolding subspace_def by auto
huffman@44521
   968
qed
huffman@44521
   969
wenzelm@49522
   970
lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
huffman@44521
   971
proof -
huffman@44521
   972
  have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
huffman@44521
   973
    unfolding span_union span_singleton
huffman@44521
   974
    apply safe
huffman@44521
   975
    apply (rule_tac x=k in exI, simp)
huffman@44521
   976
    apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
haftmann@54230
   977
    apply auto
huffman@44521
   978
    done
wenzelm@49522
   979
  then show ?thesis by simp
huffman@44521
   980
qed
huffman@44521
   981
huffman@44133
   982
lemma span_breakdown:
wenzelm@53406
   983
  assumes bS: "b \<in> S"
wenzelm@53406
   984
    and aS: "a \<in> span S"
huffman@44521
   985
  shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
huffman@44521
   986
  using assms span_insert [of b "S - {b}"]
huffman@44521
   987
  by (simp add: insert_absorb)
huffman@44133
   988
wenzelm@53406
   989
lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
huffman@44521
   990
  by (simp add: span_insert)
huffman@44133
   991
huffman@44133
   992
text {* Hence some "reversal" results. *}
huffman@44133
   993
huffman@44133
   994
lemma in_span_insert:
wenzelm@49711
   995
  assumes a: "a \<in> span (insert b S)"
wenzelm@49711
   996
    and na: "a \<notin> span S"
huffman@44133
   997
  shows "b \<in> span (insert a S)"
wenzelm@49663
   998
proof -
huffman@55910
   999
  from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
huffman@55910
  1000
    unfolding span_insert by fast
wenzelm@53406
  1001
  show ?thesis
wenzelm@53406
  1002
  proof (cases "k = 0")
wenzelm@53406
  1003
    case True
huffman@55910
  1004
    with k have "a \<in> span S" by simp
huffman@55910
  1005
    with na show ?thesis by simp
wenzelm@53406
  1006
  next
wenzelm@53406
  1007
    case False
huffman@55910
  1008
    from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
huffman@44133
  1009
      by (rule span_mul)
huffman@55910
  1010
    then have "b - inverse k *\<^sub>R a \<in> span S"
huffman@55910
  1011
      using `k \<noteq> 0` by (simp add: scaleR_diff_right)
huffman@55910
  1012
    then show ?thesis
huffman@55910
  1013
      unfolding span_insert by fast
wenzelm@53406
  1014
  qed
huffman@44133
  1015
qed
huffman@44133
  1016
huffman@44133
  1017
lemma in_span_delete:
huffman@44133
  1018
  assumes a: "a \<in> span S"
wenzelm@53716
  1019
    and na: "a \<notin> span (S - {b})"
huffman@44133
  1020
  shows "b \<in> span (insert a (S - {b}))"
huffman@44133
  1021
  apply (rule in_span_insert)
huffman@44133
  1022
  apply (rule set_rev_mp)
huffman@44133
  1023
  apply (rule a)
huffman@44133
  1024
  apply (rule span_mono)
huffman@44133
  1025
  apply blast
huffman@44133
  1026
  apply (rule na)
huffman@44133
  1027
  done
huffman@44133
  1028
huffman@44133
  1029
text {* Transitivity property. *}
huffman@44133
  1030
huffman@44521
  1031
lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
huffman@44521
  1032
  unfolding span_def by (rule hull_redundant)
huffman@44521
  1033
huffman@44133
  1034
lemma span_trans:
wenzelm@53406
  1035
  assumes x: "x \<in> span S"
wenzelm@53406
  1036
    and y: "y \<in> span (insert x S)"
huffman@44133
  1037
  shows "y \<in> span S"
huffman@44521
  1038
  using assms by (simp only: span_redundant)
huffman@44133
  1039
huffman@44133
  1040
lemma span_insert_0[simp]: "span (insert 0 S) = span S"
huffman@44521
  1041
  by (simp only: span_redundant span_0)
huffman@44133
  1042
huffman@44133
  1043
text {* An explicit expansion is sometimes needed. *}
huffman@44133
  1044
huffman@44133
  1045
lemma span_explicit:
huffman@44133
  1046
  "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
huffman@44133
  1047
  (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
wenzelm@49663
  1048
proof -
wenzelm@53406
  1049
  {
wenzelm@53406
  1050
    fix x
huffman@55910
  1051
    assume "?h x"
huffman@55910
  1052
    then obtain S u where "finite S" and "S \<subseteq> P" and "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
huffman@44133
  1053
      by blast
huffman@55910
  1054
    then have "x \<in> span P"
huffman@55910
  1055
      by (auto intro: span_setsum span_mul span_superset)
wenzelm@53406
  1056
  }
huffman@44133
  1057
  moreover
huffman@55910
  1058
  have "\<forall>x \<in> span P. ?h x"
wenzelm@49522
  1059
  proof (rule span_induct_alt')
huffman@55910
  1060
    show "?h 0"
huffman@55910
  1061
      by (rule exI[where x="{}"], simp)
huffman@44133
  1062
  next
huffman@44133
  1063
    fix c x y
wenzelm@53406
  1064
    assume x: "x \<in> P"
huffman@55910
  1065
    assume hy: "?h y"
huffman@44133
  1066
    from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
huffman@44133
  1067
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
huffman@44133
  1068
    let ?S = "insert x S"
wenzelm@49522
  1069
    let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
wenzelm@53406
  1070
    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
wenzelm@53406
  1071
      by blast+
wenzelm@53406
  1072
    have "?Q ?S ?u (c*\<^sub>R x + y)"
wenzelm@53406
  1073
    proof cases
wenzelm@53406
  1074
      assume xS: "x \<in> S"
huffman@55910
  1075
      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"
huffman@55910
  1076
        using xS by (simp add: setsum.remove [OF fS xS] insert_absorb)
huffman@44133
  1077
      also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
huffman@55910
  1078
        by (simp add: setsum.remove [OF fS xS] algebra_simps)
huffman@44133
  1079
      also have "\<dots> = c*\<^sub>R x + y"
huffman@44133
  1080
        by (simp add: add_commute u)
huffman@44133
  1081
      finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
wenzelm@53406
  1082
      then show ?thesis using th0 by blast
wenzelm@53406
  1083
    next
wenzelm@53406
  1084
      assume xS: "x \<notin> S"
wenzelm@49522
  1085
      have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
wenzelm@49522
  1086
        unfolding u[symmetric]
wenzelm@49522
  1087
        apply (rule setsum_cong2)
wenzelm@53406
  1088
        using xS
wenzelm@53406
  1089
        apply auto
wenzelm@49522
  1090
        done
wenzelm@53406
  1091
      show ?thesis using fS xS th0
huffman@55910
  1092
        by (simp add: th00 add_commute cong del: if_weak_cong)
wenzelm@53406
  1093
    qed
huffman@55910
  1094
    then show "?h (c*\<^sub>R x + y)"
huffman@55910
  1095
      by fast
huffman@44133
  1096
  qed
huffman@44133
  1097
  ultimately show ?thesis by blast
huffman@44133
  1098
qed
huffman@44133
  1099
huffman@44133
  1100
lemma dependent_explicit:
wenzelm@49522
  1101
  "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))"
wenzelm@49522
  1102
  (is "?lhs = ?rhs")
wenzelm@49522
  1103
proof -
wenzelm@53406
  1104
  {
wenzelm@53406
  1105
    assume dP: "dependent P"
huffman@44133
  1106
    then obtain a S u where aP: "a \<in> P" and fS: "finite S"
huffman@44133
  1107
      and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
huffman@44133
  1108
      unfolding dependent_def span_explicit by blast
huffman@44133
  1109
    let ?S = "insert a S"
huffman@44133
  1110
    let ?u = "\<lambda>y. if y = a then - 1 else u y"
huffman@44133
  1111
    let ?v = a
wenzelm@53406
  1112
    from aP SP have aS: "a \<notin> S"
wenzelm@53406
  1113
      by blast
wenzelm@53406
  1114
    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
wenzelm@53406
  1115
      by auto
huffman@44133
  1116
    have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
huffman@44133
  1117
      using fS aS
huffman@55910
  1118
      apply simp
huffman@44133
  1119
      apply (subst (2) ua[symmetric])
huffman@44133
  1120
      apply (rule setsum_cong2)
wenzelm@49522
  1121
      apply auto
wenzelm@49522
  1122
      done
huffman@55910
  1123
    with th0 have ?rhs by fast
wenzelm@49522
  1124
  }
huffman@44133
  1125
  moreover
wenzelm@53406
  1126
  {
wenzelm@53406
  1127
    fix S u v
wenzelm@49522
  1128
    assume fS: "finite S"
wenzelm@53406
  1129
      and SP: "S \<subseteq> P"
wenzelm@53406
  1130
      and vS: "v \<in> S"
wenzelm@53406
  1131
      and uv: "u v \<noteq> 0"
wenzelm@49522
  1132
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
huffman@44133
  1133
    let ?a = v
huffman@44133
  1134
    let ?S = "S - {v}"
huffman@44133
  1135
    let ?u = "\<lambda>i. (- u i) / u v"
wenzelm@53406
  1136
    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
wenzelm@53406
  1137
      using fS SP vS by auto
wenzelm@53406
  1138
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
wenzelm@53406
  1139
      setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
wenzelm@49522
  1140
      using fS vS uv by (simp add: setsum_diff1 divide_inverse field_simps)
wenzelm@53406
