src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
author webertj
Fri Oct 19 15:12:52 2012 +0200 (2012-10-19)
changeset 49962 a8cc904a6820
parent 49644 343bfcbad2ec
child 50526 899c9c4e4a4c
permissions -rw-r--r--
Renamed {left,right}_distrib to distrib_{right,left}.
hoelzl@37489
     1
hoelzl@37489
     2
header {*Instanciates the finite cartesian product of euclidean spaces as a euclidean space.*}
hoelzl@37489
     3
hoelzl@37489
     4
theory Cartesian_Euclidean_Space
hoelzl@37489
     5
imports Finite_Cartesian_Product Integration
hoelzl@37489
     6
begin
hoelzl@37489
     7
hoelzl@37489
     8
lemma delta_mult_idempotent:
wenzelm@49644
     9
  "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)"
wenzelm@49644
    10
  by (cases "k=a") auto
hoelzl@37489
    11
hoelzl@37489
    12
lemma setsum_Plus:
hoelzl@37489
    13
  "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
hoelzl@37489
    14
    (\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))"
hoelzl@37489
    15
  unfolding Plus_def
hoelzl@37489
    16
  by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)
hoelzl@37489
    17
hoelzl@37489
    18
lemma setsum_UNIV_sum:
hoelzl@37489
    19
  fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
hoelzl@37489
    20
  shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
hoelzl@37489
    21
  apply (subst UNIV_Plus_UNIV [symmetric])
hoelzl@37489
    22
  apply (rule setsum_Plus [OF finite finite])
hoelzl@37489
    23
  done
hoelzl@37489
    24
hoelzl@37489
    25
lemma setsum_mult_product:
hoelzl@37489
    26
  "setsum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
hoelzl@37489
    27
  unfolding sumr_group[of h B A, unfolded atLeast0LessThan, symmetric]
hoelzl@37489
    28
proof (rule setsum_cong, simp, rule setsum_reindex_cong)
wenzelm@49644
    29
  fix i
wenzelm@49644
    30
  show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
hoelzl@37489
    31
  show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
hoelzl@37489
    32
  proof safe
hoelzl@37489
    33
    fix j assume "j \<in> {i * B..<i * B + B}"
wenzelm@49644
    34
    then show "j \<in> (\<lambda>j. j + i * B) ` {..<B}"
hoelzl@37489
    35
      by (auto intro!: image_eqI[of _ _ "j - i * B"])
hoelzl@37489
    36
  qed simp
hoelzl@37489
    37
qed simp
hoelzl@37489
    38
wenzelm@49644
    39
hoelzl@37489
    40
subsection{* Basic componentwise operations on vectors. *}
hoelzl@37489
    41
huffman@44136
    42
instantiation vec :: (times, finite) times
hoelzl@37489
    43
begin
wenzelm@49644
    44
wenzelm@49644
    45
definition "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
wenzelm@49644
    46
instance ..
wenzelm@49644
    47
hoelzl@37489
    48
end
hoelzl@37489
    49
huffman@44136
    50
instantiation vec :: (one, finite) one
hoelzl@37489
    51
begin
wenzelm@49644
    52
wenzelm@49644
    53
definition "1 \<equiv> (\<chi> i. 1)"
wenzelm@49644
    54
instance ..
wenzelm@49644
    55
hoelzl@37489
    56
end
hoelzl@37489
    57
huffman@44136
    58
instantiation vec :: (ord, finite) ord
hoelzl@37489
    59
begin
wenzelm@49644
    60
wenzelm@49644
    61
definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"
wenzelm@49644
    62
definition "x < y \<longleftrightarrow> (\<forall>i. x$i < y$i)"
wenzelm@49644
    63
instance ..
wenzelm@49644
    64
hoelzl@37489
    65
end
hoelzl@37489
    66
hoelzl@37489
    67
text{* The ordering on one-dimensional vectors is linear. *}
hoelzl@37489
    68
wenzelm@49197
    69
class cart_one =
wenzelm@49197
    70
  assumes UNIV_one: "card (UNIV \<Colon> 'a set) = Suc 0"
hoelzl@37489
    71
begin
wenzelm@49197
    72
wenzelm@49197
    73
subclass finite
wenzelm@49197
    74
proof
wenzelm@49197
    75
  from UNIV_one show "finite (UNIV :: 'a set)"
wenzelm@49197
    76
    by (auto intro!: card_ge_0_finite)
wenzelm@49197
    77
qed
wenzelm@49197
    78
hoelzl@37489
    79
end
hoelzl@37489
    80
wenzelm@49197
    81
instantiation vec :: (linorder, cart_one) linorder
wenzelm@49197
    82
begin
wenzelm@49197
    83
wenzelm@49197
    84
instance
wenzelm@49197
    85
proof
wenzelm@49197
    86
  obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"
wenzelm@49197
    87
  proof -
wenzelm@49197
    88
    have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)
wenzelm@49197
    89
    then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)
wenzelm@49197
    90
    then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto
wenzelm@49197
    91
    then show thesis by (auto intro: that)
wenzelm@49197
    92
  qed
wenzelm@49197
    93
wenzelm@49197
    94
  note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
wenzelm@49197
    95
  fix x y z :: "'a^'b::cart_one"
wenzelm@49197
    96
  show "x \<le> x" "(x < y) = (x \<le> y \<and> \<not> y \<le> x)" "x \<le> y \<or> y \<le> x" by auto
wenzelm@49197
    97
  { assume "x\<le>y" "y\<le>z" then show "x\<le>z" by auto }
wenzelm@49197
    98
  { assume "x\<le>y" "y\<le>x" then show "x=y" by auto }
wenzelm@49197
    99
qed
wenzelm@49197
   100
wenzelm@49197
   101
end
hoelzl@37489
   102
hoelzl@37489
   103
text{* Constant Vectors *} 
hoelzl@37489
   104
hoelzl@37489
   105
definition "vec x = (\<chi> i. x)"
hoelzl@37489
   106
hoelzl@37489
   107
text{* Also the scalar-vector multiplication. *}
hoelzl@37489
   108
hoelzl@37489
   109
definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
hoelzl@37489
   110
  where "c *s x = (\<chi> i. c * (x$i))"
hoelzl@37489
   111
wenzelm@49644
   112
hoelzl@37489
   113
subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
hoelzl@37489
   114
hoelzl@37489
   115
method_setup vector = {*
hoelzl@37489
   116
let
hoelzl@37489
   117
  val ss1 = HOL_basic_ss addsimps [@{thm setsum_addf} RS sym,
wenzelm@49644
   118
    @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
wenzelm@49644
   119
    @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
hoelzl@37489
   120
  val ss2 = @{simpset} addsimps
huffman@44136
   121
             [@{thm plus_vec_def}, @{thm times_vec_def},
huffman@44136
   122
              @{thm minus_vec_def}, @{thm uminus_vec_def},
huffman@44136
   123
              @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
huffman@44136
   124
              @{thm scaleR_vec_def},
huffman@44136
   125
              @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}]
wenzelm@49644
   126
  fun vector_arith_tac ths =
wenzelm@49644
   127
    simp_tac ss1
wenzelm@49644
   128
    THEN' (fn i => rtac @{thm setsum_cong2} i
hoelzl@37489
   129
         ORELSE rtac @{thm setsum_0'} i
huffman@44136
   130
         ORELSE simp_tac (HOL_basic_ss addsimps [@{thm vec_eq_iff}]) i)
wenzelm@49644
   131
    (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
wenzelm@49644
   132
    THEN' asm_full_simp_tac (ss2 addsimps ths)
wenzelm@49644
   133
in
hoelzl@37489
   134
  Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths)))
wenzelm@49644
   135
end
wenzelm@42814
   136
*} "lift trivial vector statements to real arith statements"
hoelzl@37489
   137
huffman@44136
   138
lemma vec_0[simp]: "vec 0 = 0" by (vector zero_vec_def)
huffman@44136
   139
lemma vec_1[simp]: "vec 1 = 1" by (vector one_vec_def)
hoelzl@37489
   140
hoelzl@37489
   141
lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
hoelzl@37489
   142
hoelzl@37489
   143
lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
hoelzl@37489
   144
hoelzl@37489
   145
lemma vec_add: "vec(x + y) = vec x + vec y"  by (vector vec_def)
hoelzl@37489
   146
lemma vec_sub: "vec(x - y) = vec x - vec y" by (vector vec_def)
hoelzl@37489
   147
lemma vec_cmul: "vec(c * x) = c *s vec x " by (vector vec_def)
hoelzl@37489
   148
lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def)
hoelzl@37489
   149
wenzelm@49644
   150
lemma vec_setsum:
wenzelm@49644
   151
  assumes "finite S"
hoelzl@37489
   152
  shows "vec(setsum f S) = setsum (vec o f) S"
wenzelm@49644
   153
  using assms
wenzelm@49644
   154
proof induct
wenzelm@49644
   155
  case empty
wenzelm@49644
   156
  then show ?case by simp
wenzelm@49644
   157
next
wenzelm@49644
   158
  case insert
wenzelm@49644
   159
  then show ?case by (auto simp add: vec_add)
wenzelm@49644
   160
qed
hoelzl@37489
   161
hoelzl@37489
   162
text{* Obvious "component-pushing". *}
hoelzl@37489
   163
hoelzl@37489
   164
lemma vec_component [simp]: "vec x $ i = x"
hoelzl@37489
   165
  by (vector vec_def)
hoelzl@37489
   166
hoelzl@37489
   167
lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
hoelzl@37489
   168
  by vector
hoelzl@37489
   169
hoelzl@37489
   170
lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
hoelzl@37489
   171
  by vector
hoelzl@37489
   172
hoelzl@37489
   173
lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
hoelzl@37489
   174
hoelzl@37489
   175
lemmas vector_component =
hoelzl@37489
   176
  vec_component vector_add_component vector_mult_component
hoelzl@37489
   177
  vector_smult_component vector_minus_component vector_uminus_component
hoelzl@37489
   178
  vector_scaleR_component cond_component
hoelzl@37489
   179
wenzelm@49644
   180
hoelzl@37489
   181
subsection {* Some frequently useful arithmetic lemmas over vectors. *}
hoelzl@37489
   182
huffman@44136
   183
instance vec :: (semigroup_mult, finite) semigroup_mult
huffman@44136
   184
  by default (vector mult_assoc)
hoelzl@37489
   185
huffman@44136
   186
instance vec :: (monoid_mult, finite) monoid_mult
huffman@44136
   187
  by default vector+
hoelzl@37489
   188
huffman@44136
   189
instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
huffman@44136
   190
  by default (vector mult_commute)
hoelzl@37489
   191
huffman@44136
   192
instance vec :: (ab_semigroup_idem_mult, finite) ab_semigroup_idem_mult
huffman@44136
   193
  by default (vector mult_idem)
hoelzl@37489
   194
huffman@44136
   195
instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
huffman@44136
   196
  by default vector
hoelzl@37489
   197
huffman@44136
   198
instance vec :: (semiring, finite) semiring
huffman@44136
   199
  by default (vector field_simps)+
hoelzl@37489
   200
huffman@44136
   201
instance vec :: (semiring_0, finite) semiring_0
huffman@44136
   202
  by default (vector field_simps)+
huffman@44136
   203
instance vec :: (semiring_1, finite) semiring_1
huffman@44136
   204
  by default vector
huffman@44136
   205
instance vec :: (comm_semiring, finite) comm_semiring
huffman@44136
   206
  by default (vector field_simps)+
hoelzl@37489
   207
huffman@44136
   208
instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
huffman@44136
   209
instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
huffman@44136
   210
instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
huffman@44136
   211
instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
huffman@44136
   212
instance vec :: (ring, finite) ring ..
huffman@44136
   213
instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
huffman@44136
   214
instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
hoelzl@37489
   215
huffman@44136
   216
instance vec :: (ring_1, finite) ring_1 ..
hoelzl@37489
   217
huffman@44136
   218
instance vec :: (real_algebra, finite) real_algebra
wenzelm@49644
   219
  by default (simp_all add: vec_eq_iff)
hoelzl@37489
   220
huffman@44136
   221
instance vec :: (real_algebra_1, finite) real_algebra_1 ..
hoelzl@37489
   222
wenzelm@49644
   223
lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
wenzelm@49644
   224
proof (induct n)
wenzelm@49644
   225
  case 0
wenzelm@49644
   226
  then show ?case by vector
wenzelm@49644
   227
next
wenzelm@49644
   228
  case Suc
wenzelm@49644
   229
  then show ?case by vector
wenzelm@49644
   230
qed
hoelzl@37489
   231
wenzelm@49644
   232
lemma one_index[simp]: "(1 :: 'a::one ^'n)$i = 1"
wenzelm@49644
   233
  by vector
hoelzl@37489
   234
huffman@44136
   235
instance vec :: (semiring_char_0, finite) semiring_char_0
haftmann@38621
   236
proof
haftmann@38621
   237
  fix m n :: nat
haftmann@38621
   238
  show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
huffman@44136
   239
    by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
hoelzl@37489
   240
qed
hoelzl@37489
   241
huffman@47108
   242
instance vec :: (numeral, finite) numeral ..
huffman@47108
   243
instance vec :: (semiring_numeral, finite) semiring_numeral ..
huffman@47108
   244
huffman@47108
   245
lemma numeral_index [simp]: "numeral w $ i = numeral w"
wenzelm@49644
   246
  by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
huffman@47108
   247
huffman@47108
   248
lemma neg_numeral_index [simp]: "neg_numeral w $ i = neg_numeral w"
huffman@47108
   249
  by (simp only: neg_numeral_def vector_uminus_component numeral_index)
huffman@47108
   250
huffman@44136
   251
instance vec :: (comm_ring_1, finite) comm_ring_1 ..
huffman@44136
   252
instance vec :: (ring_char_0, finite) ring_char_0 ..
hoelzl@37489
   253
hoelzl@37489
   254
lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
hoelzl@37489
   255
  by (vector mult_assoc)
hoelzl@37489
   256
lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
hoelzl@37489
   257
  by (vector field_simps)
hoelzl@37489
   258
lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
hoelzl@37489
   259
  by (vector field_simps)
hoelzl@37489
   260
lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
hoelzl@37489
   261
lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
hoelzl@37489
   262
lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
hoelzl@37489
   263
  by (vector field_simps)
hoelzl@37489
   264
lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
hoelzl@37489
   265
lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
huffman@47108
   266
lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
hoelzl@37489
   267
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
hoelzl@37489
   268
lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
hoelzl@37489
   269
  by (vector field_simps)
hoelzl@37489
   270
hoelzl@37489
   271
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
huffman@44136
   272
  by (simp add: vec_eq_iff)
hoelzl@37489
   273
hoelzl@37489
   274
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
hoelzl@37489
   275
lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
hoelzl@37489
   276
  by vector
hoelzl@37489
   277
lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
hoelzl@37489
   278
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
hoelzl@37489
   279
lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
hoelzl@37489
   280
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
hoelzl@37489
   281
lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
hoelzl@37489
   282
  by (metis vector_mul_lcancel)
hoelzl@37489
   283
lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
hoelzl@37489
   284
  by (metis vector_mul_rcancel)
hoelzl@37489
   285
hoelzl@37489
   286
lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x"
huffman@44136
   287
  apply (simp add: norm_vec_def)
hoelzl@37489
   288
  apply (rule member_le_setL2, simp_all)
hoelzl@37489
   289
  done
hoelzl@37489
   290
hoelzl@37489
   291
lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x$i\<bar> <= e"
hoelzl@37489
   292
  by (metis component_le_norm_cart order_trans)
hoelzl@37489
   293
hoelzl@37489
   294
lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
hoelzl@37489
   295
  by (metis component_le_norm_cart basic_trans_rules(21))
hoelzl@37489
   296
hoelzl@37489
   297
lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
huffman@44136
   298
  by (simp add: norm_vec_def setL2_le_setsum)
hoelzl@37489
   299
hoelzl@37489
   300
lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
huffman@44136
   301
  unfolding scaleR_vec_def vector_scalar_mult_def by simp
hoelzl@37489
   302
hoelzl@37489
   303
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
hoelzl@37489
   304
  unfolding dist_norm scalar_mult_eq_scaleR
hoelzl@37489
   305
  unfolding scaleR_right_diff_distrib[symmetric] by simp
hoelzl@37489
   306
hoelzl@37489
   307
lemma setsum_component [simp]:
hoelzl@37489
   308
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
hoelzl@37489
   309
  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
wenzelm@49644
   310
proof (cases "finite S")
wenzelm@49644
   311
  case True
wenzelm@49644
   312
  then show ?thesis by induct simp_all
wenzelm@49644
   313
next
wenzelm@49644
   314
  case False
wenzelm@49644
   315
  then show ?thesis by simp
wenzelm@49644
   316
qed
hoelzl@37489
   317
hoelzl@37489
   318
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
huffman@44136
   319
  by (simp add: vec_eq_iff)
hoelzl@37489
   320
hoelzl@37489
   321
lemma setsum_cmul:
hoelzl@37489
   322
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
hoelzl@37489
   323
  shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
huffman@44136
   324
  by (simp add: vec_eq_iff setsum_right_distrib)
hoelzl@37489
   325
hoelzl@37489
   326
(* TODO: use setsum_norm_allsubsets_bound *)
hoelzl@37489
   327
lemma setsum_norm_allsubsets_bound_cart:
hoelzl@37489
   328
  fixes f:: "'a \<Rightarrow> real ^'n"
hoelzl@37489
   329
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
hoelzl@37489
   330
  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
wenzelm@49644
   331
proof -
hoelzl@37489
   332
  let ?d = "real CARD('n)"
hoelzl@37489
   333
  let ?nf = "\<lambda>x. norm (f x)"
hoelzl@37489
   334
  let ?U = "UNIV :: 'n set"
hoelzl@37489
   335
  have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x $ i\<bar>) P) ?U"
hoelzl@37489
   336
    by (rule setsum_commute)
hoelzl@37489
   337
  have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
hoelzl@37489
   338
  have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
wenzelm@49644
   339
    apply (rule setsum_mono)
wenzelm@49644
   340
    apply (rule norm_le_l1_cart)
wenzelm@49644
   341
    done
hoelzl@37489
   342
  also have "\<dots> \<le> 2 * ?d * e"
hoelzl@37489
   343
    unfolding th0 th1
hoelzl@37489
   344
  proof(rule setsum_bounded)
hoelzl@37489
   345
    fix i assume i: "i \<in> ?U"
hoelzl@37489
   346
    let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
hoelzl@37489
   347
    let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
hoelzl@37489
   348
    have thp: "P = ?Pp \<union> ?Pn" by auto
hoelzl@37489
   349
    have thp0: "?Pp \<inter> ?Pn ={}" by auto
hoelzl@37489
   350
    have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
hoelzl@37489
   351
    have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
hoelzl@37489
   352
      using component_le_norm_cart[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
hoelzl@37489
   353
      by (auto intro: abs_le_D1)
hoelzl@37489
   354
    have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
hoelzl@37489
   355
      using component_le_norm_cart[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
hoelzl@37489
   356
      by (auto simp add: setsum_negf intro: abs_le_D1)
hoelzl@37489
   357
    have "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn"
hoelzl@37489
   358
      apply (subst thp)
hoelzl@37489
   359
      apply (rule setsum_Un_zero)
wenzelm@49644
   360
      using fP thp0 apply auto
wenzelm@49644
   361
      done
hoelzl@37489
   362
    also have "\<dots> \<le> 2*e" using Pne Ppe by arith
hoelzl@37489
   363
    finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
hoelzl@37489
   364
  qed
hoelzl@37489
   365
  finally show ?thesis .
