src/HOL/Analysis/Cartesian_Euclidean_Space.thy
 author paulson Sat Apr 14 09:23:00 2018 +0100 (16 months ago) changeset 67979 53323937ee25 parent 67971 e9f66b35d636 child 67981 349c639e593c permissions -rw-r--r--
new material about vec, real^1, etc.
```     1 section \<open>Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space\<close>
```
```     2
```
```     3 theory Cartesian_Euclidean_Space
```
```     4 imports Finite_Cartesian_Product Derivative
```
```     5 begin
```
```     6
```
```     7 lemma subspace_special_hyperplane: "subspace {x. x \$ k = 0}"
```
```     8   by (simp add: subspace_def)
```
```     9
```
```    10 lemma sum_mult_product:
```
```    11   "sum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
```
```    12   unfolding sum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]
```
```    13 proof (rule sum.cong, simp, rule sum.reindex_cong)
```
```    14   fix i
```
```    15   show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
```
```    16   show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
```
```    17   proof safe
```
```    18     fix j assume "j \<in> {i * B..<i * B + B}"
```
```    19     then show "j \<in> (\<lambda>j. j + i * B) ` {..<B}"
```
```    20       by (auto intro!: image_eqI[of _ _ "j - i * B"])
```
```    21   qed simp
```
```    22 qed simp
```
```    23
```
```    24 subsection\<open>Basic componentwise operations on vectors\<close>
```
```    25
```
```    26 instantiation vec :: (times, finite) times
```
```    27 begin
```
```    28
```
```    29 definition "( * ) \<equiv> (\<lambda> x y.  (\<chi> i. (x\$i) * (y\$i)))"
```
```    30 instance ..
```
```    31
```
```    32 end
```
```    33
```
```    34 instantiation vec :: (one, finite) one
```
```    35 begin
```
```    36
```
```    37 definition "1 \<equiv> (\<chi> i. 1)"
```
```    38 instance ..
```
```    39
```
```    40 end
```
```    41
```
```    42 instantiation vec :: (ord, finite) ord
```
```    43 begin
```
```    44
```
```    45 definition "x \<le> y \<longleftrightarrow> (\<forall>i. x\$i \<le> y\$i)"
```
```    46 definition "x < (y::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
```
```    47 instance ..
```
```    48
```
```    49 end
```
```    50
```
```    51 text\<open>The ordering on one-dimensional vectors is linear.\<close>
```
```    52
```
```    53 class cart_one =
```
```    54   assumes UNIV_one: "card (UNIV :: 'a set) = Suc 0"
```
```    55 begin
```
```    56
```
```    57 subclass finite
```
```    58 proof
```
```    59   from UNIV_one show "finite (UNIV :: 'a set)"
```
```    60     by (auto intro!: card_ge_0_finite)
```
```    61 qed
```
```    62
```
```    63 end
```
```    64
```
```    65 instance vec:: (order, finite) order
```
```    66   by standard (auto simp: less_eq_vec_def less_vec_def vec_eq_iff
```
```    67       intro: order.trans order.antisym order.strict_implies_order)
```
```    68
```
```    69 instance vec :: (linorder, cart_one) linorder
```
```    70 proof
```
```    71   obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"
```
```    72   proof -
```
```    73     have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)
```
```    74     then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)
```
```    75     then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto
```
```    76     then show thesis by (auto intro: that)
```
```    77   qed
```
```    78   fix x y :: "'a^'b::cart_one"
```
```    79   note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
```
```    80   show "x \<le> y \<or> y \<le> x" by auto
```
```    81 qed
```
```    82
```
```    83 text\<open>Constant Vectors\<close>
```
```    84
```
```    85 definition "vec x = (\<chi> i. x)"
```
```    86
```
```    87 lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b"
```
```    88   by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis)
```
```    89
```
```    90 text\<open>Also the scalar-vector multiplication.\<close>
```
```    91
```
```    92 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
```
```    93   where "c *s x = (\<chi> i. c * (x\$i))"
```
```    94
```
```    95
```
```    96 subsection \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space\<close>
```
```    97
```
```    98 lemma sum_cong_aux:
```
```    99   "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> sum f A = sum g A"
```
```   100   by (auto intro: sum.cong)
```
```   101
```
```   102 hide_fact (open) sum_cong_aux
```
```   103
```
```   104 method_setup vector = \<open>
```
```   105 let
```
```   106   val ss1 =
```
```   107     simpset_of (put_simpset HOL_basic_ss @{context}
```
```   108       addsimps [@{thm sum.distrib} RS sym,
```
```   109       @{thm sum_subtractf} RS sym, @{thm sum_distrib_left},
```
```   110       @{thm sum_distrib_right}, @{thm sum_negf} RS sym])
```
```   111   val ss2 =
```
```   112     simpset_of (@{context} addsimps
```
```   113              [@{thm plus_vec_def}, @{thm times_vec_def},
```
```   114               @{thm minus_vec_def}, @{thm uminus_vec_def},
```
```   115               @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
```
```   116               @{thm scaleR_vec_def},
```
```   117               @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}])
```
```   118   fun vector_arith_tac ctxt ths =
```
```   119     simp_tac (put_simpset ss1 ctxt)
```
```   120     THEN' (fn i => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.sum_cong_aux} i
```
```   121          ORELSE resolve_tac ctxt @{thms sum.neutral} i
```