  1141
    also have "\<dots> = ?a"
wenzelm@53406
  1142
      unfolding scaleR_right.setsum [symmetric] u using uv by simp
wenzelm@53406
  1143
    finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
huffman@44133
  1144
    with th0 have ?lhs
huffman@44133
  1145
      unfolding dependent_def span_explicit
huffman@44133
  1146
      apply -
huffman@44133
  1147
      apply (rule bexI[where x= "?a"])
huffman@44133
  1148
      apply (simp_all del: scaleR_minus_left)
huffman@44133
  1149
      apply (rule exI[where x= "?S"])
wenzelm@49522
  1150
      apply (auto simp del: scaleR_minus_left)
wenzelm@49522
  1151
      done
wenzelm@49522
  1152
  }
huffman@44133
  1153
  ultimately show ?thesis by blast
huffman@44133
  1154
qed
huffman@44133
  1155
huffman@44133
  1156
huffman@44133
  1157
lemma span_finite:
huffman@44133
  1158
  assumes fS: "finite S"
huffman@44133
  1159
  shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
huffman@44133
  1160
  (is "_ = ?rhs")
wenzelm@49522
  1161
proof -
wenzelm@53406
  1162
  {
wenzelm@53406
  1163
    fix y
wenzelm@49711
  1164
    assume y: "y \<in> span S"
wenzelm@53406
  1165
    from y obtain S' u where fS': "finite S'"
wenzelm@53406
  1166
      and SS': "S' \<subseteq> S"
wenzelm@53406
  1167
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
wenzelm@53406
  1168
      unfolding span_explicit by blast
huffman@44133
  1169
    let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
huffman@44133
  1170
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
huffman@44133
  1171
      using SS' fS by (auto intro!: setsum_mono_zero_cong_right)
wenzelm@49522
  1172
    then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
wenzelm@53406
  1173
    then have "y \<in> ?rhs" by auto
wenzelm@53406
  1174
  }
huffman@44133
  1175
  moreover
wenzelm@53406
  1176
  {
wenzelm@53406
  1177
    fix y u
wenzelm@49522
  1178
    assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
wenzelm@53406
  1179
    then have "y \<in> span S" using fS unfolding span_explicit by auto
wenzelm@53406
  1180
  }
huffman@44133
  1181
  ultimately show ?thesis by blast
huffman@44133
  1182
qed
huffman@44133
  1183
huffman@44133
  1184
text {* This is useful for building a basis step-by-step. *}
huffman@44133
  1185
huffman@44133
  1186
lemma independent_insert:
wenzelm@53406
  1187
  "independent (insert a S) \<longleftrightarrow>
wenzelm@53406
  1188
    (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
wenzelm@53406
  1189
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53406
  1190
proof (cases "a \<in> S")
wenzelm@53406
  1191
  case True
wenzelm@53406
  1192
  then show ?thesis
wenzelm@53406
  1193
    using insert_absorb[OF True] by simp
wenzelm@53406
  1194
next
wenzelm@53406
  1195
  case False
wenzelm@53406
  1196
  show ?thesis
wenzelm@53406
  1197
  proof
wenzelm@53406
  1198
    assume i: ?lhs
wenzelm@53406
  1199
    then show ?rhs
wenzelm@53406
  1200
      using False
wenzelm@53406
  1201
      apply simp
wenzelm@53406
  1202
      apply (rule conjI)
wenzelm@53406
  1203
      apply (rule independent_mono)
wenzelm@53406
  1204
      apply assumption
wenzelm@53406
  1205
      apply blast
wenzelm@53406
  1206
      apply (simp add: dependent_def)
wenzelm@53406
  1207
      done
wenzelm@53406
  1208
  next
wenzelm@53406
  1209
    assume i: ?rhs
wenzelm@53406
  1210
    show ?lhs
wenzelm@53406
  1211
      using i False
wenzelm@53406
  1212
      apply (auto simp add: dependent_def)
lp15@55775
  1213
      by (metis in_span_insert insert_Diff insert_Diff_if insert_iff)
wenzelm@53406
  1214
  qed
huffman@44133
  1215
qed
huffman@44133
  1216
huffman@44133
  1217
text {* The degenerate case of the Exchange Lemma. *}
huffman@44133
  1218
huffman@44133
  1219
lemma spanning_subset_independent:
wenzelm@49711
  1220
  assumes BA: "B \<subseteq> A"
wenzelm@49711
  1221
    and iA: "independent A"
wenzelm@49522
  1222
    and AsB: "A \<subseteq> span B"
huffman@44133
  1223
  shows "A = B"
huffman@44133
  1224
proof
wenzelm@49663
  1225
  show "B \<subseteq> A" by (rule BA)
wenzelm@49663
  1226
huffman@44133
  1227
  from span_mono[OF BA] span_mono[OF AsB]
huffman@44133
  1228
  have sAB: "span A = span B" unfolding span_span by blast
huffman@44133
  1229
wenzelm@53406
  1230
  {
wenzelm@53406
  1231
    fix x
wenzelm@53406
  1232
    assume x: "x \<in> A"
huffman@44133
  1233
    from iA have th0: "x \<notin> span (A - {x})"
huffman@44133
  1234
      unfolding dependent_def using x by blast
wenzelm@53406
  1235
    from x have xsA: "x \<in> span A"
wenzelm@53406
  1236
      by (blast intro: span_superset)
huffman@44133
  1237
    have "A - {x} \<subseteq> A" by blast
wenzelm@53406
  1238
    then have th1: "span (A - {x}) \<subseteq> span A"
wenzelm@53406
  1239
      by (metis span_mono)
wenzelm@53406
  1240
    {
wenzelm@53406
  1241
      assume xB: "x \<notin> B"
wenzelm@53406
  1242
      from xB BA have "B \<subseteq> A - {x}"
wenzelm@53406
  1243
        by blast
wenzelm@53406
  1244
      then have "span B \<subseteq> span (A - {x})"
wenzelm@53406
  1245
        by (metis span_mono)
wenzelm@53406
  1246
      with th1 th0 sAB have "x \<notin> span A"
wenzelm@53406
  1247
        by blast
wenzelm@53406
  1248
      with x have False
wenzelm@53406
  1249
        by (metis span_superset)
wenzelm@53406
  1250
    }
wenzelm@53406
  1251
    then have "x \<in> B" by blast
wenzelm@53406
  1252
  }
huffman@44133
  1253
  then show "A \<subseteq> B" by blast
huffman@44133
  1254
qed
huffman@44133
  1255
huffman@44133
  1256
text {* The general case of the Exchange Lemma, the key to what follows. *}
huffman@44133
  1257
huffman@44133
  1258
lemma exchange_lemma:
wenzelm@49711
  1259
  assumes f:"finite t"
wenzelm@49711
  1260
    and i: "independent s"
wenzelm@49711
  1261
    and sp: "s \<subseteq> span t"
wenzelm@53406
  1262
  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'"
wenzelm@49663
  1263
  using f i sp
wenzelm@49522
  1264
proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
huffman@44133
  1265
  case less
huffman@44133
  1266
  note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
wenzelm@53406
  1267
  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'"
huffman@44133
  1268
  let ?ths = "\<exists>t'. ?P t'"
wenzelm@53406
  1269
  {
lp15@55775
  1270
    assume "s \<subseteq> t"
lp15@55775
  1271
    then have ?ths
lp15@55775
  1272
      by (metis ft Un_commute sp sup_ge1)
wenzelm@53406
  1273
  }
huffman@44133
  1274
  moreover
wenzelm@53406
  1275
  {
wenzelm@53406
  1276
    assume st: "t \<subseteq> s"
wenzelm@53406
  1277
    from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
wenzelm@53406
  1278
    have ?ths
lp15@55775
  1279
      by (metis Un_absorb sp)
wenzelm@53406
  1280
  }
huffman@44133
  1281
  moreover
wenzelm@53406
  1282
  {
wenzelm@53406
  1283
    assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
wenzelm@53406
  1284
    from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
wenzelm@53406
  1285
      by blast
wenzelm@53406
  1286
    from b have "t - {b} - s \<subset> t - s"
wenzelm@53406
  1287
      by blast
wenzelm@53406
  1288
    then have cardlt: "card (t - {b} - s) < card (t - s)"
wenzelm@53406
  1289
      using ft by (auto intro: psubset_card_mono)
wenzelm@53406
  1290
    from b ft have ct0: "card t \<noteq> 0"
wenzelm@53406
  1291
      by auto
wenzelm@53406
  1292
    have ?ths
wenzelm@53406
  1293
    proof cases
wenzelm@53716
  1294
      assume stb: "s \<subseteq> span (t - {b})"
wenzelm@53716
  1295
      from ft have ftb: "finite (t - {b})"
wenzelm@53406
  1296
        by auto
huffman@44133
  1297
      from less(1)[OF cardlt ftb s stb]
wenzelm@53716
  1298
      obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
wenzelm@49522
  1299
        and fu: "finite u" by blast
huffman@44133
  1300
      let ?w = "insert b u"
wenzelm@53406
  1301
      have th0: "s \<subseteq> insert b u"
wenzelm@53406
  1302
        using u by blast
wenzelm@53406
  1303
      from u(3) b have "u \<subseteq> s \<union> t"
wenzelm@53406
  1304
        by blast
wenzelm@53406
  1305
      then have th1: "insert b u \<subseteq> s \<union> t"
wenzelm@53406
  1306
        using u b by blast
wenzelm@53406
  1307
      have bu: "b \<notin> u"
wenzelm@53406
  1308
        using b u by blast
wenzelm@53406
  1309
      from u(1) ft b have "card u = (card t - 1)"
wenzelm@53406
  1310
        by auto
wenzelm@49522
  1311
      then have th2: "card (insert b u) = card t"
huffman@44133
  1312
        using card_insert_disjoint[OF fu bu] ct0 by auto
huffman@44133
  1313
      from u(4) have "s \<subseteq> span u" .
wenzelm@53406
  1314
      also have "\<dots> \<subseteq> span (insert b u)"
wenzelm@53406
  1315
        by (rule span_mono) blast
huffman@44133
  1316
      finally have th3: "s \<subseteq> span (insert b u)" .
