src/HOL/Analysis/Cartesian_Euclidean_Space.thy
author paulson <lp15@cam.ac.uk>
Sun Apr 15 21:41:40 2018 +0100 (14 months ago)
changeset 67986 b65c4a6a015e
parent 67982 7643b005b29a
child 68038 20b713cff87a
child 68072 493b818e8e10
permissions -rw-r--r--
quite a few more results about negligibility, etc., and a bit of tidying up
nipkow@67968
     1
section \<open>Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space\<close>
hoelzl@37489
     2
hoelzl@37489
     3
theory Cartesian_Euclidean_Space
immler@67685
     4
imports Finite_Cartesian_Product Derivative
hoelzl@37489
     5
begin
hoelzl@37489
     6
lp15@67982
     7
lemma norm_le_componentwise:
lp15@67982
     8
   "(\<And>b. b \<in> Basis \<Longrightarrow> abs(x \<bullet> b) \<le> abs(y \<bullet> b)) \<Longrightarrow> norm x \<le> norm y"
lp15@67982
     9
  by (auto simp: norm_le euclidean_inner [of x x] euclidean_inner [of y y] abs_le_square_iff power2_eq_square intro!: sum_mono)
lp15@67982
    10
lp15@67982
    11
lemma norm_le_componentwise_cart:
lp15@67982
    12
  fixes x :: "real^'n"
lp15@67982
    13
  shows "(\<And>i. abs(x$i) \<le> abs(y$i)) \<Longrightarrow> norm x \<le> norm y"
lp15@67982
    14
  unfolding cart_eq_inner_axis
lp15@67982
    15
  by (rule norm_le_componentwise) (metis axis_index)
lp15@67982
    16
  
lp15@63016
    17
lemma subspace_special_hyperplane: "subspace {x. x $ k = 0}"
lp15@63016
    18
  by (simp add: subspace_def)
lp15@63016
    19
nipkow@64267
    20
lemma sum_mult_product:
nipkow@64267
    21
  "sum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
nipkow@64267
    22
  unfolding sum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]
nipkow@64267
    23
proof (rule sum.cong, simp, rule sum.reindex_cong)
wenzelm@49644
    24
  fix i
wenzelm@49644
    25
  show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
hoelzl@37489
    26
  show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
hoelzl@37489
    27
  proof safe
hoelzl@37489
    28
    fix j assume "j \<in> {i * B..<i * B + B}"
wenzelm@49644
    29
    then show "j \<in> (\<lambda>j. j + i * B) ` {..<B}"
hoelzl@37489
    30
      by (auto intro!: image_eqI[of _ _ "j - i * B"])
hoelzl@37489
    31
  qed simp
hoelzl@37489
    32
qed simp
hoelzl@37489
    33
nipkow@67968
    34
subsection\<open>Basic componentwise operations on vectors\<close>
hoelzl@37489
    35
huffman@44136
    36
instantiation vec :: (times, finite) times
hoelzl@37489
    37
begin
wenzelm@49644
    38
nipkow@67399
    39
definition "( * ) \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
wenzelm@49644
    40
instance ..
wenzelm@49644
    41
hoelzl@37489
    42
end
hoelzl@37489
    43
huffman@44136
    44
instantiation vec :: (one, finite) one
hoelzl@37489
    45
begin
wenzelm@49644
    46
wenzelm@49644
    47
definition "1 \<equiv> (\<chi> i. 1)"
wenzelm@49644
    48
instance ..
wenzelm@49644
    49
hoelzl@37489
    50
end
hoelzl@37489
    51
huffman@44136
    52
instantiation vec :: (ord, finite) ord
hoelzl@37489
    53
begin
wenzelm@49644
    54
wenzelm@49644
    55
definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"
immler@54776
    56
definition "x < (y::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
wenzelm@49644
    57
instance ..
wenzelm@49644
    58
hoelzl@37489
    59
end
hoelzl@37489
    60
wenzelm@60420
    61
text\<open>The ordering on one-dimensional vectors is linear.\<close>
hoelzl@37489
    62
wenzelm@49197
    63
class cart_one =
wenzelm@61076
    64
  assumes UNIV_one: "card (UNIV :: 'a set) = Suc 0"
hoelzl@37489
    65
begin
wenzelm@49197
    66
wenzelm@49197
    67
subclass finite
wenzelm@49197
    68
proof
wenzelm@49197
    69
  from UNIV_one show "finite (UNIV :: 'a set)"
wenzelm@49197
    70
    by (auto intro!: card_ge_0_finite)
wenzelm@49197
    71
qed
wenzelm@49197
    72
hoelzl@37489
    73
end
hoelzl@37489
    74
immler@54776
    75
instance vec:: (order, finite) order
wenzelm@61169
    76
  by standard (auto simp: less_eq_vec_def less_vec_def vec_eq_iff
immler@54776
    77
      intro: order.trans order.antisym order.strict_implies_order)
wenzelm@49197
    78
immler@54776
    79
instance vec :: (linorder, cart_one) linorder
wenzelm@49197
    80
proof
wenzelm@49197
    81
  obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"
wenzelm@49197
    82
  proof -
wenzelm@49197
    83
    have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)
wenzelm@49197
    84
    then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)
wenzelm@49197
    85
    then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto
wenzelm@49197
    86
    then show thesis by (auto intro: that)
wenzelm@49197
    87
  qed
immler@54776
    88
  fix x y :: "'a^'b::cart_one"
wenzelm@49197
    89
  note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
immler@54776
    90
  show "x \<le> y \<or> y \<le> x" by auto
wenzelm@49197
    91
qed
wenzelm@49197
    92
wenzelm@60420
    93
text\<open>Constant Vectors\<close>
hoelzl@37489
    94
hoelzl@37489
    95
definition "vec x = (\<chi> i. x)"
hoelzl@37489
    96
immler@56188
    97
lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b"
immler@56188
    98
  by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis)
immler@56188
    99
wenzelm@60420
   100
text\<open>Also the scalar-vector multiplication.\<close>
hoelzl@37489
   101
hoelzl@37489
   102
definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
hoelzl@37489
   103
  where "c *s x = (\<chi> i. c * (x$i))"
hoelzl@37489
   104
wenzelm@49644
   105
nipkow@67968
   106
subsection \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space\<close>
hoelzl@37489
   107
nipkow@64267
   108
lemma sum_cong_aux:
nipkow@64267
   109
  "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> sum f A = sum g A"
nipkow@64267
   110
  by (auto intro: sum.cong)
haftmann@57418
   111
nipkow@64267
   112
hide_fact (open) sum_cong_aux
haftmann@57418
   113
wenzelm@60420
   114
method_setup vector = \<open>
hoelzl@37489
   115
let
wenzelm@51717
   116
  val ss1 =
wenzelm@51717
   117
    simpset_of (put_simpset HOL_basic_ss @{context}
nipkow@64267
   118
      addsimps [@{thm sum.distrib} RS sym,
nipkow@64267
   119
      @{thm sum_subtractf} RS sym, @{thm sum_distrib_left},
nipkow@64267
   120
      @{thm sum_distrib_right}, @{thm sum_negf} RS sym])
wenzelm@51717
   121
  val ss2 =
wenzelm@51717
   122
    simpset_of (@{context} addsimps
huffman@44136
   123
             [@{thm plus_vec_def}, @{thm times_vec_def},
huffman@44136
   124
              @{thm minus_vec_def}, @{thm uminus_vec_def},
huffman@44136
   125
              @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
huffman@44136
   126
              @{thm scaleR_vec_def},
wenzelm@51717
   127
              @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}])
wenzelm@51717
   128
  fun vector_arith_tac ctxt ths =
wenzelm@51717
   129
    simp_tac (put_simpset ss1 ctxt)
nipkow@64267
   130
    THEN' (fn i => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.sum_cong_aux} i
nipkow@64267
   131
         ORELSE resolve_tac ctxt @{thms sum.neutral} i
wenzelm@51717
   132
         ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i)
wenzelm@49644
   133
    (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
wenzelm@51717
   134
    THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths)
wenzelm@49644
   135
in
wenzelm@51717
   136
  Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths))
wenzelm@49644
   137
end
wenzelm@60420
   138
\<close> "lift trivial vector statements to real arith statements"
hoelzl@37489
   139
wenzelm@57865
   140
lemma vec_0[simp]: "vec 0 = 0" by vector
wenzelm@57865
   141
lemma vec_1[simp]: "vec 1 = 1" by vector
hoelzl@37489
   142
hoelzl@37489
   143
lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
hoelzl@37489
   144
hoelzl@37489
   145
lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
hoelzl@37489
   146
wenzelm@57865
   147
lemma vec_add: "vec(x + y) = vec x + vec y"  by vector
wenzelm@57865
   148
lemma vec_sub: "vec(x - y) = vec x - vec y" by vector
wenzelm@57865
   149
lemma vec_cmul: "vec(c * x) = c *s vec x " by vector
wenzelm@57865
   150
lemma vec_neg: "vec(- x) = - vec x " by vector
hoelzl@37489
   151
lp15@67979
   152
lemma vec_scaleR: "vec(c * x) = c *\<^sub>R vec x"
lp15@67979
   153
  by vector
lp15@67979
   154
nipkow@64267
   155
lemma vec_sum:
wenzelm@49644
   156
  assumes "finite S"
nipkow@64267
   157
  shows "vec(sum f S) = sum (vec \<circ> f) S"
wenzelm@49644
   158
  using assms
wenzelm@49644
   159
proof induct
wenzelm@49644
   160
  case empty
wenzelm@49644
   161
  then show ?case by simp
wenzelm@49644
   162
next
wenzelm@49644
   163
  case insert
wenzelm@49644
   164
  then show ?case by (auto simp add: vec_add)
wenzelm@49644
   165
qed
hoelzl@37489
   166
wenzelm@60420
   167
text\<open>Obvious "component-pushing".\<close>
hoelzl@37489
   168
hoelzl@37489
   169
lemma vec_component [simp]: "vec x $ i = x"
wenzelm@57865
   170
  by vector
hoelzl@37489
   171
hoelzl@37489
   172
lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
hoelzl@37489
   173
  by vector
hoelzl@37489
   174
hoelzl@37489
   175
lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
hoelzl@37489
   176
  by vector
hoelzl@37489
   177
hoelzl@37489
   178
lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
hoelzl@37489
   179
hoelzl@37489
   180
lemmas vector_component =
hoelzl@37489
   181
  vec_component vector_add_component vector_mult_component
hoelzl@37489
   182
  vector_smult_component vector_minus_component vector_uminus_component
hoelzl@37489
   183
  vector_scaleR_component cond_component
hoelzl@37489
   184
wenzelm@49644
   185
nipkow@67968
   186
subsection \<open>Some frequently useful arithmetic lemmas over vectors\<close>
hoelzl@37489
   187
huffman@44136
   188
instance vec :: (semigroup_mult, finite) semigroup_mult
wenzelm@61169
   189
  by standard (vector mult.assoc)
hoelzl@37489
   190
huffman@44136
   191
instance vec :: (monoid_mult, finite) monoid_mult
wenzelm@61169
   192
  by standard vector+
hoelzl@37489
   193
huffman@44136
   194
instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
wenzelm@61169
   195
  by standard (vector mult.commute)
hoelzl@37489
   196
huffman@44136
   197
instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
wenzelm@61169
   198
  by standard vector
hoelzl@37489
   199
huffman@44136
   200
instance vec :: (semiring, finite) semiring
wenzelm@61169
   201
  by standard (vector field_simps)+
hoelzl@37489
   202
huffman@44136
   203
instance vec :: (semiring_0, finite) semiring_0
wenzelm@61169
   204
  by standard (vector field_simps)+
huffman@44136
   205
instance vec :: (semiring_1, finite) semiring_1
wenzelm@61169
   206
  by standard vector
huffman@44136
   207
instance vec :: (comm_semiring, finite) comm_semiring
wenzelm@61169
   208
  by standard (vector field_simps)+
hoelzl@37489
   209
huffman@44136
   210
instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
huffman@44136
   211
instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
huffman@44136
   212
instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
huffman@44136
   213
instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
huffman@44136
   214
instance vec :: (ring, finite) ring ..
huffman@44136
   215
instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
huffman@44136
   216
instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
hoelzl@37489
   217
huffman@44136
   218
instance vec :: (ring_1, finite) ring_1 ..
hoelzl@37489
   219
huffman@44136
   220
instance vec :: (real_algebra, finite) real_algebra
wenzelm@61169
   221
  by standard (simp_all add: vec_eq_iff)
hoelzl@37489
   222
huffman@44136
   223
instance vec :: (real_algebra_1, finite) real_algebra_1 ..
hoelzl@37489
   224
wenzelm@49644
   225
lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
wenzelm@49644
   226
proof (induct n)
wenzelm@49644
   227
  case 0
wenzelm@49644
   228
  then show ?case by vector
wenzelm@49644
   229
next
wenzelm@49644
   230
  case Suc
wenzelm@49644
   231
  then show ?case by vector
wenzelm@49644
   232
qed
hoelzl@37489
   233
haftmann@54489
   234
lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) $ i = 1"
haftmann@54489
   235
  by vector
haftmann@54489
   236
haftmann@54489
   237
lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) $ i = - 1"
wenzelm@49644
   238
  by vector
hoelzl@37489
   239
huffman@44136
   240
instance vec :: (semiring_char_0, finite) semiring_char_0
haftmann@38621
   241
proof
haftmann@38621
   242
  fix m n :: nat
haftmann@38621
   243
  show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
huffman@44136
   244
    by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
hoelzl@37489
   245
qed
hoelzl@37489
   246
huffman@47108
   247
instance vec :: (numeral, finite) numeral ..