```   122          ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i)
```
```   123     (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
```
```   124     THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths)
```
```   125 in
```
```   126   Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths))
```
```   127 end
```
```   128 \<close> "lift trivial vector statements to real arith statements"
```
```   129
```
```   130 lemma vec_0[simp]: "vec 0 = 0" by vector
```
```   131 lemma vec_1[simp]: "vec 1 = 1" by vector
```
```   132
```
```   133 lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
```
```   134
```
```   135 lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
```
```   136
```
```   137 lemma vec_add: "vec(x + y) = vec x + vec y"  by vector
```
```   138 lemma vec_sub: "vec(x - y) = vec x - vec y" by vector
```
```   139 lemma vec_cmul: "vec(c * x) = c *s vec x " by vector
```
```   140 lemma vec_neg: "vec(- x) = - vec x " by vector
```
```   141
```
```   142 lemma vec_scaleR: "vec(c * x) = c *\<^sub>R vec x"
```
```   143   by vector
```
```   144
```
```   145 lemma vec_sum:
```
```   146   assumes "finite S"
```
```   147   shows "vec(sum f S) = sum (vec \<circ> f) S"
```
```   148   using assms
```
```   149 proof induct
```
```   150   case empty
```
```   151   then show ?case by simp
```
```   152 next
```
```   153   case insert
```
```   154   then show ?case by (auto simp add: vec_add)
```
```   155 qed
```
```   156
```
```   157 text\<open>Obvious "component-pushing".\<close>
```
```   158
```
```   159 lemma vec_component [simp]: "vec x \$ i = x"
```
```   160   by vector
```
```   161
```
```   162 lemma vector_mult_component [simp]: "(x * y)\$i = x\$i * y\$i"
```
```   163   by vector
```
```   164
```
```   165 lemma vector_smult_component [simp]: "(c *s y)\$i = c * (y\$i)"
```
```   166   by vector
```
```   167
```
```   168 lemma cond_component: "(if b then x else y)\$i = (if b then x\$i else y\$i)" by vector
```
```   169
```
```   170 lemmas vector_component =
```
```   171   vec_component vector_add_component vector_mult_component
```
```   172   vector_smult_component vector_minus_component vector_uminus_component
```
```   173   vector_scaleR_component cond_component
```
```   174
```
```   175
```
```   176 subsection \<open>Some frequently useful arithmetic lemmas over vectors\<close>
```
```   177
```
```   178 instance vec :: (semigroup_mult, finite) semigroup_mult
```
```   179   by standard (vector mult.assoc)
```
```   180
```
```   181 instance vec :: (monoid_mult, finite) monoid_mult
```
```   182   by standard vector+
```
```   183
```
```   184 instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
```
```   185   by standard (vector mult.commute)
```
```   186
```
```   187 instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
```
```   188   by standard vector
```
```   189
```
```   190 instance vec :: (semiring, finite) semiring
```
```   191   by standard (vector field_simps)+
```
```   192
```
```   193 instance vec :: (semiring_0, finite) semiring_0
```
```   194   by standard (vector field_simps)+
```
```   195 instance vec :: (semiring_1, finite) semiring_1
```
```   196   by standard vector
```
```   197 instance vec :: (comm_semiring, finite) comm_semiring
```
```   198   by standard (vector field_simps)+
```
```   199
```
```   200 instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
```
```   201 instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
```
```   202 instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
```
```   203 instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
```
```   204 instance vec :: (ring, finite) ring ..
```
```   205 instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
```
```   206 instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
```
```   207
```
```   208 instance vec :: (ring_1, finite) ring_1 ..
```
```   209
```
```   210 instance vec :: (real_algebra, finite) real_algebra
```
```   211   by standard (simp_all add: vec_eq_iff)
```
```   212
```
```   213 instance vec :: (real_algebra_1, finite) real_algebra_1 ..
```
```   214
```
```   215 lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)\$i = of_nat n"
```
```   216 proof (induct n)
```
```   217   case 0
```
```   218   then show ?case by vector
```
```   219 next
```
```   220   case Suc
```
```   221   then show ?case by vector
```
```   222 qed
```
```   223
```
```   224 lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) \$ i = 1"
```
```   225   by vector
```
```   226
```
```   227 lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) \$ i = - 1"
```
```   228   by vector
```
```   229
```
```   230 instance vec :: (semiring_char_0, finite) semiring_char_0
```
```   231 proof
```
```   232   fix m n :: nat
```
```   233   show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
```
```   234     by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
```
```   235 qed
```
```   236
```
```   237 instance vec :: (numeral, finite) numeral ..
```
```   238 instance vec :: (semiring_numeral, finite) semiring_numeral ..
```
```   239
```
```   240 lemma numeral_index [simp]: "numeral w \$ i = numeral w"
```
```   241   by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
```
```   242
```
```   243 lemma neg_numeral_index [simp]: "- numeral w \$ i = - numeral w"
```
```   244   by (simp only: vector_uminus_component numeral_index)
```
```   245
```
```   246 instance vec :: (comm_ring_1, finite) comm_ring_1 ..
```
```   247 instance vec :: (ring_char_0, finite) ring_char_0 ..