wenzelm@53406
  1317
      from th0 th1 th2 th3 fu have th: "?P ?w"
wenzelm@53406
  1318
        by blast
wenzelm@53406
  1319
      from th show ?thesis by blast
wenzelm@53406
  1320
    next
wenzelm@53716
  1321
      assume stb: "\<not> s \<subseteq> span (t - {b})"
wenzelm@53406
  1322
      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
wenzelm@53406
  1323
        by blast
wenzelm@53406
  1324
      have ab: "a \<noteq> b"
wenzelm@53406
  1325
        using a b by blast
wenzelm@53406
  1326
      have at: "a \<notin> t"
wenzelm@53406
  1327
        using a ab span_superset[of a "t- {b}"] by auto
huffman@44133
  1328
      have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
huffman@44133
  1329
        using cardlt ft a b by auto
wenzelm@53406
  1330
      have ft': "finite (insert a (t - {b}))"
wenzelm@53406
  1331
        using ft by auto
wenzelm@53406
  1332
      {
wenzelm@53406
  1333
        fix x
wenzelm@53406
  1334
        assume xs: "x \<in> s"
wenzelm@53406
  1335
        have t: "t \<subseteq> insert b (insert a (t - {b}))"
wenzelm@53406
  1336
          using b by auto
wenzelm@53406
  1337
        from b(1) have "b \<in> span t"
wenzelm@53406
  1338
          by (simp add: span_superset)
wenzelm@53406
  1339
        have bs: "b \<in> span (insert a (t - {b}))"
wenzelm@53406
  1340
          apply (rule in_span_delete)
wenzelm@53406
  1341
          using a sp unfolding subset_eq
wenzelm@53406
  1342
          apply auto
wenzelm@53406
  1343
          done
wenzelm@53406
  1344
        from xs sp have "x \<in> span t"
wenzelm@53406
  1345
          by blast
wenzelm@53406
  1346
        with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
wenzelm@53406
  1347
        from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
wenzelm@53406
  1348
      }
wenzelm@53406
  1349
      then have sp': "s \<subseteq> span (insert a (t - {b}))"
wenzelm@53406
  1350
        by blast
wenzelm@53406
  1351
      from less(1)[OF mlt ft' s sp'] obtain u where u:
wenzelm@53716
  1352
        "card u = card (insert a (t - {b}))"
wenzelm@53716
  1353
        "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
wenzelm@53406
  1354
        "s \<subseteq> span u" by blast
wenzelm@53406
  1355
      from u a b ft at ct0 have "?P u"
wenzelm@53406
  1356
        by auto
wenzelm@53406
  1357
      then show ?thesis by blast
wenzelm@53406
  1358
    qed
huffman@44133
  1359
  }
wenzelm@49522
  1360
  ultimately show ?ths by blast
huffman@44133
  1361
qed
huffman@44133
  1362
huffman@44133
  1363
text {* This implies corresponding size bounds. *}
huffman@44133
  1364
huffman@44133
  1365
lemma independent_span_bound:
wenzelm@53406
  1366
  assumes f: "finite t"
wenzelm@53406
  1367
    and i: "independent s"
wenzelm@53406
  1368
    and sp: "s \<subseteq> span t"
huffman@44133
  1369
  shows "finite s \<and> card s \<le> card t"
huffman@44133
  1370
  by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
huffman@44133
  1371
huffman@44133
  1372
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
wenzelm@49522
  1373
proof -
wenzelm@53406
  1374
  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
wenzelm@53406
  1375
    by auto
huffman@44133
  1376
  show ?thesis unfolding eq
huffman@44133
  1377
    apply (rule finite_imageI)
huffman@44133
  1378
    apply (rule finite)
huffman@44133
  1379
    done
huffman@44133
  1380
qed
huffman@44133
  1381
wenzelm@53406
  1382
wenzelm@53406
  1383
subsection {* Euclidean Spaces as Typeclass *}
huffman@44133
  1384
hoelzl@50526
  1385
lemma independent_Basis: "independent Basis"
hoelzl@50526
  1386
  unfolding dependent_def
hoelzl@50526
  1387
  apply (subst span_finite)
hoelzl@50526
  1388
  apply simp
huffman@44133
  1389
  apply clarify
hoelzl@50526
  1390
  apply (drule_tac f="inner a" in arg_cong)
hoelzl@50526
  1391
  apply (simp add: inner_Basis inner_setsum_right eq_commute)
hoelzl@50526
  1392
  done
hoelzl@50526
  1393
huffman@53939
  1394
lemma span_Basis [simp]: "span Basis = UNIV"
huffman@53939
  1395
  unfolding span_finite [OF finite_Basis]
huffman@53939
  1396
  by (fast intro: euclidean_representation)
huffman@44133
  1397
hoelzl@50526
  1398
lemma in_span_Basis: "x \<in> span Basis"
hoelzl@50526
  1399
  unfolding span_Basis ..
hoelzl@50526
  1400
hoelzl@50526
  1401
lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
hoelzl@50526
  1402
  by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
hoelzl@50526
  1403
hoelzl@50526
  1404
lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
hoelzl@50526
  1405
  by (metis Basis_le_norm order_trans)
hoelzl@50526
  1406
hoelzl@50526
  1407
lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
huffman@53595
  1408
  by (metis Basis_le_norm le_less_trans)
hoelzl@50526
  1409
hoelzl@50526
  1410
lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
hoelzl@50526
  1411
  apply (subst euclidean_representation[of x, symmetric])
huffman@44176
  1412
  apply (rule order_trans[OF norm_setsum])
wenzelm@49522
  1413
  apply (auto intro!: setsum_mono)
wenzelm@49522
  1414
  done
huffman@44133
  1415
huffman@44133
  1416
lemma setsum_norm_allsubsets_bound:
huffman@44133
  1417
  fixes f:: "'a \<Rightarrow> 'n::euclidean_space"
wenzelm@53406
  1418
  assumes fP: "finite P"
wenzelm@53406
  1419
    and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
hoelzl@50526
  1420
  shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
wenzelm@49522
  1421
proof -
hoelzl@50526
  1422
  have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
hoelzl@50526
  1423
    by (rule setsum_mono) (rule norm_le_l1)
hoelzl@50526
  1424
  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>)"
huffman@44133
  1425
    by (rule setsum_commute)
hoelzl@50526
  1426
  also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
wenzelm@49522
  1427
  proof (rule setsum_bounded)
wenzelm@53406
  1428
    fix i :: 'n
wenzelm@53406
  1429
    assume i: "i \<in> Basis"
wenzelm@53406
  1430
    have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
hoelzl@50526
  1431
      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)"
wenzelm@55136
  1432
      by (simp add: abs_real_def setsum_cases[OF fP] setsum_negf norm_triangle_ineq4 inner_setsum_left del: real_norm_def)
wenzelm@53406
  1433
    also have "\<dots> \<le> e + e"
wenzelm@53406
  1434
      unfolding real_norm_def
hoelzl@50526
  1435
      by (intro add_mono norm_bound_Basis_le i fPs) auto
hoelzl@50526
  1436
    finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
huffman@44133
  1437
  qed
hoelzl@50526
  1438
  also have "\<dots> = 2 * real DIM('n) * e"
hoelzl@50526
  1439
    by (simp add: real_of_nat_def)
huffman@44133
  1440
  finally show ?thesis .
huffman@44133
  1441
qed
huffman@44133
  1442
wenzelm@53406
  1443
huffman@44133
  1444
subsection {* Linearity and Bilinearity continued *}
huffman@44133
  1445
huffman@44133
  1446
lemma linear_bounded:
huffman@44133
  1447
  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1448
  assumes lf: "linear f"
huffman@44133
  1449
  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1450
proof
hoelzl@50526
  1451
  let ?B = "\<Sum>b\<in>Basis. norm (f b)"
huffman@53939
  1452
  show "\<forall>x. norm (f x) \<le> ?B * norm x"
huffman@53939
  1453
  proof
wenzelm@53406
  1454
    fix x :: 'a
hoelzl@50526
  1455
    let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
hoelzl@50526
  1456
    have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
hoelzl@50526
  1457
      unfolding euclidean_representation ..
hoelzl@50526
  1458
    also have "\<dots> = norm (setsum ?g Basis)"
huffman@53939
  1459
      by (simp add: linear_setsum [OF lf] linear_cmul [OF lf])
hoelzl@50526
  1460
    finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
huffman@53939
  1461
    have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
huffman@53939
  1462
    proof
wenzelm@53406
  1463
      fix i :: 'a
wenzelm@53406
  1464
      assume i: "i \<in> Basis"
hoelzl@50526
  1465
      from Basis_le_norm[OF i, of x]
huffman@53939
  1466
      show "norm (?g i) \<le> norm (f i) * norm x"
wenzelm@49663
  1467
        unfolding norm_scaleR
hoelzl@50526
  1468
        apply (subst mult_commute)
wenzelm@49663
  1469
        apply (rule mult_mono)
wenzelm@49663
  1470
        apply (auto simp add: field_simps)
wenzelm@53406
  1471
        done
huffman@53939
  1472
    qed
hoelzl@50526
  1473
    from setsum_norm_le[of _ ?g, OF th]
huffman@53939
  1474
    show "norm (f x) \<le> ?B * norm x"
wenzelm@53406
  1475
      unfolding th0 setsum_left_distrib by metis
huffman@53939
  1476
  qed
huffman@44133
  1477
qed
huffman@44133
  1478
huffman@44133
  1479
lemma linear_conv_bounded_linear:
huffman@44133
  1480
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1481
  shows "linear f \<longleftrightarrow> bounded_linear f"
huffman@44133
  1482
proof
huffman@44133
  1483
  assume "linear f"
huffman@53939
  1484
  then interpret f: linear f .
huffman@44133
  1485
  show "bounded_linear f"
huffman@44133
  1486
  proof
huffman@44133
  1487
    have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
huffman@44133
  1488
      using `linear f` by (rule linear_bounded)
wenzelm@49522
  1489
    then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
huffman@44133
  1490
      by (simp add: mult_commute)
huffman@44133
  1491
  qed
huffman@44133
  1492
next
huffman@44133
  1493
  assume "bounded_linear f"
huffman@44133
  1494
  then interpret f: bounded_linear f .
huffman@53939
  1495
  show "linear f" ..
huffman@53939
  1496
qed
huffman@53939
  1497
huffman@53939
  1498
lemma linear_bounded_pos:
huffman@53939
  1499
  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@53939
  1500
  assumes lf: "linear f"
huffman@53939
  1501
  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1502
proof -
huffman@53939
  1503
  have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
huffman@53939
  1504
    using lf unfolding linear_conv_bounded_linear
huffman@53939
  1505
    by (rule bounded_linear.pos_bounded)
huffman@53939
  1506
  then show ?thesis
huffman@53939
  1507
    by (simp only: mult_commute)
huffman@44133
  1508
qed
huffman@44133
  1509
wenzelm@49522
  1510
lemma bounded_linearI':
wenzelm@49522
  1511
  fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@53406
  1512
  assumes "\<And>x y. f (x + y) = f x + f y"
wenzelm@53406
  1513
    and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
wenzelm@49522
  1514
  shows "bounded_linear f"
wenzelm@53406
  1515
  unfolding linear_conv_bounded_linear[symmetric]
wenzelm@49522
  1516
  by (rule linearI[OF assms])
huffman@44133
  1517
huffman@44133
  1518
lemma bilinear_bounded:
huffman@44133
  1519
  fixes h:: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
huffman@44133
  1520
  assumes bh: "bilinear h"
huffman@44133
  1521
  shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
hoelzl@50526
  1522
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
wenzelm@53406
  1523
  fix x :: 'm
wenzelm@53406
  1524
  fix y :: 'n
wenzelm@53406
  1525
  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))"
wenzelm@53406
  1526
    apply (subst euclidean_representation[where 'a='m])
wenzelm@53406
  1527
    apply (subst euclidean_representation[where 'a='n])
hoelzl@50526
  1528
    apply rule
hoelzl@50526
  1529
    done
wenzelm@53406
  1530
  also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
hoelzl@50526
  1531
    unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
hoelzl@50526
  1532
  finally have th: "norm (h x y) = \<dots>" .