huffman@47108
   248
instance vec :: (semiring_numeral, finite) semiring_numeral ..
huffman@47108
   249
huffman@47108
   250
lemma numeral_index [simp]: "numeral w $ i = numeral w"
wenzelm@49644
   251
  by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
huffman@47108
   252
haftmann@54489
   253
lemma neg_numeral_index [simp]: "- numeral w $ i = - numeral w"
haftmann@54489
   254
  by (simp only: vector_uminus_component numeral_index)
huffman@47108
   255
huffman@44136
   256
instance vec :: (comm_ring_1, finite) comm_ring_1 ..
huffman@44136
   257
instance vec :: (ring_char_0, finite) ring_char_0 ..
hoelzl@37489
   258
hoelzl@37489
   259
lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
haftmann@57512
   260
  by (vector mult.assoc)
hoelzl@37489
   261
lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
hoelzl@37489
   262
  by (vector field_simps)
hoelzl@37489
   263
lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
hoelzl@37489
   264
  by (vector field_simps)
hoelzl@37489
   265
lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
hoelzl@37489
   266
lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
hoelzl@37489
   267
lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
hoelzl@37489
   268
  by (vector field_simps)
hoelzl@37489
   269
lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
hoelzl@37489
   270
lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
huffman@47108
   271
lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
hoelzl@37489
   272
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
hoelzl@37489
   273
lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
hoelzl@37489
   274
  by (vector field_simps)
hoelzl@37489
   275
hoelzl@37489
   276
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
huffman@44136
   277
  by (simp add: vec_eq_iff)
hoelzl@37489
   278
lp15@67979
   279
lemma linear_vec [simp]: "linear vec"
lp15@67979
   280
  by (simp add: linearI vec_add vec_eq_iff)
lp15@67979
   281
lp15@67979
   282
lemma differentiable_vec:
lp15@67979
   283
  fixes S :: "'a::euclidean_space set"
lp15@67979
   284
  shows "vec differentiable_on S"
lp15@67979
   285
  by (simp add: linear_linear bounded_linear_imp_differentiable_on)
lp15@67979
   286
lp15@67979
   287
lemma continuous_vec [continuous_intros]:
lp15@67979
   288
  fixes x :: "'a::euclidean_space"
lp15@67979
   289
  shows "isCont vec x"
lp15@67979
   290
  apply (clarsimp simp add: continuous_def LIM_def dist_vec_def L2_set_def)
lp15@67979
   291
  apply (rule_tac x="r / sqrt (real CARD('b))" in exI)
lp15@67979
   292
  by (simp add: mult.commute pos_less_divide_eq real_sqrt_mult)
lp15@67979
   293
lp15@67979
   294
lemma box_vec_eq_empty [simp]:
lp15@67979
   295
  shows "cbox (vec a) (vec b) = {} \<longleftrightarrow> cbox a b = {}"
lp15@67979
   296
        "box (vec a) (vec b) = {} \<longleftrightarrow> box a b = {}"
lp15@67979
   297
  by (auto simp: Basis_vec_def mem_box box_eq_empty inner_axis)
lp15@67979
   298
hoelzl@37489
   299
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
lp15@67683
   300
lp15@67683
   301
lemma norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
lp15@67683
   302
  by (simp add: inner_axis' norm_eq_1)
lp15@67683
   303
hoelzl@37489
   304
lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
hoelzl@37489
   305
  by vector
lp15@67683
   306
hoelzl@37489
   307
lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
hoelzl@37489
   308
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
lp15@67683
   309
hoelzl@37489
   310
lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
hoelzl@37489
   311
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
lp15@67683
   312
hoelzl@37489
   313
lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
hoelzl@37489
   314
  by (metis vector_mul_lcancel)
lp15@67683
   315
hoelzl@37489
   316
lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
hoelzl@37489
   317
  by (metis vector_mul_rcancel)
hoelzl@37489
   318
lp15@67979
   319
lemma component_le_norm_cart: "\<bar>x$i\<bar> \<le> norm x"
huffman@44136
   320
  apply (simp add: norm_vec_def)
nipkow@67155
   321
  apply (rule member_le_L2_set, simp_all)
hoelzl@37489
   322
  done
hoelzl@37489
   323
lp15@67979
   324
lemma norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x$i\<bar> \<le> e"
hoelzl@37489
   325
  by (metis component_le_norm_cart order_trans)
hoelzl@37489
   326
hoelzl@37489
   327
lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
huffman@53595
   328
  by (metis component_le_norm_cart le_less_trans)
hoelzl@37489
   329
lp15@67979
   330
lemma norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
nipkow@67155
   331
  by (simp add: norm_vec_def L2_set_le_sum)
hoelzl@37489
   332
lp15@67969
   333
lemma scalar_mult_eq_scaleR [simp]: "c *s x = c *\<^sub>R x"
huffman@44136
   334
  unfolding scaleR_vec_def vector_scalar_mult_def by simp
hoelzl@37489
   335
hoelzl@37489
   336
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
hoelzl@37489
   337
  unfolding dist_norm scalar_mult_eq_scaleR
hoelzl@37489
   338
  unfolding scaleR_right_diff_distrib[symmetric] by simp
hoelzl@37489
   339
nipkow@64267
   340
lemma sum_component [simp]:
hoelzl@37489
   341
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
nipkow@64267
   342
  shows "(sum f S)$i = sum (\<lambda>x. (f x)$i) S"
wenzelm@49644
   343
proof (cases "finite S")
wenzelm@49644
   344
  case True
wenzelm@49644
   345
  then show ?thesis by induct simp_all
wenzelm@49644
   346
next
wenzelm@49644
   347
  case False
wenzelm@49644
   348
  then show ?thesis by simp
wenzelm@49644
   349
qed
hoelzl@37489
   350
nipkow@64267
   351
lemma sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)$i ) S)"
huffman@44136
   352
  by (simp add: vec_eq_iff)
hoelzl@37489
   353
nipkow@64267
   354
lemma sum_cmul:
hoelzl@37489
   355
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
nipkow@64267
   356
  shows "sum (\<lambda>x. c *s f x) S = c *s sum f S"
nipkow@64267
   357
  by (simp add: vec_eq_iff sum_distrib_left)
hoelzl@37489
   358
nipkow@64267
   359
lemma sum_norm_allsubsets_bound_cart:
hoelzl@37489
   360
  fixes f:: "'a \<Rightarrow> real ^'n"
nipkow@64267
   361
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
nipkow@64267
   362
  shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
nipkow@64267
   363
  using sum_norm_allsubsets_bound[OF assms]
wenzelm@57865
   364
  by simp
hoelzl@37489
   365
lp15@62397
   366
subsection\<open>Closures and interiors of halfspaces\<close>
lp15@62397
   367
lp15@62397
   368
lemma interior_halfspace_le [simp]:
lp15@62397
   369
  assumes "a \<noteq> 0"
lp15@62397
   370
    shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
lp15@62397
   371
proof -
lp15@62397
   372
  have *: "a \<bullet> x < b" if x: "x \<in> S" and S: "S \<subseteq> {x. a \<bullet> x \<le> b}" and "open S" for S x
lp15@62397
   373
  proof -
lp15@62397
   374
    obtain e where "e>0" and e: "cball x e \<subseteq> S"
lp15@62397
   375
      using \<open>open S\<close> open_contains_cball x by blast
lp15@62397
   376
    then have "x + (e / norm a) *\<^sub>R a \<in> cball x e"
lp15@62397
   377
      by (simp add: dist_norm)
lp15@62397
   378
    then have "x + (e / norm a) *\<^sub>R a \<in> S"
lp15@62397
   379
      using e by blast
lp15@62397
   380
    then have "x + (e / norm a) *\<^sub>R a \<in> {x. a \<bullet> x \<le> b}"
lp15@62397
   381
      using S by blast
lp15@62397
   382
    moreover have "e * (a \<bullet> a) / norm a > 0"
lp15@62397
   383
      by (simp add: \<open>0 < e\<close> assms)
lp15@62397
   384
    ultimately show ?thesis
lp15@62397
   385
      by (simp add: algebra_simps)
lp15@62397
   386
  qed
lp15@62397
   387
  show ?thesis
lp15@62397
   388
    by (rule interior_unique) (auto simp: open_halfspace_lt *)
lp15@62397
   389
qed
lp15@62397
   390
lp15@62397
   391
lemma interior_halfspace_ge [simp]:
lp15@62397
   392
   "a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"
lp15@62397
   393
using interior_halfspace_le [of "-a" "-b"] by simp
lp15@62397
   394
lp15@62397
   395
lemma interior_halfspace_component_le [simp]:
wenzelm@67731
   396
     "interior {x. x$k \<le> a} = {x :: (real^'n). x$k < a}" (is "?LE")
lp15@62397
   397
  and interior_halfspace_component_ge [simp]:
wenzelm@67731
   398
     "interior {x. x$k \<ge> a} = {x :: (real^'n). x$k > a}" (is "?GE")
lp15@62397
   399
proof -
lp15@62397
   400
  have "axis k (1::real) \<noteq> 0"
lp15@62397
   401
    by (simp add: axis_def vec_eq_iff)
lp15@62397
   402
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
lp15@62397
   403
    by (simp add: cart_eq_inner_axis inner_commute)
lp15@62397
   404
  ultimately show ?LE ?GE
lp15@62397
   405
    using interior_halfspace_le [of "axis k (1::real)" a]
lp15@62397
   406
          interior_halfspace_ge [of "axis k (1::real)" a] by auto
lp15@62397
   407
qed
lp15@62397
   408
lp15@62397
   409
lemma closure_halfspace_lt [simp]:
lp15@62397
   410
  assumes "a \<noteq> 0"
lp15@62397
   411
    shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
lp15@62397
   412
proof -
lp15@62397
   413
  have [simp]: "-{x. a \<bullet> x < b} = {x. a \<bullet> x \<ge> b}"
lp15@62397
   414
    by (force simp:)
lp15@62397
   415
  then show ?thesis
lp15@62397
   416
    using interior_halfspace_ge [of a b] assms
lp15@62397
   417
    by (force simp: closure_interior)
lp15@62397
   418
qed
lp15@62397
   419
lp15@62397
   420
lemma closure_halfspace_gt [simp]:
lp15@62397
   421
   "a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"
lp15@62397
   422
using closure_halfspace_lt [of "-a" "-b"] by simp
lp15@62397
   423
lp15@62397
   424
lemma closure_halfspace_component_lt [simp]:
wenzelm@67731
   425
     "closure {x. x$k < a} = {x :: (real^'n). x$k \<le> a}" (is "?LE")
lp15@62397
   426
  and closure_halfspace_component_gt [simp]:
wenzelm@67731
   427
     "closure {x. x$k > a} = {x :: (real^'n). x$k \<ge> a}" (is "?GE")
lp15@62397
   428
proof -
lp15@62397
   429
  have "axis k (1::real) \<noteq> 0"
lp15@62397
   430
    by (simp add: axis_def vec_eq_iff)
lp15@62397
   431
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
lp15@62397
   432
    by (simp add: cart_eq_inner_axis inner_commute)
lp15@62397
   433
  ultimately show ?LE ?GE
lp15@62397
   434
    using closure_halfspace_lt [of "axis k (1::real)" a]
lp15@62397
   435
          closure_halfspace_gt [of "axis k (1::real)" a] by auto
lp15@62397
   436
qed
lp15@62397
   437
lp15@62397
   438
lemma interior_hyperplane [simp]:
lp15@62397
   439
  assumes "a \<noteq> 0"
lp15@62397
   440
    shows "interior {x. a \<bullet> x = b} = {}"
lp15@62397
   441
proof -
lp15@62397
   442
  have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
lp15@62397
   443
    by (force simp:)
lp15@62397
   444
  then show ?thesis
lp15@62397
   445
    by (auto simp: assms)
lp15@62397
   446
qed
lp15@62397
   447
lp15@62397
   448
lemma frontier_halfspace_le:
lp15@62397
   449
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   450
    shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
lp15@62397
   451
proof (cases "a = 0")
lp15@62397
   452
  case True with assms show ?thesis by simp
lp15@62397
   453
next
lp15@62397
   454
  case False then show ?thesis
lp15@62397
   455
    by (force simp: frontier_def closed_halfspace_le)
lp15@62397
   456
qed
lp15@62397
   457
lp15@62397
   458
lemma frontier_halfspace_ge:
lp15@62397
   459
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   460
    shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"
lp15@62397
   461
proof (cases "a = 0")
lp15@62397
   462
  case True with assms show ?thesis by simp
lp15@62397
   463
next
lp15@62397
   464
  case False then show ?thesis
lp15@62397
   465
    by (force simp: frontier_def closed_halfspace_ge)
lp15@62397
   466
qed
lp15@62397
   467
lp15@62397
   468
lemma frontier_halfspace_lt:
lp15@62397
   469
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   470
    shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"
lp15@62397
   471
proof (cases "a = 0")
lp15@62397
   472
  case True with assms show ?thesis by simp
lp15@62397
   473
next
lp15@62397
   474
  case False then show ?thesis
lp15@62397
   475
    by (force simp: frontier_def interior_open open_halfspace_lt)
lp15@62397
   476
qed
lp15@62397
   477
lp15@62397
   478
lemma frontier_halfspace_gt:
lp15@62397
   479
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   480
    shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"
lp15@62397
   481
proof (cases "a = 0")
lp15@62397
   482
  case True with assms show ?thesis by simp
lp15@62397
   483
next
lp15@62397
   484
  case False then show ?thesis
lp15@62397
   485
    by (force simp: frontier_def interior_open open_halfspace_gt)
lp15@62397
   486
qed
lp15@62397
   487
lp15@62397
   488
lemma interior_standard_hyperplane:
wenzelm@67731
   489
   "interior {x :: (real^'n). x$k = a} = {}"
lp15@62397
   490
proof -
lp15@62397
   491
  have "axis k (1::real) \<noteq> 0"
lp15@62397
   492
    by (simp add: axis_def vec_eq_iff)
lp15@62397
   493
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
lp15@62397
   494
    by (simp add: cart_eq_inner_axis inner_commute)
lp15@62397
   495
  ultimately show ?thesis
lp15@62397
   496
    using interior_hyperplane [of "axis k (1::real)" a]
lp15@62397
   497
    by force
lp15@62397
   498
qed
lp15@62397
   499
wenzelm@60420
   500
subsection \<open>Matrix operations\<close>
hoelzl@37489
   501
wenzelm@60420
   502
text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close>
hoelzl@37489
   503
immler@67962
   504
definition map_matrix::"('a \<Rightarrow> 'b) \<Rightarrow> (('a, 'i::finite)vec, 'j::finite) vec \<Rightarrow> (('b, 'i)vec, 'j) vec" where
immler@67962
   505
  "map_matrix f x = (\<chi> i j. f (x $ i $ j))"
immler@67962
   506
immler@67962
   507
lemma nth_map_matrix[simp]: "map_matrix f x $ i $ j = f (x $ i $ j)"
immler@67962
   508
  by (simp add: map_matrix_def)
immler@67962
   509
wenzelm@49644
   510
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
wenzelm@49644
   511
    (infixl "**" 70)
nipkow@64267
   512
  where "m ** m' == (\<chi> i j. sum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
hoelzl@37489
   513
wenzelm@49644
   514
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
wenzelm@49644
   515
    (infixl "*v" 70)
nipkow@64267
   516
  where "m *v x \<equiv> (\<chi> i. sum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