```
```   248
```
```   249 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
```
```   250   by (vector mult.assoc)
```
```   251 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
```
```   252   by (vector field_simps)
```
```   253 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
```
```   254   by (vector field_simps)
```
```   255 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
```
```   256 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
```
```   257 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
```
```   258   by (vector field_simps)
```
```   259 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
```
```   260 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
```
```   261 lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
```
```   262 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
```
```   263 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
```
```   264   by (vector field_simps)
```
```   265
```
```   266 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
```
```   267   by (simp add: vec_eq_iff)
```
```   268
```
```   269 lemma linear_vec [simp]: "linear vec"
```
```   270   by (simp add: linearI vec_add vec_eq_iff)
```
```   271
```
```   272 lemma differentiable_vec:
```
```   273   fixes S :: "'a::euclidean_space set"
```
```   274   shows "vec differentiable_on S"
```
```   275   by (simp add: linear_linear bounded_linear_imp_differentiable_on)
```
```   276
```
```   277 lemma continuous_vec [continuous_intros]:
```
```   278   fixes x :: "'a::euclidean_space"
```
```   279   shows "isCont vec x"
```
```   280   apply (clarsimp simp add: continuous_def LIM_def dist_vec_def L2_set_def)
```
```   281   apply (rule_tac x="r / sqrt (real CARD('b))" in exI)
```
```   282   by (simp add: mult.commute pos_less_divide_eq real_sqrt_mult)
```
```   283
```
```   284 lemma box_vec_eq_empty [simp]:
```
```   285   shows "cbox (vec a) (vec b) = {} \<longleftrightarrow> cbox a b = {}"
```
```   286         "box (vec a) (vec b) = {} \<longleftrightarrow> box a b = {}"
```
```   287   by (auto simp: Basis_vec_def mem_box box_eq_empty inner_axis)
```
```   288
```
```   289 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
```
```   290
```
```   291 lemma norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
```
```   292   by (simp add: inner_axis' norm_eq_1)
```
```   293
```
```   294 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
```
```   295   by vector
```
```   296
```
```   297 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
```
```   298   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
```
```   299
```
```   300 lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
```
```   301   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
```
```   302
```
```   303 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
```
```   304   by (metis vector_mul_lcancel)
```
```   305
```
```   306 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
```
```   307   by (metis vector_mul_rcancel)
```
```   308
```
```   309 lemma component_le_norm_cart: "\<bar>x\$i\<bar> \<le> norm x"
```
```   310   apply (simp add: norm_vec_def)
```
```   311   apply (rule member_le_L2_set, simp_all)
```
```   312   done
```
```   313
```
```   314 lemma norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x\$i\<bar> \<le> e"
```
```   315   by (metis component_le_norm_cart order_trans)
```
```   316
```
```   317 lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x\$i\<bar> < e"
```
```   318   by (metis component_le_norm_cart le_less_trans)
```
```   319
```
```   320 lemma norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x\$i\<bar>) UNIV"
```
```   321   by (simp add: norm_vec_def L2_set_le_sum)
```
```   322
```
```   323 lemma scalar_mult_eq_scaleR [simp]: "c *s x = c *\<^sub>R x"
```
```   324   unfolding scaleR_vec_def vector_scalar_mult_def by simp
```
```   325
```
```   326 lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
```
```   327   unfolding dist_norm scalar_mult_eq_scaleR
```
```   328   unfolding scaleR_right_diff_distrib[symmetric] by simp
```
```   329
```
```   330 lemma sum_component [simp]:
```
```   331   fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
```
```   332   shows "(sum f S)\$i = sum (\<lambda>x. (f x)\$i) S"
```
```   333 proof (cases "finite S")
```
```   334   case True
```
```   335   then show ?thesis by induct simp_all
```
```   336 next
```
```   337   case False
```
```   338   then show ?thesis by simp
```
```   339 qed
```
```   340
```
```   341 lemma sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)\$i ) S)"
```
```   342   by (simp add: vec_eq_iff)
```
```   343
```
```   344 lemma sum_cmul:
```
```   345   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
```
```   346   shows "sum (\<lambda>x. c *s f x) S = c *s sum f S"
```
```   347   by (simp add: vec_eq_iff sum_distrib_left)
```
```   348
```
```   349 lemma sum_norm_allsubsets_bound_cart:
```
```   350   fixes f:: "'a \<Rightarrow> real ^'n"
```
```   351   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
```
```   352   shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
```
```   353   using sum_norm_allsubsets_bound[OF assms]
```
```   354   by simp
```
```   355
```
```   356 subsection\<open>Closures and interiors of halfspaces\<close>
```
```   357
```
```   358 lemma interior_halfspace_le [simp]:
```
```   359   assumes "a \<noteq> 0"
```
```   360     shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
```
```   361 proof -
```
```   362   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
```
```   363   proof -
```
```   364     obtain e where "e>0" and e: "cball x e \<subseteq> S"
```
```   365       using \<open>open S\<close> open_contains_cball x by blast
```
```   366     then have "x + (e / norm a) *\<^sub>R a \<in> cball x e"
```
```   367       by (simp add: dist_norm)
```
```   368     then have "x + (e / norm a) *\<^sub>R a \<in> S"
```
```   369       using e by blast
```
```   370     then have "x + (e / norm a) *\<^sub>R a \<in> {x. a \<bullet> x \<le> b}"
```
```   371       using S by blast
```
```   372     moreover have "e * (a \<bullet> a) / norm a > 0"
```
```   373       by (simp add: \<open>0 < e\<close> assms)
```
```   374     ultimately show ?thesis
```
```   375       by (simp add: algebra_simps)
```
```   376   qed