hoelzl@50526
  1533
  show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
wenzelm@53406
  1534
    apply (auto simp add: setsum_left_distrib th setsum_cartesian_product)
wenzelm@53406
  1535
    apply (rule setsum_norm_le)
wenzelm@53406
  1536
    apply simp
wenzelm@53406
  1537
    apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
wenzelm@53406
  1538
      field_simps simp del: scaleR_scaleR)
wenzelm@53406
  1539
    apply (rule mult_mono)
wenzelm@53406
  1540
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  1541
    apply (rule mult_mono)
wenzelm@53406
  1542
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  1543
    done
huffman@44133
  1544
qed
huffman@44133
  1545
huffman@44133
  1546
lemma bilinear_conv_bounded_bilinear:
huffman@44133
  1547
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@44133
  1548
  shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
huffman@44133
  1549
proof
huffman@44133
  1550
  assume "bilinear h"
huffman@44133
  1551
  show "bounded_bilinear h"
huffman@44133
  1552
  proof
wenzelm@53406
  1553
    fix x y z
wenzelm@53406
  1554
    show "h (x + y) z = h x z + h y z"
huffman@53600
  1555
      using `bilinear h` unfolding bilinear_def linear_iff by simp
huffman@44133
  1556
  next
wenzelm@53406
  1557
    fix x y z
wenzelm@53406
  1558
    show "h x (y + z) = h x y + h x z"
huffman@53600
  1559
      using `bilinear h` unfolding bilinear_def linear_iff by simp
huffman@44133
  1560
  next
wenzelm@53406
  1561
    fix r x y
wenzelm@53406
  1562
    show "h (scaleR r x) y = scaleR r (h x y)"
huffman@53600
  1563
      using `bilinear h` unfolding bilinear_def linear_iff
huffman@44133
  1564
      by simp
huffman@44133
  1565
  next
wenzelm@53406
  1566
    fix r x y
wenzelm@53406
  1567
    show "h x (scaleR r y) = scaleR r (h x y)"
huffman@53600
  1568
      using `bilinear h` unfolding bilinear_def linear_iff
huffman@44133
  1569
      by simp
huffman@44133
  1570
  next
huffman@44133
  1571
    have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
huffman@44133
  1572
      using `bilinear h` by (rule bilinear_bounded)
wenzelm@49522
  1573
    then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
huffman@44133
  1574
      by (simp add: mult_ac)
huffman@44133
  1575
  qed
huffman@44133
  1576
next
huffman@44133
  1577
  assume "bounded_bilinear h"
huffman@44133
  1578
  then interpret h: bounded_bilinear h .
huffman@44133
  1579
  show "bilinear h"
huffman@44133
  1580
    unfolding bilinear_def linear_conv_bounded_linear
wenzelm@49522
  1581
    using h.bounded_linear_left h.bounded_linear_right by simp
huffman@44133
  1582
qed
huffman@44133
  1583
huffman@53939
  1584
lemma bilinear_bounded_pos:
huffman@53939
  1585
  fixes h:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@53939
  1586
  assumes bh: "bilinear h"
huffman@53939
  1587
  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
huffman@53939
  1588
proof -
huffman@53939
  1589
  have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
huffman@53939
  1590
    using bh [unfolded bilinear_conv_bounded_bilinear]
huffman@53939
  1591
    by (rule bounded_bilinear.pos_bounded)
huffman@53939
  1592
  then show ?thesis
huffman@53939
  1593
    by (simp only: mult_ac)
huffman@53939
  1594
qed
huffman@53939
  1595
wenzelm@49522
  1596
huffman@44133
  1597
subsection {* We continue. *}
huffman@44133
  1598
huffman@44133
  1599
lemma independent_bound:
wenzelm@53716
  1600
  fixes S :: "'a::euclidean_space set"
wenzelm@53716
  1601
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
hoelzl@50526
  1602
  using independent_span_bound[OF finite_Basis, of S] by auto
huffman@44133
  1603
wenzelm@49663
  1604
lemma dependent_biggerset:
wenzelm@53406
  1605
  "(finite (S::('a::euclidean_space) set) \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
huffman@44133
  1606
  by (metis independent_bound not_less)
huffman@44133
  1607
huffman@44133
  1608
text {* Hence we can create a maximal independent subset. *}
huffman@44133
  1609
huffman@44133
  1610
lemma maximal_independent_subset_extend:
wenzelm@53406
  1611
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1612
  assumes sv: "S \<subseteq> V"
wenzelm@49663
  1613
    and iS: "independent S"
huffman@44133
  1614
  shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
huffman@44133
  1615
  using sv iS
wenzelm@49522
  1616
proof (induct "DIM('a) - card S" arbitrary: S rule: less_induct)
huffman@44133
  1617
  case less
huffman@44133
  1618
  note sv = `S \<subseteq> V` and i = `independent S`
huffman@44133
  1619
  let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
huffman@44133
  1620
  let ?ths = "\<exists>x. ?P x"
huffman@44133
  1621
  let ?d = "DIM('a)"
wenzelm@53406
  1622
  show ?ths
wenzelm@53406
  1623
  proof (cases "V \<subseteq> span S")
wenzelm@53406
  1624
    case True
wenzelm@53406
  1625
    then show ?thesis
wenzelm@53406
  1626
      using sv i by blast
wenzelm@53406
  1627
  next
wenzelm@53406
  1628
    case False
wenzelm@53406
  1629
    then obtain a where a: "a \<in> V" "a \<notin> span S"
wenzelm@53406
  1630
      by blast
wenzelm@53406
  1631
    from a have aS: "a \<notin> S"
wenzelm@53406
  1632
      by (auto simp add: span_superset)
wenzelm@53406
  1633
    have th0: "insert a S \<subseteq> V"
wenzelm@53406
  1634
      using a sv by blast
huffman@44133
  1635
    from independent_insert[of a S]  i a
wenzelm@53406
  1636
    have th1: "independent (insert a S)"
wenzelm@53406
  1637
      by auto
huffman@44133
  1638
    have mlt: "?d - card (insert a S) < ?d - card S"
wenzelm@49522
  1639
      using aS a independent_bound[OF th1] by auto
huffman@44133
  1640
huffman@44133
  1641
    from less(1)[OF mlt th0 th1]
huffman@44133
  1642
    obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
huffman@44133
  1643
      by blast
huffman@44133
  1644
    from B have "?P B" by auto
wenzelm@53406
  1645
    then show ?thesis by blast
wenzelm@53406
  1646
  qed
huffman@44133
  1647
qed
huffman@44133
  1648
huffman@44133
  1649
lemma maximal_independent_subset:
huffman@44133
  1650
  "\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
wenzelm@49522
  1651
  by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"]
wenzelm@49522
  1652
    empty_subsetI independent_empty)
huffman@44133
  1653
huffman@44133
  1654
huffman@44133
  1655
text {* Notion of dimension. *}
huffman@44133
  1656
wenzelm@53406
  1657
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
huffman@44133
  1658
wenzelm@49522
  1659
lemma basis_exists:
wenzelm@49522
  1660
  "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
wenzelm@49522
  1661
  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)"]
wenzelm@49522
  1662
  using maximal_independent_subset[of V] independent_bound
wenzelm@49522
  1663
  by auto
huffman@44133
  1664
huffman@44133
  1665
text {* Consequences of independence or spanning for cardinality. *}
huffman@44133
  1666
wenzelm@53406
  1667
lemma independent_card_le_dim:
wenzelm@53406
  1668
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1669
  assumes "B \<subseteq> V"
wenzelm@53406
  1670
    and "independent B"
wenzelm@49522
  1671
  shows "card B \<le> dim V"
huffman@44133
  1672
proof -
huffman@44133
  1673
  from basis_exists[of V] `B \<subseteq> V`
wenzelm@53406
  1674
  obtain B' where "independent B'"
wenzelm@53406
  1675
    and "B \<subseteq> span B'"
wenzelm@53406
  1676
    and "card B' = dim V"
wenzelm@53406
  1677
    by blast
huffman@44133
  1678
  with independent_span_bound[OF _ `independent B` `B \<subseteq> span B'`] independent_bound[of B']
huffman@44133
  1679
  show ?thesis by auto
huffman@44133
  1680
qed
huffman@44133
  1681
wenzelm@49522
  1682
lemma span_card_ge_dim:
wenzelm@53406
  1683
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1684
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
huffman@44133
  1685
  by (metis basis_exists[of V] independent_span_bound subset_trans)
huffman@44133
  1686
huffman@44133
  1687
lemma basis_card_eq_dim:
wenzelm@53406
  1688
  fixes V :: "'a::euclidean_space set"
wenzelm@53406
  1689
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
huffman@44133
  1690
  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
huffman@44133
  1691
wenzelm@53406
  1692
lemma dim_unique:
wenzelm@53406
  1693
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1694
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
huffman@44133
  1695
  by (metis basis_card_eq_dim)
huffman@44133
  1696
huffman@44133
  1697
text {* More lemmas about dimension. *}
huffman@44133
  1698
wenzelm@53406
  1699
lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
hoelzl@50526
  1700
  using independent_Basis
hoelzl@50526
  1701
  by (intro dim_unique[of Basis]) auto
huffman@44133
  1702
huffman@44133
  1703
lemma dim_subset:
wenzelm@53406
  1704
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1705
  shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  1706
  using basis_exists[of T] basis_exists[of S]
huffman@44133
  1707
  by (metis independent_card_le_dim subset_trans)
huffman@44133
  1708
wenzelm@53406
  1709
lemma dim_subset_UNIV:
wenzelm@53406
  1710
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1711
  shows "dim S \<le> DIM('a)"
huffman@44133
  1712
  by (metis dim_subset subset_UNIV dim_UNIV)
huffman@44133
  1713
huffman@44133
  1714
text {* Converses to those. *}
huffman@44133
  1715
huffman@44133
  1716
lemma card_ge_dim_independent:
wenzelm@53406
  1717
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1718
  assumes BV: "B \<subseteq> V"
wenzelm@53406
  1719
    and iB: "independent B"
wenzelm@53406
  1720
    and dVB: "dim V \<le> card B"
huffman@44133
  1721
  shows "V \<subseteq> span B"
wenzelm@53406
  1722
proof
wenzelm@53406
  1723
  fix a
wenzelm@53406
  1724
  assume aV: "a \<in> V"
wenzelm@53406
  1725
  {
wenzelm@53406
  1726
    assume aB: "a \<notin> span B"
wenzelm@53406
  1727
    then have iaB: "independent (insert a B)"
wenzelm@53406
  1728
      using iB aV BV by (simp add: independent_insert)
wenzelm@53406
  1729
    from aV BV have th0: "insert a B \<subseteq> V"
wenzelm@53406
  1730
      by blast
wenzelm@53406
  1731
    from aB have "a \<notin>B"
wenzelm@53406
  1732
      by (auto simp add: span_superset)
wenzelm@53406
  1733
    with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
wenzelm@53406
  1734
    have False by auto
wenzelm@53406
  1735
  }
wenzelm@53406
  1736
  then show "a \<in> span B" by blast
huffman@44133
  1737
qed
huffman@44133
  1738
huffman@44133
  1739
lemma card_le_dim_spanning:
wenzelm@49663
  1740
  assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
wenzelm@49663
  1741
    and VB: "V \<subseteq> span B"
wenzelm@49663
  1742
    and fB: "finite B"
wenzelm@49663
  1743
    and dVB: "dim V \<ge> card B"
huffman@44133
  1744
  shows "independent B"
wenzelm@49522
  1745
proof -
wenzelm@53406
  1746
  {
wenzelm@53406
  1747
    fix a
wenzelm@53716
  1748
    assume a: "a \<in> B" "a \<in> span (B - {a})"
wenzelm@53406
  1749
    from a fB have c0: "card B \<noteq> 0"
wenzelm@53406
  1750
      by auto
wenzelm@53716
  1751
    from a fB have cb: "card (B - {a}) = card B - 1"
wenzelm@53406
  1752
      by auto
wenzelm@53716
  1753
    from BV a have th0: "B - {a} \<subseteq> V"
wenzelm@53406
  1754
      by blast
wenzelm@53406
  1755
    {
wenzelm@53406
  1756
      fix x
wenzelm@53406
  1757
      assume x: "x \<in> V"
wenzelm@53716
  1758
      from a have eq: "insert a (B - {a}) = B"
wenzelm@53406
  1759
        by blast
wenzelm@53406
  1760
      from x VB have x': "x \<in> span B"
wenzelm@53406
  1761
        by blast
huffman@44133
  1762
      from span_trans[OF a(2), unfolded eq, OF x']
wenzelm@53716
  1763
      have "x \<in> span (B - {a})" .