hoelzl@37489
   517
wenzelm@49644
   518
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
wenzelm@49644
   519
    (infixl "v*" 70)
nipkow@64267
   520
  where "v v* m == (\<chi> j. sum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
hoelzl@37489
   521
hoelzl@37489
   522
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
hoelzl@63332
   523
definition transpose where
hoelzl@37489
   524
  "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
hoelzl@37489
   525
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
hoelzl@37489
   526
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
hoelzl@37489
   527
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
hoelzl@37489
   528
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
hoelzl@37489
   529
hoelzl@37489
   530
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
hoelzl@37489
   531
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
nipkow@64267
   532
  by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
hoelzl@37489
   533
lp15@67673
   534
lemma matrix_mul_lid [simp]:
hoelzl@37489
   535
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   536
  shows "mat 1 ** A = A"
hoelzl@37489
   537
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   538
  apply vector
nipkow@64267
   539
  apply (auto simp only: if_distrib cond_application_beta sum.delta'[OF finite]
wenzelm@49644
   540
    mult_1_left mult_zero_left if_True UNIV_I)
wenzelm@49644
   541
  done
hoelzl@37489
   542
lp15@67673
   543
lemma matrix_mul_rid [simp]:
hoelzl@37489
   544
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   545
  shows "A ** mat 1 = A"
hoelzl@37489
   546
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   547
  apply vector
nipkow@64267
   548
  apply (auto simp only: if_distrib cond_application_beta sum.delta[OF finite]
wenzelm@49644
   549
    mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
wenzelm@49644
   550
  done
hoelzl@37489
   551
hoelzl@37489
   552
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
nipkow@64267
   553
  apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc)
haftmann@66804
   554
  apply (subst sum.swap)
hoelzl@37489
   555
  apply simp
hoelzl@37489
   556
  done
hoelzl@37489
   557
hoelzl@37489
   558
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
wenzelm@49644
   559
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def
nipkow@64267
   560
    sum_distrib_left sum_distrib_right mult.assoc)
haftmann@66804
   561
  apply (subst sum.swap)
hoelzl@37489
   562
  apply simp
hoelzl@37489
   563
  done
hoelzl@37489
   564
lp15@67673
   565
lemma matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
hoelzl@37489
   566
  apply (vector matrix_vector_mult_def mat_def)
nipkow@64267
   567
  apply (simp add: if_distrib cond_application_beta sum.delta' cong del: if_weak_cong)
wenzelm@49644
   568
  done
hoelzl@37489
   569
wenzelm@49644
   570
lemma matrix_transpose_mul:
wenzelm@49644
   571
    "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
haftmann@57512
   572
  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
hoelzl@37489
   573
hoelzl@37489
   574
lemma matrix_eq:
hoelzl@37489
   575
  fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
hoelzl@37489
   576
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@37489
   577
  apply auto
huffman@44136
   578
  apply (subst vec_eq_iff)
hoelzl@37489
   579
  apply clarify
hoelzl@50526
   580
  apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
hoelzl@50526
   581
  apply (erule_tac x="axis ia 1" in allE)
hoelzl@37489
   582
  apply (erule_tac x="i" in allE)
hoelzl@50526
   583
  apply (auto simp add: if_distrib cond_application_beta axis_def
nipkow@64267
   584
    sum.delta[OF finite] cong del: if_weak_cong)
wenzelm@49644
   585
  done
hoelzl@37489
   586
wenzelm@49644
   587
lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
huffman@44136
   588
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   589
hoelzl@37489
   590
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
nipkow@64267
   591
  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps)
haftmann@66804
   592
  apply (subst sum.swap)
wenzelm@49644
   593
  apply simp
wenzelm@49644
   594
  done
hoelzl@37489
   595
lp15@67673
   596
lemma transpose_mat [simp]: "transpose (mat n) = mat n"
hoelzl@37489
   597
  by (vector transpose_def mat_def)
hoelzl@37489
   598
lp15@67683
   599
lemma transpose_transpose [simp]: "transpose(transpose A) = A"
hoelzl@37489
   600
  by (vector transpose_def)
hoelzl@37489
   601
lp15@67673
   602
lemma row_transpose [simp]:
hoelzl@37489
   603
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   604
  shows "row i (transpose A) = column i A"
huffman@44136
   605
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   606
lp15@67673
   607
lemma column_transpose [simp]:
hoelzl@37489
   608
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   609
  shows "column i (transpose A) = row i A"
huffman@44136
   610
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   611
lp15@67683
   612
lemma rows_transpose [simp]: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
wenzelm@49644
   613
  by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
hoelzl@37489
   614
lp15@67683
   615
lemma columns_transpose [simp]: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
wenzelm@49644
   616
  by (metis transpose_transpose rows_transpose)
hoelzl@37489
   617
lp15@67673
   618
lemma matrix_mult_transpose_dot_column:
lp15@67673
   619
  fixes A :: "real^'n^'n"
lp15@67673
   620
  shows "transpose A ** A = (\<chi> i j. (column i A) \<bullet> (column j A))"
lp15@67673
   621
  by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)
lp15@67673
   622
lp15@67673
   623
lemma matrix_mult_transpose_dot_row:
lp15@67673
   624
  fixes A :: "real^'n^'n"
lp15@67673
   625
  shows "A ** transpose A = (\<chi> i j. (row i A) \<bullet> (row j A))"
lp15@67673
   626
  by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)
lp15@67673
   627
wenzelm@60420
   628
text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
hoelzl@37489
   629
hoelzl@37489
   630
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
huffman@44136
   631
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   632
lp15@67673
   633
lemma matrix_mult_sum:
nipkow@64267
   634
  "(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
haftmann@57512
   635
  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
hoelzl@37489
   636
hoelzl@37489
   637
lemma vector_componentwise:
hoelzl@50526
   638
  "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
nipkow@64267
   639
  by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)
hoelzl@50526
   640
nipkow@64267
   641
lemma basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
nipkow@64267
   642
  by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
hoelzl@37489
   643
lp15@63938
   644
lemma linear_componentwise_expansion:
hoelzl@37489
   645
  fixes f:: "real ^'m \<Rightarrow> real ^ _"
hoelzl@37489
   646
  assumes lf: "linear f"
nipkow@64267
   647
  shows "(f x)$j = sum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
wenzelm@49644
   648
proof -
hoelzl@37489
   649
  let ?M = "(UNIV :: 'm set)"
hoelzl@37489
   650
  let ?N = "(UNIV :: 'n set)"
nipkow@64267
   651
  have "?rhs = (sum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
nipkow@64267
   652
    unfolding sum_component by simp
wenzelm@49644
   653
  then show ?thesis
nipkow@64267
   654
    unfolding linear_sum_mul[OF lf, symmetric]
hoelzl@50526
   655
    unfolding scalar_mult_eq_scaleR[symmetric]
hoelzl@50526
   656
    unfolding basis_expansion
hoelzl@50526
   657
    by simp
hoelzl@37489
   658
qed
hoelzl@37489
   659
lp15@67719
   660
subsection\<open>Inverse matrices  (not necessarily square)\<close>
hoelzl@37489
   661
wenzelm@49644
   662
definition
wenzelm@49644
   663
  "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   664
wenzelm@49644
   665
definition
wenzelm@49644
   666
  "matrix_inv(A:: 'a::semiring_1^'n^'m) =
wenzelm@49644
   667
    (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   668
wenzelm@60420
   669
text\<open>Correspondence between matrices and linear operators.\<close>
hoelzl@37489
   670
wenzelm@49644
   671
definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
hoelzl@50526
   672
  where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
hoelzl@37489
   673
lp15@67986
   674
lemma matrix_id_mat_1: "matrix id = mat 1"
lp15@67986
   675
  by (simp add: mat_def matrix_def axis_def)
lp15@67986
   676
lp15@67986
   677
lemma matrix_scaleR: "(matrix (( *\<^sub>R) r)) = mat r"
lp15@67986
   678
  by (simp add: mat_def matrix_def axis_def if_distrib cong: if_cong)
lp15@67986
   679
hoelzl@37489
   680
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
huffman@53600
   681
  by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
nipkow@64267
   682
      field_simps sum_distrib_left sum.distrib)
hoelzl@37489
   683
lp15@67683
   684
lemma
lp15@67683
   685
  fixes A :: "real^'n^'m"
lp15@67683
   686
  shows matrix_vector_mult_linear_continuous_at [continuous_intros]: "isCont (( *v) A) z"
lp15@67683
   687
    and matrix_vector_mult_linear_continuous_on [continuous_intros]: "continuous_on S (( *v) A)"
lp15@67683
   688
  by (simp_all add: linear_linear linear_continuous_at linear_continuous_on matrix_vector_mul_linear)
lp15@67683
   689
lp15@67673
   690
lemma matrix_vector_mult_add_distrib [algebra_simps]:
immler@67728
   691
  "A *v (x + y) = A *v x + A *v y"
immler@67728
   692
  by (vector matrix_vector_mult_def sum.distrib distrib_left)
lp15@67673
   693
lp15@67673
   694
lemma matrix_vector_mult_diff_distrib [algebra_simps]:
immler@67728
   695
  fixes A :: "'a::ring_1^'n^'m"
lp15@67673
   696
  shows "A *v (x - y) = A *v x - A *v y"
immler@67728
   697
  by (vector matrix_vector_mult_def sum_subtractf right_diff_distrib)
lp15@67673
   698
lp15@67673
   699
lemma matrix_vector_mult_scaleR[algebra_simps]:
lp15@67673
   700
  fixes A :: "real^'n^'m"
lp15@67673
   701
  shows "A *v (c *\<^sub>R x) = c *\<^sub>R (A *v x)"
lp15@67673
   702
  using linear_iff matrix_vector_mul_linear by blast
lp15@67673
   703
lp15@67673
   704
lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
lp15@67673
   705
  by (simp add: matrix_vector_mult_def vec_eq_iff)
lp15@67673
   706
lp15@67673
   707
lemma matrix_vector_mult_0 [simp]: "0 *v w = 0"
lp15@67673
   708
  by (simp add: matrix_vector_mult_def vec_eq_iff)
lp15@67673
   709
lp15@67673
   710
lemma matrix_vector_mult_add_rdistrib [algebra_simps]:
immler@67728
   711
  "(A + B) *v x = (A *v x) + (B *v x)"
immler@67728
   712
  by (vector matrix_vector_mult_def sum.distrib distrib_right)
lp15@67673
   713
lp15@67673
   714
lemma matrix_vector_mult_diff_rdistrib [algebra_simps]:
immler@67728
   715
  fixes A :: "'a :: ring_1^'n^'m"
lp15@67673
   716
  shows "(A - B) *v x = (A *v x) - (B *v x)"
immler@67728
   717
  by (vector matrix_vector_mult_def sum_subtractf left_diff_distrib)
lp15@67673
   718
wenzelm@49644
   719
lemma matrix_works:
wenzelm@49644
   720
  assumes lf: "linear f"
wenzelm@49644
   721
  shows "matrix f *v x = f (x::real ^ 'n)"
haftmann@57512
   722
  apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)
lp15@63938
   723
  by (simp add: linear_componentwise_expansion lf)
hoelzl@37489
   724
wenzelm@49644
   725
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
wenzelm@49644
   726
  by (simp add: ext matrix_works)
hoelzl@37489
   727
lp15@67683
   728
declare matrix_vector_mul [symmetric, simp]
lp15@67683
   729
lp15@67673
   730
lemma matrix_of_matrix_vector_mul [simp]: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
hoelzl@37489
   731
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
hoelzl@37489
   732
hoelzl@37489
   733
lemma matrix_compose:
hoelzl@37489
   734
  assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
wenzelm@49644
   735
    and lg: "linear (g::real^'m \<Rightarrow> real^_)"
wenzelm@61736
   736
  shows "matrix (g \<circ> f) = matrix g ** matrix f"
hoelzl@37489
   737
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
wenzelm@49644
   738
  by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
hoelzl@37489
   739
wenzelm@49644
   740
lemma matrix_vector_column:
nipkow@64267
   741
  "(A::'a::comm_semiring_1^'n^_) *v x = sum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
haftmann@57512
   742
  by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
hoelzl@37489
   743
hoelzl@37489
   744
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
hoelzl@37489
   745
  apply (rule adjoint_unique)
wenzelm@49644
   746
  apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
nipkow@64267
   747
    sum_distrib_right sum_distrib_left)
haftmann@66804
   748
  apply (subst sum.swap)
haftmann@57514
   749
  apply (auto simp add: ac_simps)
hoelzl@37489
   750
  done
hoelzl@37489
   751
hoelzl@37489
   752
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
hoelzl@37489
   753
  shows "matrix(adjoint f) = transpose(matrix f)"
hoelzl@37489
   754
  apply (subst matrix_vector_mul[OF lf])
wenzelm@49644
   755
  unfolding adjoint_matrix matrix_of_matrix_vector_mul
wenzelm@49644
   756
  apply rule
wenzelm@49644
   757
  done
wenzelm@49644
   758
lp15@67981
   759
lemma inj_matrix_vector_mult:
lp15@67981
   760
  fixes A::"'a::field^'n^'m"
lp15@67981
   761
  assumes "invertible A"
lp15@67981
   762
  shows "inj (( *v) A)"
lp15@67981
   763
  by (metis assms inj_on_inverseI invertible_def matrix_vector_mul_assoc matrix_vector_mul_lid)
lp15@67981
   764
hoelzl@37489
   765
nipkow@67968
   766
subsection\<open>Some bounds on components etc. relative to operator norm\<close>
lp15@67719
   767
lp15@67719
   768
lemma norm_column_le_onorm:
lp15@67719
   769
  fixes A :: "real^'n^'m"
lp15@67719
   770
  shows "norm(column i A) \<le> onorm(( *v) A)"
lp15@67719
   771
proof -
lp15@67719
   772
  have bl: "bounded_linear (( *v) A)"
lp15@67719
   773
    by (simp add: linear_linear matrix_vector_mul_linear)
lp15@67719
   774
  have "norm (\<chi> j. A $ j $ i) \<le> norm (A *v axis i 1)"
lp15@67719
   775
    by (simp add: matrix_mult_dot cart_eq_inner_axis)
lp15@67719
   776
  also have "\<dots> \<le> onorm (( *v) A)"
lp15@67982
   777
    using onorm [OF bl, of "axis i 1"] by auto
lp15@67719
   778
  finally have "norm (\<chi> j. A $ j $ i) \<le> onorm (( *v) A)" .