hoelzl@37489
   366
qed
hoelzl@37489
   367
hoelzl@37489
   368
lemma if_distr: "(if P then f else g) $ i = (if P then f $ i else g $ i)" by simp
hoelzl@37489
   369
hoelzl@37489
   370
lemma split_dimensions'[consumes 1]:
huffman@44129
   371
  assumes "k < DIM('a::euclidean_space^'b)"
wenzelm@49644
   372
  obtains i j where "i < CARD('b::finite)"
wenzelm@49644
   373
    and "j < DIM('a::euclidean_space)"
wenzelm@49644
   374
    and "k = j + i * DIM('a::euclidean_space)"
wenzelm@49644
   375
  using split_times_into_modulo[OF assms[simplified]] .
hoelzl@37489
   376
hoelzl@37489
   377
lemma cart_euclidean_bound[intro]:
huffman@44129
   378
  assumes j:"j < DIM('a::euclidean_space)"
huffman@44129
   379
  shows "j + \<pi>' (i::'b::finite) * DIM('a) < CARD('b) * DIM('a::euclidean_space)"
hoelzl@37489
   380
  using linear_less_than_times[OF pi'_range j, of i] .
hoelzl@37489
   381
huffman@44129
   382
lemma (in euclidean_space) forall_CARD_DIM:
hoelzl@37489
   383
  "(\<forall>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<forall>(i::'b::finite) j. j<DIM('a) \<longrightarrow> P (j + \<pi>' i * DIM('a)))"
hoelzl@37489
   384
   (is "?l \<longleftrightarrow> ?r")
hoelzl@37489
   385
proof (safe elim!: split_times_into_modulo)
wenzelm@49644
   386
  fix i :: 'b and j
wenzelm@49644
   387
  assume "j < DIM('a)"
hoelzl@37489
   388
  note linear_less_than_times[OF pi'_range[of i] this]
hoelzl@37489
   389
  moreover assume "?l"
hoelzl@37489
   390
  ultimately show "P (j + \<pi>' i * DIM('a))" by auto
hoelzl@37489
   391
next
wenzelm@49644
   392
  fix i j
wenzelm@49644
   393
  assume "i < CARD('b)" "j < DIM('a)" and "?r"
hoelzl@37489
   394
  from `?r`[rule_format, OF `j < DIM('a)`, of "\<pi> i"] `i < CARD('b)`
hoelzl@37489
   395
  show "P (j + i * DIM('a))" by simp
hoelzl@37489
   396
qed
hoelzl@37489
   397
huffman@44129
   398
lemma (in euclidean_space) exists_CARD_DIM:
hoelzl@37489
   399
  "(\<exists>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<exists>i::'b::finite. \<exists>j<DIM('a). P (j + \<pi>' i * DIM('a)))"
hoelzl@37489
   400
  using forall_CARD_DIM[where 'b='b, of "\<lambda>x. \<not> P x"] by blast
hoelzl@37489
   401
hoelzl@37489
   402
lemma forall_CARD:
hoelzl@37489
   403
  "(\<forall>i<CARD('b). P i) \<longleftrightarrow> (\<forall>i::'b::finite. P (\<pi>' i))"
hoelzl@37489
   404
  using forall_CARD_DIM[where 'a=real, of P] by simp
hoelzl@37489
   405
hoelzl@37489
   406
lemma exists_CARD:
hoelzl@37489
   407
  "(\<exists>i<CARD('b). P i) \<longleftrightarrow> (\<exists>i::'b::finite. P (\<pi>' i))"
hoelzl@37489
   408
  using exists_CARD_DIM[where 'a=real, of P] by simp
hoelzl@37489
   409
hoelzl@37489
   410
lemmas cart_simps = forall_CARD_DIM exists_CARD_DIM forall_CARD exists_CARD
hoelzl@37489
   411
hoelzl@37489
   412
lemma cart_euclidean_nth[simp]:
huffman@44136
   413
  fixes x :: "('a::euclidean_space, 'b::finite) vec"
hoelzl@37489
   414
  assumes j:"j < DIM('a)"
hoelzl@37489
   415
  shows "x $$ (j + \<pi>' i * DIM('a)) = x $ i $$ j"
huffman@44136
   416
  unfolding euclidean_component_def inner_vec_def basis_eq_pi'[OF j] if_distrib cond_application_beta
hoelzl@37489
   417
  by (simp add: setsum_cases)
hoelzl@37489
   418
hoelzl@37489
   419
lemma real_euclidean_nth:
hoelzl@37489
   420
  fixes x :: "real^'n"
hoelzl@37489
   421
  shows "x $$ \<pi>' i = (x $ i :: real)"
hoelzl@37489
   422
  using cart_euclidean_nth[where 'a=real, of 0 x i] by simp
hoelzl@37489
   423
hoelzl@37489
   424
lemmas nth_conv_component = real_euclidean_nth[symmetric]
hoelzl@37489
   425
hoelzl@37489
   426
lemma mult_split_eq:
hoelzl@37489
   427
  fixes A :: nat assumes "x < A" "y < A"
hoelzl@37489
   428
  shows "x + i * A = y + j * A \<longleftrightarrow> x = y \<and> i = j"
hoelzl@37489
   429
proof
hoelzl@37489
   430
  assume *: "x + i * A = y + j * A"
hoelzl@37489
   431
  { fix x y i j assume "i < j" "x < A" and *: "x + i * A = y + j * A"
hoelzl@37489
   432
    hence "x + i * A < Suc i * A" using `x < A` by simp
hoelzl@37489
   433
    also have "\<dots> \<le> j * A" using `i < j` unfolding mult_le_cancel2 by simp
hoelzl@37489
   434
    also have "\<dots> \<le> y + j * A" by simp
hoelzl@37489
   435
    finally have "i = j" using * by simp }
hoelzl@37489
   436
  note eq = this
hoelzl@37489
   437
hoelzl@37489
   438
  have "i = j"
hoelzl@37489
   439
  proof (cases rule: linorder_cases)
wenzelm@49644
   440
    assume "i < j"
wenzelm@49644
   441
    from eq[OF this `x < A` *] show "i = j" by simp
hoelzl@37489
   442
  next
wenzelm@49644
   443
    assume "j < i"
wenzelm@49644
   444
    from eq[OF this `y < A` *[symmetric]] show "i = j" by simp
hoelzl@37489
   445
  qed simp
hoelzl@37489
   446
  thus "x = y \<and> i = j" using * by simp
hoelzl@37489
   447
qed simp
hoelzl@37489
   448
huffman@44136
   449
instance vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
hoelzl@37489
   450
proof
hoelzl@37489
   451
  fix x y::"'a^'b"
huffman@44136
   452
  show "(x \<le> y) = (\<forall>i<DIM(('a, 'b) vec). x $$ i \<le> y $$ i)"
huffman@44136
   453
    unfolding less_eq_vec_def apply(subst eucl_le) by (simp add: cart_simps)
huffman@44136
   454
  show"(x < y) = (\<forall>i<DIM(('a, 'b) vec). x $$ i < y $$ i)"
huffman@44136
   455
    unfolding less_vec_def apply(subst eucl_less) by (simp add: cart_simps)
hoelzl@37489
   456
qed
hoelzl@37489
   457
wenzelm@49644
   458
hoelzl@37489
   459
subsection{* Basis vectors in coordinate directions. *}
hoelzl@37489
   460
hoelzl@37489
   461
definition "cart_basis k = (\<chi> i. if i = k then 1 else 0)"
hoelzl@37489
   462
hoelzl@37489
   463
lemma basis_component [simp]: "cart_basis k $ i = (if k=i then 1 else 0)"
hoelzl@37489
   464
  unfolding cart_basis_def by simp
hoelzl@37489
   465
hoelzl@37489
   466
lemma norm_basis[simp]:
hoelzl@37489
   467
  shows "norm (cart_basis k :: real ^'n) = 1"
huffman@44136
   468
  apply (simp add: cart_basis_def norm_eq_sqrt_inner) unfolding inner_vec_def
hoelzl@37489
   469
  apply (vector delta_mult_idempotent)
wenzelm@49644
   470
  using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"] apply auto
wenzelm@49644
   471
  done
hoelzl@37489
   472
hoelzl@37489
   473
lemma norm_basis_1: "norm(cart_basis 1 :: real ^'n::{finite,one}) = 1"
hoelzl@37489
   474
  by (rule norm_basis)
hoelzl@37489
   475
hoelzl@37489
   476
lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c"
hoelzl@37489
   477
  by (rule exI[where x="c *\<^sub>R cart_basis arbitrary"]) simp
hoelzl@37489
   478
wenzelm@49644
   479
lemma vector_choose_dist:
wenzelm@49644
   480
  assumes e: "0 <= e"
hoelzl@37489
   481
  shows "\<exists>(y::real^'n). dist x y = e"
wenzelm@49644
   482
proof -
wenzelm@49644
   483
  from vector_choose_size[OF e] obtain c:: "real ^'n" where "norm c = e"
hoelzl@37489
   484
    by blast
hoelzl@37489
   485
  then have "dist x (x - c) = e" by (simp add: dist_norm)
hoelzl@37489
   486
  then show ?thesis by blast
hoelzl@37489
   487
qed
hoelzl@37489
   488
hoelzl@37489
   489
lemma basis_inj[intro]: "inj (cart_basis :: 'n \<Rightarrow> real ^'n)"
huffman@44136
   490
  by (simp add: inj_on_def vec_eq_iff)
hoelzl@37489
   491
wenzelm@49644
   492
lemma basis_expansion: "setsum (\<lambda>i. (x$i) *s cart_basis i) UNIV = (x::('a::ring_1) ^'n)"
wenzelm@49644
   493
  (is "?lhs = ?rhs" is "setsum ?f ?S = _")
wenzelm@49644
   494
  by (auto simp add: vec_eq_iff
wenzelm@49644
   495
      if_distrib setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
hoelzl@37489
   496
hoelzl@37489
   497
lemma smult_conv_scaleR: "c *s x = scaleR c x"
huffman@44136
   498
  unfolding vector_scalar_mult_def scaleR_vec_def by simp
hoelzl@37489
   499
wenzelm@49644
   500
lemma basis_expansion': "setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) UNIV = x"
hoelzl@37489
   501
  by (rule basis_expansion [where 'a=real, unfolded smult_conv_scaleR])
hoelzl@37489
   502
hoelzl@37489
   503
lemma basis_expansion_unique:
hoelzl@37489
   504
  "setsum (\<lambda>i. f i *s cart_basis i) UNIV = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i. f i = x$i)"
huffman@44136
   505
  by (simp add: vec_eq_iff setsum_delta if_distrib cong del: if_weak_cong)
hoelzl@37489
   506
wenzelm@49644
   507
lemma dot_basis: "cart_basis i \<bullet> x = x$i" "x \<bullet> (cart_basis i) = (x$i)"
huffman@44136
   508
  by (auto simp add: inner_vec_def cart_basis_def cond_application_beta if_distrib setsum_delta
hoelzl@37489
   509
           cong del: if_weak_cong)
hoelzl@37489
   510
hoelzl@37489
   511
lemma inner_basis:
hoelzl@37489
   512
  fixes x :: "'a::{real_inner, real_algebra_1} ^ 'n"
hoelzl@37489
   513
  shows "inner (cart_basis i) x = inner 1 (x $ i)"
hoelzl@37489
   514
    and "inner x (cart_basis i) = inner (x $ i) 1"
huffman@44136
   515
  unfolding inner_vec_def cart_basis_def
hoelzl@37489
   516
  by (auto simp add: cond_application_beta if_distrib setsum_delta cong del: if_weak_cong)
hoelzl@37489
   517
hoelzl@37489
   518
lemma basis_eq_0: "cart_basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
huffman@44136
   519
  by (auto simp add: vec_eq_iff)
hoelzl@37489
   520
wenzelm@49644
   521
lemma basis_nonzero: "cart_basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
hoelzl@37489
   522
  by (simp add: basis_eq_0)
hoelzl@37489
   523
hoelzl@37489
   524
text {* some lemmas to map between Eucl and Cart *}
hoelzl@37489
   525
lemma basis_real_n[simp]:"(basis (\<pi>' i)::real^'a) = cart_basis i"
huffman@44136
   526
  unfolding basis_vec_def using pi'_range[where 'n='a]
huffman@44166
   527
  by (auto simp: vec_eq_iff axis_def)
hoelzl@37489
   528
hoelzl@37489
   529
subsection {* Orthogonality on cartesian products *}
hoelzl@37489
   530
wenzelm@49644
   531
lemma orthogonal_basis: "orthogonal (cart_basis i) x \<longleftrightarrow> x$i = (0::real)"
huffman@44136
   532
  by (auto simp add: orthogonal_def inner_vec_def cart_basis_def if_distrib
hoelzl@37489
   533
                     cond_application_beta setsum_delta cong del: if_weak_cong)
hoelzl@37489
   534
wenzelm@49644
   535
lemma orthogonal_basis_basis: "orthogonal (cart_basis i :: real^'n) (cart_basis j) \<longleftrightarrow> i \<noteq> j"
hoelzl@37489
   536
  unfolding orthogonal_basis[of i] basis_component[of j] by simp
hoelzl@37489
   537
hoelzl@37489
   538
subsection {* Linearity on cartesian products *}
hoelzl@37489
   539
hoelzl@37489
   540
lemma linear_vmul_component:
wenzelm@49644
   541
  assumes "linear f"
hoelzl@37489
   542
  shows "linear (\<lambda>x. f x $ k *\<^sub>R v)"
wenzelm@49644
   543
  using assms by (auto simp add: linear_def algebra_simps)
hoelzl@37489
   544
hoelzl@37489
   545
wenzelm@49644
   546
subsection {* Adjoints on cartesian products *}
hoelzl@37489
   547
hoelzl@37489
   548
text {* TODO: The following lemmas about adjoints should hold for any
hoelzl@37489
   549
Hilbert space (i.e. complete inner product space).