wenzelm@53406
  1764
    }
wenzelm@53716
  1765
    then have th1: "V \<subseteq> span (B - {a})"
wenzelm@53406
  1766
      by blast
wenzelm@53716
  1767
    have th2: "finite (B - {a})"
wenzelm@53406
  1768
      using fB by auto
huffman@44133
  1769
    from span_card_ge_dim[OF th0 th1 th2]
wenzelm@53716
  1770
    have c: "dim V \<le> card (B - {a})" .
wenzelm@53406
  1771
    from c c0 dVB cb have False by simp
wenzelm@53406
  1772
  }
wenzelm@53406
  1773
  then show ?thesis
wenzelm@53406
  1774
    unfolding dependent_def by blast
huffman@44133
  1775
qed
huffman@44133
  1776
wenzelm@53406
  1777
lemma card_eq_dim:
wenzelm@53406
  1778
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1779
  shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
wenzelm@49522
  1780
  by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
huffman@44133
  1781
huffman@44133
  1782
text {* More general size bound lemmas. *}
huffman@44133
  1783
huffman@44133
  1784
lemma independent_bound_general:
wenzelm@53406
  1785
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1786
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
huffman@44133
  1787
  by (metis independent_card_le_dim independent_bound subset_refl)
huffman@44133
  1788
wenzelm@49522
  1789
lemma dependent_biggerset_general:
wenzelm@53406
  1790
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1791
  shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
huffman@44133
  1792
  using independent_bound_general[of S] by (metis linorder_not_le)
huffman@44133
  1793
wenzelm@53406
  1794
lemma dim_span:
wenzelm@53406
  1795
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1796
  shows "dim (span S) = dim S"
wenzelm@49522
  1797
proof -
huffman@44133
  1798
  have th0: "dim S \<le> dim (span S)"
huffman@44133
  1799
    by (auto simp add: subset_eq intro: dim_subset span_superset)
huffman@44133
  1800
  from basis_exists[of S]
wenzelm@53406
  1801
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
wenzelm@53406
  1802
    by blast
wenzelm@53406
  1803
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  1804
    using independent_bound by blast+
wenzelm@53406
  1805
  have bSS: "B \<subseteq> span S"
wenzelm@53406
  1806
    using B(1) by (metis subset_eq span_inc)
wenzelm@53406
  1807
  have sssB: "span S \<subseteq> span B"
wenzelm@53406
  1808
    using span_mono[OF B(3)] by (simp add: span_span)
huffman@44133
  1809
  from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
wenzelm@49522
  1810
    using fB(2) by arith
huffman@44133
  1811
qed
huffman@44133
  1812
wenzelm@53406
  1813
lemma subset_le_dim:
wenzelm@53406
  1814
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1815
  shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  1816
  by (metis dim_span dim_subset)
huffman@44133
  1817
wenzelm@53406
  1818
lemma span_eq_dim:
wenzelm@53406
  1819
  fixes S:: "'a::euclidean_space set"
wenzelm@53406
  1820
  shows "span S = span T \<Longrightarrow> dim S = dim T"
huffman@44133
  1821
  by (metis dim_span)
huffman@44133
  1822
huffman@44133
  1823
lemma spans_image:
wenzelm@49663
  1824
  assumes lf: "linear f"
wenzelm@49663
  1825
    and VB: "V \<subseteq> span B"
huffman@44133
  1826
  shows "f ` V \<subseteq> span (f ` B)"
wenzelm@49522
  1827
  unfolding span_linear_image[OF lf] by (metis VB image_mono)
huffman@44133
  1828
huffman@44133
  1829
lemma dim_image_le:
huffman@44133
  1830
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
wenzelm@49663
  1831
  assumes lf: "linear f"
wenzelm@49663
  1832
  shows "dim (f ` S) \<le> dim (S)"
wenzelm@49522
  1833
proof -
huffman@44133
  1834
  from basis_exists[of S] obtain B where
huffman@44133
  1835
    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
wenzelm@53406
  1836
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  1837
    using independent_bound by blast+
huffman@44133
  1838
  have "dim (f ` S) \<le> card (f ` B)"
huffman@44133
  1839
    apply (rule span_card_ge_dim)
wenzelm@53406
  1840
    using lf B fB
wenzelm@53406
  1841
    apply (auto simp add: span_linear_image spans_image subset_image_iff)
wenzelm@49522
  1842
    done
wenzelm@53406
  1843
  also have "\<dots> \<le> dim S"
wenzelm@53406
  1844
    using card_image_le[OF fB(1)] fB by simp
huffman@44133
  1845
  finally show ?thesis .
huffman@44133
  1846
qed
huffman@44133
  1847
huffman@44133
  1848
text {* Relation between bases and injectivity/surjectivity of map. *}
huffman@44133
  1849
huffman@44133
  1850
lemma spanning_surjective_image:
huffman@44133
  1851
  assumes us: "UNIV \<subseteq> span S"
wenzelm@53406
  1852
    and lf: "linear f"
wenzelm@53406
  1853
    and sf: "surj f"
huffman@44133
  1854
  shows "UNIV \<subseteq> span (f ` S)"
wenzelm@49663
  1855
proof -
wenzelm@53406
  1856
  have "UNIV \<subseteq> f ` UNIV"
wenzelm@53406
  1857
    using sf by (auto simp add: surj_def)
wenzelm@53406
  1858
  also have " \<dots> \<subseteq> span (f ` S)"
wenzelm@53406
  1859
    using spans_image[OF lf us] .
wenzelm@53406
  1860
  finally show ?thesis .