lp15@67719
   779
  then show ?thesis
lp15@67719
   780
    unfolding column_def .
lp15@67719
   781
qed
lp15@67719
   782
lp15@67719
   783
lemma matrix_component_le_onorm:
lp15@67719
   784
  fixes A :: "real^'n^'m"
lp15@67719
   785
  shows "\<bar>A $ i $ j\<bar> \<le> onorm(( *v) A)"
lp15@67719
   786
proof -
lp15@67719
   787
  have "\<bar>A $ i $ j\<bar> \<le> norm (\<chi> n. (A $ n $ j))"
lp15@67719
   788
    by (metis (full_types, lifting) component_le_norm_cart vec_lambda_beta)
lp15@67719
   789
  also have "\<dots> \<le> onorm (( *v) A)"
lp15@67719
   790
    by (metis (no_types) column_def norm_column_le_onorm)
lp15@67719
   791
  finally show ?thesis .
lp15@67719
   792
qed
lp15@67719
   793
lp15@67719
   794
lemma component_le_onorm:
lp15@67719
   795
  fixes f :: "real^'m \<Rightarrow> real^'n"
lp15@67719
   796
  shows "linear f \<Longrightarrow> \<bar>matrix f $ i $ j\<bar> \<le> onorm f"
lp15@67719
   797
  by (metis matrix_component_le_onorm matrix_vector_mul)
hoelzl@37489
   798
lp15@67719
   799
lemma onorm_le_matrix_component_sum:
lp15@67719
   800
  fixes A :: "real^'n^'m"
lp15@67719
   801
  shows "onorm(( *v) A) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>)"
lp15@67719
   802
proof (rule onorm_le)
lp15@67719
   803
  fix x
lp15@67719
   804
  have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
lp15@67719
   805
    by (rule norm_le_l1_cart)
lp15@67719
   806
  also have "\<dots> \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)"
lp15@67719
   807
  proof (rule sum_mono)
lp15@67719
   808
    fix i
lp15@67719
   809
    have "\<bar>(A *v x) $ i\<bar> \<le> \<bar>\<Sum>j\<in>UNIV. A $ i $ j * x $ j\<bar>"
lp15@67719
   810
      by (simp add: matrix_vector_mult_def)
lp15@67719
   811
    also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j * x $ j\<bar>)"
lp15@67719
   812
      by (rule sum_abs)
lp15@67719
   813
    also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)"
lp15@67719
   814
      by (rule sum_mono) (simp add: abs_mult component_le_norm_cart mult_left_mono)
lp15@67719
   815
    finally show "\<bar>(A *v x) $ i\<bar> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)" .
lp15@67719
   816
  qed
lp15@67719
   817
  finally show "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>) * norm x"
lp15@67719
   818
    by (simp add: sum_distrib_right)
lp15@67719
   819
qed
lp15@67719
   820
lp15@67719
   821
lemma onorm_le_matrix_component:
lp15@67719
   822
  fixes A :: "real^'n^'m"
lp15@67719
   823
  assumes "\<And>i j. abs(A$i$j) \<le> B"
lp15@67719
   824
  shows "onorm(( *v) A) \<le> real (CARD('m)) * real (CARD('n)) * B"
lp15@67719
   825
proof (rule onorm_le)
wenzelm@67731
   826
  fix x :: "real^'n::_"
lp15@67719
   827
  have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
lp15@67719
   828
    by (rule norm_le_l1_cart)
lp15@67719
   829
  also have "\<dots> \<le> (\<Sum>i::'m \<in>UNIV. real (CARD('n)) * B * norm x)"
lp15@67719
   830
  proof (rule sum_mono)
lp15@67719
   831
    fix i
lp15@67719
   832
    have "\<bar>(A *v x) $ i\<bar> \<le> norm(A $ i) * norm x"
lp15@67719
   833
      by (simp add: matrix_mult_dot Cauchy_Schwarz_ineq2)
lp15@67719
   834
    also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>) * norm x"
lp15@67719
   835
      by (simp add: mult_right_mono norm_le_l1_cart)
lp15@67719
   836
    also have "\<dots> \<le> real (CARD('n)) * B * norm x"
lp15@67719
   837
      by (simp add: assms sum_bounded_above mult_right_mono)
lp15@67719
   838
    finally show "\<bar>(A *v x) $ i\<bar> \<le> real (CARD('n)) * B * norm x" .
lp15@67719
   839
  qed
lp15@67719
   840
  also have "\<dots> \<le> CARD('m) * real (CARD('n)) * B * norm x"
lp15@67719
   841
    by simp
lp15@67719
   842
  finally show "norm (A *v x) \<le> CARD('m) * real (CARD('n)) * B * norm x" .
lp15@67719
   843
qed
lp15@67719
   844
lp15@67719
   845
subsection \<open>lambda skolemization on cartesian products\<close>
hoelzl@37489
   846
hoelzl@37489
   847
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
hoelzl@37494
   848
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   849
proof -
hoelzl@37489
   850
  let ?S = "(UNIV :: 'n set)"
wenzelm@49644
   851
  { assume H: "?rhs"
wenzelm@49644
   852
    then have ?lhs by auto }
hoelzl@37489
   853
  moreover
wenzelm@49644
   854
  { assume H: "?lhs"
hoelzl@37489
   855
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
hoelzl@37489
   856
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
wenzelm@49644
   857
    { fix i
hoelzl@37489
   858
      from f have "P i (f i)" by metis
hoelzl@37494
   859
      then have "P i (?x $ i)" by auto
hoelzl@37489
   860
    }
hoelzl@37489
   861
    hence "\<forall>i. P i (?x$i)" by metis
hoelzl@37489
   862
    hence ?rhs by metis }
hoelzl@37489
   863
  ultimately show ?thesis by metis
hoelzl@37489
   864
qed
hoelzl@37489
   865
lp15@67719
   866
lemma rational_approximation:
lp15@67719
   867
  assumes "e > 0"
lp15@67719
   868
  obtains r::real where "r \<in> \<rat>" "\<bar>r - x\<bar> < e"
lp15@67719
   869
  using Rats_dense_in_real [of "x - e/2" "x + e/2"] assms by auto
lp15@67719
   870
lp15@67719
   871
lemma matrix_rational_approximation:
lp15@67719
   872
  fixes A :: "real^'n^'m"
lp15@67719
   873
  assumes "e > 0"
lp15@67719
   874
  obtains B where "\<And>i j. B$i$j \<in> \<rat>" "onorm(\<lambda>x. (A - B) *v x) < e"
lp15@67719
   875
proof -
lp15@67719
   876
  have "\<forall>i j. \<exists>q \<in> \<rat>. \<bar>q - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))"
lp15@67719
   877
    using assms by (force intro: rational_approximation [of "e / (2 * CARD('m) * CARD('n))"])
lp15@67719
   878
  then obtain B where B: "\<And>i j. B$i$j \<in> \<rat>" and Bclo: "\<And>i j. \<bar>B$i$j - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))"
lp15@67719
   879
    by (auto simp: lambda_skolem Bex_def)
lp15@67719
   880
  show ?thesis
lp15@67719
   881
  proof
lp15@67719
   882
    have "onorm (( *v) (A - B)) \<le> real CARD('m) * real CARD('n) *
lp15@67719
   883
    (e / (2 * real CARD('m) * real CARD('n)))"
lp15@67719
   884
      apply (rule onorm_le_matrix_component)
lp15@67719
   885
      using Bclo by (simp add: abs_minus_commute less_imp_le)
lp15@67719
   886
    also have "\<dots> < e"
lp15@67719
   887
      using \<open>0 < e\<close> by (simp add: divide_simps)
lp15@67719
   888
    finally show "onorm (( *v) (A - B)) < e" .
lp15@67719
   889
  qed (use B in auto)
lp15@67719
   890
qed
lp15@67719
   891
hoelzl@37489
   892
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
hoelzl@50526
   893
  unfolding inner_simps scalar_mult_eq_scaleR by auto
hoelzl@37489
   894
hoelzl@37489
   895
lemma left_invertible_transpose:
hoelzl@37489
   896
  "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
hoelzl@37489
   897
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   898
hoelzl@37489
   899
lemma right_invertible_transpose:
hoelzl@37489
   900
  "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
hoelzl@37489
   901
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   902
hoelzl@37489
   903
lemma matrix_left_invertible_injective:
lp15@67986
   904
  fixes A :: "real^'n^'m"
lp15@67986
   905
  shows "(\<exists>B. B ** A = mat 1) \<longleftrightarrow> inj (( *v) A)"
lp15@67986
   906
proof safe
lp15@67986
   907
  fix B
lp15@67986
   908
  assume B: "B ** A = mat 1"
lp15@67986
   909
  show "inj (( *v) A)"
lp15@67986
   910
    unfolding inj_on_def
lp15@67986
   911
      by (metis B matrix_vector_mul_assoc matrix_vector_mul_lid)
lp15@67986
   912
next
lp15@67986
   913
  assume "inj (( *v) A)"
lp15@67986
   914
  with linear_injective_left_inverse[OF matrix_vector_mul_linear]
lp15@67986
   915
  obtain g where "linear g" and g: "g \<circ> ( *v) A = id"
lp15@67986
   916
    by blast
lp15@67986
   917
  have "matrix g ** A = mat 1"
lp15@67986
   918
    by (metis \<open>linear g\<close> g matrix_compose matrix_id_mat_1 matrix_of_matrix_vector_mul matrix_vector_mul_linear)
lp15@67986
   919
  then show "\<exists>B. B ** A = mat 1"
lp15@67986
   920
    by metis
hoelzl@37489
   921
qed
hoelzl@37489
   922
hoelzl@37489
   923
lemma matrix_left_invertible_ker:
hoelzl@37489
   924
  "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
hoelzl@37489
   925
  unfolding matrix_left_invertible_injective
hoelzl@37489
   926
  using linear_injective_0[OF matrix_vector_mul_linear, of A]
hoelzl@37489
   927
  by (simp add: inj_on_def)
hoelzl@37489
   928
hoelzl@37489
   929
lemma matrix_right_invertible_surjective:
wenzelm@49644
   930
  "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
wenzelm@49644
   931
proof -
wenzelm@49644
   932
  { fix B :: "real ^'m^'n"
wenzelm@49644
   933
    assume AB: "A ** B = mat 1"
wenzelm@49644
   934
    { fix x :: "real ^ 'm"
hoelzl@37489
   935
      have "A *v (B *v x) = x"
wenzelm@49644
   936
        by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
nipkow@67399
   937
    hence "surj (( *v) A)" unfolding surj_def by metis }
hoelzl@37489
   938
  moreover
nipkow@67399
   939
  { assume sf: "surj (( *v) A)"
hoelzl@37489
   940
    from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
nipkow@67399
   941
    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "( *v) A \<circ> g = id"
hoelzl@37489
   942
      by blast
hoelzl@37489
   943
hoelzl@37489
   944
    have "A ** (matrix g) = mat 1"
hoelzl@37489
   945
      unfolding matrix_eq  matrix_vector_mul_lid
hoelzl@37489
   946
        matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   947
      using g(2) unfolding o_def fun_eq_iff id_def
hoelzl@37489
   948
      .