hoelzl@37489
   550
(see \url{http://en.wikipedia.org/wiki/Hermitian_adjoint})
hoelzl@37489
   551
*}
hoelzl@37489
   552
hoelzl@37489
   553
lemma adjoint_works_lemma:
hoelzl@37489
   554
  fixes f:: "real ^'n \<Rightarrow> real ^'m"
hoelzl@37489
   555
  assumes lf: "linear f"
hoelzl@37489
   556
  shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
wenzelm@49644
   557
proof -
hoelzl@37489
   558
  let ?N = "UNIV :: 'n set"
hoelzl@37489
   559
  let ?M = "UNIV :: 'm set"
hoelzl@37489
   560
  have fN: "finite ?N" by simp
hoelzl@37489
   561
  have fM: "finite ?M" by simp
wenzelm@49644
   562
  { fix y:: "real ^ 'm"
hoelzl@37489
   563
    let ?w = "(\<chi> i. (f (cart_basis i) \<bullet> y)) :: real ^ 'n"
wenzelm@49644
   564
    { fix x
hoelzl@37489
   565
      have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) ?N) \<bullet> y"
hoelzl@37489
   566
        by (simp only: basis_expansion')
hoelzl@37489
   567
      also have "\<dots> = (setsum (\<lambda>i. (x$i) *\<^sub>R f (cart_basis i)) ?N) \<bullet> y"
hoelzl@37489
   568
        unfolding linear_setsum[OF lf fN]
hoelzl@37489
   569
        by (simp add: linear_cmul[OF lf])
hoelzl@37489
   570
      finally have "f x \<bullet> y = x \<bullet> ?w"
wenzelm@49644
   571
        by (simp add: inner_vec_def setsum_left_distrib
wenzelm@49644
   572
            setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps)
wenzelm@49644
   573
    }
hoelzl@37489
   574
  }
wenzelm@49644
   575
  then show ?thesis
wenzelm@49644
   576
    unfolding adjoint_def some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
hoelzl@37489
   577
    using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
hoelzl@37489
   578
    by metis
hoelzl@37489
   579
qed
hoelzl@37489
   580
hoelzl@37489
   581
lemma adjoint_works:
hoelzl@37489
   582
  fixes f:: "real ^'n \<Rightarrow> real ^'m"
hoelzl@37489
   583
  assumes lf: "linear f"
hoelzl@37489
   584
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@37489
   585
  using adjoint_works_lemma[OF lf] by metis
hoelzl@37489
   586
hoelzl@37489
   587
lemma adjoint_linear:
hoelzl@37489
   588
  fixes f:: "real ^'n \<Rightarrow> real ^'m"
hoelzl@37489
   589
  assumes lf: "linear f"
hoelzl@37489
   590
  shows "linear (adjoint f)"
hoelzl@37489
   591
  unfolding linear_def vector_eq_ldot[where 'a="real^'n", symmetric] apply safe
hoelzl@37489
   592
  unfolding inner_simps smult_conv_scaleR adjoint_works[OF lf] by auto
hoelzl@37489
   593
hoelzl@37489
   594
lemma adjoint_clauses:
hoelzl@37489
   595
  fixes f:: "real ^'n \<Rightarrow> real ^'m"
hoelzl@37489
   596
  assumes lf: "linear f"
hoelzl@37489
   597
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
wenzelm@49644
   598
    and "adjoint f y \<bullet> x = y \<bullet> f x"
hoelzl@37489
   599
  by (simp_all add: adjoint_works[OF lf] inner_commute)
hoelzl@37489
   600
hoelzl@37489
   601
lemma adjoint_adjoint:
hoelzl@37489
   602
  fixes f:: "real ^'n \<Rightarrow> real ^'m"
hoelzl@37489
   603
  assumes lf: "linear f"
hoelzl@37489
   604
  shows "adjoint (adjoint f) = f"
hoelzl@37489
   605
  by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
hoelzl@37489
   606
hoelzl@37489
   607
hoelzl@37489
   608
subsection {* Matrix operations *}
hoelzl@37489
   609
hoelzl@37489
   610
text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
hoelzl@37489
   611
wenzelm@49644
   612
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
wenzelm@49644
   613
    (infixl "**" 70)
hoelzl@37489
   614
  where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
hoelzl@37489
   615
wenzelm@49644
   616
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
wenzelm@49644
   617
    (infixl "*v" 70)
hoelzl@37489
   618
  where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
hoelzl@37489
   619
wenzelm@49644
   620
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
wenzelm@49644
   621
    (infixl "v*" 70)
hoelzl@37489
   622
  where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
hoelzl@37489
   623
hoelzl@37489
   624
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
hoelzl@37489
   625
definition transpose where 
hoelzl@37489
   626
  "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
hoelzl@37489
   627
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
hoelzl@37489
   628
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
hoelzl@37489
   629
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
hoelzl@37489
   630
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
hoelzl@37489
   631
hoelzl@37489
   632
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
hoelzl@37489
   633
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
hoelzl@37489
   634
  by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps)
hoelzl@37489
   635
hoelzl@37489
   636
lemma matrix_mul_lid:
hoelzl@37489
   637
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   638
  shows "mat 1 ** A = A"
hoelzl@37489
   639
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   640
  apply vector
wenzelm@49644
   641
  apply (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite]
wenzelm@49644
   642
    mult_1_left mult_zero_left if_True UNIV_I)
wenzelm@49644
   643
  done
hoelzl@37489
   644
hoelzl@37489
   645
hoelzl@37489
   646
lemma matrix_mul_rid:
hoelzl@37489
   647
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   648
  shows "A ** mat 1 = A"
hoelzl@37489
   649
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   650
  apply vector
wenzelm@49644
   651
  apply (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite]
wenzelm@49644
   652
    mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
wenzelm@49644
   653
  done
hoelzl@37489
   654
hoelzl@37489
   655
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
hoelzl@37489
   656
  apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
hoelzl@37489
   657
  apply (subst setsum_commute)
hoelzl@37489
   658
  apply simp
hoelzl@37489
   659
  done
hoelzl@37489
   660
hoelzl@37489
   661
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
wenzelm@49644
   662
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def
wenzelm@49644
   663
    setsum_right_distrib setsum_left_distrib mult_assoc)
hoelzl@37489
   664
  apply (subst setsum_commute)
hoelzl@37489
   665
  apply simp
hoelzl@37489
   666
  done
hoelzl@37489
   667
hoelzl@37489
   668
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
hoelzl@37489
   669
  apply (vector matrix_vector_mult_def mat_def)
wenzelm@49644
   670
  apply (simp add: if_distrib cond_application_beta setsum_delta' cong del: if_weak_cong)
wenzelm@49644
   671
  done
hoelzl@37489
   672
wenzelm@49644
   673
lemma matrix_transpose_mul:
wenzelm@49644
   674
    "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
huffman@44136
   675
  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute)
hoelzl@37489
   676
hoelzl@37489
   677
lemma matrix_eq:
hoelzl@37489
   678
  fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
hoelzl@37489
   679
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@37489
   680
  apply auto
huffman@44136
   681
  apply (subst vec_eq_iff)
hoelzl@37489
   682
  apply clarify
huffman@44136
   683
  apply (clarsimp simp add: matrix_vector_mult_def cart_basis_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
hoelzl@37489
   684
  apply (erule_tac x="cart_basis ia" in allE)
hoelzl@37489
   685
  apply (erule_tac x="i" in allE)
wenzelm@49644
   686
  apply (auto simp add: cart_basis_def if_distrib cond_application_beta
wenzelm@49644
   687
    setsum_delta[OF finite] cong del: if_weak_cong)
wenzelm@49644
   688
  done
hoelzl@37489
   689
wenzelm@49644
   690
lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
huffman@44136
   691
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   692
hoelzl@37489
   693
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
huffman@44136
   694
  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
hoelzl@37489
   695
  apply (subst setsum_commute)
wenzelm@49644
   696
  apply simp
wenzelm@49644
   697
  done
hoelzl@37489
   698
hoelzl@37489
   699
lemma transpose_mat: "transpose (mat n) = mat n"
hoelzl@37489
   700
  by (vector transpose_def mat_def)
hoelzl@37489
   701
hoelzl@37489
   702
lemma transpose_transpose: "transpose(transpose A) = A"
hoelzl@37489
   703
  by (vector transpose_def)
hoelzl@37489
   704
hoelzl@37489
   705
lemma row_transpose:
hoelzl@37489
   706
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   707
  shows "row i (transpose A) = column i A"
huffman@44136
   708
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   709
hoelzl@37489
   710
lemma column_transpose:
hoelzl@37489
   711
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   712
  shows "column i (transpose A) = row i A"
huffman@44136
   713
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   714
hoelzl@37489
   715
lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
wenzelm@49644
   716
  by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
hoelzl@37489
   717
wenzelm@49644
   718
lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
wenzelm@49644
   719
  by (metis transpose_transpose rows_transpose)
hoelzl@37489
   720
hoelzl@37489
   721
text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
hoelzl@37489
   722
hoelzl@37489
   723
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
huffman@44136
   724
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   725
wenzelm@49644
   726
lemma matrix_mult_vsum:
wenzelm@49644
   727
  "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
huffman@44136
   728
  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
hoelzl@37489
   729
hoelzl@37489
   730
lemma vector_componentwise:
hoelzl@37489
   731
  "(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (cart_basis i :: 'a^'n)$j) (UNIV :: 'n set))"
hoelzl@37489
   732
  apply (subst basis_expansion[symmetric])
wenzelm@49644
   733
  apply (vector vec_eq_iff setsum_component)
wenzelm@49644
   734
  done
hoelzl@37489
   735
hoelzl@37489
   736
lemma linear_componentwise:
hoelzl@37489
   737
  fixes f:: "real ^'m \<Rightarrow> real ^ _"
hoelzl@37489
   738
  assumes lf: "linear f"
hoelzl@37489
   739
  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (cart_basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
wenzelm@49644
   740
proof -
hoelzl@37489
   741
  let ?M = "(UNIV :: 'm set)"
hoelzl@37489
   742
  let ?N = "(UNIV :: 'n set)"
hoelzl@37489
   743
  have fM: "finite ?M" by simp
hoelzl@37489
   744
  have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (cart_basis i) ) ?M)$j"
hoelzl@37489
   745
    unfolding vector_smult_component[symmetric] smult_conv_scaleR
hoelzl@37489
   746
    unfolding setsum_component[of "(\<lambda>i.(x$i) *\<^sub>R f (cart_basis i :: real^'m))" ?M]
hoelzl@37489
   747
    ..
wenzelm@49644
   748
  then show ?thesis
wenzelm@49644
   749
    unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion' ..
hoelzl@37489
   750
qed
hoelzl@37489
   751
hoelzl@37489
   752
text{* Inverse matrices  (not necessarily square) *}
hoelzl@37489
   753
wenzelm@49644
   754
definition
wenzelm@49644
   755
  "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   756
wenzelm@49644
   757
definition
wenzelm@49644
   758
  "matrix_inv(A:: 'a::semiring_1^'n^'m) =
wenzelm@49644
   759
    (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   760
hoelzl@37489
   761
text{* Correspondence between matrices and linear operators. *}
hoelzl@37489
   762
wenzelm@49644
   763
definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
wenzelm@49644
   764
  where "matrix f = (\<chi> i j. (f(cart_basis j))$i)"
hoelzl@37489
   765
hoelzl@37489
   766
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
wenzelm@49644
   767
  by (simp add: linear_def matrix_vector_mult_def vec_eq_iff
wenzelm@49644
   768
      field_simps setsum_right_distrib setsum_addf)
hoelzl@37489
   769
wenzelm@49644
   770
lemma matrix_works:
wenzelm@49644
   771
  assumes lf: "linear f"
wenzelm@49644
   772
  shows "matrix f *v x = f (x::real ^ 'n)"
wenzelm@49644
   773
  apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
wenzelm@49644
   774
  apply clarify
wenzelm@49644
   775
  apply (rule linear_componentwise[OF lf, symmetric])
wenzelm@49644
   776
  done
hoelzl@37489
   777
wenzelm@49644
   778
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
wenzelm@49644
   779
  by (simp add: ext matrix_works)
hoelzl@37489
   780
hoelzl@37489
   781
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
hoelzl@37489
   782
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
hoelzl@37489
   783
hoelzl@37489
   784
lemma matrix_compose:
hoelzl@37489
   785
  assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
wenzelm@49644
   786
    and lg: "linear (g::real^'m \<Rightarrow> real^_)"
hoelzl@37489
   787
  shows "matrix (g o f) = matrix g ** matrix f"
hoelzl@37489
   788
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
wenzelm@49644
   789
  by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
hoelzl@37489
   790
wenzelm@49644
   791
lemma matrix_vector_column:
wenzelm@49644
   792
  "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
huffman@44136
   793
  by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult_commute)
hoelzl@37489
   794
hoelzl@37489
   795
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
hoelzl@37489
   796
  apply (rule adjoint_unique)
wenzelm@49644
   797
  apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
wenzelm@49644
   798
    setsum_left_distrib setsum_right_distrib)
hoelzl@37489
   799
  apply (subst setsum_commute)
hoelzl@37489
   800
  apply (auto simp add: mult_ac)
hoelzl@37489
   801
  done
hoelzl@37489
   802
hoelzl@37489
   803
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
hoelzl@37489
   804
  shows "matrix(adjoint f) = transpose(matrix f)"
hoelzl@37489
   805
  apply (subst matrix_vector_mul[OF lf])
wenzelm@49644
   806
  unfolding adjoint_matrix matrix_of_matrix_vector_mul
wenzelm@49644
   807
  apply rule
wenzelm@49644
   808
  done
wenzelm@49644
   809
hoelzl@37489
   810
huffman@44360
   811
subsection {* lambda skolemization on cartesian products *}
hoelzl@37489
   812
hoelzl@37489
   813
(* FIXME: rename do choice_cart *)
hoelzl@37489
   814
hoelzl@37489
   815
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
hoelzl@37494
   816
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   817
proof -
hoelzl@37489
   818
  let ?S = "(UNIV :: 'n set)"
wenzelm@49644
   819
  { assume H: "?rhs"
wenzelm@49644
   820
    then have ?lhs by auto }
hoelzl@37489
   821
  moreover
wenzelm@49644
   822
  { assume H: "?lhs"
hoelzl@37489
   823
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
hoelzl@37489
   824
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
wenzelm@49644
   825
    { fix i
hoelzl@37489
   826
      from f have "P i (f i)" by metis
hoelzl@37494
   827
      then have "P i (?x $ i)" by auto
hoelzl@37489
   828
    }
hoelzl@37489
   829
    hence "\<forall>i. P i (?x$i)" by metis
hoelzl@37489
   830
    hence ?rhs by metis }
hoelzl@37489
   831
  ultimately show ?thesis by metis
hoelzl@37489
   832
qed
hoelzl@37489
   833
wenzelm@49644
   834
hoelzl@37489
   835
subsection {* Standard bases are a spanning set, and obviously finite. *}
hoelzl@37489
   836
hoelzl@37489
   837
lemma span_stdbasis:"span {cart_basis i :: real^'n | i. i \<in> (UNIV :: 'n set)} = UNIV"
wenzelm@49644
   838
  apply (rule set_eqI)
wenzelm@49644
   839
  apply auto
wenzelm@49644
   840
  apply (subst basis_expansion'[symmetric])
wenzelm@49644
   841
  apply (rule span_setsum)
wenzelm@49644
   842
  apply simp
wenzelm@49644
   843
  apply auto
wenzelm@49644
   844
  apply (rule span_mul)
wenzelm@49644
   845
  apply (rule span_superset)
wenzelm@49644
   846
  apply auto
wenzelm@49644
   847
  done
hoelzl@37489
   848
hoelzl@37489
   849
lemma finite_stdbasis: "finite {cart_basis i ::real^'n |i. i\<in> (UNIV:: 'n set)}" (is "finite ?S")
wenzelm@49644
   850
proof -
wenzelm@49644
   851
  have "?S = cart_basis ` UNIV" by blast
wenzelm@49644
   852
  then show ?thesis by auto
hoelzl@37489
   853
qed
hoelzl@37489
   854
hoelzl@37489
   855
lemma card_stdbasis: "card {cart_basis i ::real^'n |i. i\<in> (UNIV :: 'n set)} = CARD('n)" (is "card ?S = _")
wenzelm@49644
   856
proof -
wenzelm@49644
   857
  have "?S = cart_basis ` UNIV" by blast
wenzelm@49644
   858
  then show ?thesis using card_image[OF basis_inj] by simp
hoelzl@37489
   859
qed
hoelzl@37489
   860
hoelzl@37489
   861
lemma independent_stdbasis_lemma:
hoelzl@37489
   862
  assumes x: "(x::real ^ 'n) \<in> span (cart_basis ` S)"
wenzelm@49644
   863
    and iS: "i \<notin> S"
hoelzl@37489
   864
  shows "(x$i) = 0"
wenzelm@49644
   865
proof -
hoelzl@37489
   866
  let ?U = "UNIV :: 'n set"
hoelzl@37489
   867
  let ?B = "cart_basis ` S"
huffman@44170
   868
  let ?P = "{(x::real^_). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0}"
wenzelm@49644
   869
  { fix x::"real^_" assume xS: "x\<in> ?B"
wenzelm@49644
   870
    from xS have "x \<in> ?P" by auto }
wenzelm@49644
   871
  moreover
wenzelm@49644
   872
  have "subspace ?P"
wenzelm@49644
   873
    by (auto simp add: subspace_def)
wenzelm@49644
   874
  ultimately show ?thesis
wenzelm@49644
   875
    using x span_induct[of x ?B ?P] iS by blast
hoelzl@37489
   876
qed
hoelzl@37489
   877
hoelzl@37489
   878
lemma independent_stdbasis: "independent {cart_basis i ::real^'n |i. i\<in> (UNIV :: 'n set)}"
wenzelm@49644
   879
proof -
hoelzl@37489
   880
  let ?I = "UNIV :: 'n set"
hoelzl@37489
   881
  let ?b = "cart_basis :: _ \<Rightarrow> real ^'n"
hoelzl@37489
   882
  let ?B = "?b ` ?I"
wenzelm@49644
   883
  have eq: "{?b i|i. i \<in> ?I} = ?B" by auto
wenzelm@49644
   884
  { assume d: "dependent ?B"
hoelzl@37489
   885
    then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"
hoelzl@37489
   886
      unfolding dependent_def by auto
hoelzl@37489
   887
    have eq1: "?B - {?b k} = ?B - ?b ` {k}"  by simp
hoelzl@37489
   888
    have eq2: "?B - {?b k} = ?b ` (?I - {k})"
hoelzl@37489
   889
      unfolding eq1
hoelzl@37489
   890
      apply (rule inj_on_image_set_diff[symmetric])
wenzelm@49644
   891
      apply (rule basis_inj) using k(1)
wenzelm@49644
   892
      apply auto
wenzelm@49644
   893
      done
hoelzl@37489
   894
    from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
hoelzl@37489
   895
    from independent_stdbasis_lemma[OF th0, of k, simplified]
wenzelm@49644
   896
    have False by simp }
hoelzl@37489
   897
  then show ?thesis unfolding eq dependent_def ..