```
```   377   show ?thesis
```
```   378     by (rule interior_unique) (auto simp: open_halfspace_lt *)
```
```   379 qed
```
```   380
```
```   381 lemma interior_halfspace_ge [simp]:
```
```   382    "a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"
```
```   383 using interior_halfspace_le [of "-a" "-b"] by simp
```
```   384
```
```   385 lemma interior_halfspace_component_le [simp]:
```
```   386      "interior {x. x\$k \<le> a} = {x :: (real^'n). x\$k < a}" (is "?LE")
```
```   387   and interior_halfspace_component_ge [simp]:
```
```   388      "interior {x. x\$k \<ge> a} = {x :: (real^'n). x\$k > a}" (is "?GE")
```
```   389 proof -
```
```   390   have "axis k (1::real) \<noteq> 0"
```
```   391     by (simp add: axis_def vec_eq_iff)
```
```   392   moreover have "axis k (1::real) \<bullet> x = x\$k" for x
```
```   393     by (simp add: cart_eq_inner_axis inner_commute)
```
```   394   ultimately show ?LE ?GE
```
```   395     using interior_halfspace_le [of "axis k (1::real)" a]
```
```   396           interior_halfspace_ge [of "axis k (1::real)" a] by auto
```
```   397 qed
```
```   398
```
```   399 lemma closure_halfspace_lt [simp]:
```
```   400   assumes "a \<noteq> 0"
```
```   401     shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
```
```   402 proof -
```
```   403   have [simp]: "-{x. a \<bullet> x < b} = {x. a \<bullet> x \<ge> b}"
```
```   404     by (force simp:)
```
```   405   then show ?thesis
```
```   406     using interior_halfspace_ge [of a b] assms
```
```   407     by (force simp: closure_interior)
```
```   408 qed
```
```   409
```
```   410 lemma closure_halfspace_gt [simp]:
```
```   411    "a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"
```
```   412 using closure_halfspace_lt [of "-a" "-b"] by simp
```
```   413
```
```   414 lemma closure_halfspace_component_lt [simp]:
```
```   415      "closure {x. x\$k < a} = {x :: (real^'n). x\$k \<le> a}" (is "?LE")
```
```   416   and closure_halfspace_component_gt [simp]:
```
```   417      "closure {x. x\$k > a} = {x :: (real^'n). x\$k \<ge> a}" (is "?GE")
```
```   418 proof -
```
```   419   have "axis k (1::real) \<noteq> 0"
```
```   420     by (simp add: axis_def vec_eq_iff)
```
```   421   moreover have "axis k (1::real) \<bullet> x = x\$k" for x
```
```   422     by (simp add: cart_eq_inner_axis inner_commute)
```
```   423   ultimately show ?LE ?GE
```
```   424     using closure_halfspace_lt [of "axis k (1::real)" a]
```
```   425           closure_halfspace_gt [of "axis k (1::real)" a] by auto
```
```   426 qed
```
```   427
```
```   428 lemma interior_hyperplane [simp]:
```
```   429   assumes "a \<noteq> 0"
```
```   430     shows "interior {x. a \<bullet> x = b} = {}"
```
```   431 proof -
```
```   432   have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
```
```   433     by (force simp:)
```
```   434   then show ?thesis
```
```   435     by (auto simp: assms)
```
```   436 qed
```
```   437
```
```   438 lemma frontier_halfspace_le:
```
```   439   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
```
```   440     shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
```
```   441 proof (cases "a = 0")
```
```   442   case True with assms show ?thesis by simp
```
```   443 next
```
```   444   case False then show ?thesis
```
```   445     by (force simp: frontier_def closed_halfspace_le)
```
```   446 qed
```
```   447
```
```   448 lemma frontier_halfspace_ge:
```
```   449   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
```
```   450     shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"
```
```   451 proof (cases "a = 0")
```
```   452   case True with assms show ?thesis by simp
```
```   453 next
```
```   454   case False then show ?thesis
```
```   455     by (force simp: frontier_def closed_halfspace_ge)
```
```   456 qed
```
```   457
```
```   458 lemma frontier_halfspace_lt:
```
```   459   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
```
```   460     shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"
```
```   461 proof (cases "a = 0")
```
```   462   case True with assms show ?thesis by simp
```
```   463 next
```
```   464   case False then show ?thesis
```
```   465     by (force simp: frontier_def interior_open open_halfspace_lt)
```
```   466 qed
```
```   467
```
```   468 lemma frontier_halfspace_gt:
```
```   469   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
```
```   470     shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"
```
```   471 proof (cases "a = 0")
```
```   472   case True with assms show ?thesis by simp
```
```   473 next
```
```   474   case False then show ?thesis
```
```   475     by (force simp: frontier_def interior_open open_halfspace_gt)
```
```   476 qed
```
```   477
```
```   478 lemma interior_standard_hyperplane:
```
```   479    "interior {x :: (real^'n). x\$k = a} = {}"
```
```   480 proof -
```
```   481   have "axis k (1::real) \<noteq> 0"
```
```   482     by (simp add: axis_def vec_eq_iff)
```
```   483   moreover have "axis k (1::real) \<bullet> x = x\$k" for x
```
```   484     by (simp add: cart_eq_inner_axis inner_commute)
```
```   485   ultimately show ?thesis
```
```   486     using interior_hyperplane [of "axis k (1::real)" a]
```
```   487     by force
```
```   488 qed
```
```   489
```
```   490 subsection \<open>Matrix operations\<close>
```
```   491
```
```   492 text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close>
```
```   493
```
```   494 definition map_matrix::"('a \<Rightarrow> 'b) \<Rightarrow> (('a, 'i::finite)vec, 'j::finite) vec \<Rightarrow> (('b, 'i)vec, 'j) vec" where
```
```   495   "map_matrix f x = (\<chi> i j. f (x \$ i \$ j))"
```
```   496
```
```   497 lemma nth_map_matrix[simp]: "map_matrix f x \$ i \$ j = f (x \$ i \$ j)"
```
```   498   by (simp add: map_matrix_def)
```
```   499
```
```   500 definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
```
```   501     (infixl "**" 70)
```
```   502   where "m ** m' == (\<chi> i j. sum (\<lambda>k. ((m\$i)\$k) * ((m'\$k)\$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
```
```   503
```
```   504 definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
```
```   505     (infixl "*v" 70)
```
```   506   where "m *v x \<equiv> (\<chi> i. sum (\<lambda>j. ((m\$i)\$j) * (x\$j)) (UNIV ::'n set)) :: 'a^'m"
```
```   507
```