hoelzl@37489
   949
    hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
hoelzl@37489
   950
  }
hoelzl@37489
   951
  ultimately show ?thesis unfolding surj_def by blast
hoelzl@37489
   952
qed
hoelzl@37489
   953
hoelzl@37489
   954
lemma matrix_left_invertible_independent_columns:
hoelzl@37489
   955
  fixes A :: "real^'n^'m"
wenzelm@49644
   956
  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
nipkow@64267
   957
      (\<forall>c. sum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
wenzelm@49644
   958
    (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   959
proof -
hoelzl@37489
   960
  let ?U = "UNIV :: 'n set"
wenzelm@49644
   961
  { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
wenzelm@49644
   962
    { fix c i
nipkow@64267
   963
      assume c: "sum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
hoelzl@37489
   964
      let ?x = "\<chi> i. c i"
hoelzl@37489
   965
      have th0:"A *v ?x = 0"
hoelzl@37489
   966
        using c
lp15@67673
   967
        unfolding matrix_mult_sum vec_eq_iff
hoelzl@37489
   968
        by auto
hoelzl@37489
   969
      from k[rule_format, OF th0] i
huffman@44136
   970
      have "c i = 0" by (vector vec_eq_iff)}
wenzelm@49644
   971
    hence ?rhs by blast }
hoelzl@37489
   972
  moreover
wenzelm@49644
   973
  { assume H: ?rhs
wenzelm@49644
   974
    { fix x assume x: "A *v x = 0"
hoelzl@37489
   975
      let ?c = "\<lambda>i. ((x$i ):: real)"
lp15@67673
   976
      from H[rule_format, of ?c, unfolded matrix_mult_sum[symmetric], OF x]
wenzelm@49644
   977
      have "x = 0" by vector }
wenzelm@49644
   978
  }
hoelzl@37489
   979
  ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
hoelzl@37489
   980
qed
hoelzl@37489
   981
hoelzl@37489
   982
lemma matrix_right_invertible_independent_rows:
hoelzl@37489
   983
  fixes A :: "real^'n^'m"
wenzelm@49644
   984
  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
nipkow@64267
   985
    (\<forall>c. sum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
hoelzl@37489
   986
  unfolding left_invertible_transpose[symmetric]
hoelzl@37489
   987
    matrix_left_invertible_independent_columns
hoelzl@37489
   988
  by (simp add: column_transpose)
hoelzl@37489
   989
hoelzl@37489
   990
lemma matrix_right_invertible_span_columns:
wenzelm@49644
   991
  "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
wenzelm@49644
   992
    span (columns A) = UNIV" (is "?lhs = ?rhs")
wenzelm@49644
   993
proof -
hoelzl@37489
   994
  let ?U = "UNIV :: 'm set"
hoelzl@37489
   995
  have fU: "finite ?U" by simp
nipkow@64267
   996
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y)"
lp15@67673
   997
    unfolding matrix_right_invertible_surjective matrix_mult_sum surj_def
wenzelm@49644
   998
    apply (subst eq_commute)
wenzelm@49644
   999
    apply rule
wenzelm@49644
  1000
    done
hoelzl@37489
  1001
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
wenzelm@49644
  1002
  { assume h: ?lhs
wenzelm@49644
  1003
    { fix x:: "real ^'n"
wenzelm@49644
  1004
      from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
nipkow@64267
  1005
        where y: "sum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
wenzelm@49644
  1006
      have "x \<in> span (columns A)"
wenzelm@49644
  1007
        unfolding y[symmetric]
nipkow@64267
  1008
        apply (rule span_sum)
hoelzl@50526
  1009
        unfolding scalar_mult_eq_scaleR
wenzelm@49644
  1010
        apply (rule span_mul)
wenzelm@49644
  1011
        apply (rule span_superset)
wenzelm@49644
  1012
        unfolding columns_def
wenzelm@49644
  1013
        apply blast
wenzelm@49644
  1014
        done
wenzelm@49644
  1015
    }
wenzelm@49644
  1016
    then have ?rhs unfolding rhseq by blast }
hoelzl@37489
  1017
  moreover
wenzelm@49644
  1018
  { assume h:?rhs
nipkow@64267
  1019
    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y"
wenzelm@49644
  1020
    { fix y
wenzelm@49644
  1021
      have "?P y"
hoelzl@50526
  1022
      proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
nipkow@64267
  1023
        show "\<exists>x::real ^ 'm. sum (\<lambda>i. (x$i) *s column i A) ?U = 0"
hoelzl@37489
  1024
          by (rule exI[where x=0], simp)
hoelzl@37489
  1025
      next
wenzelm@49644
  1026
        fix c y1 y2
wenzelm@49644
  1027
        assume y1: "y1 \<in> columns A" and y2: "?P y2"
hoelzl@37489
  1028
        from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
hoelzl@37489
  1029
          unfolding columns_def by blast
hoelzl@37489
  1030
        from y2 obtain x:: "real ^'m" where
nipkow@64267
  1031
          x: "sum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
hoelzl@37489
  1032
        let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
hoelzl@37489
  1033
        show "?P (c*s y1 + y2)"
webertj@49962
  1034
        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
  1035
          fix j
wenzelm@49644
  1036
          have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
  1037
              else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
wenzelm@49644
  1038
            using i(1) by (simp add: field_simps)
nipkow@64267
  1039
          have "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
nipkow@64267
  1040
              else (x$xa) * ((column xa A$j))) ?U = sum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"
nipkow@64267
  1041
            apply (rule sum.cong[OF refl])
wenzelm@49644
  1042
            using th apply blast
wenzelm@49644
  1043
            done
nipkow@64267
  1044
          also have "\<dots> = sum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
nipkow@64267
  1045
            by (simp add: sum.distrib)
nipkow@64267
  1046
          also have "\<dots> = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
nipkow@64267
  1047
            unfolding sum.delta[OF fU]
wenzelm@49644
  1048
            using i(1) by simp
nipkow@64267
  1049
          finally show "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
nipkow@64267
  1050
            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
wenzelm@49644
  1051
        qed
wenzelm@49644
  1052
      next
wenzelm@49644
  1053
        show "y \<in> span (columns A)"
wenzelm@49644
  1054
          unfolding h by blast
wenzelm@49644
  1055
      qed
wenzelm@49644
  1056
    }
wenzelm@49644
  1057
    then have ?lhs unfolding lhseq ..
wenzelm@49644
  1058
  }
hoelzl@37489
  1059
  ultimately show ?thesis by blast
hoelzl@37489
  1060
qed
hoelzl@37489
  1061
hoelzl@37489
  1062
lemma matrix_left_invertible_span_rows:
hoelzl@37489
  1063
  "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
hoelzl@37489
  1064
  unfolding right_invertible_transpose[symmetric]
hoelzl@37489
  1065
  unfolding columns_transpose[symmetric]
hoelzl@37489
  1066
  unfolding matrix_right_invertible_span_columns
wenzelm@49644
  1067
  ..
hoelzl@37489
  1068
wenzelm@60420
  1069
text \<open>The same result in terms of square matrices.\<close>
hoelzl@37489
  1070
hoelzl@37489
  1071
lemma matrix_left_right_inverse:
hoelzl@37489
  1072
  fixes A A' :: "real ^'n^'n"
hoelzl@37489
  1073
  shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
wenzelm@49644
  1074
proof -
wenzelm@49644
  1075
  { fix A A' :: "real ^'n^'n"
wenzelm@49644
  1076
    assume AA': "A ** A' = mat 1"
nipkow@67399
  1077
    have sA: "surj (( *v) A)"
hoelzl@37489
  1078
      unfolding surj_def
hoelzl@37489
  1079
      apply clarify
hoelzl@37489
  1080
      apply (rule_tac x="(A' *v y)" in exI)
wenzelm@49644
  1081
      apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
wenzelm@49644
  1082
      done
hoelzl@37489
  1083
    from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
hoelzl@37489
  1084
    obtain f' :: "real ^'n \<Rightarrow> real ^'n"
hoelzl@37489
  1085
      where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
hoelzl@37489
  1086
    have th: "matrix f' ** A = mat 1"
wenzelm@49644
  1087
      by (simp add: matrix_eq matrix_works[OF f'(1)]
wenzelm@49644
  1088
          matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
hoelzl@37489
  1089
    hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
wenzelm@49644
  1090
    hence "matrix f' = A'"
wenzelm@49644
  1091
      by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
hoelzl@37489
  1092
    hence "matrix f' ** A = A' ** A" by simp
wenzelm@49644
  1093
    hence "A' ** A = mat 1" by (simp add: th)
wenzelm@49644
  1094
  }
hoelzl@37489
  1095
  then show ?thesis by blast
hoelzl@37489
  1096
qed
hoelzl@37489
  1097
wenzelm@60420
  1098
text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
hoelzl@37489
  1099
hoelzl@37489
  1100
definition "rowvector v = (\<chi> i j. (v$j))"
hoelzl@37489
  1101
hoelzl@37489
  1102
definition "columnvector v = (\<chi> i j. (v$i))"
hoelzl@37489
  1103
wenzelm@49644
  1104
lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
huffman@44136
  1105
  by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
hoelzl@37489
  1106
hoelzl@37489
  1107
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
huffman@44136
  1108
  by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
hoelzl@37489
  1109
wenzelm@49644
  1110
lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
hoelzl@37489
  1111
  by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
hoelzl@37489
  1112
wenzelm@49644
  1113
lemma dot_matrix_product:
wenzelm@49644
  1114
  "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
huffman@44136
  1115
  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
hoelzl@37489
  1116
hoelzl@37489
  1117
lemma dot_matrix_vector_mul:
hoelzl@37489
  1118
  fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
hoelzl@37489
  1119
  shows "(A *v x) \<bullet> (B *v y) =
hoelzl@37489
  1120
      (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
wenzelm@49644
  1121
  unfolding dot_matrix_product transpose_columnvector[symmetric]
wenzelm@49644
  1122
    dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
hoelzl@37489
  1123
wenzelm@61945
  1124
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}"
hoelzl@50526
  1125
  by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
hoelzl@37489
  1126
wenzelm@49644
  1127
lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
hoelzl@50526
  1128
  using Basis_le_infnorm[of "axis i 1" x]
hoelzl@50526
  1129
  by (simp add: Basis_vec_def axis_eq_axis inner_axis)
hoelzl@37489
  1130
hoelzl@63334
  1131
lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
huffman@44647
  1132
  unfolding continuous_def by (rule tendsto_vec_nth)
huffman@44213
  1133
hoelzl@63334
  1134
lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
huffman@44647
  1135
  unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
huffman@44213
  1136
hoelzl@63334
  1137
lemma continuous_on_vec_lambda[continuous_intros]:
hoelzl@63334
  1138
  "(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)"
hoelzl@63334
  1139
  unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)
hoelzl@63334
  1140
hoelzl@37489
  1141
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
hoelzl@63332
  1142
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
huffman@44213
  1143
hoelzl@37489
  1144
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
wenzelm@49644
  1145
  unfolding bounded_def
wenzelm@49644
  1146
  apply clarify
wenzelm@49644
  1147
  apply (rule_tac x="x $ i" in exI)
wenzelm@49644
  1148
  apply (rule_tac x="e" in exI)
wenzelm@49644
  1149
  apply clarify
wenzelm@49644
  1150
  apply (rule order_trans [OF dist_vec_nth_le], simp)
wenzelm@49644
  1151
  done
hoelzl@37489
  1152
hoelzl@37489
  1153
lemma compact_lemma_cart:
hoelzl@37489
  1154
  fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
hoelzl@50998
  1155
  assumes f: "bounded (range f)"
eberlm@66447
  1156
  shows "\<exists>l r. strict_mono r \<and>
hoelzl@37489
  1157
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
immler@62127
  1158
    (is "?th d")
immler@62127
  1159
proof -
immler@62127
  1160
  have "\<forall>d' \<subseteq> d. ?th d'"
immler@62127
  1161
    by (rule compact_lemma_general[where unproj=vec_lambda])
immler@62127
  1162
      (auto intro!: f bounded_component_cart simp: vec_lambda_eta)
immler@62127
  1163
  then show "?th d" by simp
hoelzl@37489
  1164
qed
hoelzl@37489
  1165
huffman@44136
  1166