hoelzl@37489
   898
qed
hoelzl@37489
   899
hoelzl@37489
   900
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
hoelzl@37489
   901
  unfolding inner_simps smult_conv_scaleR by auto
hoelzl@37489
   902
hoelzl@37489
   903
lemma linear_eq_stdbasis_cart:
hoelzl@37489
   904
  assumes lf: "linear (f::real^'m \<Rightarrow> _)" and lg: "linear g"
wenzelm@49644
   905
    and fg: "\<forall>i. f (cart_basis i) = g(cart_basis i)"
hoelzl@37489
   906
  shows "f = g"
wenzelm@49644
   907
proof -
hoelzl@37489
   908
  let ?U = "UNIV :: 'm set"
hoelzl@37489
   909
  let ?I = "{cart_basis i:: real^'m|i. i \<in> ?U}"
wenzelm@49644
   910
  { fix x assume x: "x \<in> (UNIV :: (real^'m) set)"
hoelzl@37489
   911
    from equalityD2[OF span_stdbasis]
hoelzl@37489
   912
    have IU: " (UNIV :: (real^'m) set) \<subseteq> span ?I" by blast
hoelzl@37489
   913
    from linear_eq[OF lf lg IU] fg x
wenzelm@49644
   914
    have "f x = g x" unfolding Ball_def mem_Collect_eq by metis
wenzelm@49644
   915
  }
huffman@44457
   916
  then show ?thesis by auto
hoelzl@37489
   917
qed
hoelzl@37489
   918
hoelzl@37489
   919
lemma bilinear_eq_stdbasis_cart:
hoelzl@37489
   920
  assumes bf: "bilinear (f:: real^'m \<Rightarrow> real^'n \<Rightarrow> _)"
wenzelm@49644
   921
    and bg: "bilinear g"
wenzelm@49644
   922
    and fg: "\<forall>i j. f (cart_basis i) (cart_basis j) = g (cart_basis i) (cart_basis j)"
hoelzl@37489
   923
  shows "f = g"
wenzelm@49644
   924
proof -
wenzelm@49644
   925
  from fg have th: "\<forall>x \<in> {cart_basis i| i. i\<in> (UNIV :: 'm set)}.
wenzelm@49644
   926
      \<forall>y\<in>  {cart_basis j |j. j \<in> (UNIV :: 'n set)}. f x y = g x y"
wenzelm@49644
   927
    by blast
wenzelm@49644
   928
  from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th]
wenzelm@49644
   929
  show ?thesis by blast
hoelzl@37489
   930
qed
hoelzl@37489
   931
hoelzl@37489
   932
lemma left_invertible_transpose:
hoelzl@37489
   933
  "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
hoelzl@37489
   934
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   935
hoelzl@37489
   936
lemma right_invertible_transpose:
hoelzl@37489
   937
  "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
hoelzl@37489
   938
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   939
hoelzl@37489
   940
lemma matrix_left_invertible_injective:
wenzelm@49644
   941
  "(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
wenzelm@49644
   942
proof -
wenzelm@49644
   943
  { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
hoelzl@37489
   944
    from xy have "B*v (A *v x) = B *v (A*v y)" by simp
hoelzl@37489
   945
    hence "x = y"
wenzelm@49644
   946
      unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
hoelzl@37489
   947
  moreover
wenzelm@49644
   948
  { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
hoelzl@37489
   949
    hence i: "inj (op *v A)" unfolding inj_on_def by auto
hoelzl@37489
   950
    from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
hoelzl@37489
   951
    obtain g where g: "linear g" "g o op *v A = id" by blast
hoelzl@37489
   952
    have "matrix g ** A = mat 1"
hoelzl@37489
   953
      unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   954
      using g(2) by (simp add: fun_eq_iff)
wenzelm@49644
   955
    then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
hoelzl@37489
   956
  ultimately show ?thesis by blast
hoelzl@37489
   957
qed
hoelzl@37489
   958
hoelzl@37489
   959
lemma matrix_left_invertible_ker:
hoelzl@37489
   960
  "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
hoelzl@37489
   961
  unfolding matrix_left_invertible_injective
hoelzl@37489
   962
  using linear_injective_0[OF matrix_vector_mul_linear, of A]
hoelzl@37489
   963
  by (simp add: inj_on_def)
hoelzl@37489
   964
hoelzl@37489
   965
lemma matrix_right_invertible_surjective:
wenzelm@49644
   966
  "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
wenzelm@49644
   967
proof -
wenzelm@49644
   968
  { fix B :: "real ^'m^'n"
wenzelm@49644
   969
    assume AB: "A ** B = mat 1"
wenzelm@49644
   970
    { fix x :: "real ^ 'm"
hoelzl@37489
   971
      have "A *v (B *v x) = x"
wenzelm@49644
   972
        by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
hoelzl@37489
   973
    hence "surj (op *v A)" unfolding surj_def by metis }
hoelzl@37489
   974
  moreover
wenzelm@49644
   975
  { assume sf: "surj (op *v A)"
hoelzl@37489
   976
    from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
hoelzl@37489
   977
    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
hoelzl@37489
   978
      by blast
hoelzl@37489
   979
hoelzl@37489
   980
    have "A ** (matrix g) = mat 1"
hoelzl@37489
   981
      unfolding matrix_eq  matrix_vector_mul_lid
hoelzl@37489
   982
        matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   983
      using g(2) unfolding o_def fun_eq_iff id_def
hoelzl@37489
   984
      .
hoelzl@37489
   985
    hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
hoelzl@37489
   986
  }
hoelzl@37489
   987
  ultimately show ?thesis unfolding surj_def by blast
hoelzl@37489
   988
qed
hoelzl@37489
   989
hoelzl@37489
   990
lemma matrix_left_invertible_independent_columns:
hoelzl@37489
   991
  fixes A :: "real^'n^'m"
wenzelm@49644
   992
  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
wenzelm@49644
   993
      (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
wenzelm@49644
   994
    (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   995
proof -
hoelzl@37489
   996
  let ?U = "UNIV :: 'n set"
wenzelm@49644
   997
  { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
wenzelm@49644
   998
    { fix c i
wenzelm@49644
   999
      assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
hoelzl@37489
  1000
      let ?x = "\<chi> i. c i"
hoelzl@37489
  1001
      have th0:"A *v ?x = 0"
hoelzl@37489
  1002
        using c
huffman@44136
  1003
        unfolding matrix_mult_vsum vec_eq_iff
hoelzl@37489
  1004
        by auto
hoelzl@37489
  1005
      from k[rule_format, OF th0] i
huffman@44136
  1006
      have "c i = 0" by (vector vec_eq_iff)}
wenzelm@49644
  1007
    hence ?rhs by blast }
hoelzl@37489
  1008
  moreover
wenzelm@49644
  1009
  { assume H: ?rhs
wenzelm@49644
  1010
    { fix x assume x: "A *v x = 0"
hoelzl@37489
  1011
      let ?c = "\<lambda>i. ((x$i ):: real)"
hoelzl@37489
  1012
      from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
wenzelm@49644
  1013
      have "x = 0" by vector }
wenzelm@49644
  1014
  }
hoelzl@37489
  1015
  ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
hoelzl@37489
  1016
qed
hoelzl@37489
  1017
hoelzl@37489
  1018
lemma matrix_right_invertible_independent_rows:
hoelzl@37489
  1019
  fixes A :: "real^'n^'m"
wenzelm@49644
  1020
  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
wenzelm@49644
  1021
    (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
hoelzl@37489
  1022
  unfolding left_invertible_transpose[symmetric]
hoelzl@37489
  1023
    matrix_left_invertible_independent_columns
hoelzl@37489
  1024
  by (simp add: column_transpose)
hoelzl@37489
  1025
hoelzl@37489
  1026
lemma matrix_right_invertible_span_columns:
wenzelm@49644
  1027
  "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
wenzelm@49644
  1028
    span (columns A) = UNIV" (is "?lhs = ?rhs")
wenzelm@49644
  1029
proof -
hoelzl@37489
  1030
  let ?U = "UNIV :: 'm set"
hoelzl@37489
  1031
  have fU: "finite ?U" by simp
hoelzl@37489
  1032
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
hoelzl@37489
  1033
    unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
wenzelm@49644
  1034
    apply (subst eq_commute)
wenzelm@49644
  1035
    apply rule
wenzelm@49644
  1036
    done
hoelzl@37489
  1037
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
wenzelm@49644
  1038
  { assume h: ?lhs
wenzelm@49644
  1039
    { fix x:: "real ^'n"
wenzelm@49644
  1040
      from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
wenzelm@49644
  1041
        where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
wenzelm@49644
  1042
      have "x \<in> span (columns A)"
wenzelm@49644
  1043
        unfolding y[symmetric]
wenzelm@49644
  1044
        apply (rule span_setsum[OF fU])
wenzelm@49644
  1045
        apply clarify
wenzelm@49644
  1046
        unfolding smult_conv_scaleR
wenzelm@49644
  1047
        apply (rule span_mul)
wenzelm@49644
  1048
        apply (rule span_superset)
wenzelm@49644
  1049
        unfolding columns_def
wenzelm@49644
  1050
        apply blast
wenzelm@49644
  1051
        done
wenzelm@49644
  1052
    }
wenzelm@49644
  1053
    then have ?rhs unfolding rhseq by blast }
hoelzl@37489
  1054
  moreover
wenzelm@49644
  1055
  { assume h:?rhs
hoelzl@37489
  1056
    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
wenzelm@49644
  1057
    { fix y
wenzelm@49644
  1058
      have "?P y"
wenzelm@49644
  1059
      proof (rule span_induct_alt[of ?P "columns A", folded smult_conv_scaleR])
hoelzl@37489
  1060
        show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
hoelzl@37489
  1061
          by (rule exI[where x=0], simp)
hoelzl@37489
  1062
      next
wenzelm@49644
  1063
        fix c y1 y2
wenzelm@49644
  1064
        assume y1: "y1 \<in> columns A" and y2: "?P y2"
hoelzl@37489
  1065
        from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
hoelzl@37489
  1066
          unfolding columns_def by blast
hoelzl@37489
  1067
        from y2 obtain x:: "real ^'m" where
hoelzl@37489
  1068
          x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
hoelzl@37489
  1069
        let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
hoelzl@37489
  1070
        show "?P (c*s y1 + y2)"
webertj@49962
  1071
        proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)
wenzelm@49644
  1072
          fix j
wenzelm@49644
  1073
          have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
  1074
              else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
wenzelm@49644
  1075
            using i(1) by (simp add: field_simps)
wenzelm@49644
  1076
          have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
  1077
              else (x$xa) * ((column xa A$j))) ?U = setsum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"
wenzelm@49644
  1078
            apply (rule setsum_cong[OF refl])
wenzelm@49644
  1079
            using th apply blast
wenzelm@49644
  1080
            done
wenzelm@49644
  1081
          also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
wenzelm@49644
  1082
            by (simp add: setsum_addf)
wenzelm@49644
  1083
          also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
wenzelm@49644
  1084
            unfolding setsum_delta[OF fU]
wenzelm@49644
  1085
            using i(1) by simp
wenzelm@49644
  1086
          finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
  1087
            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
wenzelm@49644
  1088
        qed
wenzelm@49644
  1089
      next
wenzelm@49644
  1090
        show "y \<in> span (columns A)"
wenzelm@49644
  1091
          unfolding h by blast
wenzelm@49644
  1092
      qed
wenzelm@49644
  1093
    }
wenzelm@49644
  1094
    then have ?lhs unfolding lhseq ..
wenzelm@49644
  1095
  }
hoelzl@37489
  1096
  ultimately show ?thesis by blast
hoelzl@37489
  1097
qed
hoelzl@37489
  1098
hoelzl@37489
  1099
lemma matrix_left_invertible_span_rows:
hoelzl@37489
  1100
  "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
hoelzl@37489
  1101
  unfolding right_invertible_transpose[symmetric]
hoelzl@37489
  1102
  unfolding columns_transpose[symmetric]
hoelzl@37489
  1103
  unfolding matrix_right_invertible_span_columns
wenzelm@49644
  1104
  ..
hoelzl@37489
  1105
hoelzl@37489
  1106
text {* The same result in terms of square matrices. *}
hoelzl@37489
  1107
hoelzl@37489
  1108
lemma matrix_left_right_inverse:
hoelzl@37489
  1109
  fixes A A' :: "real ^'n^'n"
hoelzl@37489
  1110
  shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
wenzelm@49644
  1111
proof -
wenzelm@49644
  1112
  { fix A A' :: "real ^'n^'n"
wenzelm@49644
  1113
    assume AA': "A ** A' = mat 1"
hoelzl@37489
  1114
    have sA: "surj (op *v A)"
hoelzl@37489
  1115
      unfolding surj_def
hoelzl@37489
  1116
      apply clarify
hoelzl@37489
  1117
      apply (rule_tac x="(A' *v y)" in exI)
wenzelm@49644
  1118
      apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
wenzelm@49644
  1119
      done
hoelzl@37489
  1120
    from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
hoelzl@37489
  1121
    obtain f' :: "real ^'n \<Rightarrow> real ^'n"
hoelzl@37489
  1122
      where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
hoelzl@37489
  1123
    have th: "matrix f' ** A = mat 1"
wenzelm@49644
  1124
      by (simp add: matrix_eq matrix_works[OF f'(1)]
wenzelm@49644
  1125
          matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
hoelzl@37489
  1126
    hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
wenzelm@49644
  1127
    hence "matrix f' = A'"
wenzelm@49644
  1128
      by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
hoelzl@37489
  1129
    hence "matrix f' ** A = A' ** A" by simp
wenzelm@49644
  1130
    hence "A' ** A = mat 1" by (simp add: th)
wenzelm@49644
  1131
  }
hoelzl@37489
  1132
  then show ?thesis by blast
hoelzl@37489
  1133
qed
hoelzl@37489
  1134
hoelzl@37489
  1135
text {* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *}
hoelzl@37489
  1136
hoelzl@37489
  1137
definition "rowvector v = (\<chi> i j. (v$j))"
hoelzl@37489
  1138
hoelzl@37489
  1139
definition "columnvector v = (\<chi> i j. (v$i))"
hoelzl@37489
  1140
wenzelm@49644
  1141
lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
huffman@44136
  1142
  by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
hoelzl@37489
  1143
hoelzl@37489
  1144
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
huffman@44136
  1145
  by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
hoelzl@37489
  1146
wenzelm@49644
  1147
lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
hoelzl@37489
  1148
  by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
hoelzl@37489
  1149
wenzelm@49644
  1150
lemma dot_matrix_product:
wenzelm@49644
  1151
  "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
huffman@44136
  1152
  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
hoelzl@37489
  1153
hoelzl@37489
  1154
lemma dot_matrix_vector_mul:
hoelzl@37489
  1155
  fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
hoelzl@37489
  1156
  shows "(A *v x) \<bullet> (B *v y) =
hoelzl@37489
  1157
      (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
wenzelm@49644
  1158
  unfolding dot_matrix_product transpose_columnvector[symmetric]
wenzelm@49644
  1159
    dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
hoelzl@37489
  1160
hoelzl@37489
  1161
hoelzl@37489
  1162
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in> (UNIV :: 'n set)}"
hoelzl@37489
  1163
  unfolding infnorm_def apply(rule arg_cong[where f=Sup]) apply safe
hoelzl@37489
  1164
  apply(rule_tac x="\<pi> i" in exI) defer
wenzelm@49644
  1165
  apply(rule_tac x="\<pi>' i" in exI)
wenzelm@49644
  1166
  unfolding nth_conv_component
wenzelm@49644
  1167
  using pi'_range apply auto
wenzelm@49644
  1168
  done
hoelzl@37489
  1169
wenzelm@49644
  1170
lemma infnorm_set_image_cart: "{abs(x$i) |i. i\<in> (UNIV :: _ set)} =
hoelzl@37489
  1171
  (\<lambda>i. abs(x$i)) ` (UNIV)" by blast
hoelzl@37489
  1172
hoelzl@37489
  1173
lemma infnorm_set_lemma_cart:
wenzelm@49644
  1174
  "finite {abs((x::'a::abs ^'n)$i) |i. i\<in> (UNIV :: 'n set)}"
wenzelm@49644
  1175
  "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"
wenzelm@49644
  1176
  unfolding infnorm_set_image_cart by auto
hoelzl@37489
  1177
wenzelm@49644
  1178
lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
hoelzl@37489
  1179
  unfolding nth_conv_component
hoelzl@37489
  1180
  using component_le_infnorm[of x] .