```   508 definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
```
```   509     (infixl "v*" 70)
```
```   510   where "v v* m == (\<chi> j. sum (\<lambda>i. ((m\$i)\$j) * (v\$i)) (UNIV :: 'm set)) :: 'a^'n"
```
```   511
```
```   512 definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
```
```   513 definition transpose where
```
```   514   "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A\$j)\$i))"
```
```   515 definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A\$i)\$j))"
```
```   516 definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A\$i)\$j))"
```
```   517 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
```
```   518 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
```
```   519
```
```   520 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
```
```   521 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
```
```   522   by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
```
```   523
```
```   524 lemma matrix_mul_lid [simp]:
```
```   525   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
```
```   526   shows "mat 1 ** A = A"
```
```   527   apply (simp add: matrix_matrix_mult_def mat_def)
```
```   528   apply vector
```
```   529   apply (auto simp only: if_distrib cond_application_beta sum.delta'[OF finite]
```
```   530     mult_1_left mult_zero_left if_True UNIV_I)
```
```   531   done
```
```   532
```
```   533
```
```   534 lemma matrix_mul_rid [simp]:
```
```   535   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
```
```   536   shows "A ** mat 1 = A"
```
```   537   apply (simp add: matrix_matrix_mult_def mat_def)
```
```   538   apply vector
```
```   539   apply (auto simp only: if_distrib cond_application_beta sum.delta[OF finite]
```
```   540     mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
```
```   541   done
```
```   542
```
```   543 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
```
```   544   apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc)
```
```   545   apply (subst sum.swap)
```
```   546   apply simp
```
```   547   done
```
```   548
```
```   549 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
```
```   550   apply (vector matrix_matrix_mult_def matrix_vector_mult_def
```
```   551     sum_distrib_left sum_distrib_right mult.assoc)
```
```   552   apply (subst sum.swap)
```
```   553   apply simp
```
```   554   done
```
```   555
```
```   556 lemma matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
```
```   557   apply (vector matrix_vector_mult_def mat_def)
```
```   558   apply (simp add: if_distrib cond_application_beta sum.delta' cong del: if_weak_cong)
```
```   559   done
```
```   560
```
```   561 lemma matrix_transpose_mul:
```
```   562     "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
```
```   563   by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
```
```   564
```
```   565 lemma matrix_eq:
```
```   566   fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
```
```   567   shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   568   apply auto
```
```   569   apply (subst vec_eq_iff)
```
```   570   apply clarify
```
```   571   apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
```
```   572   apply (erule_tac x="axis ia 1" in allE)
```
```   573   apply (erule_tac x="i" in allE)
```
```   574   apply (auto simp add: if_distrib cond_application_beta axis_def
```
```   575     sum.delta[OF finite] cong del: if_weak_cong)
```
```   576   done
```
```   577
```
```   578 lemma matrix_vector_mul_component: "((A::real^_^_) *v x)\$k = (A\$k) \<bullet> x"
```
```   579   by (simp add: matrix_vector_mult_def inner_vec_def)
```
```   580
```
```   581 lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
```
```   582   apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps)
```
```   583   apply (subst sum.swap)
```
```   584   apply simp
```
```   585   done
```
```   586
```
```   587 lemma transpose_mat [simp]: "transpose (mat n) = mat n"
```
```   588   by (vector transpose_def mat_def)
```
```   589
```
```   590 lemma transpose_transpose [simp]: "transpose(transpose A) = A"
```
```   591   by (vector transpose_def)
```
```   592
```
```   593 lemma row_transpose [simp]:
```
```   594   fixes A:: "'a::semiring_1^_^_"
```
```   595   shows "row i (transpose A) = column i A"
```
```   596   by (simp add: row_def column_def transpose_def vec_eq_iff)
```
```   597
```
```   598 lemma column_transpose [simp]:
```
```   599   fixes A:: "'a::semiring_1^_^_"
```
```   600   shows "column i (transpose A) = row i A"
```
```   601   by (simp add: row_def column_def transpose_def vec_eq_iff)
```
```   602
```
```   603 lemma rows_transpose [simp]: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
```
```   604   by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
```
```   605
```
```   606 lemma columns_transpose [simp]: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
```
```   607   by (metis transpose_transpose rows_transpose)
```
```   608
```
```   609 lemma matrix_mult_transpose_dot_column:
```
```   610   fixes A :: "real^'n^'n"
```
```   611   shows "transpose A ** A = (\<chi> i j. (column i A) \<bullet> (column j A))"
```
```   612   by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)
```
```   613
```
```   614 lemma matrix_mult_transpose_dot_row:
```
```   615   fixes A :: "real^'n^'n"
```
```   616   shows "A ** transpose A = (\<chi> i j. (row i A) \<bullet> (row j A))"
```
```   617   by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)
```
```   618
```
```   619 text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
```
```   620
```
```   621 lemma matrix_mult_dot: "A *v x = (\<chi> i. A\$i \<bullet> x)"
```
```   622   by (simp add: matrix_vector_mult_def inner_vec_def)
```
```   623
```
```   624 lemma matrix_mult_sum:
```
```   625   "(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x\$i) *s column i A) (UNIV:: 'n set)"
```
```   626   by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
```
```   627
```
```   628 lemma vector_componentwise:
```
```   629   "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x\$i) * (axis i 1 :: 'a^'n) \$ j)"
```
```   630   by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)
```
```   631
```
```   632 lemma basis_expansion: "sum (\<lambda>i. (x\$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
```