instance vec :: (heine_borel, finite) heine_borel
hoelzl@37489
  1167
proof
hoelzl@50998
  1168
  fix f :: "nat \<Rightarrow> 'a ^ 'b"
hoelzl@50998
  1169
  assume f: "bounded (range f)"
eberlm@66447
  1170
  then obtain l r where r: "strict_mono r"
wenzelm@49644
  1171
      and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@50998
  1172
    using compact_lemma_cart [OF f] by blast
hoelzl@37489
  1173
  let ?d = "UNIV::'b set"
hoelzl@37489
  1174
  { fix e::real assume "e>0"
hoelzl@37489
  1175
    hence "0 < e / (real_of_nat (card ?d))"
wenzelm@49644
  1176
      using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
hoelzl@37489
  1177
    with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
hoelzl@37489
  1178
      by simp
hoelzl@37489
  1179
    moreover
wenzelm@49644
  1180
    { fix n
wenzelm@49644
  1181
      assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
hoelzl@37489
  1182
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
nipkow@67155
  1183
        unfolding dist_vec_def using zero_le_dist by (rule L2_set_le_sum)
hoelzl@37489
  1184
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
nipkow@64267
  1185
        by (rule sum_strict_mono) (simp_all add: n)
hoelzl@37489
  1186
      finally have "dist (f (r n)) l < e" by simp
hoelzl@37489
  1187
    }
hoelzl@37489
  1188
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
lp15@61810
  1189
      by (rule eventually_mono)
hoelzl@37489
  1190
  }
wenzelm@61973
  1191
  hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp
eberlm@66447
  1192
  with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
hoelzl@37489
  1193
qed
hoelzl@37489
  1194
wenzelm@49644
  1195
lemma interval_cart:
immler@54775
  1196
  fixes a :: "real^'n"
immler@54775
  1197
  shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
immler@56188
  1198
    and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
immler@56188
  1199
  by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
hoelzl@37489
  1200
lp15@67673
  1201
lemma mem_box_cart:
immler@54775
  1202
  fixes a :: "real^'n"
immler@54775
  1203
  shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
immler@56188
  1204
    and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
wenzelm@49644
  1205
  using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
hoelzl@37489
  1206
wenzelm@49644
  1207
lemma interval_eq_empty_cart:
wenzelm@49644
  1208
  fixes a :: "real^'n"
immler@54775
  1209
  shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
immler@56188
  1210
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
wenzelm@49644
  1211
proof -
immler@54775
  1212
  { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>box a b"
lp15@67673
  1213
    hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_box_cart by auto
hoelzl@37489
  1214
    hence "a$i < b$i" by auto
wenzelm@49644
  1215
    hence False using as by auto }
hoelzl@37489
  1216
  moreover
hoelzl@37489
  1217
  { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
hoelzl@37489
  1218
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
  1219
    { fix i
hoelzl@37489
  1220
      have "a$i < b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
  1221
      hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
hoelzl@37489
  1222
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
  1223
        by auto }
lp15@67673
  1224
    hence "box a b \<noteq> {}" using mem_box_cart(1)[of "?x" a b] by auto }
hoelzl@37489
  1225
  ultimately show ?th1 by blast
hoelzl@37489
  1226
immler@56188
  1227
  { fix i x assume as:"b$i < a$i" and x:"x\<in>cbox a b"
lp15@67673
  1228
    hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_box_cart by auto
hoelzl@37489
  1229
    hence "a$i \<le> b$i" by auto
wenzelm@49644
  1230
    hence False using as by auto }
hoelzl@37489
  1231
  moreover
hoelzl@37489
  1232
  { assume as:"\<forall>i. \<not> (b$i < a$i)"
hoelzl@37489
  1233
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
  1234
    { fix i
hoelzl@37489
  1235
      have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
  1236
      hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
hoelzl@37489
  1237
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
  1238
        by auto }
lp15@67673
  1239
    hence "cbox a b \<noteq> {}" using mem_box_cart(2)[of "?x" a b] by auto  }
hoelzl@37489
  1240
  ultimately show ?th2 by blast
hoelzl@37489
  1241
qed
hoelzl@37489
  1242
wenzelm@49644
  1243
lemma interval_ne_empty_cart:
wenzelm@49644
  1244
  fixes a :: "real^'n"
immler@56188
  1245
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
immler@54775
  1246
    and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
hoelzl@37489
  1247
  unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
hoelzl@37489
  1248
    (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
  1249
wenzelm@49644
  1250
lemma subset_interval_imp_cart:
wenzelm@49644
  1251
  fixes a :: "real^'n"
immler@56188
  1252
  shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
immler@56188
  1253
    and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
immler@56188
  1254
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
immler@54775
  1255
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b"
lp15@67673
  1256
  unfolding subset_eq[unfolded Ball_def] unfolding mem_box_cart
hoelzl@37489
  1257
  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
  1258
wenzelm@49644
  1259
lemma interval_sing:
wenzelm@49644
  1260
  fixes a :: "'a::linorder^'n"
wenzelm@49644
  1261
  shows "{a .. a} = {a} \<and> {a<..<a} = {}"
wenzelm@49644
  1262
  apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
wenzelm@49644
  1263
  done
hoelzl@37489
  1264
wenzelm@49644
  1265
lemma subset_interval_cart:
wenzelm@49644
  1266
  fixes a :: "real^'n"
immler@56188
  1267
  shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
immler@56188
  1268
    and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
immler@56188
  1269
    and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
immler@54775
  1270
    and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
immler@56188
  1271
  using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
  1272
wenzelm@49644
  1273
lemma disjoint_interval_cart:
wenzelm@49644
  1274
  fixes a::"real^'n"
immler@56188
  1275
  shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
immler@56188
  1276
    and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
immler@56188
  1277
    and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
immler@54775
  1278
    and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
hoelzl@50526
  1279
  using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
  1280
lp15@67719
  1281
lemma Int_interval_cart:
immler@54775
  1282
  fixes a :: "real^'n"
immler@56188
  1283
  shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
lp15@63945
  1284
  unfolding Int_interval
immler@56188
  1285
  by (auto simp: mem_box less_eq_vec_def)
immler@56188
  1286
    (auto simp: Basis_vec_def inner_axis)
hoelzl@37489
  1287
wenzelm@49644
  1288
lemma closed_interval_left_cart:
wenzelm@49644
  1289
  fixes b :: "real^'n"
hoelzl@37489
  1290
  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
hoelzl@63332
  1291
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1292
wenzelm@49644
  1293
lemma closed_interval_right_cart:
wenzelm@49644
  1294
  fixes a::"real^'n"
hoelzl@37489
  1295
  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
hoelzl@63332
  1296
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1297
wenzelm@49644
  1298
lemma is_interval_cart:
wenzelm@49644
  1299
  "is_interval (s::(real^'n) set) \<longleftrightarrow>
wenzelm@49644
  1300
    (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
hoelzl@50526
  1301
  by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
hoelzl@37489
  1302
wenzelm@49644
  1303
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
hoelzl@63332
  1304
  by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1305
wenzelm@49644
  1306
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
hoelzl@63332
  1307
  by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1308
wenzelm@49644
  1309
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
hoelzl@63332
  1310
  by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
wenzelm@49644
  1311
wenzelm@49644
  1312
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
hoelzl@63332
  1313
  by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1314
wenzelm@49644
  1315
lemma Lim_component_le_cart:
wenzelm@49644
  1316
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
  1317
  assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"
hoelzl@37489
  1318
  shows "l$i \<le> b"
hoelzl@50526
  1319
  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
hoelzl@37489
  1320
wenzelm@49644
  1321
lemma Lim_component_ge_cart:
wenzelm@49644
  1322
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
  1323
  assumes "(f \<longlongrightarrow> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
hoelzl@37489
  1324
  shows "b \<le> l$i"
hoelzl@50526
  1325
  by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
hoelzl@37489
  1326
wenzelm@49644
  1327
lemma Lim_component_eq_cart:
wenzelm@49644
  1328
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
  1329
  assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
hoelzl@37489
  1330
  shows "l$i = b"
wenzelm@49644
  1331
  using ev[unfolded order_eq_iff eventually_conj_iff] and
wenzelm@49644
  1332
    Lim_component_ge_cart[OF net, of b i] and
hoelzl@37489
  1333
    Lim_component_le_cart[OF net, of i b] by auto
hoelzl@37489
  1334
wenzelm@49644
  1335
lemma connected_ivt_component_cart:
wenzelm@49644
  1336
  fixes x :: "real^'n"
wenzelm@49644
  1337
  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$k = a)"
hoelzl@50526
  1338
  using connected_ivt_hyperplane[of s x y "axis k 1" a]
hoelzl@50526
  1339
  by (auto simp add: inner_axis inner_commute)
hoelzl@37489
  1340
wenzelm@49644
  1341
lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
hoelzl@37489
  1342
  unfolding subspace_def by auto
hoelzl@37489
  1343
hoelzl@37489
  1344
lemma closed_substandard_cart:
huffman@44213
  1345
  "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
  1346
proof -
huffman@44213
  1347
  { fix i::'n
huffman@44213
  1348
    have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
hoelzl@63332
  1349
      by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) }
huffman@44213
  1350
  thus ?thesis
huffman@44213
  1351
    unfolding Collect_all_eq by (simp add: closed_INT)
hoelzl@37489
  1352
qed
hoelzl@37489
  1353
wenzelm@49644
  1354
lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
wenzelm@49644
  1355
  (is "dim ?A = _")
wenzelm@49644
  1356
proof -
hoelzl@50526
  1357
  let ?a = "\<lambda>x. axis x 1 :: real^'n"
hoelzl@50526
  1358
  have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
hoelzl@50526
  1359
    by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
hoelzl@50526
  1360
  have "?a ` d \<subseteq> Basis"
hoelzl@50526
  1361
    by (auto simp: Basis_vec_def)
wenzelm@49644
  1362
  thus ?thesis
hoelzl@50526
  1363
    using dim_substandard[of "?a ` d"] card_image[of ?a d]
hoelzl@50526
  1364
    by (auto simp: axis_eq_axis inj_on_def *)
hoelzl@37489
  1365
qed
hoelzl@37489
  1366
lp15@67719
  1367
lemma dim_subset_UNIV_cart:
lp15@67719
  1368
  fixes S :: "(real^'n) set"
lp15@67719
  1369
  shows "dim S \<le> CARD('n)"
lp15@67719
  1370
  by (metis dim_subset_UNIV DIM_cart DIM_real mult.right_neutral)
lp15@67719
  1371
hoelzl@37489
  1372
lemma affinity_inverses:
hoelzl@37489
  1373
  assumes m0: "m \<noteq> (0::'a::field)"
wenzelm@61736
  1374
  shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
wenzelm@61736
  1375
  "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
hoelzl@37489
  1376
  using m0
haftmann@54230
  1377
  apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
haftmann@54230
  1378
  apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
wenzelm@49644
  1379
  done
hoelzl@37489
  1380
hoelzl@37489
  1381
lemma vector_affinity_eq:
hoelzl@37489
  1382
  assumes m0: "(m::'a::field) \<noteq> 0"
hoelzl@37489
  1383
  shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
hoelzl@37489
  1384
proof
hoelzl@37489
  1385