hoelzl@37489
  1181
wenzelm@49644
  1182
lemma continuous_component: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
huffman@44647
  1183
  unfolding continuous_def by (rule tendsto_vec_nth)
huffman@44213
  1184
wenzelm@49644
  1185
lemma continuous_on_component: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
huffman@44647
  1186
  unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
huffman@44213
  1187
hoelzl@37489
  1188
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
huffman@44233
  1189
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
huffman@44213
  1190
hoelzl@37489
  1191
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
wenzelm@49644
  1192
  unfolding bounded_def
wenzelm@49644
  1193
  apply clarify
wenzelm@49644
  1194
  apply (rule_tac x="x $ i" in exI)
wenzelm@49644
  1195
  apply (rule_tac x="e" in exI)
wenzelm@49644
  1196
  apply clarify
wenzelm@49644
  1197
  apply (rule order_trans [OF dist_vec_nth_le], simp)
wenzelm@49644
  1198
  done
hoelzl@37489
  1199
hoelzl@37489
  1200
lemma compact_lemma_cart:
hoelzl@37489
  1201
  fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
hoelzl@37489
  1202
  assumes "bounded s" and "\<forall>n. f n \<in> s"
hoelzl@37489
  1203
  shows "\<forall>d.
hoelzl@37489
  1204
        \<exists>l r. subseq r \<and>
hoelzl@37489
  1205
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
hoelzl@37489
  1206
proof
wenzelm@49644
  1207
  fix d :: "'n set"
wenzelm@49644
  1208
  have "finite d" by simp
hoelzl@37489
  1209
  thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
hoelzl@37489
  1210
      (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
wenzelm@49644
  1211
  proof (induct d)
wenzelm@49644
  1212
    case empty
wenzelm@49644
  1213
    thus ?case unfolding subseq_def by auto
wenzelm@49644
  1214
  next
wenzelm@49644
  1215
    case (insert k d)
wenzelm@49644
  1216
    have s': "bounded ((\<lambda>x. x $ k) ` s)"
wenzelm@49644
  1217
      using `bounded s` by (rule bounded_component_cart)
wenzelm@49644
  1218
    obtain l1::"'a^'n" and r1 where r1:"subseq r1"
wenzelm@49644
  1219
      and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
hoelzl@37489
  1220
      using insert(3) by auto
wenzelm@49644
  1221
    have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` s"
wenzelm@49644
  1222
      using `\<forall>n. f n \<in> s` by simp
wenzelm@49644
  1223
    obtain l2 r2 where r2: "subseq r2"
wenzelm@49644
  1224
      and lr2: "((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
hoelzl@37489
  1225
      using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
wenzelm@49644
  1226
    def r \<equiv> "r1 \<circ> r2"
wenzelm@49644
  1227
    have r: "subseq r"
hoelzl@37489
  1228
      using r1 and r2 unfolding r_def o_def subseq_def by auto
hoelzl@37489
  1229
    moreover
hoelzl@37489
  1230
    def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
wenzelm@49644
  1231
    { fix e :: real assume "e > 0"
wenzelm@49644
  1232
      from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
wenzelm@49644
  1233
        by blast
wenzelm@49644
  1234
      from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially"
wenzelm@49644
  1235
        by (rule tendstoD)
hoelzl@37489
  1236
      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
hoelzl@37489
  1237
        by (rule eventually_subseq)
hoelzl@37489
  1238
      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@37489
  1239
        using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
hoelzl@37489
  1240
    }
hoelzl@37489
  1241
    ultimately show ?case by auto
hoelzl@37489
  1242
  qed
hoelzl@37489
  1243
qed
hoelzl@37489
  1244
huffman@44136
  1245
instance vec :: (heine_borel, finite) heine_borel
hoelzl@37489
  1246
proof
hoelzl@37489
  1247
  fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"
hoelzl@37489
  1248
  assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
hoelzl@37489
  1249
  then obtain l r where r: "subseq r"
wenzelm@49644
  1250
      and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@37489
  1251
    using compact_lemma_cart [OF s f] by blast
hoelzl@37489
  1252
  let ?d = "UNIV::'b set"
hoelzl@37489
  1253
  { fix e::real assume "e>0"
hoelzl@37489
  1254
    hence "0 < e / (real_of_nat (card ?d))"
wenzelm@49644
  1255
      using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
hoelzl@37489
  1256
    with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
hoelzl@37489
  1257
      by simp
hoelzl@37489
  1258
    moreover
wenzelm@49644
  1259
    { fix n
wenzelm@49644
  1260
      assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
hoelzl@37489
  1261
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
huffman@44136
  1262
        unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
hoelzl@37489
  1263
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
hoelzl@37489
  1264
        by (rule setsum_strict_mono) (simp_all add: n)
hoelzl@37489
  1265
      finally have "dist (f (r n)) l < e" by simp
hoelzl@37489
  1266
    }
hoelzl@37489
  1267
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
hoelzl@37489
  1268
      by (rule eventually_elim1)
hoelzl@37489
  1269
  }
wenzelm@49644
  1270
  hence "((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
hoelzl@37489
  1271
  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
hoelzl@37489
  1272
qed
hoelzl@37489
  1273
wenzelm@49644
  1274
lemma interval_cart:
wenzelm@49644
  1275
  fixes a :: "'a::ord^'n"
wenzelm@49644
  1276
  shows "{a <..< b} = {x::'a^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
wenzelm@49644
  1277
    and "{a .. b} = {x::'a^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
huffman@44136
  1278
  by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
hoelzl@37489
  1279
wenzelm@49644
  1280
lemma mem_interval_cart:
wenzelm@49644
  1281
  fixes a :: "'a::ord^'n"
wenzelm@49644
  1282
  shows "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
wenzelm@49644
  1283
    and "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
wenzelm@49644
  1284
  using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
hoelzl@37489
  1285
wenzelm@49644
  1286
lemma interval_eq_empty_cart:
wenzelm@49644
  1287
  fixes a :: "real^'n"
wenzelm@49644
  1288
  shows "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
wenzelm@49644
  1289
    and "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
wenzelm@49644
  1290
proof -
hoelzl@37489
  1291
  { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>{a <..< b}"
hoelzl@37489
  1292
    hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto
hoelzl@37489
  1293
    hence "a$i < b$i" by auto
wenzelm@49644
  1294
    hence False using as by auto }
hoelzl@37489
  1295
  moreover
hoelzl@37489
  1296
  { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
hoelzl@37489
  1297
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
  1298
    { fix i
hoelzl@37489
  1299
      have "a$i < b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
  1300
      hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
hoelzl@37489
  1301
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
  1302
        by auto }
wenzelm@49644
  1303
    hence "{a <..< b} \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
hoelzl@37489
  1304
  ultimately show ?th1 by blast
hoelzl@37489
  1305
hoelzl@37489
  1306
  { fix i x assume as:"b$i < a$i" and x:"x\<in>{a .. b}"
hoelzl@37489
  1307
    hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto
hoelzl@37489
  1308
    hence "a$i \<le> b$i" by auto
wenzelm@49644
  1309
    hence False using as by auto }
hoelzl@37489
  1310
  moreover
hoelzl@37489
  1311
  { assume as:"\<forall>i. \<not> (b$i < a$i)"
hoelzl@37489
  1312
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
  1313
    { fix i
hoelzl@37489
  1314
      have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
  1315
      hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
hoelzl@37489
  1316
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
  1317
        by auto }
hoelzl@37489
  1318
    hence "{a .. b} \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
hoelzl@37489
  1319
  ultimately show ?th2 by blast
hoelzl@37489
  1320
qed
hoelzl@37489
  1321
wenzelm@49644
  1322
lemma interval_ne_empty_cart:
wenzelm@49644
  1323
  fixes a :: "real^'n"
wenzelm@49644
  1324
  shows "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
wenzelm@49644
  1325
    and "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
hoelzl@37489
  1326
  unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
hoelzl@37489
  1327
    (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
  1328
wenzelm@49644
  1329
lemma subset_interval_imp_cart:
wenzelm@49644
  1330
  fixes a :: "real^'n"
wenzelm@49644
  1331
  shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}"
wenzelm@49644
  1332
    and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}"
wenzelm@49644
  1333
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}"
wenzelm@49644
  1334
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
hoelzl@37489
  1335
  unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
hoelzl@37489
  1336
  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
  1337
wenzelm@49644
  1338
lemma interval_sing:
wenzelm@49644
  1339
  fixes a :: "'a::linorder^'n"
wenzelm@49644
  1340
  shows "{a .. a} = {a} \<and> {a<..<a} = {}"
wenzelm@49644
  1341
  apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
wenzelm@49644
  1342
  apply (simp add: order_eq_iff)
wenzelm@49644
  1343
  apply (auto simp add: not_less less_imp_le)
wenzelm@49644
  1344
  done
hoelzl@37489
  1345
wenzelm@49644
  1346
lemma interval_open_subset_closed_cart:
wenzelm@49644
  1347
  fixes a :: "'a::preorder^'n"
wenzelm@49644
  1348
  shows "{a<..<b} \<subseteq> {a .. b}"
wenzelm@49644
  1349
proof (simp add: subset_eq, rule)
hoelzl@37489
  1350
  fix x
wenzelm@49644
  1351
  assume x: "x \<in>{a<..<b}"
hoelzl@37489
  1352
  { fix i
hoelzl@37489
  1353
    have "a $ i \<le> x $ i"
hoelzl@37489
  1354
      using x order_less_imp_le[of "a$i" "x$i"]
huffman@44136
  1355
      by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
hoelzl@37489
  1356
  }
hoelzl@37489
  1357
  moreover
hoelzl@37489
  1358
  { fix i
hoelzl@37489
  1359
    have "x $ i \<le> b $ i"
hoelzl@37489
  1360
      using x order_less_imp_le[of "x$i" "b$i"]
huffman@44136
  1361
      by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
hoelzl@37489
  1362
  }
hoelzl@37489
  1363
  ultimately
hoelzl@37489
  1364
  show "a \<le> x \<and> x \<le> b"
huffman@44136
  1365
    by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
hoelzl@37489
  1366
qed
hoelzl@37489
  1367
wenzelm@49644
  1368
lemma subset_interval_cart:
wenzelm@49644
  1369
  fixes a :: "real^'n"
wenzelm@49644
  1370
  shows "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
wenzelm@49644
  1371
    and "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
wenzelm@49644
  1372
    and "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
wenzelm@49644
  1373
    and "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
hoelzl@37489
  1374
  using subset_interval[of c d a b] by (simp_all add: cart_simps real_euclidean_nth)
hoelzl@37489
  1375
wenzelm@49644
  1376
lemma disjoint_interval_cart:
wenzelm@49644
  1377
  fixes a::"real^'n"
wenzelm@49644
  1378
  shows "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
wenzelm@49644
  1379
    and "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
wenzelm@49644
  1380
    and "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
wenzelm@49644
  1381
    and "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
hoelzl@37489
  1382
  using disjoint_interval[of a b c d] by (simp_all add: cart_simps real_euclidean_nth)
hoelzl@37489
  1383
wenzelm@49644
  1384
lemma inter_interval_cart:
wenzelm@49644
  1385
  fixes a :: "'a::linorder^'n"
wenzelm@49644
  1386
  shows "{a .. b} \<inter> {c .. d} =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
nipkow@39302
  1387
  unfolding set_eq_iff and Int_iff and mem_interval_cart
hoelzl@37489
  1388
  by auto
hoelzl@37489
  1389
wenzelm@49644
  1390
lemma closed_interval_left_cart:
wenzelm@49644
  1391
  fixes b :: "real^'n"
hoelzl@37489
  1392
  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
huffman@44233
  1393
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
hoelzl@37489
  1394
wenzelm@49644
  1395
lemma closed_interval_right_cart:
wenzelm@49644
  1396
  fixes a::"real^'n"
hoelzl@37489
  1397
  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
huffman@44233
  1398
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
hoelzl@37489
  1399
wenzelm@49644
  1400
lemma is_interval_cart:
wenzelm@49644
  1401
  "is_interval (s::(real^'n) set) \<longleftrightarrow>
wenzelm@49644
  1402
    (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
wenzelm@49644
  1403
  by (simp add: is_interval_def Ball_def cart_simps real_euclidean_nth)
hoelzl@37489
  1404
wenzelm@49644
  1405
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
huffman@44233
  1406
  by (simp add: closed_Collect_le)
hoelzl@37489
  1407
wenzelm@49644
  1408
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
huffman@44233
  1409
  by (simp add: closed_Collect_le)
hoelzl@37489
  1410
wenzelm@49644
  1411
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
wenzelm@49644
  1412
  by (simp add: open_Collect_less)
wenzelm@49644
  1413
wenzelm@49644
  1414
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
huffman@44233
  1415
  by (simp add: open_Collect_less)
hoelzl@37489
  1416
wenzelm@49644
  1417
lemma Lim_component_le_cart:
wenzelm@49644
  1418
  fixes f :: "'a \<Rightarrow> real^'n"
hoelzl@37489
  1419
  assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)$i \<le> b) net"
hoelzl@37489
  1420
  shows "l$i \<le> b"
wenzelm@49644
  1421
proof -
wenzelm@49644
  1422
  { fix x
wenzelm@49644
  1423
    have "x \<in> {x::real^'n. inner (cart_basis i) x \<le> b} \<longleftrightarrow> x$i \<le> b"
wenzelm@49644
  1424
      unfolding inner_basis by auto }
wenzelm@49644
  1425
  then show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<le> b}" f net l]
hoelzl@37489
  1426
    using closed_halfspace_le[of "(cart_basis i)::real^'n" b] and assms(1,2,3) by auto
hoelzl@37489
  1427
qed
hoelzl@37489
  1428
wenzelm@49644
  1429
lemma Lim_component_ge_cart:
wenzelm@49644
  1430
  fixes f :: "'a \<Rightarrow> real^'n"
hoelzl@37489
  1431
  assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
hoelzl@37489
  1432
  shows "b \<le> l$i"
wenzelm@49644
  1433
proof -
wenzelm@49644
  1434
  { fix x
wenzelm@49644
  1435
    have "x \<in> {x::real^'n. inner (cart_basis i) x \<ge> b} \<longleftrightarrow> x$i \<ge> b"
wenzelm@49644
  1436
      unfolding inner_basis by auto }
wenzelm@49644
  1437
  then show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<ge> b}" f net l]
hoelzl@37489
  1438
    using closed_halfspace_ge[of b "(cart_basis i)::real^'n"] and assms(1,2,3) by auto
hoelzl@37489
  1439
qed
hoelzl@37489
  1440
wenzelm@49644
  1441
lemma Lim_component_eq_cart:
wenzelm@49644
  1442
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@49644