```   633   by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
```
```   634
```
```   635 lemma linear_componentwise_expansion:
```
```   636   fixes f:: "real ^'m \<Rightarrow> real ^ _"
```
```   637   assumes lf: "linear f"
```
```   638   shows "(f x)\$j = sum (\<lambda>i. (x\$i) * (f (axis i 1)\$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
```
```   639 proof -
```
```   640   let ?M = "(UNIV :: 'm set)"
```
```   641   let ?N = "(UNIV :: 'n set)"
```
```   642   have "?rhs = (sum (\<lambda>i.(x\$i) *\<^sub>R f (axis i 1) ) ?M)\$j"
```
```   643     unfolding sum_component by simp
```
```   644   then show ?thesis
```
```   645     unfolding linear_sum_mul[OF lf, symmetric]
```
```   646     unfolding scalar_mult_eq_scaleR[symmetric]
```
```   647     unfolding basis_expansion
```
```   648     by simp
```
```   649 qed
```
```   650
```
```   651 subsection\<open>Inverse matrices  (not necessarily square)\<close>
```
```   652
```
```   653 definition
```
```   654   "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
```
```   655
```
```   656 definition
```
```   657   "matrix_inv(A:: 'a::semiring_1^'n^'m) =
```
```   658     (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
```
```   659
```
```   660 text\<open>Correspondence between matrices and linear operators.\<close>
```
```   661
```
```   662 definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
```
```   663   where "matrix f = (\<chi> i j. (f(axis j 1))\$i)"
```
```   664
```
```   665 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
```
```   666   by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
```
```   667       field_simps sum_distrib_left sum.distrib)
```
```   668
```
```   669 lemma
```
```   670   fixes A :: "real^'n^'m"
```
```   671   shows matrix_vector_mult_linear_continuous_at [continuous_intros]: "isCont (( *v) A) z"
```
```   672     and matrix_vector_mult_linear_continuous_on [continuous_intros]: "continuous_on S (( *v) A)"
```
```   673   by (simp_all add: linear_linear linear_continuous_at linear_continuous_on matrix_vector_mul_linear)
```
```   674
```
```   675 lemma matrix_vector_mult_add_distrib [algebra_simps]:
```
```   676   "A *v (x + y) = A *v x + A *v y"
```
```   677   by (vector matrix_vector_mult_def sum.distrib distrib_left)
```
```   678
```
```   679 lemma matrix_vector_mult_diff_distrib [algebra_simps]:
```
```   680   fixes A :: "'a::ring_1^'n^'m"
```
```   681   shows "A *v (x - y) = A *v x - A *v y"
```
```   682   by (vector matrix_vector_mult_def sum_subtractf right_diff_distrib)
```
```   683
```
```   684 lemma matrix_vector_mult_scaleR[algebra_simps]:
```
```   685   fixes A :: "real^'n^'m"
```
```   686   shows "A *v (c *\<^sub>R x) = c *\<^sub>R (A *v x)"
```
```   687   using linear_iff matrix_vector_mul_linear by blast
```
```   688
```
```   689 lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
```
```   690   by (simp add: matrix_vector_mult_def vec_eq_iff)
```
```   691
```
```   692 lemma matrix_vector_mult_0 [simp]: "0 *v w = 0"
```
```   693   by (simp add: matrix_vector_mult_def vec_eq_iff)
```
```   694
```
```   695 lemma matrix_vector_mult_add_rdistrib [algebra_simps]:
```
```   696   "(A + B) *v x = (A *v x) + (B *v x)"
```
```   697   by (vector matrix_vector_mult_def sum.distrib distrib_right)
```
```   698
```
```   699 lemma matrix_vector_mult_diff_rdistrib [algebra_simps]:
```
```   700   fixes A :: "'a :: ring_1^'n^'m"
```
```   701   shows "(A - B) *v x = (A *v x) - (B *v x)"
```
```   702   by (vector matrix_vector_mult_def sum_subtractf left_diff_distrib)
```
```   703
```
```   704 lemma matrix_works:
```
```   705   assumes lf: "linear f"
```
```   706   shows "matrix f *v x = f (x::real ^ 'n)"
```
```   707   apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)
```
```   708   by (simp add: linear_componentwise_expansion lf)
```
```   709
```
```   710 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
```
```   711   by (simp add: ext matrix_works)
```
```   712
```
```   713 declare matrix_vector_mul [symmetric, simp]
```
```   714
```
```   715 lemma matrix_of_matrix_vector_mul [simp]: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
```
```   716   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
```
```   717
```
```   718 lemma matrix_compose:
```
```   719   assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
```
```   720     and lg: "linear (g::real^'m \<Rightarrow> real^_)"
```
```   721   shows "matrix (g \<circ> f) = matrix g ** matrix f"
```
```   722   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
```
```   723   by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
```
```   724
```
```   725 lemma matrix_vector_column:
```
```   726   "(A::'a::comm_semiring_1^'n^_) *v x = sum (\<lambda>i. (x\$i) *s ((transpose A)\$i)) (UNIV:: 'n set)"
```
```   727   by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
```
```   728
```
```   729 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
```
```   730   apply (rule adjoint_unique)
```
```   731   apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
```
```   732     sum_distrib_right sum_distrib_left)
```
```   733   apply (subst sum.swap)
```
```   734   apply (auto simp add: ac_simps)
```
```   735   done
```
```   736
```
```   737 lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
```
```   738   shows "matrix(adjoint f) = transpose(matrix f)"
```
```   739   apply (subst matrix_vector_mul[OF lf])
```
```   740   unfolding adjoint_matrix matrix_of_matrix_vector_mul
```
```   741   apply rule
```
```   742   done
```
```   743
```
```   744
```
```   745 subsection\<open>Some bounds on components etc. relative to operator norm\<close>
```
```   746
```
```   747 lemma norm_column_le_onorm:
```
```   748   fixes A :: "real^'n^'m"
```
```   749   shows "norm(column i A) \<le> onorm(( *v) A)"
```
```   750 proof -
```
```   751   have bl: "bounded_linear (( *v) A)"
```
```   752     by (simp add: linear_linear matrix_vector_mul_linear)
```
```   753   have "norm (\<chi> j. A \$ j \$ i) \<le> norm (A *v axis i 1)"
```
```   754     by (simp add: matrix_mult_dot cart_eq_inner_axis)
```
```   755   also have "\<dots> \<le> onorm (( *v) A)"
```
```   756     using onorm [OF bl, of "axis i 1"] by (auto simp: axis_in_Basis)
```
```   757   finally have "norm (\<chi> j. A \$ j \$ i) \<le> onorm (( *v) A)" .