huffman@44133
  1861
qed
huffman@44133
  1862
huffman@44133
  1863
lemma independent_injective_image:
wenzelm@49663
  1864
  assumes iS: "independent S"
wenzelm@49663
  1865
    and lf: "linear f"
wenzelm@49663
  1866
    and fi: "inj f"
huffman@44133
  1867
  shows "independent (f ` S)"
wenzelm@49663
  1868
proof -
wenzelm@53406
  1869
  {
wenzelm@53406
  1870
    fix a
wenzelm@49663
  1871
    assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
wenzelm@53406
  1872
    have eq: "f ` S - {f a} = f ` (S - {a})"
wenzelm@53406
  1873
      using fi by (auto simp add: inj_on_def)
wenzelm@53716
  1874
    from a have "f a \<in> f ` span (S - {a})"
wenzelm@53406
  1875
      unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
wenzelm@53716
  1876
    then have "a \<in> span (S - {a})"
wenzelm@53406
  1877
      using fi by (auto simp add: inj_on_def)
wenzelm@53406
  1878
    with a(1) iS have False
wenzelm@53406
  1879
      by (simp add: dependent_def)
wenzelm@53406
  1880
  }
wenzelm@53406
  1881
  then show ?thesis
wenzelm@53406
  1882
    unfolding dependent_def by blast
huffman@44133
  1883
qed
huffman@44133
  1884
huffman@44133
  1885
text {* Picking an orthogonal replacement for a spanning set. *}
huffman@44133
  1886
wenzelm@53406
  1887
(* FIXME : Move to some general theory ?*)
huffman@44133
  1888
definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
huffman@44133
  1889
wenzelm@53406
  1890
lemma vector_sub_project_orthogonal:
wenzelm@53406
  1891
  fixes b x :: "'a::euclidean_space"
wenzelm@53406
  1892
  shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
huffman@44133
  1893
  unfolding inner_simps by auto
huffman@44133
  1894
huffman@44528
  1895
lemma pairwise_orthogonal_insert:
huffman@44528
  1896
  assumes "pairwise orthogonal S"
wenzelm@49522
  1897
    and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
huffman@44528
  1898
  shows "pairwise orthogonal (insert x S)"
huffman@44528
  1899
  using assms unfolding pairwise_def
huffman@44528
  1900
  by (auto simp add: orthogonal_commute)
huffman@44528
  1901
huffman@44133
  1902
lemma basis_orthogonal:
wenzelm@53406
  1903
  fixes B :: "'a::real_inner set"
huffman@44133
  1904
  assumes fB: "finite B"
huffman@44133
  1905
  shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
huffman@44133
  1906
  (is " \<exists>C. ?P B C")
wenzelm@49522
  1907
  using fB
wenzelm@49522
  1908
proof (induct rule: finite_induct)
wenzelm@49522
  1909
  case empty
wenzelm@53406
  1910
  then show ?case
wenzelm@53406
  1911
    apply (rule exI[where x="{}"])
wenzelm@53406
  1912
    apply (auto simp add: pairwise_def)
wenzelm@53406
  1913
    done
huffman@44133
  1914
next
wenzelm@49522
  1915
  case (insert a B)
huffman@44133
  1916
  note fB = `finite B` and aB = `a \<notin> B`
huffman@44133
  1917
  from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
huffman@44133
  1918
  obtain C where C: "finite C" "card C \<le> card B"
huffman@44133
  1919
    "span C = span B" "pairwise orthogonal C" by blast
huffman@44133
  1920
  let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
huffman@44133
  1921
  let ?C = "insert ?a C"
wenzelm@53406
  1922
  from C(1) have fC: "finite ?C"
wenzelm@53406
  1923
    by simp
wenzelm@49522
  1924
  from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
wenzelm@49522
  1925
    by (simp add: card_insert_if)
wenzelm@53406
  1926
  {
wenzelm@53406
  1927
    fix x k
wenzelm@49522
  1928
    have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
wenzelm@49522
  1929
      by (simp add: field_simps)
huffman@44133
  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"
huffman@44133
  1931
      apply (simp only: scaleR_right_diff_distrib th0)
huffman@44133
  1932
      apply (rule span_add_eq)
huffman@44133
  1933
      apply (rule span_mul)
huffman@56196
  1934
      apply (rule span_setsum)
huffman@44133
  1935
      apply clarify
huffman@44133
  1936
      apply (rule span_mul)
wenzelm@49522
  1937
      apply (rule span_superset)
wenzelm@49522
  1938
      apply assumption
wenzelm@53406
  1939
      done
wenzelm@53406
  1940
  }
huffman@44133
  1941
  then have SC: "span ?C = span (insert a B)"
huffman@44133
  1942
    unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
wenzelm@53406
  1943
  {
wenzelm@53406
  1944
    fix y
wenzelm@53406
  1945
    assume yC: "y \<in> C"
wenzelm@53406
  1946
    then have Cy: "C = insert y (C - {y})"
wenzelm@53406
  1947
      by blast
wenzelm@53406
  1948
    have fth: "finite (C - {y})"
wenzelm@53406
  1949
      using C by simp
huffman@44528
  1950
    have "orthogonal ?a y"
huffman@44528
  1951
      unfolding orthogonal_def
haftmann@54230
  1952
      unfolding inner_diff inner_setsum_left right_minus_eq
huffman@44528
  1953
      unfolding setsum_diff1' [OF `finite C` `y \<in> C`]
huffman@44528
  1954
      apply (clarsimp simp add: inner_commute[of y a])
huffman@44528
  1955
      apply (rule setsum_0')
huffman@44528
  1956
      apply clarsimp
huffman@44528
  1957
      apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
wenzelm@53406
  1958
      using `y \<in> C` by auto
wenzelm@53406
  1959
  }
huffman@44528
  1960
  with `pairwise orthogonal C` have CPO: "pairwise orthogonal ?C"
huffman@44528
  1961
    by (rule pairwise_orthogonal_insert)
wenzelm@53406
  1962
  from fC cC SC CPO have "?P (insert a B) ?C"
wenzelm@53406
  1963
    by blast
huffman@44133
  1964
  then show ?case by blast
huffman@44133
  1965
qed
huffman@44133
  1966
huffman@44133
  1967
lemma orthogonal_basis_exists:
huffman@44133
  1968
  fixes V :: "('a::euclidean_space) set"
huffman@44133
  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"
wenzelm@49663
  1970
proof -
wenzelm@49522
  1971
  from basis_exists[of V] obtain B where
wenzelm@53406
  1972
    B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
wenzelm@53406
  1973
    by blast
wenzelm@53406
  1974
  from B have fB: "finite B" "card B = dim V"
wenzelm@53406
  1975
    using independent_bound by auto
huffman@44133
  1976
  from basis_orthogonal[OF fB(1)] obtain C where
wenzelm@53406
  1977
    C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
wenzelm@53406
  1978
    by blast
wenzelm@53406
  1979
  from C B have CSV: "C \<subseteq> span V"
wenzelm@53406
  1980
    by (metis span_inc span_mono subset_trans)
wenzelm@53406
  1981
  from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
wenzelm@53406
  1982
    by (simp add: span_span)
huffman@44133
  1983
  from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
wenzelm@53406
  1984
  have iC: "independent C"
huffman@44133
  1985
    by (simp add: dim_span)
wenzelm@53406
  1986
  from C fB have "card C \<le> dim V"
wenzelm@53406
  1987
    by simp
wenzelm@53406
  1988
  moreover have "dim V \<le> card C"
wenzelm@53406
  1989
    using span_card_ge_dim[OF CSV SVC C(1)]
wenzelm@53406
  1990
    by (simp add: dim_span)
wenzelm@53406
  1991
  ultimately have CdV: "card C = dim V"
wenzelm@53406
  1992
    using C(1) by simp
wenzelm@53406
  1993
  from C B CSV CdV iC show ?thesis
wenzelm@53406
  1994
    by auto
huffman@44133
  1995
qed
huffman@44133
  1996
huffman@44133
  1997
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
huffman@44133
  1998
  using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
wenzelm@49522
  1999
  by (auto simp add: span_span)
huffman@44133
  2000
huffman@44133
  2001
text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
huffman@44133
  2002
wenzelm@49522
  2003
lemma span_not_univ_orthogonal:
wenzelm@53406
  2004
  fixes S :: "'a::euclidean_space set"
huffman@44133
  2005
  assumes sU: "span S \<noteq> UNIV"
huffman@44133
  2006
  shows "\<exists>(a::'a). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
wenzelm@49522
  2007
proof -
wenzelm@53406
  2008
  from sU obtain a where a: "a \<notin> span S"
wenzelm@53406
  2009
    by blast
huffman@44133
  2010
  from orthogonal_basis_exists obtain B where
huffman@44133
  2011
    B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
huffman@44133
  2012
    by blast
wenzelm@53406
  2013
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  2014
    using independent_bound by auto
huffman@44133
  2015
  from span_mono[OF B(2)] span_mono[OF B(3)]
wenzelm@53406
  2016
  have sSB: "span S = span B"
wenzelm@53406
  2017
    by (simp add: span_span)
huffman@44133
  2018
  let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
huffman@44133
  2019
  have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
huffman@44133
  2020
    unfolding sSB
huffman@56196
  2021
    apply (rule span_setsum)
huffman@44133
  2022
    apply clarsimp
huffman@44133
  2023
    apply (rule span_mul)
wenzelm@49522
  2024
    apply (rule span_superset)
wenzelm@49522
  2025
    apply assumption
wenzelm@49522
  2026
    done
wenzelm@53406
  2027
  with a have a0:"?a  \<noteq> 0"
wenzelm@53406
  2028
    by auto
huffman@44133
  2029
  have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
wenzelm@49522
  2030
  proof (rule span_induct')
wenzelm@49522
  2031
    show "subspace {x. ?a \<bullet> x = 0}"
wenzelm@49522
  2032
      by (auto simp add: subspace_def inner_add)
wenzelm@49522
  2033
  next
wenzelm@53406
  2034
    {
wenzelm@53406
  2035
      fix x
wenzelm@53406
  2036
      assume x: "x \<in> B"
wenzelm@53406
  2037
      from x have B': "B = insert x (B - {x})"
wenzelm@53406
  2038
        by blast
wenzelm@53406
  2039
      have fth: "finite (B - {x})"
wenzelm@53406
  2040
        using fB by simp
huffman@44133
  2041
      have "?a \<bullet> x = 0"
wenzelm@53406
  2042
        apply (subst B')
wenzelm@53406
  2043
        using fB fth
huffman@44133
  2044
        unfolding setsum_clauses(2)[OF fth]
huffman@44133
  2045
        apply simp unfolding inner_simps
huffman@44527
  2046
        apply (clarsimp simp add: inner_add inner_setsum_left)
huffman@44133
  2047
        apply (rule setsum_0', rule ballI)
huffman@44133
  2048
        unfolding inner_commute
wenzelm@49711
  2049
        apply (auto simp add: x field_simps
wenzelm@49711
  2050
          intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
wenzelm@53406
  2051
        done
wenzelm@53406
  2052
    }
wenzelm@53406
  2053
    then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
wenzelm@53406
  2054
      by blast
huffman@44133
  2055
  qed
wenzelm@53406
  2056
  with a0 show ?thesis
wenzelm@53406
  2057
    unfolding sSB by (auto intro: exI[where x="?a"])
huffman@44133
  2058
qed
huffman@44133
  2059
huffman@44133
  2060
lemma span_not_univ_subset_hyperplane:
wenzelm@53406
  2061
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2062
  assumes SU: "span S \<noteq> UNIV"
huffman@44133
  2063
  shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
huffman@44133
  2064
  using span_not_univ_orthogonal[OF SU] by auto
huffman@44133
  2065
wenzelm@49663
  2066
lemma lowdim_subset_hyperplane:
wenzelm@53406
  2067
  fixes S :: "'a::euclidean_space set"
huffman@44133
  2068
  assumes d: "dim S < DIM('a)"
huffman@44133
  2069
  shows "\<exists>(a::'a). a  \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
wenzelm@49522
  2070
proof -
wenzelm@53406
  2071
  {
wenzelm@53406
  2072
    assume "span S = UNIV"
wenzelm@53406
  2073
    then have "dim (span S) = dim (UNIV :: ('a) set)"
wenzelm@53406
  2074
      by simp
wenzelm@53406
  2075
    then have "dim S = DIM('a)"
wenzelm@53406
  2076
      by (simp add: dim_span dim_UNIV)
wenzelm@53406
  2077
    with d have False by arith
wenzelm@53406
  2078
  }
wenzelm@53406
  2079
  then have th: "span S \<noteq> UNIV"
wenzelm@53406
  2080
    by blast
huffman@44133
  2081
  from span_not_univ_subset_hyperplane[OF th] show ?thesis .