  1443
  assumes net: "(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
hoelzl@37489
  1444
  shows "l$i = b"
wenzelm@49644
  1445
  using ev[unfolded order_eq_iff eventually_conj_iff] and
wenzelm@49644
  1446
    Lim_component_ge_cart[OF net, of b i] and
hoelzl@37489
  1447
    Lim_component_le_cart[OF net, of i b] by auto
hoelzl@37489
  1448
wenzelm@49644
  1449
lemma connected_ivt_component_cart:
wenzelm@49644
  1450
  fixes x :: "real^'n"
wenzelm@49644
  1451
  shows "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s.  z$k = a)"
wenzelm@49644
  1452
  using connected_ivt_hyperplane[of s x y "(cart_basis k)::real^'n" a]
wenzelm@49644
  1453
  by (auto simp add: inner_basis)
hoelzl@37489
  1454
wenzelm@49644
  1455
lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
hoelzl@37489
  1456
  unfolding subspace_def by auto
hoelzl@37489
  1457
hoelzl@37489
  1458
lemma closed_substandard_cart:
huffman@44213
  1459
  "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
  1460
proof -
huffman@44213
  1461
  { fix i::'n
huffman@44213
  1462
    have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
  1463
      by (cases "P i") (simp_all add: closed_Collect_eq) }
huffman@44213
  1464
  thus ?thesis
huffman@44213
  1465
    unfolding Collect_all_eq by (simp add: closed_INT)
hoelzl@37489
  1466
qed
hoelzl@37489
  1467
wenzelm@49644
  1468
lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
wenzelm@49644
  1469
  (is "dim ?A = _")
wenzelm@49644
  1470
proof -
wenzelm@49644
  1471
  have *: "{x. \<forall>i<DIM((real, 'n) vec). i \<notin> \<pi>' ` d \<longrightarrow> x $$ i = 0} =
wenzelm@49644
  1472
      {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0}"
wenzelm@49644
  1473
    apply safe
wenzelm@49644
  1474
    apply (erule_tac x="\<pi>' i" in allE) defer
wenzelm@49644
  1475
    apply (erule_tac x="\<pi> i" in allE)
wenzelm@49644
  1476
    unfolding image_iff real_euclidean_nth[symmetric]
wenzelm@49644
  1477
    apply (auto simp: pi'_inj[THEN inj_eq])
wenzelm@49644
  1478
    done
wenzelm@49644
  1479
  have " \<pi>' ` d \<subseteq> {..<DIM((real, 'n) vec)}"
wenzelm@49644
  1480
    using pi'_range[where 'n='n] by auto
wenzelm@49644
  1481
  thus ?thesis
wenzelm@49644
  1482
    using dim_substandard[of "\<pi>' ` d", where 'a="real^'n"] 
wenzelm@49644
  1483
    unfolding * using card_image[of "\<pi>'" d] using pi'_inj unfolding inj_on_def
wenzelm@49644
  1484
    by auto
hoelzl@37489
  1485
qed
hoelzl@37489
  1486
hoelzl@37489
  1487
lemma affinity_inverses:
hoelzl@37489
  1488
  assumes m0: "m \<noteq> (0::'a::field)"
hoelzl@37489
  1489
  shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
hoelzl@37489
  1490
  "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
hoelzl@37489
  1491
  using m0
wenzelm@49644
  1492
  apply (auto simp add: fun_eq_iff vector_add_ldistrib)
wenzelm@49644
  1493
  apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1[symmetric])
wenzelm@49644
  1494
  done
hoelzl@37489
  1495
hoelzl@37489
  1496
lemma vector_affinity_eq:
hoelzl@37489
  1497
  assumes m0: "(m::'a::field) \<noteq> 0"
hoelzl@37489
  1498
  shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
hoelzl@37489
  1499
proof
hoelzl@37489
  1500
  assume h: "m *s x + c = y"
hoelzl@37489
  1501
  hence "m *s x = y - c" by (simp add: field_simps)
hoelzl@37489
  1502
  hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
hoelzl@37489
  1503
  then show "x = inverse m *s y + - (inverse m *s c)"
hoelzl@37489
  1504
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1505
next
hoelzl@37489
  1506
  assume h: "x = inverse m *s y + - (inverse m *s c)"
hoelzl@37489
  1507
  show "m *s x + c = y" unfolding h diff_minus[symmetric]
hoelzl@37489
  1508
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1509
qed
hoelzl@37489
  1510
hoelzl@37489
  1511
lemma vector_eq_affinity:
wenzelm@49644
  1512
    "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
hoelzl@37489
  1513
  using vector_affinity_eq[where m=m and x=x and y=y and c=c]
hoelzl@37489
  1514
  by metis
hoelzl@37489
  1515
hoelzl@37489
  1516
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<chi>\<chi> i. d)"
hoelzl@37489
  1517
  apply(subst euclidean_eq)
wenzelm@49644
  1518
proof safe
wenzelm@49644
  1519
  case goal1
wenzelm@49644
  1520
  hence *: "(basis i::real^'n) = cart_basis (\<pi> i)"
wenzelm@49644
  1521
    unfolding basis_real_n[symmetric] by auto
hoelzl@37489
  1522
  have "((\<chi> i. d)::real^'n) $$ i = d" unfolding euclidean_component_def *
hoelzl@37489
  1523
    unfolding dot_basis by auto
hoelzl@37489
  1524
  thus ?case using goal1 by auto
hoelzl@37489
  1525
qed
hoelzl@37489
  1526
wenzelm@49644
  1527
huffman@44360
  1528
subsection "Convex Euclidean Space"
hoelzl@37489
  1529
hoelzl@37489
  1530
lemma Cart_1:"(1::real^'n) = (\<chi>\<chi> i. 1)"
hoelzl@37489
  1531
  apply(subst euclidean_eq)
wenzelm@49644
  1532
proof safe
wenzelm@49644
  1533
  case goal1
wenzelm@49644
  1534
  thus ?case
wenzelm@49644
  1535
    using nth_conv_component[THEN sym,where i1="\<pi> i" and x1="1::real^'n"] by auto
hoelzl@37489
  1536
qed
hoelzl@37489
  1537
hoelzl@37489
  1538
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
hoelzl@37489
  1539
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
hoelzl@37489
  1540
huffman@44136
  1541
lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta basis_component vector_uminus_component
hoelzl@37489
  1542
hoelzl@37489
  1543
lemma convex_box_cart:
hoelzl@37489
  1544
  assumes "\<And>i. convex {x. P i x}"
hoelzl@37489
  1545
  shows "convex {x. \<forall>i. P i (x$i)}"
hoelzl@37489
  1546
  using assms unfolding convex_def by auto
hoelzl@37489
  1547
hoelzl@37489
  1548
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
hoelzl@37489
  1549
  by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)
hoelzl@37489
  1550
hoelzl@37489
  1551
lemma unit_interval_convex_hull_cart:
hoelzl@37489
  1552
  "{0::real^'n .. 1} = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" (is "?int = convex hull ?points")
hoelzl@37489
  1553
  unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"]
nipkow@39302
  1554
  apply(rule arg_cong[where f="\<lambda>x. convex hull x"]) apply(rule set_eqI) unfolding mem_Collect_eq
hoelzl@37489
  1555
  apply safe apply(erule_tac x="\<pi>' i" in allE) unfolding nth_conv_component defer
wenzelm@49644
  1556
  apply(erule_tac x="\<pi> i" in allE)
wenzelm@49644
  1557
  apply auto
wenzelm@49644
  1558
  done
hoelzl@37489
  1559
hoelzl@37489
  1560
lemma cube_convex_hull_cart:
wenzelm@49644
  1561
  assumes "0 < d"
wenzelm@49644
  1562
  obtains s::"(real^'n) set"
wenzelm@49644
  1563
    where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s"
wenzelm@49644
  1564
proof -
wenzelm@49644
  1565
  obtain s where s: "finite s" "{x - (\<chi>\<chi> i. d)..x + (\<chi>\<chi> i. d)} = convex hull s"
wenzelm@49644
  1566
    by (rule cube_convex_hull [OF assms])
wenzelm@49644
  1567
  show thesis
wenzelm@49644
  1568
    apply(rule that[OF s(1)]) unfolding s(2)[symmetric] const_vector_cart ..
hoelzl@37489
  1569
qed
hoelzl@37489
  1570
hoelzl@37489
  1571
lemma std_simplex_cart:
hoelzl@37489
  1572
  "(insert (0::real^'n) { cart_basis i | i. i\<in>UNIV}) =
wenzelm@49644
  1573
    (insert 0 { basis i | i. i<DIM((real,'n) vec)})"
wenzelm@49644
  1574
  apply (rule arg_cong[where f="\<lambda>s. (insert 0 s)"])
wenzelm@49644
  1575
  unfolding basis_real_n[symmetric]
wenzelm@49644
  1576
  apply safe
wenzelm@49644
  1577
  apply (rule_tac x="\<pi>' i" in exI) defer
wenzelm@49644
  1578
  apply (rule_tac x="\<pi> i" in exI) using pi'_range[where 'n='n]
wenzelm@49644
  1579
  apply auto
wenzelm@49644
  1580
  done
wenzelm@49644
  1581
hoelzl@37489
  1582
hoelzl@37489
  1583
subsection "Brouwer Fixpoint"
hoelzl@37489
  1584
hoelzl@37489
  1585
lemma kuhn_labelling_lemma_cart:
hoelzl@37489
  1586
  assumes "(\<forall>x::real^_. P x \<longrightarrow> P (f x))"  "\<forall>x. P x \<longrightarrow> (\<forall>i. Q i \<longrightarrow> 0 \<le> x$i \<and> x$i \<le> 1)"
hoelzl@37489
  1587
  shows "\<exists>l. (\<forall>x i. l x i \<le> (1::nat)) \<and>
hoelzl@37489
  1588
             (\<forall>x i. P x \<and> Q i \<and> (x$i = 0) \<longrightarrow> (l x i = 0)) \<and>
hoelzl@37489
  1589
             (\<forall>x i. P x \<and> Q i \<and> (x$i = 1) \<longrightarrow> (l x i = 1)) \<and>
hoelzl@37489
  1590
             (\<forall>x i. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x$i \<le> f(x)$i) \<and>
wenzelm@49644
  1591
             (\<forall>x i. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)$i \<le> x$i)"
wenzelm@49644
  1592
proof -
wenzelm@49644
  1593
  have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)"
wenzelm@49644
  1594
    by auto
wenzelm@49644
  1595
  have *: "\<forall>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow> (x \<noteq> 1 \<and> x \<le> y \<or> x \<noteq> 0 \<and> y \<le> x)"
wenzelm@49644
  1596
    by auto
wenzelm@49644
  1597
  show ?thesis
wenzelm@49644
  1598
    unfolding and_forall_thm apply(subst choice_iff[symmetric])+
wenzelm@49644
  1599
  proof (rule, rule)
wenzelm@49644
  1600
    case goal1
hoelzl@37489
  1601
    let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x $ xa = 0 \<longrightarrow> y = (0::nat)) \<and>
wenzelm@49644
  1602
        (P x \<and> Q xa \<and> x $ xa = 1 \<longrightarrow> y = 1) \<and>
wenzelm@49644
  1603
        (P x \<and> Q xa \<and> y = 0 \<longrightarrow> x $ xa \<le> f x $ xa) \<and>
wenzelm@49644
  1604
        (P x \<and> Q xa \<and> y = 1 \<longrightarrow> f x $ xa \<le> x $ xa)"
wenzelm@49644
  1605
    { assume "P x" "Q xa"
wenzelm@49644
  1606
      hence "0 \<le> f x $ xa \<and> f x $ xa \<le> 1"
wenzelm@49644
  1607
        using assms(2)[rule_format,of "f x" xa]
wenzelm@49644
  1608
        apply (drule_tac assms(1)[rule_format])
wenzelm@49644
  1609
        apply auto
wenzelm@49644
  1610
        done
wenzelm@49644
  1611
    }
wenzelm@49644
  1612
    hence "?R 0 \<or> ?R 1" by auto
wenzelm@49644
  1613
    thus ?case by auto
wenzelm@49644
  1614
  qed
wenzelm@49644
  1615
qed 
hoelzl@37489
  1616
hoelzl@37489
  1617
lemma interval_bij_cart:"interval_bij = (\<lambda> (a,b) (u,v) (x::real^'n).
hoelzl@37489
  1618
    (\<chi> i. u$i + (x$i - a$i) / (b$i - a$i) * (v$i - u$i))::real^'n)"
hoelzl@37489
  1619
  unfolding interval_bij_def apply(rule ext)+ apply safe
huffman@44136
  1620
  unfolding vec_eq_iff vec_lambda_beta unfolding nth_conv_component
wenzelm@49644
  1621
  apply rule
wenzelm@49644
  1622
  apply (subst euclidean_lambda_beta)
wenzelm@49644
  1623
  using pi'_range apply auto
wenzelm@49644
  1624
  done
hoelzl@37489
  1625
hoelzl@37489
  1626
lemma interval_bij_affine_cart:
hoelzl@37489
  1627
 "interval_bij (a,b) (u,v) = (\<lambda>x. (\<chi> i. (v$i - u$i) / (b$i - a$i) * x$i) +
hoelzl@37489
  1628
            (\<chi> i. u$i - (v$i - u$i) / (b$i - a$i) * a$i)::real^'n)"
wenzelm@49644
  1629
  apply rule
wenzelm@49644
  1630
  unfolding vec_eq_iff interval_bij_cart vector_component_simps
wenzelm@49644
  1631
  apply (auto simp add: field_simps add_divide_distrib[symmetric]) 
wenzelm@49644
  1632
  done
wenzelm@49644
  1633
hoelzl@37489
  1634
hoelzl@37489
  1635
subsection "Derivative"
hoelzl@37489
  1636
wenzelm@49644
  1637
lemma has_derivative_vmul_component_cart:
wenzelm@49644
  1638
  fixes c :: "real^'a \<Rightarrow> real^'b" and v :: "real^'c"
hoelzl@37489
  1639
  assumes "(c has_derivative c') net"
huffman@44140
  1640
  shows "((\<lambda>x. c(x)$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$k *\<^sub>R v)) net"
wenzelm@49644
  1641
  unfolding nth_conv_component by (intro has_derivative_intros assms)
hoelzl@37489
  1642
wenzelm@49644
  1643
lemma differentiable_at_imp_differentiable_on:
wenzelm@49644
  1644
  "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
hoelzl@37489
  1645
  unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI)
hoelzl@37489
  1646
hoelzl@37489
  1647
definition "jacobian f net = matrix(frechet_derivative f net)"
hoelzl@37489
  1648
wenzelm@49644
  1649
lemma jacobian_works:
wenzelm@49644
  1650
  "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
wenzelm@49644
  1651
    (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
wenzelm@49644
  1652
  apply rule
wenzelm@49644
  1653
  unfolding jacobian_def
wenzelm@49644
  1654
  apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
wenzelm@49644
  1655
  apply (rule differentiableI)
wenzelm@49644
  1656
  apply assumption
wenzelm@49644
  1657
  unfolding frechet_derivative_works
wenzelm@49644
  1658
  apply assumption
wenzelm@49644
  1659
  done
hoelzl@37489
  1660
hoelzl@37489
  1661
wenzelm@49644
  1662
subsection {* Component of the differential must be zero if it exists at a local
wenzelm@49644
  1663
  maximum or minimum for that corresponding component. *}
hoelzl@37489
  1664
wenzelm@49644
  1665
lemma differential_zero_maxmin_component:
wenzelm@49644
  1666
  fixes f::"real^'a \<Rightarrow> real^'b"
wenzelm@49644
  1667
  assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
wenzelm@49644
  1668
    "f differentiable (at x)" shows "jacobian f (at x) $ k = 0"
wenzelm@49644
  1669
(* FIXME: reuse proof of generic differential_zero_maxmin_component*)
wenzelm@49644
  1670
proof (rule ccontr)
wenzelm@49644
  1671
  def D \<equiv> "jacobian f (at x)"
wenzelm@49644
  1672
  assume "jacobian f (at x) $ k \<noteq> 0"
huffman@44136
  1673
  then obtain j where j:"D$k$j \<noteq> 0" unfolding vec_eq_iff D_def by auto
wenzelm@49644
  1674
  hence *: "abs (jacobian f (at x) $ k $ j) / 2 > 0"
wenzelm@49644
  1675
    unfolding D_def by auto
hoelzl@37489
  1676
  note as = assms(3)[unfolded jacobian_works has_derivative_at_alt]
hoelzl@37489
  1677
  guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this
hoelzl@37489
  1678
  guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this
wenzelm@49644
  1679
  { fix c
wenzelm@49644
  1680
    assume "abs c \<le> d" 
wenzelm@49644
  1681
    hence *:"norm (x + c *\<^sub>R cart_basis j - x) < e'"
wenzelm@49644
  1682
      using norm_basis[of j] d by auto
wenzelm@49644
  1683
    have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le>
wenzelm@49644
  1684
        norm (f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j))" 
wenzelm@49644
  1685
      by (rule component_le_norm_cart)
wenzelm@49644
  1686
    also have "\<dots> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"
wenzelm@49644
  1687
      using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j]
wenzelm@49644
  1688
      unfolding D_def[symmetric] by auto
wenzelm@49644
  1689
    finally
wenzelm@49644
  1690
    have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le>
wenzelm@49644
  1691
      \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" by simp
wenzelm@49644
  1692
    hence "\<bar>f (x + c *\<^sub>R cart_basis j) $ k - f x $ k - c * D $ k $ j\<bar> \<le>
wenzelm@49644
  1693
      \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"
wenzelm@49644
  1694
      unfolding vector_component_simps matrix_vector_mul_component
wenzelm@49644
  1695
      unfolding smult_conv_scaleR[symmetric] 
wenzelm@49644
  1696