```
```   758   then show ?thesis
```
```   759     unfolding column_def .
```
```   760 qed
```
```   761
```
```   762 lemma matrix_component_le_onorm:
```
```   763   fixes A :: "real^'n^'m"
```
```   764   shows "\<bar>A \$ i \$ j\<bar> \<le> onorm(( *v) A)"
```
```   765 proof -
```
```   766   have "\<bar>A \$ i \$ j\<bar> \<le> norm (\<chi> n. (A \$ n \$ j))"
```
```   767     by (metis (full_types, lifting) component_le_norm_cart vec_lambda_beta)
```
```   768   also have "\<dots> \<le> onorm (( *v) A)"
```
```   769     by (metis (no_types) column_def norm_column_le_onorm)
```
```   770   finally show ?thesis .
```
```   771 qed
```
```   772
```
```   773 lemma component_le_onorm:
```
```   774   fixes f :: "real^'m \<Rightarrow> real^'n"
```
```   775   shows "linear f \<Longrightarrow> \<bar>matrix f \$ i \$ j\<bar> \<le> onorm f"
```
```   776   by (metis matrix_component_le_onorm matrix_vector_mul)
```
```   777
```
```   778 lemma onorm_le_matrix_component_sum:
```
```   779   fixes A :: "real^'n^'m"
```
```   780   shows "onorm(( *v) A) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A \$ i \$ j\<bar>)"
```
```   781 proof (rule onorm_le)
```
```   782   fix x
```
```   783   have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) \$ i\<bar>)"
```
```   784     by (rule norm_le_l1_cart)
```
```   785   also have "\<dots> \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A \$ i \$ j\<bar> * norm x)"
```
```   786   proof (rule sum_mono)
```
```   787     fix i
```
```   788     have "\<bar>(A *v x) \$ i\<bar> \<le> \<bar>\<Sum>j\<in>UNIV. A \$ i \$ j * x \$ j\<bar>"
```
```   789       by (simp add: matrix_vector_mult_def)
```
```   790     also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A \$ i \$ j * x \$ j\<bar>)"
```
```   791       by (rule sum_abs)
```
```   792     also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A \$ i \$ j\<bar> * norm x)"
```
```   793       by (rule sum_mono) (simp add: abs_mult component_le_norm_cart mult_left_mono)
```
```   794     finally show "\<bar>(A *v x) \$ i\<bar> \<le> (\<Sum>j\<in>UNIV. \<bar>A \$ i \$ j\<bar> * norm x)" .
```
```   795   qed
```
```   796   finally show "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A \$ i \$ j\<bar>) * norm x"
```
```   797     by (simp add: sum_distrib_right)
```
```   798 qed
```
```   799
```
```   800 lemma onorm_le_matrix_component:
```
```   801   fixes A :: "real^'n^'m"
```
```   802   assumes "\<And>i j. abs(A\$i\$j) \<le> B"
```
```   803   shows "onorm(( *v) A) \<le> real (CARD('m)) * real (CARD('n)) * B"
```
```   804 proof (rule onorm_le)
```
```   805   fix x :: "real^'n::_"
```
```   806   have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) \$ i\<bar>)"
```
```   807     by (rule norm_le_l1_cart)
```
```   808   also have "\<dots> \<le> (\<Sum>i::'m \<in>UNIV. real (CARD('n)) * B * norm x)"
```
```   809   proof (rule sum_mono)
```
```   810     fix i
```
```   811     have "\<bar>(A *v x) \$ i\<bar> \<le> norm(A \$ i) * norm x"
```
```   812       by (simp add: matrix_mult_dot Cauchy_Schwarz_ineq2)
```
```   813     also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A \$ i \$ j\<bar>) * norm x"
```
```   814       by (simp add: mult_right_mono norm_le_l1_cart)
```
```   815     also have "\<dots> \<le> real (CARD('n)) * B * norm x"
```
```   816       by (simp add: assms sum_bounded_above mult_right_mono)
```
```   817     finally show "\<bar>(A *v x) \$ i\<bar> \<le> real (CARD('n)) * B * norm x" .
```
```   818   qed
```
```   819   also have "\<dots> \<le> CARD('m) * real (CARD('n)) * B * norm x"
```
```   820     by simp
```
```   821   finally show "norm (A *v x) \<le> CARD('m) * real (CARD('n)) * B * norm x" .