  assume h: "m *s x + c = y"
hoelzl@37489
  1386
  hence "m *s x = y - c" by (simp add: field_simps)
hoelzl@37489
  1387
  hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
hoelzl@37489
  1388
  then show "x = inverse m *s y + - (inverse m *s c)"
hoelzl@37489
  1389
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1390
next
hoelzl@37489
  1391
  assume h: "x = inverse m *s y + - (inverse m *s c)"
haftmann@54230
  1392
  show "m *s x + c = y" unfolding h
hoelzl@37489
  1393
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1394
qed
hoelzl@37489
  1395
hoelzl@37489
  1396
lemma vector_eq_affinity:
wenzelm@49644
  1397
    "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
hoelzl@37489
  1398
  using vector_affinity_eq[where m=m and x=x and y=y and c=c]
hoelzl@37489
  1399
  by metis
hoelzl@37489
  1400
hoelzl@50526
  1401
lemma vector_cart:
hoelzl@50526
  1402
  fixes f :: "real^'n \<Rightarrow> real"
hoelzl@50526
  1403
  shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
hoelzl@50526
  1404
  unfolding euclidean_eq_iff[where 'a="real^'n"]
hoelzl@50526
  1405
  by simp (simp add: Basis_vec_def inner_axis)
hoelzl@63332
  1406
hoelzl@50526
  1407
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
hoelzl@50526
  1408
  by (rule vector_cart)
wenzelm@49644
  1409
huffman@44360
  1410
subsection "Convex Euclidean Space"
hoelzl@37489
  1411
hoelzl@50526
  1412
lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
hoelzl@50526
  1413
  using const_vector_cart[of 1] by (simp add: one_vec_def)
hoelzl@37489
  1414
hoelzl@37489
  1415
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
hoelzl@37489
  1416
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
hoelzl@37489
  1417
hoelzl@50526
  1418
lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
hoelzl@37489
  1419
hoelzl@37489
  1420
lemma convex_box_cart:
hoelzl@37489
  1421
  assumes "\<And>i. convex {x. P i x}"
hoelzl@37489
  1422
  shows "convex {x. \<forall>i. P i (x$i)}"
hoelzl@37489
  1423
  using assms unfolding convex_def by auto
hoelzl@37489
  1424
hoelzl@37489
  1425
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
hoelzl@63334
  1426
  by (rule convex_box_cart) (simp add: atLeast_def[symmetric])
hoelzl@37489
  1427
hoelzl@37489
  1428
lemma unit_interval_convex_hull_cart:
immler@56188
  1429
  "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}"
immler@56188
  1430
  unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
hoelzl@50526
  1431
  by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
hoelzl@37489
  1432
hoelzl@37489
  1433
lemma cube_convex_hull_cart:
wenzelm@49644
  1434
  assumes "0 < d"
wenzelm@49644
  1435
  obtains s::"(real^'n) set"
immler@56188
  1436
    where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
wenzelm@49644
  1437
proof -
wenzelm@55522
  1438
  from assms obtain s where "finite s"
nipkow@67399
  1439
    and "cbox (x - sum (( *\<^sub>R) d) Basis) (x + sum (( *\<^sub>R) d) Basis) = convex hull s"
wenzelm@55522
  1440
    by (rule cube_convex_hull)
wenzelm@55522
  1441
  with that[of s] show thesis
wenzelm@55522
  1442
    by (simp add: const_vector_cart)
hoelzl@37489
  1443
qed
hoelzl@37489
  1444
hoelzl@37489
  1445
hoelzl@37489
  1446
subsection "Derivative"
hoelzl@37489
  1447
hoelzl@37489
  1448
definition "jacobian f net = matrix(frechet_derivative f net)"
hoelzl@37489
  1449
wenzelm@49644
  1450
lemma jacobian_works:
wenzelm@49644
  1451
  "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
lp15@67986
  1452
    (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net" (is "?lhs = ?rhs")
lp15@67986
  1453
proof
lp15@67986
  1454
  assume ?lhs then show ?rhs
lp15@67986
  1455
    by (simp add: frechet_derivative_works has_derivative_linear jacobian_def)
lp15@67986
  1456
next
lp15@67986
  1457
  assume ?rhs then show ?lhs
lp15@67986
  1458
    by (rule differentiableI)
lp15@67986
  1459
qed
hoelzl@37489
  1460
hoelzl@37489
  1461
wenzelm@60420
  1462
subsection \<open>Component of the differential must be zero if it exists at a local
nipkow@67968
  1463
  maximum or minimum for that corresponding component\<close>
hoelzl@37489
  1464
hoelzl@50526
  1465
lemma differential_zero_maxmin_cart:
wenzelm@49644
  1466
  fixes f::"real^'a \<Rightarrow> real^'b"
wenzelm@49644
  1467
  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))"
hoelzl@50526
  1468
    "f differentiable (at x)"
hoelzl@50526
  1469
  shows "jacobian f (at x) $ k = 0"
hoelzl@50526
  1470
  using differential_zero_maxmin_component[of "axis k 1" e x f] assms
hoelzl@50526
  1471
    vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
hoelzl@50526
  1472
  by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
wenzelm@49644
  1473
wenzelm@60420
  1474
subsection \<open>Lemmas for working on @{typ "real^1"}\<close>
hoelzl@37489
  1475
hoelzl@37489
  1476
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
wenzelm@49644
  1477
  by (metis (full_types) num1_eq_iff)
hoelzl@37489
  1478
hoelzl@37489
  1479
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
wenzelm@49644
  1480
  by auto (metis (full_types) num1_eq_iff)
hoelzl@37489
  1481
hoelzl@37489
  1482
lemma exhaust_2:
wenzelm@49644
  1483
  fixes x :: 2
wenzelm@49644
  1484
  shows "x = 1 \<or> x = 2"
hoelzl@37489
  1485
proof (induct x)
hoelzl@37489
  1486
  case (of_int z)
lp15@67979
  1487
  then have "0 \<le> z" and "z < 2" by simp_all
hoelzl@37489
  1488
  then have "z = 0 | z = 1" by arith
hoelzl@37489
  1489
  then show ?case by auto
hoelzl@37489
  1490
qed
hoelzl@37489
  1491
hoelzl@37489
  1492
lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
hoelzl@37489
  1493
  by (metis exhaust_2)
hoelzl@37489
  1494
hoelzl@37489
  1495
lemma exhaust_3:
wenzelm@49644
  1496
  fixes x :: 3
wenzelm@49644
  1497
  shows "x = 1 \<or> x = 2 \<or> x = 3"
hoelzl@37489
  1498
proof (induct x)
hoelzl@37489
  1499
  case (of_int z)
lp15@67979
  1500
  then have "0 \<le> z" and "z < 3" by simp_all
hoelzl@37489
  1501
  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
hoelzl@37489
  1502
  then show ?case by auto
hoelzl@37489
  1503
qed
hoelzl@37489
  1504
hoelzl@37489
  1505
lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
hoelzl@37489
  1506
  by (metis exhaust_3)
hoelzl@37489
  1507
hoelzl@37489
  1508
lemma UNIV_1 [simp]: "UNIV = {1::1}"
hoelzl@37489
  1509
  by (auto simp add: num1_eq_iff)
hoelzl@37489
  1510
hoelzl@37489
  1511
lemma UNIV_2: "UNIV = {1::2, 2::2}"
hoelzl@37489
  1512
  using exhaust_2 by auto
hoelzl@37489
  1513
hoelzl@37489
  1514
lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
hoelzl@37489
  1515
  using exhaust_3 by auto
hoelzl@37489
  1516
nipkow@64267
  1517
lemma sum_1: "sum f (UNIV::1 set) = f 1"
hoelzl@37489
  1518
  unfolding UNIV_1 by simp
hoelzl@37489
  1519
nipkow@64267
  1520
lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2"
hoelzl@37489
  1521
  unfolding UNIV_2 by simp
hoelzl@37489
  1522
nipkow@64267
  1523
lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3"
haftmann@57514
  1524
  unfolding UNIV_3 by (simp add: ac_simps)
hoelzl@37489
  1525
lp15@67979
  1526
lemma num1_eqI:
lp15@67979
  1527
  fixes a::num1 shows "a = b"
lp15@67979
  1528
  by (metis (full_types) UNIV_1 UNIV_I empty_iff insert_iff)
lp15@67979
  1529
lp15@67979
  1530
lemma num1_eq1 [simp]:
lp15@67979
  1531
  fixes a::num1 shows "a = 1"
lp15@67979
  1532
  by (rule num1_eqI)
lp15@67979
  1533
wenzelm@49644
  1534
instantiation num1 :: cart_one
wenzelm@49644
  1535
begin
wenzelm@49644
  1536
wenzelm@49644
  1537
instance
wenzelm@49644
  1538
proof
hoelzl@37489
  1539
  show "CARD(1) = Suc 0" by auto
wenzelm@49644
  1540
qed
wenzelm@49644
  1541
wenzelm@49644
  1542
end
hoelzl@37489
  1543
lp15@67979
  1544
instantiation num1 :: linorder begin
lp15@67979
  1545
definition "a < b \<longleftrightarrow> Rep_num1 a < Rep_num1 b"
lp15@67979
  1546
definition "a \<le> b \<longleftrightarrow> Rep_num1 a \<le> Rep_num1 b"
lp15@67979
  1547
instance
lp15@67979
  1548
  by intro_classes (auto simp: less_eq_num1_def less_num1_def intro: num1_eqI)
lp15@67979
  1549
end
lp15@67979
  1550
lp15@67979
  1551
instance num1 :: wellorder
lp15@67979
  1552
  by intro_classes (auto simp: less_eq_num1_def less_num1_def)
lp15@67979
  1553
nipkow@67968
  1554
subsection\<open>The collapse of the general concepts to dimension one\<close>
hoelzl@37489
  1555
hoelzl@37489
  1556
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
huffman@44136
  1557
  by (simp add: vec_eq_iff)
hoelzl@37489
  1558
hoelzl@37489
  1559
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
hoelzl@37489
  1560
  apply auto
hoelzl@37489
  1561
  apply (erule_tac x= "x$1" in allE)
hoelzl@37489
  1562
  apply (simp only: vector_one[symmetric])
hoelzl@37489
  1563
  done
hoelzl@37489
  1564
hoelzl@37489
  1565
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
huffman@44136
  1566
  by (simp add: norm_vec_def)
hoelzl@37489
  1567
lp15@67979
  1568
lemma dist_vector_1:
lp15@67979
  1569
  fixes x :: "'a::real_normed_vector^1"
lp15@67979
  1570
  shows "dist x y = dist (x$1) (y$1)"
lp15@67979
  1571
  by (simp add: dist_norm norm_vector_1)
lp15@67979
  1572
wenzelm@61945
  1573
lemma norm_real: "norm(x::real ^ 1) = \<bar>x$1\<bar>"
hoelzl@37489
  1574
  by (simp add: norm_vector_1)
hoelzl@37489
  1575
wenzelm@61945
  1576
lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x$1) - (y$1)\<bar>"
hoelzl@37489
  1577
  by (auto simp add: norm_real dist_norm)
hoelzl@37489
  1578
lp15@67986
  1579
subsection\<open> Rank of a matrix\<close>
lp15@67986
  1580
lp15@67986
  1581
text\<open>Equivalence of row and column rank is taken from George Mackiw's paper, Mathematics Magazine 1995, p. 285.\<close>
lp15@67986
  1582
lp15@67986
  1583
lemma matrix_vector_mult_in_columnspace:
lp15@67986
  1584
  fixes A :: "real^'n^'m"
lp15@67986
  1585
  shows "(A *v x) \<in> span(columns A)"
lp15@67986
  1586
  apply (simp add: matrix_vector_column columns_def transpose_def column_def)
lp15@67986
  1587
  apply (intro span_sum span_mul)
lp15@67986
  1588
  apply (force intro: span_superset)
lp15@67986
  1589
  done
lp15@67986
  1590
lp15@67986
  1591
lemma orthogonal_nullspace_rowspace:
lp15@67986
  1592
  fixes A :: "real^'n^'m"
lp15@67986
  1593
  assumes 0: "A *v x = 0" and y: "y \<in> span(rows A)"
lp15@67986
  1594
  shows "orthogonal x y"
lp15@67986
  1595
proof (rule span_induct [OF y])
lp15@67986
  1596
  show "subspace {a. orthogonal x a}"
lp15@67986
  1597
    by (simp add: subspace_orthogonal_to_vector)
lp15@67986
  1598
next
lp15@67986
  1599
  fix v
lp15@67986
  1600
  assume "v \<in> rows A"
lp15@67986
  1601
  then obtain i where "v = row i A"
lp15@67986
  1602
    by (auto simp: rows_def)
lp15@67986
  1603
  with 0 show "orthogonal x v"
lp15@67986
  1604
    unfolding orthogonal_def inner_vec_def matrix_vector_mult_def row_def
lp15@67986
  1605
    by (simp add: mult.commute) (metis (no_types) vec_lambda_beta zero_index)
lp15@67986
  1606
qed
lp15@67986
  1607
lp15@67986
  1608
lemma nullspace_inter_rowspace:
lp15@67986
  1609
  fixes A :: "real^'n^'m"
lp15@67986
  1610
  shows "A *v x = 0 \<and> x \<in> span(rows A) \<longleftrightarrow> x = 0"
lp15@67986
  1611
  using orthogonal_nullspace_rowspace orthogonal_self by auto
lp15@67986
  1612
lp15@67986
  1613
lemma matrix_vector_mul_injective_on_rowspace:
lp15@67986
  1614
  fixes A :: "real^'n^'m"
lp15@67986
  1615
  shows "\<lbrakk>A *v x = A *v y; x \<in> span(rows A); y \<in> span(rows A)\<rbrakk> \<Longrightarrow> x = y"
lp15@67986
  1616
  using nullspace_inter_rowspace [of A "x-y"]
lp15@67986
  1617
  by (metis eq_iff_diff_eq_0 matrix_vector_mult_diff_distrib span_diff)
lp15@67986
  1618
lp15@67986
  1619
definition rank :: "real^'n^'m=>nat"
lp15@67986
  1620
  where "rank A \<equiv> dim(columns A)"
lp15@67986
  1621
lp15@67986
  1622
lemma dim_rows_le_dim_columns:
lp15@67986
  1623
  fixes A :: "real^'n^'m"
lp15@67986
  1624
  shows "dim(rows A) \<le> dim(columns A)"
lp15@67986
  1625
proof -
lp15@67986
  1626
  have "dim (span (rows A)) \<le> dim (span (columns A))"
lp15@67986
  1627
  proof -
lp15@67986
  1628
    obtain B where "independent B" "span(rows A) \<subseteq> span B"
lp15@67986
  1629
              and B: "B \<subseteq> span(rows A)""card B = dim (span(rows A))"
lp15@67986
  1630
      using basis_exists [of "span(rows A)"] by blast