huffman@44133
  2082
qed
huffman@44133
  2083
huffman@44133
  2084
text {* We can extend a linear basis-basis injection to the whole set. *}
huffman@44133
  2085
huffman@44133
  2086
lemma linear_indep_image_lemma:
wenzelm@49663
  2087
  assumes lf: "linear f"
wenzelm@49663
  2088
    and fB: "finite B"
wenzelm@49522
  2089
    and ifB: "independent (f ` B)"
wenzelm@49663
  2090
    and fi: "inj_on f B"
wenzelm@49663
  2091
    and xsB: "x \<in> span B"
wenzelm@49522
  2092
    and fx: "f x = 0"
huffman@44133
  2093
  shows "x = 0"
huffman@44133
  2094
  using fB ifB fi xsB fx
wenzelm@49522
  2095
proof (induct arbitrary: x rule: finite_induct[OF fB])
wenzelm@49663
  2096
  case 1
wenzelm@49663
  2097
  then show ?case by auto
huffman@44133
  2098
next
huffman@44133
  2099
  case (2 a b x)
huffman@44133
  2100
  have fb: "finite b" using "2.prems" by simp
huffman@44133
  2101
  have th0: "f ` b \<subseteq> f ` (insert a b)"
wenzelm@53406
  2102
    apply (rule image_mono)
wenzelm@53406
  2103
    apply blast
wenzelm@53406
  2104
    done
huffman@44133
  2105
  from independent_mono[ OF "2.prems"(2) th0]
huffman@44133
  2106
  have ifb: "independent (f ` b)"  .
huffman@44133
  2107
  have fib: "inj_on f b"
huffman@44133
  2108
    apply (rule subset_inj_on [OF "2.prems"(3)])
wenzelm@49522
  2109
    apply blast
wenzelm@49522
  2110
    done
huffman@44133
  2111
  from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
wenzelm@53406
  2112
  obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
wenzelm@53406
  2113
    by blast
huffman@44133
  2114
  have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
huffman@44133
  2115
    unfolding span_linear_image[OF lf]
huffman@44133
  2116
    apply (rule imageI)
wenzelm@53716
  2117
    using k span_mono[of "b - {a}" b]
wenzelm@53406
  2118
    apply blast
wenzelm@49522
  2119
    done
wenzelm@49522
  2120
  then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
huffman@44133
  2121
    by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
wenzelm@49522
  2122
  then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
huffman@44133
  2123
    using "2.prems"(5) by simp
wenzelm@53406
  2124
  have xsb: "x \<in> span b"
wenzelm@53406
  2125
  proof (cases "k = 0")
wenzelm@53406
  2126
    case True
wenzelm@53716
  2127
    with k have "x \<in> span (b - {a})" by simp
wenzelm@53716
  2128
    then show ?thesis using span_mono[of "b - {a}" b]
wenzelm@53406
  2129
      by blast
wenzelm@53406
  2130
  next
wenzelm@53406
  2131
    case False
wenzelm@53406
  2132
    with span_mul[OF th, of "- 1/ k"]
huffman@44133
  2133
    have th1: "f a \<in> span (f ` b)"
huffman@44133
  2134
      by auto
huffman@44133
  2135
    from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
huffman@44133
  2136
    have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
huffman@44133
  2137
    from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
huffman@44133
  2138
    have "f a \<notin> span (f ` b)" using tha
huffman@44133
  2139
      using "2.hyps"(2)
huffman@44133
  2140
      "2.prems"(3) by auto
huffman@44133
  2141
    with th1 have False by blast
wenzelm@53406
  2142
    then show ?thesis by blast
wenzelm@53406
  2143
  qed
wenzelm@53406
  2144
  from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
huffman@44133
  2145
qed
huffman@44133
  2146
huffman@44133
  2147
text {* We can extend a linear mapping from basis. *}
huffman@44133
  2148
huffman@44133
  2149
lemma linear_independent_extend_lemma:
huffman@44133
  2150
  fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
wenzelm@53406
  2151
  assumes fi: "finite B"
wenzelm@53406
  2152
    and ib: "independent B"
wenzelm@53406
  2153
  shows "\<exists>g.
wenzelm@53406
  2154
    (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y) \<and>
wenzelm@53406
  2155
    (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
wenzelm@53406
  2156
    (\<forall>x\<in> B. g x = f x)"
wenzelm@49663
  2157
  using ib fi
wenzelm@49522
  2158
proof (induct rule: finite_induct[OF fi])
wenzelm@49663
  2159
  case 1
wenzelm@49663
  2160
  then show ?case by auto
huffman@44133
  2161
next
huffman@44133
  2162
  case (2 a b)
huffman@44133
  2163
  from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
huffman@44133
  2164
    by (simp_all add: independent_insert)
huffman@44133
  2165
  from "2.hyps"(3)[OF ibf] obtain g where
huffman@44133
  2166
    g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
huffman@44133
  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
huffman@44133
  2168
  let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
wenzelm@53406
  2169
  {
wenzelm@53406
  2170
    fix z
wenzelm@53406
  2171
    assume z: "z \<in> span (insert a b)"
huffman@44133
  2172
    have th0: "z - ?h z *\<^sub>R a \<in> span b"
huffman@44133
  2173
      apply (rule someI_ex)
huffman@44133
  2174
      unfolding span_breakdown_eq[symmetric]
wenzelm@53406
  2175
      apply (rule z)
wenzelm@53406
  2176
      done
wenzelm@53406
  2177
    {
wenzelm@53406
  2178
      fix k
wenzelm@53406
  2179
      assume k: "z - k *\<^sub>R a \<in> span b"
huffman@44133
  2180
      have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
huffman@44133
  2181
        by (simp add: field_simps scaleR_left_distrib [symmetric])
wenzelm@53406
  2182
      from span_sub[OF th0 k] have khz: "(k - ?h z) *\<^sub>R a \<in> span b"
wenzelm@53406
  2183
        by (simp add: eq)
wenzelm@53406
  2184
      {
wenzelm@53406
  2185
        assume "k \<noteq> ?h z"
wenzelm@53406
  2186
        then have k0: "k - ?h z \<noteq> 0" by simp
huffman@44133
  2187
        from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
huffman@44133
  2188
        have "a \<in> span b" by simp
huffman@44133
  2189
        with "2.prems"(1) "2.hyps"(2) have False
wenzelm@53406
  2190
          by (auto simp add: dependent_def)
wenzelm@53406
  2191
      }
wenzelm@53406
  2192
      then have "k = ?h z" by blast
wenzelm@53406
  2193
    }
wenzelm@53406
  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)"
wenzelm@53406
  2195
      by blast
wenzelm@53406
  2196
  }
huffman@44133
  2197
  note h = this
huffman@44133
  2198
  let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)"
wenzelm@53406
  2199
  {
wenzelm@53406
  2200
    fix x y
wenzelm@53406
  2201
    assume x: "x \<in> span (insert a b)"
wenzelm@53406
  2202
      and y: "y \<in> span (insert a b)"
huffman@44133
  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)"
huffman@44133
  2204
      by (simp add: algebra_simps)
huffman@44133
  2205
    have addh: "?h (x + y) = ?h x + ?h y"
huffman@44133
  2206
      apply (rule conjunct2[OF h, rule_format, symmetric])
huffman@44133
  2207
      apply (rule span_add[OF x y])
huffman@44133
  2208
      unfolding tha
wenzelm@53406
  2209
      apply (metis span_add x y conjunct1[OF h, rule_format])
wenzelm@53406
  2210
      done
huffman@44133
  2211
    have "?g (x + y) = ?g x + ?g y"
huffman@44133
  2212
      unfolding addh tha
huffman@44133
  2213
      g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
huffman@44133
  2214
      by (simp add: scaleR_left_distrib)}
huffman@44133
  2215
  moreover
wenzelm@53406
  2216
  {
wenzelm@53406
  2217
    fix x :: "'a"
wenzelm@53406
  2218
    fix c :: real
wenzelm@49522
  2219
    assume x: "x \<in> span (insert a b)"
huffman@44133
  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)"
huffman@44133
  2221
      by (simp add: algebra_simps)
huffman@44133
  2222
    have hc: "?h (c *\<^sub>R x) = c * ?h x"
huffman@44133
  2223
      apply (rule conjunct2[OF h, rule_format, symmetric])
huffman@44133
  2224
      apply (metis span_mul x)
wenzelm@49522
  2225
      apply (metis tha span_mul x conjunct1[OF h])
wenzelm@49522
  2226
      done
huffman@44133
  2227
    have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
huffman@44133
  2228
      unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
wenzelm@53406
  2229
      by (simp add: algebra_simps)
wenzelm@53406
  2230
  }
huffman@44133
  2231
  moreover
wenzelm@53406
  2232
  {
wenzelm@53406
  2233
    fix x
wenzelm@53406
  2234
    assume x: "x \<in> insert a b"
wenzelm@53406
  2235
    {
wenzelm@53406
  2236
      assume xa: "x = a"
huffman@44133
  2237
      have ha1: "1 = ?h a"
huffman@44133
  2238
        apply (rule conjunct2[OF h, rule_format])
huffman@44133
  2239
        apply (metis span_superset insertI1)
huffman@44133
  2240
        using conjunct1[OF h, OF span_superset, OF insertI1]
wenzelm@49522
  2241
        apply (auto simp add: span_0)
wenzelm@49522
  2242
        done
huffman@44133
  2243
      from xa ha1[symmetric] have "?g x = f x"
huffman@44133
  2244
        apply simp
huffman@44133
  2245
        using g(2)[rule_format, OF span_0, of 0]
wenzelm@49522
  2246
        apply simp
wenzelm@53406
  2247
        done
wenzelm@53406
  2248
    }
huffman@44133
  2249
    moreover
wenzelm@53406
  2250
    {
wenzelm@53406
  2251
      assume xb: "x \<in> b"
huffman@44133
  2252
      have h0: "0 = ?h x"
huffman@44133
  2253
        apply (rule conjunct2[OF h, rule_format])
huffman@44133
  2254
        apply (metis  span_superset x)
huffman@44133
  2255
        apply simp
huffman@44133
  2256
        apply (metis span_superset xb)
huffman@44133
  2257
        done
huffman@44133
  2258
      have "?g x = f x"
wenzelm@53406
  2259
        by (simp add: h0[symmetric] g(3)[rule_format, OF xb])
wenzelm@53406
  2260
    }
wenzelm@53406
  2261
    ultimately have "?g x = f x"
wenzelm@53406
  2262
      using x by blast
wenzelm@53406
  2263
  }
wenzelm@49663
  2264
  ultimately show ?case
wenzelm@49663
  2265
    apply -
wenzelm@49663
  2266
    apply (rule exI[where x="?g"])
wenzelm@49663
  2267
    apply blast
wenzelm@49663
  2268
    done
huffman@44133
  2269
qed
huffman@44133
  2270
huffman@44133
  2271
lemma linear_independent_extend:
wenzelm@53406
  2272
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2273
  assumes iB: "independent B"
huffman@44133
  2274
  shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
wenzelm@49522
  2275
proof -
huffman@44133
  2276
  from maximal_independent_subset_extend[of B UNIV] iB
wenzelm@53406
  2277
  obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C"
wenzelm@53406
  2278
    by auto
huffman@44133
  2279
huffman@44133
  2280
  from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
wenzelm@53406
  2281
  obtain g where g:
wenzelm@53406
  2282
    "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) \<and>
wenzelm@53406
  2283
     (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
wenzelm@53406
  2284
     (\<forall>x\<in> C. g x = f x)" by blast
wenzelm@53406
  2285
  from g show ?thesis
huffman@53600
  2286
    unfolding linear_iff
wenzelm@53406
  2287
    using C
wenzelm@49663
  2288
    apply clarsimp
wenzelm@49663
  2289
    apply blast
wenzelm@49663
  2290
    done
huffman@44133
  2291
qed
huffman@44133
  2292
huffman@44133
  2293
text {* Can construct an isomorphism between spaces of same dimension. *}
huffman@44133
  2294
huffman@44133
  2295
lemma subspace_isomorphism:
wenzelm@53406
  2296
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2297
    and T :: "'b::euclidean_space set"
wenzelm@53406
  2298
  assumes s: "subspace S"
wenzelm@53406
  2299
    and t: "subspace T"
wenzelm@49522
  2300
    and d: "dim S = dim T"
huffman@44133
  2301
  shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
wenzelm@49522
  2302
proof -
wenzelm@53406
  2303
  from basis_exists[of S] independent_bound
wenzelm@53406
  2304
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
wenzelm@53406
  2305
    by blast
wenzelm@53406
  2306
  from basis_exists[of T] independent_bound
wenzelm@53406
  2307
  obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C"
wenzelm@53406
  2308
    by blast
wenzelm@53406
  2309
  from B(4) C(4) card_le_inj[of B C] d
wenzelm@53406
  2310
  obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C`
wenzelm@53406
  2311
    by auto
wenzelm@53406
  2312
  from linear_independent_extend[OF B(2)]
wenzelm@53406
  2313
  obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x"
wenzelm@53406
  2314
    by blast
wenzelm@53406
  2315
  from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
huffman@44133
  2316
    by simp
wenzelm@53406
  2317
  with B(4) C(4) have ceq: "card (f ` B) = card C"
wenzelm@53406
  2318
    using d by simp
wenzelm@53406
  2319
  have "g ` B = f ` B"
wenzelm@53406
  2320
    using g(2) by (auto simp add: image_iff)
huffman@44133
  2321
  also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
huffman@44133
  2322
  finally have gBC: "g ` B = C" .