      unfolding inner_simps dot_basis smult_conv_scaleR by simp
wenzelm@49644
  1697
  } note * = this
hoelzl@37489
  1698
  have "x + d *\<^sub>R cart_basis j \<in> ball x e" "x - d *\<^sub>R cart_basis j \<in> ball x e"
hoelzl@37489
  1699
    unfolding mem_ball dist_norm using norm_basis[of j] d by auto
wenzelm@49644
  1700
  hence **: "((f (x - d *\<^sub>R cart_basis j))$k \<le> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<le> (f x)$k) \<or>
wenzelm@49644
  1701
      ((f (x - d *\<^sub>R cart_basis j))$k \<ge> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<ge> (f x)$k)"
wenzelm@49644
  1702
    using assms(2) by auto
wenzelm@49644
  1703
  have ***: "\<And>y y1 y2 d dx::real. (y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow>
wenzelm@49644
  1704
    d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
wenzelm@49644
  1705
  show False
wenzelm@49644
  1706
    apply (rule ***[OF **, where dx="d * D $ k $ j" and d="\<bar>D $ k $ j\<bar> / 2 * \<bar>d\<bar>"])
wenzelm@49644
  1707
    using *[of "-d"] and *[of d] and d[THEN conjunct1] and j
wenzelm@49644
  1708
    unfolding mult_minus_left
wenzelm@49644
  1709
    unfolding abs_mult diff_minus_eq_add scaleR_minus_left
wenzelm@49644
  1710
    unfolding algebra_simps
wenzelm@49644
  1711
    apply (auto intro: mult_pos_pos)
wenzelm@49644
  1712
    done
hoelzl@37489
  1713
qed
hoelzl@37489
  1714
wenzelm@49644
  1715
hoelzl@37494
  1716
subsection {* Lemmas for working on @{typ "real^1"} *}
hoelzl@37489
  1717
hoelzl@37489
  1718
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
wenzelm@49644
  1719
  by (metis (full_types) num1_eq_iff)
hoelzl@37489
  1720
hoelzl@37489
  1721
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
wenzelm@49644
  1722
  by auto (metis (full_types) num1_eq_iff)
hoelzl@37489
  1723
hoelzl@37489
  1724
lemma exhaust_2:
wenzelm@49644
  1725
  fixes x :: 2
wenzelm@49644
  1726
  shows "x = 1 \<or> x = 2"
hoelzl@37489
  1727
proof (induct x)
hoelzl@37489
  1728
  case (of_int z)
hoelzl@37489
  1729
  then have "0 <= z" and "z < 2" by simp_all
hoelzl@37489
  1730
  then have "z = 0 | z = 1" by arith
hoelzl@37489
  1731
  then show ?case by auto
hoelzl@37489
  1732
qed
hoelzl@37489
  1733
hoelzl@37489
  1734
lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
hoelzl@37489
  1735
  by (metis exhaust_2)
hoelzl@37489
  1736
hoelzl@37489
  1737
lemma exhaust_3:
wenzelm@49644
  1738
  fixes x :: 3
wenzelm@49644
  1739
  shows "x = 1 \<or> x = 2 \<or> x = 3"
hoelzl@37489
  1740
proof (induct x)
hoelzl@37489
  1741
  case (of_int z)
hoelzl@37489
  1742
  then have "0 <= z" and "z < 3" by simp_all
hoelzl@37489
  1743
  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
hoelzl@37489
  1744
  then show ?case by auto
hoelzl@37489
  1745
qed
hoelzl@37489
  1746
hoelzl@37489
  1747
lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
hoelzl@37489
  1748
  by (metis exhaust_3)
hoelzl@37489
  1749
hoelzl@37489
  1750
lemma UNIV_1 [simp]: "UNIV = {1::1}"
hoelzl@37489
  1751
  by (auto simp add: num1_eq_iff)
hoelzl@37489
  1752
hoelzl@37489
  1753
lemma UNIV_2: "UNIV = {1::2, 2::2}"
hoelzl@37489
  1754
  using exhaust_2 by auto
hoelzl@37489
  1755
hoelzl@37489
  1756
lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
hoelzl@37489
  1757
  using exhaust_3 by auto
hoelzl@37489
  1758
hoelzl@37489
  1759
lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
hoelzl@37489
  1760
  unfolding UNIV_1 by simp
hoelzl@37489
  1761
hoelzl@37489
  1762
lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
hoelzl@37489
  1763
  unfolding UNIV_2 by simp
hoelzl@37489
  1764
hoelzl@37489
  1765
lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
hoelzl@37489
  1766
  unfolding UNIV_3 by (simp add: add_ac)
hoelzl@37489
  1767
wenzelm@49644
  1768
instantiation num1 :: cart_one
wenzelm@49644
  1769
begin
wenzelm@49644
  1770
wenzelm@49644
  1771
instance
wenzelm@49644
  1772
proof
hoelzl@37489
  1773
  show "CARD(1) = Suc 0" by auto
wenzelm@49644
  1774
qed
wenzelm@49644
  1775
wenzelm@49644
  1776
end
hoelzl@37489
  1777
hoelzl@37489
  1778
(* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)
hoelzl@37489
  1779
hoelzl@37489
  1780
abbreviation vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x \<equiv> vec x"
hoelzl@37489
  1781
wenzelm@49644
  1782
abbreviation dest_vec1:: "'a ^1 \<Rightarrow> 'a" where "dest_vec1 x \<equiv> (x$1)"
hoelzl@37489
  1783
huffman@44167
  1784
lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x"
huffman@44167
  1785
  by (simp add: vec_eq_iff)
hoelzl@37489
  1786
hoelzl@37489
  1787
lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))"
hoelzl@37489
  1788
  by (metis vec1_dest_vec1(1))
hoelzl@37489
  1789
hoelzl@37489
  1790
lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))"
hoelzl@37489
  1791
  by (metis vec1_dest_vec1(1))
hoelzl@37489
  1792
hoelzl@37489
  1793
lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y"
hoelzl@37489
  1794
  by (metis vec1_dest_vec1(1))
hoelzl@37489
  1795
wenzelm@49644
  1796
hoelzl@37489
  1797
subsection{* The collapse of the general concepts to dimension one. *}
hoelzl@37489
  1798
hoelzl@37489
  1799
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
huffman@44136
  1800
  by (simp add: vec_eq_iff)
hoelzl@37489
  1801
hoelzl@37489
  1802
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
hoelzl@37489
  1803
  apply auto
hoelzl@37489
  1804
  apply (erule_tac x= "x$1" in allE)
hoelzl@37489
  1805
  apply (simp only: vector_one[symmetric])
hoelzl@37489
  1806
  done
hoelzl@37489
  1807
hoelzl@37489
  1808
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
huffman@44136
  1809
  by (simp add: norm_vec_def)
hoelzl@37489
  1810
hoelzl@37489
  1811
lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
hoelzl@37489
  1812
  by (simp add: norm_vector_1)
hoelzl@37489
  1813
hoelzl@37489
  1814
lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
hoelzl@37489
  1815
  by (auto simp add: norm_real dist_norm)
hoelzl@37489
  1816
wenzelm@49644
  1817
hoelzl@37489
  1818
subsection{* Explicit vector construction from lists. *}
hoelzl@37489
  1819
hoelzl@43995
  1820
definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
hoelzl@37489
  1821
hoelzl@37489
  1822
lemma vector_1: "(vector[x]) $1 = x"
hoelzl@37489
  1823
  unfolding vector_def by simp
hoelzl@37489
  1824
hoelzl@37489
  1825
lemma vector_2:
hoelzl@37489
  1826
 "(vector[x,y]) $1 = x"
hoelzl@37489
  1827
 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
hoelzl@37489
  1828
  unfolding vector_def by simp_all
hoelzl@37489
  1829
hoelzl@37489
  1830
lemma vector_3:
hoelzl@37489
  1831
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
hoelzl@37489
  1832
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
hoelzl@37489
  1833
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
hoelzl@37489
  1834
  unfolding vector_def by simp_all
hoelzl@37489
  1835
hoelzl@37489
  1836
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
hoelzl@37489
  1837
  apply auto
hoelzl@37489
  1838
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1839
  apply (subgoal_tac "vector [v$1] = v")
hoelzl@37489
  1840
  apply simp
hoelzl@37489
  1841
  apply (vector vector_def)
hoelzl@37489
  1842
  apply simp
hoelzl@37489
  1843
  done
hoelzl@37489
  1844
hoelzl@37489
  1845
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
hoelzl@37489
  1846
  apply auto
hoelzl@37489
  1847
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1848
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1849
  apply (subgoal_tac "vector [v$1, v$2] = v")
hoelzl@37489
  1850
  apply simp
hoelzl@37489
  1851
  apply (vector vector_def)
hoelzl@37489
  1852
  apply (simp add: forall_2)
hoelzl@37489
  1853
  done
hoelzl@37489
  1854
hoelzl@37489
  1855
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
hoelzl@37489
  1856
  apply auto
hoelzl@37489
  1857
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1858
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1859
  apply (erule_tac x="v$3" in allE)
hoelzl@37489
  1860
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
hoelzl@37489
  1861
  apply simp
hoelzl@37489
  1862
  apply (vector vector_def)
hoelzl@37489
  1863
  apply (simp add: forall_3)
hoelzl@37489
  1864
  done
hoelzl@37489
  1865
wenzelm@49644
  1866
lemma range_vec1[simp]:"range vec1 = UNIV"
wenzelm@49644
  1867
  apply (rule set_eqI,rule) unfolding image_iff defer
wenzelm@49644
  1868
  apply (rule_tac x="dest_vec1 x" in bexI)
wenzelm@49644
  1869
  apply auto
wenzelm@49644
  1870
  done
hoelzl@37489
  1871
hoelzl@37489
  1872
lemma dest_vec1_lambda: "dest_vec1(\<chi> i. x i) = x 1"
wenzelm@49644
  1873
  by simp
hoelzl@37489
  1874
hoelzl@37489
  1875
lemma dest_vec1_vec: "dest_vec1(vec x) = x"
wenzelm@49644
  1876
  by simp
hoelzl@37489
  1877
hoelzl@37489
  1878
lemma dest_vec1_sum: assumes fS: "finite S"
hoelzl@37489
  1879
  shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"
hoelzl@37489
  1880
  apply (induct rule: finite_induct[OF fS])
hoelzl@37489
  1881
  apply simp
hoelzl@37489
  1882
  apply auto
hoelzl@37489
  1883
  done
hoelzl@37489
  1884
hoelzl@37489
  1885
lemma norm_vec1 [simp]: "norm(vec1 x) = abs(x)"
hoelzl@37489
  1886
  by (simp add: vec_def norm_real)
hoelzl@37489
  1887
hoelzl@37489
  1888
lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)"
huffman@44167
  1889
  by (simp only: dist_real vec_component)
hoelzl@37489
  1890
lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
hoelzl@37489
  1891
  by (metis vec1_dest_vec1(1) norm_vec1)
hoelzl@37489
  1892
wenzelm@49644
  1893
lemmas vec1_dest_vec1_simps =
wenzelm@49644
  1894
  forall_vec1 vec_add[symmetric] dist_vec1 vec_sub[symmetric] vec1_dest_vec1 norm_vec1 vector_smult_component
wenzelm@49644
  1895
  vec_inj[where 'b=1] vec_cmul[symmetric] smult_conv_scaleR[symmetric] o_def dist_real_def real_norm_def
hoelzl@37489
  1896
wenzelm@49644
  1897
lemma bounded_linear_vec1: "bounded_linear (vec1::real\<Rightarrow>real^1)"
hoelzl@37489
  1898
  unfolding bounded_linear_def additive_def bounded_linear_axioms_def 
wenzelm@49644
  1899
  unfolding smult_conv_scaleR[symmetric]
wenzelm@49644
  1900
  unfolding vec1_dest_vec1_simps
wenzelm@49644
  1901
  apply (rule conjI) defer  
wenzelm@49644
  1902
  apply (rule conjI) defer
wenzelm@49644
  1903
  apply (rule_tac x=1 in exI)
wenzelm@49644
  1904
  apply auto
wenzelm@49644
  1905
  done
hoelzl@37489
  1906
hoelzl@37489
  1907
lemma linear_vmul_dest_vec1:
hoelzl@37489
  1908
  fixes f:: "real^_ \<Rightarrow> real^1"
hoelzl@37489
  1909
  shows "linear f \<Longrightarrow> linear (\<lambda>x. dest_vec1(f x) *s v)"
hoelzl@37489
  1910
  unfolding smult_conv_scaleR
hoelzl@37489
  1911
  by (rule linear_vmul_component)
hoelzl@37489
  1912
hoelzl@37489
  1913
lemma linear_from_scalars:
hoelzl@37489
  1914
  assumes lf: "linear (f::real^1 \<Rightarrow> real^_)"
hoelzl@37489
  1915
  shows "f = (\<lambda>x. dest_vec1 x *s column 1 (matrix f))"
hoelzl@37489
  1916
  unfolding smult_conv_scaleR
hoelzl@37489
  1917
  apply (rule ext)
hoelzl@37489
  1918
  apply (subst matrix_works[OF lf, symmetric])
huffman@44136
  1919
  apply (auto simp add: vec_eq_iff matrix_vector_mult_def column_def mult_commute)
hoelzl@37489
  1920
  done
hoelzl@37489
  1921
wenzelm@49644
  1922
lemma linear_to_scalars:
wenzelm@49644
  1923
  assumes lf: "linear (f::real ^'n \<Rightarrow> real^1)"
hoelzl@37489
  1924
  shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
hoelzl@37489
  1925
  apply (rule ext)
hoelzl@37489
  1926
  apply (subst matrix_works[OF lf, symmetric])
huffman@44136
  1927
  apply (simp add: vec_eq_iff matrix_vector_mult_def row_def inner_vec_def mult_commute)
hoelzl@37489
  1928
  done
hoelzl@37489
  1929
hoelzl@37489
  1930
lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
hoelzl@37489
  1931
  by (simp add: dest_vec1_eq[symmetric])
hoelzl@37489
  1932
wenzelm@49644
  1933
lemma setsum_scalars:
wenzelm@49644
  1934
  assumes fS: "finite S"
hoelzl@37489
  1935
  shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)"
hoelzl@37489
  1936
  unfolding vec_setsum[OF fS] by simp
hoelzl@37489
  1937
wenzelm@49644
  1938
lemma dest_vec1_wlog_le:
wenzelm@49644
  1939
  "(\<And>(x::'a::linorder ^ 1) y. P x y \<longleftrightarrow> P y x)
wenzelm@49644
  1940
    \<Longrightarrow> (\<And>x y. dest_vec1 x <= dest_vec1 y ==> P x y) \<Longrightarrow> P x y"
hoelzl@37489
  1941
  apply (cases "dest_vec1 x \<le> dest_vec1 y")
hoelzl@37489
  1942
  apply simp
hoelzl@37489
  1943
  apply (subgoal_tac "dest_vec1 y \<le> dest_vec1 x")
wenzelm@49644
  1944
  apply auto
hoelzl@37489
  1945
  done
hoelzl@37489
  1946
hoelzl@37489
  1947
text{* Lifting and dropping *}
hoelzl@37489
  1948
wenzelm@49644
  1949
lemma continuous_on_o_dest_vec1:
wenzelm@49644
  1950
  fixes f::"real \<Rightarrow> 'a::real_normed_vector"
wenzelm@49644
  1951
  assumes "continuous_on {a..b::real} f"
wenzelm@49644
  1952
  shows "continuous_on {vec1 a..vec1 b} (f o dest_vec1)"
hoelzl@37489
  1953
  using assms unfolding continuous_on_iff apply safe
wenzelm@49644
  1954
  apply (erule_tac x="x$1" in ballE,erule_tac x=e in allE)
wenzelm@49644
  1955
  apply safe
wenzelm@49644
  1956
  apply (rule_tac x=d in exI)
wenzelm@49644
  1957
  apply safe
wenzelm@49644
  1958
  unfolding o_def dist_real_def dist_real
wenzelm@49644
  1959
  apply (erule_tac x="dest_vec1 x'" in ballE)
wenzelm@49644
  1960
  apply (auto simp add:less_eq_vec_def)
wenzelm@49644
  1961
  done
hoelzl@37489
  1962
wenzelm@49644
  1963
lemma continuous_on_o_vec1:
wenzelm@49644
  1964
  fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
wenzelm@49644
  1965
  assumes "continuous_on {a..b} f"
wenzelm@49644
  1966
  shows "continuous_on {dest_vec1 a..dest_vec1 b} (f o vec1)"
wenzelm@49644
  1967
  using assms unfolding continuous_on_iff
wenzelm@49644
  1968
  apply safe
wenzelm@49644
  1969
  apply (erule_tac x="vec x" in ballE,erule_tac x=e in allE)
wenzelm@49644
  1970
  apply safe
wenzelm@49644
  1971
  apply (rule_tac x=d in exI)
wenzelm@49644
  1972
  apply safe
wenzelm@49644
  1973
  unfolding o_def dist_real_def dist_real
wenzelm@49644
  1974
  apply (erule_tac x="vec1 x'" in ballE)
wenzelm@49644
  1975
  apply (auto simp add:less_eq_vec_def)
wenzelm@49644
  1976
  done
hoelzl@37489
  1977
hoelzl@37489
  1978
lemma continuous_on_vec1:"continuous_on A (vec1::real\<Rightarrow>real^1)"
wenzelm@49644
  1979
  by (rule linear_continuous_on[OF bounded_linear_vec1])
hoelzl@37489
  1980
wenzelm@49644
  1981
lemma mem_interval_1:
wenzelm@49644
  1982
  fixes x :: "real^1"
wenzelm@49644
  1983
  shows "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
wenzelm@49644
  1984
    and "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
wenzelm@49644
  1985
  by (simp_all add: vec_eq_iff less_vec_def less_eq_vec_def)
hoelzl@37489
  1986
wenzelm@49644
  1987
lemma vec1_interval:
wenzelm@49644
  1988
  fixes a::"real"
wenzelm@49644
  1989
  shows "vec1 ` {a .. b} = {vec1 a .. vec1 b}"
wenzelm@49644
  1990
    and "vec1 ` {a<..<b} = {vec1 a<..<vec1 b}"
wenzelm@49644
  1991
  apply (rule_tac[!] set_eqI)
wenzelm@49644
  1992
  unfolding image_iff less_vec_def
wenzelm@49644
  1993
  unfolding mem_interval_cart