lp15@67986
  1631
    with span_subspace have eq: "span B = span(rows A)"
lp15@67986
  1632
      by auto
lp15@67986
  1633
    then have inj: "inj_on (( *v) A) (span B)"
lp15@67986
  1634
      using inj_on_def matrix_vector_mul_injective_on_rowspace by blast
lp15@67986
  1635
    then have ind: "independent (( *v) A ` B)"
lp15@67986
  1636
      by (rule independent_inj_on_image [OF \<open>independent B\<close> matrix_vector_mul_linear])
lp15@67986
  1637
    then have "finite (( *v) A ` B) \<and> card (( *v) A ` B) \<le> dim (( *v) A ` B)"
lp15@67986
  1638
      by (rule independent_bound_general)
lp15@67986
  1639
    then show ?thesis
lp15@67986
  1640
      by (metis (no_types, lifting) B ind inj eq card_image image_subset_iff independent_card_le_dim inj_on_subset matrix_vector_mult_in_columnspace)
lp15@67986
  1641
  qed
lp15@67986
  1642
  then show ?thesis
lp15@67986
  1643
    by simp
lp15@67986
  1644
qed
lp15@67986
  1645
lp15@67986
  1646
lemma rank_row:
lp15@67986
  1647
  fixes A :: "real^'n^'m"
lp15@67986
  1648
  shows "rank A = dim(rows A)"
lp15@67986
  1649
  unfolding rank_def
lp15@67986
  1650
  by (metis dim_rows_le_dim_columns columns_transpose dual_order.antisym rows_transpose)
lp15@67986
  1651
lp15@67986
  1652
lemma rank_transpose:
lp15@67986
  1653
  fixes A :: "real^'n^'m"
lp15@67986
  1654
  shows "rank(transpose A) = rank A"
lp15@67986
  1655
  by (metis rank_def rank_row rows_transpose)
lp15@67986
  1656
lp15@67986
  1657
lemma matrix_vector_mult_basis:
lp15@67986
  1658
  fixes A :: "real^'n^'m"
lp15@67986
  1659
  shows "A *v (axis k 1) = column k A"
lp15@67986
  1660
  by (simp add: cart_eq_inner_axis column_def matrix_mult_dot)
lp15@67986
  1661
lp15@67986
  1662
lemma columns_image_basis:
lp15@67986
  1663
  fixes A :: "real^'n^'m"
lp15@67986
  1664
  shows "columns A = ( *v) A ` (range (\<lambda>i. axis i 1))"
lp15@67986
  1665
  by (force simp: columns_def matrix_vector_mult_basis [symmetric])
lp15@67986
  1666
lp15@67986
  1667
lemma rank_dim_range:
lp15@67986
  1668
  fixes A :: "real^'n^'m"
lp15@67986
  1669
  shows "rank A = dim(range (\<lambda>x. A *v x))"
lp15@67986
  1670
  unfolding rank_def
lp15@67986
  1671
proof (rule span_eq_dim)
lp15@67986
  1672
  show "span (columns A) = span (range (( *v) A))"
lp15@67986
  1673
    apply (auto simp: columns_image_basis span_linear_image matrix_vector_mul_linear)
lp15@67986
  1674
    by (metis columns_image_basis matrix_vector_mul_linear matrix_vector_mult_in_columnspace span_linear_image)
lp15@67986
  1675
qed
lp15@67986
  1676
lp15@67986
  1677
lemma rank_bound:
lp15@67986
  1678
  fixes A :: "real^'n^'m"
lp15@67986
  1679
  shows "rank A \<le> min CARD('m) (CARD('n))"
lp15@67986
  1680
  by (metis (mono_tags, hide_lams) min.bounded_iff DIM_cart DIM_real dim_subset_UNIV mult.right_neutral rank_def rank_transpose)
lp15@67986
  1681
lp15@67986
  1682
lemma full_rank_injective:
lp15@67986
  1683
  fixes A :: "real^'n^'m"
lp15@67986
  1684
  shows "rank A = CARD('n) \<longleftrightarrow> inj (( *v) A)"
lp15@67986
  1685
  by (simp add: matrix_left_invertible_injective [symmetric] matrix_left_invertible_span_rows rank_row dim_eq_full [symmetric])
lp15@67986
  1686
lp15@67986
  1687
lemma full_rank_surjective:
lp15@67986
  1688
  fixes A :: "real^'n^'m"
lp15@67986
  1689
  shows "rank A = CARD('m) \<longleftrightarrow> surj (( *v) A)"
lp15@67986
  1690
  by (simp add: matrix_right_invertible_surjective [symmetric] left_invertible_transpose [symmetric]
lp15@67986
  1691
                matrix_left_invertible_injective full_rank_injective [symmetric] rank_transpose)
lp15@67986
  1692
lp15@67986
  1693
lemma rank_I: "rank(mat 1::real^'n^'n) = CARD('n)"
lp15@67986
  1694
  by (simp add: full_rank_injective inj_on_def)
lp15@67986
  1695
lp15@67986
  1696
lemma less_rank_noninjective:
lp15@67986
  1697
  fixes A :: "real^'n^'m"
lp15@67986
  1698
  shows "rank A < CARD('n) \<longleftrightarrow> \<not> inj (( *v) A)"
lp15@67986
  1699
using less_le rank_bound by (auto simp: full_rank_injective [symmetric])
lp15@67986
  1700
lp15@67986
  1701
lemma matrix_nonfull_linear_equations_eq:
lp15@67986
  1702
  fixes A :: "real^'n^'m"
lp15@67986
  1703
  shows "(\<exists>x. (x \<noteq> 0) \<and> A *v x = 0) \<longleftrightarrow> ~(rank A = CARD('n))"
lp15@67986
  1704
  by (meson matrix_left_invertible_injective full_rank_injective matrix_left_invertible_ker)
lp15@67986
  1705
lp15@67986
  1706
lemma rank_eq_0: "rank A = 0 \<longleftrightarrow> A = 0" and rank_0 [simp]: "rank 0 = 0"
lp15@67986
  1707
  by (auto simp: rank_dim_range matrix_eq)
lp15@67986
  1708
lp15@67986
  1709
lp15@67986
  1710
lemma rank_mul_le_right:
lp15@67986
  1711
  fixes A :: "real^'n^'m" and B :: "real^'p^'n"
lp15@67986
  1712
  shows "rank(A ** B) \<le> rank B"
lp15@67986
  1713
proof -
lp15@67986
  1714
  have "rank(A ** B) \<le> dim (( *v) A ` range (( *v) B))"
lp15@67986
  1715
    by (auto simp: rank_dim_range image_comp o_def matrix_vector_mul_assoc)
lp15@67986
  1716
  also have "\<dots> \<le> rank B"
lp15@67986
  1717
    by (simp add: rank_dim_range matrix_vector_mul_linear dim_image_le)
lp15@67986
  1718
  finally show ?thesis .
lp15@67986
  1719
qed
lp15@67986
  1720
lp15@67986
  1721
lemma rank_mul_le_left:
lp15@67986
  1722
  fixes A :: "real^'n^'m" and B :: "real^'p^'n"
lp15@67986
  1723
  shows "rank(A ** B) \<le> rank A"
lp15@67986
  1724
  by (metis matrix_transpose_mul rank_mul_le_right rank_transpose)
lp15@67986
  1725
lp15@67981
  1726
subsection\<open>Routine results connecting the types @{typ "real^1"} and @{typ real}\<close>
lp15@67981
  1727
lp15@67981
  1728
lemma vector_one_nth [simp]:
lp15@67981
  1729
  fixes x :: "'a^1" shows "vec (x $ 1) = x"
lp15@67981
  1730
  by (metis vec_def vector_one)
lp15@67981
  1731
lp15@67981
  1732
lemma vec_cbox_1_eq [simp]:
lp15@67981
  1733
  shows "vec ` cbox u v = cbox (vec u) (vec v ::real^1)"
lp15@67981
  1734
  by (force simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box)
lp15@67981
  1735
lp15@67981
  1736
lemma vec_nth_cbox_1_eq [simp]:
lp15@67981
  1737
  fixes u v :: "'a::euclidean_space^1"
lp15@67981
  1738
  shows "(\<lambda>x. x $ 1) ` cbox u v = cbox (u$1) (v$1)"
lp15@67981
  1739
    by (auto simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box image_iff Bex_def inner_axis) (metis vec_component)
lp15@67981
  1740
lp15@67981
  1741
lemma vec_nth_1_iff_cbox [simp]:
lp15@67981
  1742
  fixes a b :: "'a::euclidean_space"
lp15@67981
  1743
  shows "(\<lambda>x::'a^1. x $ 1) ` S = cbox a b \<longleftrightarrow> S = cbox (vec a) (vec b)"
lp15@67981
  1744
    (is "?lhs = ?rhs")
lp15@67981
  1745
proof
lp15@67981
  1746
  assume L: ?lhs show ?rhs
lp15@67981
  1747
  proof (intro equalityI subsetI)
lp15@67981
  1748
    fix x 
lp15@67981
  1749
    assume "x \<in> S"
lp15@67981
  1750
    then have "x $ 1 \<in> (\<lambda>v. v $ (1::1)) ` cbox (vec a) (vec b)"
lp15@67981
  1751
      using L by auto
lp15@67981
  1752
    then show "x \<in> cbox (vec a) (vec b)"
lp15@67981
  1753
      by (metis (no_types, lifting) imageE vector_one_nth)
lp15@67981
  1754
  next
lp15@67981
  1755
    fix x :: "'a^1"
lp15@67981
  1756
    assume "x \<in> cbox (vec a) (vec b)"
lp15@67981
  1757
    then show "x \<in> S"
lp15@67981
  1758
      by (metis (no_types, lifting) L imageE imageI vec_component vec_nth_cbox_1_eq vector_one_nth)
lp15@67981
  1759
  qed
lp15@67981
  1760
qed simp
wenzelm@49644
  1761
lp15@67979
  1762
lemma tendsto_at_within_vector_1:
lp15@67979
  1763
  fixes S :: "'a :: metric_space set"
lp15@67979
  1764
  assumes "(f \<longlongrightarrow> fx) (at x within S)"
lp15@67979
  1765
  shows "((\<lambda>y::'a^1. \<chi> i. f (y $ 1)) \<longlongrightarrow> (vec fx::'a^1)) (at (vec x) within vec ` S)"
lp15@67979
  1766
proof (rule topological_tendstoI)
lp15@67979
  1767
  fix T :: "('a^1) set"
lp15@67979
  1768
  assume "open T" "vec fx \<in> T"
lp15@67979
  1769
  have "\<forall>\<^sub>F x in at x within S. f x \<in> (\<lambda>x. x $ 1) ` T"
lp15@67979
  1770
    using \<open>open T\<close> \<open>vec fx \<in> T\<close> assms open_image_vec_nth tendsto_def by fastforce
lp15@67979
  1771
  then show "\<forall>\<^sub>F x::'a^1 in at (vec x) within vec ` S. (\<chi> i. f (x $ 1)) \<in> T"
lp15@67979
  1772
    unfolding eventually_at dist_norm [symmetric]
lp15@67979
  1773
    by (rule ex_forward)
lp15@67979
  1774
       (use \<open>open T\<close> in 
lp15@67979
  1775
         \<open>fastforce simp: dist_norm dist_vec_def L2_set_def image_iff vector_one open_vec_def\<close>)
lp15@67979
  1776
qed
lp15@67979
  1777
lp15@67979
  1778
lemma has_derivative_vector_1:
lp15@67979
  1779
  assumes der_g: "(g has_derivative (\<lambda>x. x * g' a)) (at a within S)"
lp15@67979
  1780
  shows "((\<lambda>x. vec (g (x $ 1))) has_derivative ( *\<^sub>R) (g' a))
lp15@67979
  1781
         (at ((vec a)::real^1) within vec ` S)"
lp15@67979
  1782
    using der_g
lp15@67979
  1783
    apply (auto simp: Deriv.has_derivative_within bounded_linear_scaleR_right norm_vector_1)
lp15@67979
  1784
    apply (drule tendsto_at_within_vector_1, vector)
lp15@67979
  1785
    apply (auto simp: algebra_simps eventually_at tendsto_def)
lp15@67979
  1786
    done
lp15@67979
  1787
lp15@67979
  1788
nipkow@67968
  1789
subsection\<open>Explicit vector construction from lists\<close>
hoelzl@37489
  1790
hoelzl@43995
  1791
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
  1792
hoelzl@37489
  1793
lemma vector_1: "(vector[x]) $1 = x"
hoelzl@37489
  1794
  unfolding vector_def by simp
hoelzl@37489
  1795
hoelzl@37489
  1796
lemma vector_2:
hoelzl@37489
  1797
 "(vector[x,y]) $1 = x"
hoelzl@37489
  1798
 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
hoelzl@37489
  1799
  unfolding vector_def by simp_all
hoelzl@37489
  1800
hoelzl@37489
  1801
lemma vector_3:
hoelzl@37489
  1802
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
hoelzl@37489
  1803
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
hoelzl@37489
  1804
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
hoelzl@37489
  1805
  unfolding vector_def by simp_all
hoelzl@37489
  1806
hoelzl@37489
  1807
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
lp15@67719
  1808
  by (metis vector_1 vector_one)
hoelzl@37489
  1809
hoelzl@37489
  1810
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
hoelzl@37489
  1811
  apply auto
hoelzl@37489
  1812
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1813
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1814
  apply (subgoal_tac "vector [v$1, v$2] = v")
hoelzl@37489
  1815
  apply simp
hoelzl@37489
  1816
  apply (vector vector_def)
hoelzl@37489
  1817
  apply (simp add: forall_2)
hoelzl@37489
  1818
  done
hoelzl@37489
  1819
hoelzl@37489
  1820
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
hoelzl@37489
  1821
  apply auto
hoelzl@37489
  1822
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1823
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1824
  apply (erule_tac x="v$3" in allE)
hoelzl@37489
  1825
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
hoelzl@37489
  1826
  apply simp
hoelzl@37489
  1827
  apply (vector vector_def)
hoelzl@37489
  1828
  apply (simp add: forall_3)
hoelzl@37489
  1829
  done
hoelzl@37489
  1830
hoelzl@37489
  1831
lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
wenzelm@49644
  1832
  apply (rule bounded_linearI[where K=1])
hoelzl@37489
  1833
  using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
hoelzl@37489
  1834
hoelzl@37489
  1835
lemma interval_split_cart:
hoelzl@37489
  1836
  "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
immler@56188
  1837
  "cbox a b \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
wenzelm@49644
  1838
  apply (rule_tac[!] set_eqI)
lp15@67673
  1839
  unfolding Int_iff mem_box_cart mem_Collect_eq interval_cbox_cart
wenzelm@49644
  1840
  unfolding vec_lambda_beta
wenzelm@49644
  1841
  by auto
hoelzl@37489
  1842
immler@67685
  1843
lemmas cartesian_euclidean_space_uniform_limit_intros[uniform_limit_intros] =
immler@67685
  1844
  bounded_linear.uniform_limit[OF blinfun.bounded_linear_right]
immler@67685
  1845
  bounded_linear.uniform_limit[OF bounded_linear_vec_nth]
immler@67685
  1846
  bounded_linear.uniform_limit[OF bounded_linear_component_cart]
immler@67685
  1847
hoelzl@37489
  1848
end