wenzelm@53406
  2323
  have gi: "inj_on g B"
wenzelm@53406
  2324
    using f(2) g(2) by (auto simp add: inj_on_def)
huffman@44133
  2325
  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
wenzelm@53406
  2326
  {
wenzelm@53406
  2327
    fix x y
wenzelm@53406
  2328
    assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
wenzelm@53406
  2329
    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
wenzelm@53406
  2330
      by blast+
wenzelm@53406
  2331
    from gxy have th0: "g (x - y) = 0"
wenzelm@53406
  2332
      by (simp add: linear_sub[OF g(1)])
wenzelm@53406
  2333
    have th1: "x - y \<in> span B"
wenzelm@53406
  2334
      using x' y' by (metis span_sub)
wenzelm@53406
  2335
    have "x = y"
wenzelm@53406
  2336
      using g0[OF th1 th0] by simp
wenzelm@53406
  2337
  }
huffman@44133
  2338
  then have giS: "inj_on g S"
huffman@44133
  2339
    unfolding inj_on_def by blast
wenzelm@53406
  2340
  from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)"
wenzelm@53406
  2341
    by (simp add: span_linear_image[OF g(1)])
huffman@44133
  2342
  also have "\<dots> = span C" unfolding gBC ..
huffman@44133
  2343
  also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
huffman@44133
  2344
  finally have gS: "g ` S = T" .
wenzelm@53406
  2345
  from g(1) gS giS show ?thesis
wenzelm@53406
  2346
    by blast
huffman@44133
  2347
qed
huffman@44133
  2348
huffman@44133
  2349
text {* Linear functions are equal on a subspace if they are on a spanning set. *}
huffman@44133
  2350
huffman@44133
  2351
lemma subspace_kernel:
huffman@44133
  2352
  assumes lf: "linear f"
huffman@44133
  2353
  shows "subspace {x. f x = 0}"
wenzelm@49522
  2354
  apply (simp add: subspace_def)
wenzelm@49522
  2355
  apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
wenzelm@49522
  2356
  done
huffman@44133
  2357
huffman@44133
  2358
lemma linear_eq_0_span:
huffman@44133
  2359
  assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
huffman@44133
  2360
  shows "\<forall>x \<in> span B. f x = 0"
huffman@44170
  2361
  using f0 subspace_kernel[OF lf]
huffman@44170
  2362
  by (rule span_induct')
huffman@44133
  2363
huffman@44133
  2364
lemma linear_eq_0:
wenzelm@49663
  2365
  assumes lf: "linear f"
wenzelm@49663
  2366
    and SB: "S \<subseteq> span B"
wenzelm@49663
  2367
    and f0: "\<forall>x\<in>B. f x = 0"
huffman@44133
  2368
  shows "\<forall>x \<in> S. f x = 0"
huffman@44133
  2369
  by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
huffman@44133
  2370
huffman@44133
  2371
lemma linear_eq:
wenzelm@49663
  2372
  assumes lf: "linear f"
wenzelm@49663
  2373
    and lg: "linear g"
wenzelm@49663
  2374
    and S: "S \<subseteq> span B"
wenzelm@49522
  2375
    and fg: "\<forall> x\<in> B. f x = g x"
huffman@44133
  2376
  shows "\<forall>x\<in> S. f x = g x"
wenzelm@49663
  2377
proof -
huffman@44133
  2378
  let ?h = "\<lambda>x. f x - g x"
huffman@44133
  2379
  from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
huffman@44133
  2380
  from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
huffman@44133
  2381
  show ?thesis by simp
huffman@44133
  2382
qed
huffman@44133
  2383
huffman@44133
  2384
lemma linear_eq_stdbasis:
wenzelm@49663
  2385
  assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> _)"
wenzelm@49663
  2386
    and lg: "linear g"
hoelzl@50526
  2387
    and fg: "\<forall>b\<in>Basis. f b = g b"
huffman@44133
  2388
  shows "f = g"
hoelzl@50526
  2389
  using linear_eq[OF lf lg, of _ Basis] fg by auto
huffman@44133
  2390
huffman@44133
  2391
text {* Similar results for bilinear functions. *}
huffman@44133
  2392
huffman@44133
  2393
lemma bilinear_eq:
huffman@44133
  2394
  assumes bf: "bilinear f"
wenzelm@49522
  2395
    and bg: "bilinear g"
wenzelm@53406
  2396
    and SB: "S \<subseteq> span B"
wenzelm@53406
  2397
    and TC: "T \<subseteq> span C"
wenzelm@49522
  2398
    and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
huffman@44133
  2399
  shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
wenzelm@49663
  2400
proof -
huffman@44170
  2401
  let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
huffman@44133
  2402
  from bf bg have sp: "subspace ?P"
huffman@53600
  2403
    unfolding bilinear_def linear_iff subspace_def bf bg
wenzelm@49663
  2404
    by (auto simp add: span_0 bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def
wenzelm@49663
  2405
      intro: bilinear_ladd[OF bf])
huffman@44133
  2406
huffman@44133
  2407
  have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
huffman@44170
  2408
    apply (rule span_induct' [OF _ sp])
huffman@44133
  2409
    apply (rule ballI)
huffman@44170
  2410
    apply (rule span_induct')
huffman@44170
  2411
    apply (simp add: fg)
huffman@44133
  2412
    apply (auto simp add: subspace_def)
huffman@53600
  2413
    using bf bg unfolding bilinear_def linear_iff
wenzelm@49522
  2414
    apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
wenzelm@49663
  2415
      intro: bilinear_ladd[OF bf])
wenzelm@49522
  2416
    done
wenzelm@53406
  2417
  then show ?thesis
wenzelm@53406
  2418
    using SB TC by auto
huffman@44133
  2419
qed
huffman@44133
  2420
wenzelm@49522
  2421
lemma bilinear_eq_stdbasis:
wenzelm@53406
  2422
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
huffman@44133
  2423
  assumes bf: "bilinear f"
wenzelm@49522
  2424
    and bg: "bilinear g"
hoelzl@50526
  2425
    and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
huffman@44133
  2426
  shows "f = g"
hoelzl@50526
  2427
  using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast
huffman@44133
  2428
huffman@44133
  2429
text {* Detailed theorems about left and right invertibility in general case. *}
huffman@44133
  2430
wenzelm@49522
  2431
lemma linear_injective_left_inverse:
wenzelm@53406
  2432
  fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
huffman@44133
  2433
  assumes lf: "linear f" and fi: "inj f"
huffman@44133
  2434
  shows "\<exists>g. linear g \<and> g o f = id"
wenzelm@49522
  2435
proof -
hoelzl@50526
  2436
  from linear_independent_extend[OF independent_injective_image, OF independent_Basis, OF lf fi]
wenzelm@53406
  2437
  obtain h:: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x"
wenzelm@53406
  2438
    by blast
hoelzl@50526
  2439
  from h(2) have th: "\<forall>i\<in>Basis. (h \<circ> f) i = id i"
huffman@44133
  2440
    using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
huffman@44133
  2441
    by auto
huffman@44133
  2442
  from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
huffman@44133
  2443
  have "h o f = id" .
wenzelm@53406
  2444
  then show ?thesis
wenzelm@53406
  2445
    using h(1) by blast
huffman@44133
  2446
qed
huffman@44133
  2447
wenzelm@49522
  2448
lemma linear_surjective_right_inverse:
wenzelm@53406
  2449
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
wenzelm@53406
  2450
  assumes lf: "linear f"
wenzelm@53406
  2451
    and sf: "surj f"
huffman@44133
  2452
  shows "\<exists>g. linear g \<and> f o g = id"
wenzelm@49522
  2453
proof -
hoelzl@50526
  2454
  from linear_independent_extend[OF independent_Basis[where 'a='b],of "inv f"]
wenzelm@53406
  2455
  obtain h:: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x\<in>Basis. h x = inv f x"
wenzelm@53406
  2456
    by blast
wenzelm@53406
  2457
  from h(2) have th: "\<forall>i\<in>Basis. (f o h) i = id i"
hoelzl@50526
  2458
    using sf by (auto simp add: surj_iff_all)
huffman@44133
  2459
  from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
huffman@44133
  2460
  have "f o h = id" .