wenzelm@49644
  1994
  unfolding forall_1 vec1_dest_vec1_simps
wenzelm@49644
  1995
  apply rule defer
wenzelm@49644
  1996
  apply (rule_tac x="dest_vec1 x" in bexI) prefer 3
wenzelm@49644
  1997
  apply rule defer
wenzelm@49644
  1998
  apply (rule_tac x="dest_vec1 x" in bexI)
wenzelm@49644
  1999
  apply auto
wenzelm@49644
  2000
  done
hoelzl@37489
  2001
hoelzl@37489
  2002
(* Some special cases for intervals in R^1.                                  *)
hoelzl@37489
  2003
wenzelm@49644
  2004
lemma interval_cases_1:
wenzelm@49644
  2005
  fixes x :: "real^1"
wenzelm@49644
  2006
  shows "x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"
wenzelm@49644
  2007
  unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart
wenzelm@49644
  2008
  by (auto simp del:dest_vec1_eq)
hoelzl@37489
  2009
wenzelm@49644
  2010
lemma in_interval_1:
wenzelm@49644
  2011
  fixes x :: "real^1"
wenzelm@49644
  2012
  shows "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b) \<and>
wenzelm@49644
  2013
    (x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
wenzelm@49644
  2014
  unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart
wenzelm@49644
  2015
  by (auto simp del:dest_vec1_eq)
hoelzl@37489
  2016
wenzelm@49644
  2017
lemma interval_eq_empty_1:
wenzelm@49644
  2018
  fixes a :: "real^1"
wenzelm@49644
  2019
  shows "{a .. b} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a"
wenzelm@49644
  2020
    and "{a<..<b} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
hoelzl@37489
  2021
  unfolding interval_eq_empty_cart and ex_1 by auto
hoelzl@37489
  2022
wenzelm@49644
  2023
lemma subset_interval_1:
wenzelm@49644
  2024
  fixes a :: "real^1"
wenzelm@49644
  2025
  shows "({a .. b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>
wenzelm@49644
  2026
    dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
wenzelm@49644
  2027
   "({a .. b} \<subseteq> {c<..<d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>
wenzelm@49644
  2028
    dest_vec1 c < dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b < dest_vec1 d)"
wenzelm@49644
  2029
   "({a<..<b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b \<le> dest_vec1 a \<or>
wenzelm@49644
  2030
    dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
wenzelm@49644
  2031
   "({a<..<b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
wenzelm@49644
  2032
    dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
hoelzl@37489
  2033
  unfolding subset_interval_cart[of a b c d] unfolding forall_1 by auto
hoelzl@37489
  2034
wenzelm@49644
  2035
lemma eq_interval_1:
wenzelm@49644
  2036
  fixes a :: "real^1"
wenzelm@49644
  2037
  shows "{a .. b} = {c .. d} \<longleftrightarrow>
hoelzl@37489
  2038
          dest_vec1 b < dest_vec1 a \<and> dest_vec1 d < dest_vec1 c \<or>
hoelzl@37489
  2039
          dest_vec1 a = dest_vec1 c \<and> dest_vec1 b = dest_vec1 d"
wenzelm@49644
  2040
  unfolding set_eq_subset[of "{a .. b}" "{c .. d}"]
wenzelm@49644
  2041
  unfolding subset_interval_1(1)[of a b c d]
wenzelm@49644
  2042
  unfolding subset_interval_1(1)[of c d a b]
wenzelm@49644
  2043
  by auto
hoelzl@37489
  2044
wenzelm@49644
  2045
lemma disjoint_interval_1:
wenzelm@49644
  2046
  fixes a :: "real^1"
wenzelm@49644
  2047
  shows
wenzelm@49644
  2048
    "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow>
wenzelm@49644
  2049
      dest_vec1 b < dest_vec1 a \<or> dest_vec1 d < dest_vec1 c  \<or>  dest_vec1 b < dest_vec1 c \<or> dest_vec1 d < dest_vec1 a"
wenzelm@49644
  2050
    "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow>
wenzelm@49644
  2051
      dest_vec1 b < dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
wenzelm@49644
  2052
    "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow>
wenzelm@49644
  2053
      dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d < dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
wenzelm@49644
  2054
    "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow>
wenzelm@49644
  2055
      dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
hoelzl@37489
  2056
  unfolding disjoint_interval_cart and ex_1 by auto
hoelzl@37489
  2057
wenzelm@49644
  2058
lemma open_closed_interval_1:
wenzelm@49644
  2059
  fixes a :: "real^1"
wenzelm@49644
  2060
  shows "{a<..<b} = {a .. b} - {a, b}"
wenzelm@49644
  2061
  unfolding set_eq_iff apply simp
wenzelm@49644
  2062
  unfolding less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[symmetric]
wenzelm@49644
  2063
  apply (auto simp del:dest_vec1_eq)
wenzelm@49644
  2064
  done
hoelzl@37489
  2065
wenzelm@49644
  2066
lemma closed_open_interval_1:
wenzelm@49644
  2067
  "dest_vec1 (a::real^1) \<le> dest_vec1 b ==> {a .. b} = {a<..<b} \<union> {a,b}"
wenzelm@49644
  2068
  unfolding set_eq_iff
wenzelm@49644
  2069
  apply simp
wenzelm@49644
  2070
  unfolding less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[symmetric]
wenzelm@49644
  2071
  apply (auto simp del:dest_vec1_eq)
wenzelm@49644
  2072
  done
hoelzl@37489
  2073
wenzelm@49644
  2074
lemma Lim_drop_le:
wenzelm@49644
  2075
  fixes f :: "'a \<Rightarrow> real^1"
wenzelm@49644
  2076
  shows "(f ---> l) net \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow>
wenzelm@49644
  2077
    eventually (\<lambda>x. dest_vec1 (f x) \<le> b) net ==> dest_vec1 l \<le> b"
hoelzl@37489
  2078
  using Lim_component_le_cart[of f l net 1 b] by auto
hoelzl@37489
  2079
wenzelm@49644
  2080
lemma Lim_drop_ge:
wenzelm@49644
  2081
  fixes f :: "'a \<Rightarrow> real^1"
wenzelm@49644
  2082
  shows "(f ---> l) net \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow>
wenzelm@49644
  2083
    eventually (\<lambda>x. b \<le> dest_vec1 (f x)) net ==> b \<le> dest_vec1 l"
hoelzl@37489
  2084
  using Lim_component_ge_cart[of f l net b 1] by auto
hoelzl@37489
  2085
wenzelm@49644
  2086
hoelzl@37489
  2087
text{* Also more convenient formulations of monotone convergence.                *}
hoelzl@37489
  2088
wenzelm@49644
  2089
lemma bounded_increasing_convergent:
wenzelm@49644
  2090
  fixes s :: "nat \<Rightarrow> real^1"
hoelzl@37489
  2091
  assumes "bounded {s n| n::nat. True}"  "\<forall>n. dest_vec1(s n) \<le> dest_vec1(s(Suc n))"
hoelzl@37489
  2092
  shows "\<exists>l. (s ---> l) sequentially"
wenzelm@49644
  2093
proof -
wenzelm@49644
  2094
  obtain a where a:"\<forall>n. \<bar>dest_vec1 (s n)\<bar> \<le>  a"
wenzelm@49644
  2095
    using assms(1)[unfolded bounded_iff abs_dest_vec1] by auto
hoelzl@37489
  2096
  { fix m::nat
hoelzl@37489
  2097
    have "\<And> n. n\<ge>m \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
wenzelm@49644
  2098
      apply (induct_tac n)
wenzelm@49644
  2099
      apply simp
wenzelm@49644
  2100
      using assms(2) apply (erule_tac x="na" in allE)
wenzelm@49644
  2101
      apply (auto simp add: not_less_eq_eq)
wenzelm@49644
  2102
      done
wenzelm@49644
  2103
  }
wenzelm@49644
  2104
  hence "\<forall>m n. m \<le> n \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
wenzelm@49644
  2105
    by auto
wenzelm@49644
  2106
  then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>dest_vec1 (s n) - l\<bar> < e"
wenzelm@49644
  2107
    using convergent_bounded_monotone[OF a] unfolding monoseq_def by auto
huffman@44907
  2108
  thus ?thesis unfolding LIMSEQ_def apply(rule_tac x="vec1 l" in exI)
wenzelm@49644
  2109
    unfolding dist_norm unfolding abs_dest_vec1 by auto
hoelzl@37489
  2110
qed
hoelzl@37489
  2111
wenzelm@49644
  2112
lemma dest_vec1_simps[simp]:
wenzelm@49644
  2113
  fixes a :: "real^1"
hoelzl@37489
  2114
  shows "a$1 = 0 \<longleftrightarrow> a = 0" (*"a \<le> 1 \<longleftrightarrow> dest_vec1 a \<le> 1" "0 \<le> a \<longleftrightarrow> 0 \<le> dest_vec1 a"*)
wenzelm@49644
  2115
    "a \<le> b \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 b" "dest_vec1 (1::real^1) = 1"
wenzelm@49644
  2116
  by (auto simp add: less_eq_vec_def vec_eq_iff)
hoelzl@37489
  2117
hoelzl@37489
  2118
lemma dest_vec1_inverval:
hoelzl@37489
  2119
  "dest_vec1 ` {a .. b} = {dest_vec1 a .. dest_vec1 b}"
hoelzl@37489
  2120
  "dest_vec1 ` {a<.. b} = {dest_vec1 a<.. dest_vec1 b}"
hoelzl@37489
  2121
  "dest_vec1 ` {a ..<b} = {dest_vec1 a ..<dest_vec1 b}"
hoelzl@37489
  2122
  "dest_vec1 ` {a<..<b} = {dest_vec1 a<..<dest_vec1 b}"
hoelzl@37489
  2123
  apply(rule_tac [!] equalityI)
hoelzl@37489
  2124
  unfolding subset_eq Ball_def Bex_def mem_interval_1 image_iff
hoelzl@37489
  2125
  apply(rule_tac [!] allI)apply(rule_tac [!] impI)
hoelzl@37489
  2126
  apply(rule_tac[2] x="vec1 x" in exI)apply(rule_tac[4] x="vec1 x" in exI)
hoelzl@37489
  2127
  apply(rule_tac[6] x="vec1 x" in exI)apply(rule_tac[8] x="vec1 x" in exI)
wenzelm@49644
  2128
  apply (auto simp add: less_vec_def less_eq_vec_def)
wenzelm@49644
  2129
  done
hoelzl@37489
  2130
wenzelm@49644
  2131
lemma dest_vec1_setsum:
wenzelm@49644
  2132
  assumes "finite S"
hoelzl@37489
  2133
  shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
hoelzl@37489
  2134
  using dest_vec1_sum[OF assms] by auto
hoelzl@37489
  2135
hoelzl@37489
  2136
lemma open_dest_vec1_vimage: "open S \<Longrightarrow> open (dest_vec1 -` S)"
wenzelm@49644
  2137
  unfolding open_vec_def forall_1 by auto
hoelzl@37489
  2138
hoelzl@37489
  2139
lemma tendsto_dest_vec1 [tendsto_intros]:
hoelzl@37489
  2140
  "(f ---> l) net \<Longrightarrow> ((\<lambda>x. dest_vec1 (f x)) ---> dest_vec1 l) net"
wenzelm@49644
  2141
  by (rule tendsto_vec_nth)
hoelzl@37489
  2142
wenzelm@49644
  2143
lemma continuous_dest_vec1:
wenzelm@49644
  2144
  "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
hoelzl@37489
  2145
  unfolding continuous_def by (rule tendsto_dest_vec1)
hoelzl@37489
  2146
hoelzl@37489
  2147
lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))" 
wenzelm@49644
  2148
  apply safe defer
wenzelm@49644
  2149
  apply (erule_tac x="vec1 x" in allE)
wenzelm@49644
  2150
  apply auto
wenzelm@49644
  2151
  done
hoelzl@37489
  2152
hoelzl@37489
  2153
lemma forall_of_dest_vec1: "(\<forall>v. P (\<lambda>x. dest_vec1 (v x))) \<longleftrightarrow> (\<forall>x. P x)"
wenzelm@49644
  2154
  apply rule
wenzelm@49644
  2155
  apply rule
wenzelm@49644
  2156
  apply (erule_tac x="vec1 \<circ> x" in allE)
wenzelm@49644
  2157
  unfolding o_def vec1_dest_vec1
wenzelm@49644
  2158
  apply auto
wenzelm@49644
  2159
  done
hoelzl@37489
  2160
hoelzl@37489
  2161
lemma forall_of_dest_vec1': "(\<forall>v. P (dest_vec1 v)) \<longleftrightarrow> (\<forall>x. P x)"
wenzelm@49644
  2162
  apply rule
wenzelm@49644
  2163
  apply rule
wenzelm@49644
  2164
  apply (erule_tac x="(vec1 x)" in allE) defer
wenzelm@49644
  2165
  apply rule 
wenzelm@49644
  2166
  apply (erule_tac x="dest_vec1 v" in allE)
wenzelm@49644
  2167
  unfolding o_def vec1_dest_vec1
wenzelm@49644
  2168
  apply auto
wenzelm@49644
  2169
  done
hoelzl@37489
  2170
wenzelm@49644
  2171
lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x"
wenzelm@49644
  2172
  unfolding dist_norm by auto
hoelzl@37489
  2173
wenzelm@49644
  2174
lemma bounded_linear_vec1_dest_vec1:
wenzelm@49644
  2175
  fixes f :: "real \<Rightarrow> real"
wenzelm@49644
  2176
  shows "linear (vec1 \<circ> f \<circ> dest_vec1) = bounded_linear f" (is "?l = ?r")
wenzelm@49644
  2177
proof -
wenzelm@49644
  2178
  { assume ?l
wenzelm@49644
  2179
    then have "\<exists>K. \<forall>x. norm ((vec1 \<circ> f \<circ> dest_vec1) x) \<le> K * norm x" by (rule linear_bounded)
wenzelm@49644
  2180
    then guess K ..
wenzelm@49644
  2181
    hence "\<exists>K. \<forall>x. \<bar>f x\<bar> \<le> \<bar>x\<bar> * K"
wenzelm@49644
  2182
      apply(rule_tac x=K in exI)
wenzelm@49644
  2183
      unfolding vec1_dest_vec1_simps by (auto simp add:field_simps)
wenzelm@49644
  2184
  }
wenzelm@49644
  2185
  thus ?thesis
wenzelm@49644
  2186
    unfolding linear_def bounded_linear_def additive_def bounded_linear_axioms_def o_def
wenzelm@49644
  2187
    unfolding vec1_dest_vec1_simps by auto
wenzelm@49644
  2188
qed
hoelzl@37489
  2189
wenzelm@49644
  2190
lemma vec1_le[simp]: fixes a :: real shows "vec1 a \<le> vec1 b \<longleftrightarrow> a \<le> b"
huffman@44136
  2191
  unfolding less_eq_vec_def by auto
wenzelm@49644
  2192
lemma vec1_less[simp]: fixes a :: real shows "vec1 a < vec1 b \<longleftrightarrow> a < b"
huffman@44136
  2193
  unfolding less_vec_def by auto
hoelzl@37489
  2194
hoelzl@37489
  2195
hoelzl@37489
  2196
subsection {* Derivatives on real = Derivatives on @{typ "real^1"} *}
hoelzl@37489
  2197
wenzelm@49644
  2198
lemma has_derivative_within_vec1_dest_vec1:
wenzelm@49644
  2199
  fixes f :: "real \<Rightarrow> real"
wenzelm@49644
  2200
  shows "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s)
wenzelm@49644
  2201
    = (f has_derivative f') (at x within s)"
wenzelm@49644
  2202
  unfolding has_derivative_within
wenzelm@49644
  2203
  unfolding bounded_linear_vec1_dest_vec1[unfolded linear_conv_bounded_linear]
hoelzl@37489
  2204
  unfolding o_def Lim_within Ball_def unfolding forall_vec1 
wenzelm@49644
  2205
  unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff
wenzelm@49644
  2206
  by auto
hoelzl@37489
  2207
wenzelm@49644
  2208
lemma has_derivative_at_vec1_dest_vec1:
wenzelm@49644
  2209
  fixes f :: "real \<Rightarrow> real"
wenzelm@49644
  2210
  shows "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
wenzelm@49644
  2211
  using has_derivative_within_vec1_dest_vec1[where s=UNIV, unfolded range_vec1 within_UNIV]
wenzelm@49644
  2212
  by auto
hoelzl@37489
  2213
wenzelm@49644
  2214
lemma bounded_linear_vec1':
wenzelm@49644
  2215
  fixes f :: "'a::real_normed_vector\<Rightarrow>real"
hoelzl@37489
  2216
  shows "bounded_linear f = bounded_linear (vec1 \<circ> f)"
hoelzl@37489
  2217
  unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
hoelzl@37489
  2218
  unfolding vec1_dest_vec1_simps by auto
hoelzl@37489
  2219
wenzelm@49644
  2220
lemma bounded_linear_dest_vec1:
wenzelm@49644
  2221
  fixes f :: "real\<Rightarrow>'a::real_normed_vector"
hoelzl@37489
  2222
  shows "bounded_linear f = bounded_linear (f \<circ> dest_vec1)"
hoelzl@37489
  2223
  unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
wenzelm@49644
  2224
  unfolding vec1_dest_vec1_simps
wenzelm@49644
  2225
  by auto
hoelzl@37489
  2226
wenzelm@49644
  2227
lemma has_derivative_at_vec1:
wenzelm@49644
  2228
  fixes f :: "'a::real_normed_vector\<Rightarrow>real"
wenzelm@49644
  2229
  shows "(f has_derivative f') (at x) = ((vec1 \<circ> f) has_derivative (vec1 \<circ> f')) (at x)"
wenzelm@49644
  2230
  unfolding has_derivative_at
wenzelm@49644
  2231
  unfolding bounded_linear_vec1'[unfolded linear_conv_bounded_linear]
wenzelm@49644
  2232
  unfolding o_def Lim_at
wenzelm@49644
  2233
  unfolding vec1_dest_vec1_simps dist_vec1_0
wenzelm@49644
  2234
  by auto
hoelzl@37489
  2235
wenzelm@49644
  2236
lemma has_derivative_within_dest_vec1:
wenzelm@49644
  2237
  fixes f :: "real\<Rightarrow>'a::real_normed_vector"
wenzelm@49644
  2238
  shows "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s) =
wenzelm@49644
  2239
    (f has_derivative f') (at x within s)"
wenzelm@49644
  2240
  unfolding has_derivative_within bounded_linear_dest_vec1
wenzelm@49644
  2241
  unfolding o_def Lim_w