```
```   822 qed
```
```   823
```
```   824 subsection \<open>lambda skolemization on cartesian products\<close>
```
```   825
```
```   826 lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
```
```   827    (\<exists>x::'a ^ 'n. \<forall>i. P i (x \$ i))" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   828 proof -
```
```   829   let ?S = "(UNIV :: 'n set)"
```
```   830   { assume H: "?rhs"
```
```   831     then have ?lhs by auto }
```
```   832   moreover
```
```   833   { assume H: "?lhs"
```
```   834     then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
```
```   835     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
```
```   836     { fix i
```
```   837       from f have "P i (f i)" by metis
```
```   838       then have "P i (?x \$ i)" by auto
```
```   839     }
```
```   840     hence "\<forall>i. P i (?x\$i)" by metis
```
```   841     hence ?rhs by metis }
```
```   842   ultimately show ?thesis by metis
```
```   843 qed
```
```   844
```
```   845 lemma rational_approximation:
```
```   846   assumes "e > 0"
```
```   847   obtains r::real where "r \<in> \<rat>" "\<bar>r - x\<bar> < e"
```
```   848   using Rats_dense_in_real [of "x - e/2" "x + e/2"] assms by auto
```
```   849
```
```   850 lemma matrix_rational_approximation:
```
```   851   fixes A :: "real^'n^'m"
```
```   852   assumes "e > 0"
```
```   853   obtains B where "\<And>i j. B\$i\$j \<in> \<rat>" "onorm(\<lambda>x. (A - B) *v x) < e"
```
```   854 proof -
```
```   855   have "\<forall>i j. \<exists>q \<in> \<rat>. \<bar>q - A \$ i \$ j\<bar> < e / (2 * CARD('m) * CARD('n))"
```
```   856     using assms by (force intro: rational_approximation [of "e / (2 * CARD('m) * CARD('n))"])
```
```   857   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))"
```
```   858     by (auto simp: lambda_skolem Bex_def)
```
```   859   show ?thesis
```
```   860   proof
```
```   861     have "onorm (( *v) (A - B)) \<le> real CARD('m) * real CARD('n) *
```
```   862     (e / (2 * real CARD('m) * real CARD('n)))"
```
```   863       apply (rule onorm_le_matrix_component)
```
```   864       using Bclo by (simp add: abs_minus_commute less_imp_le)
```
```   865     also have "\<dots> < e"
```
```   866       using \<open>0 < e\<close> by (simp add: divide_simps)
```
```   867     finally show "onorm (( *v) (A - B)) < e" .
```
```   868   qed (use B in auto)
```
```   869 qed
```
```   870
```
```   871 lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
```
```   872   unfolding inner_simps scalar_mult_eq_scaleR by auto
```
```   873
```
```   874 lemma left_invertible_transpose:
```
```   875   "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
```
```   876   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
```
```   877
```
```   878 lemma right_invertible_transpose:
```
```   879   "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
```
```   880   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
```
```   881
```
```   882 lemma matrix_left_invertible_injective:
```
```   883   "(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
```
```   884 proof -
```
```   885   { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
```
```   886     from xy have "B*v (A *v x) = B *v (A*v y)" by simp
```
```   887     hence "x = y"
```
```   888       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
```
```   889   moreover
```
```   890   { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
```
```   891     hence i: "inj (( *v) A)" unfolding inj_on_def by auto
```
```   892     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
```
```   893     obtain g where g: "linear g" "g \<circ> ( *v) A = id" by blast
```
```   894     have "matrix g ** A = mat 1"
```
```   895       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
```
```   896       using g(2) by (simp add: fun_eq_iff)
```
```   897     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
```
```   898   ultimately show ?thesis by blast
```
```   899 qed
```
```   900
```
```   901 lemma matrix_left_invertible_ker:
```
```   902   "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
```
```   903   unfolding matrix_left_invertible_injective
```
```   904   using linear_injective_0[OF matrix_vector_mul_linear, of A]
```
```   905   by (simp add: inj_on_def)
```
```   906
```
```   907 lemma matrix_right_invertible_surjective:
```
```   908   "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
```
```   909 proof -
```
```   910   { fix B :: "real ^'m^'n"
```
```   911     assume AB: "A ** B = mat 1"
```
```   912     { fix x :: "real ^ 'm"
```
```   913       have "A *v (B *v x) = x"
```
```   914         by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
```
```   915     hence "surj (( *v) A)" unfolding surj_def by metis }
```
```   916   moreover
```
```   917   { assume sf: "surj (( *v) A)"
```
```   918     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
```
```   919     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "( *v) A \<circ> g = id"
```
```   920       by blast
```
```   921
```
```   922     have "A ** (matrix g) = mat 1"
```
```   923       unfolding matrix_eq  matrix_vector_mul_lid
```
```   924         matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
```
```   925       using g(2) unfolding o_def fun_eq_iff id_def
```
```   926       .
```
```   927     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
```
```   928   }
```
```   929   ultimately show ?thesis unfolding surj_def by blast
```
```   930 qed
```
```   931
```
```   932 lemma matrix_left_invertible_independent_columns:
```
```   933   fixes A :: "real^'n^'m"
```
```   934   shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
```
```   935       (\<forall>c. sum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
```
```   936     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   937 proof -
```
```   938   let ?U = "UNIV :: 'n set"
```
```   939   { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
```
```   940     { fix c i
```
```   941       assume c: "sum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
```
```   942       let ?x = "\<chi> i. c i"
```
```   943       have th0:"A *v ?x = 0"
```
```   944         using c
```
```   945         unfolding matrix_mult_sum vec_eq_iff
```
```   946         by auto
```
```   947       from k[rule_format, OF th0] i
```
```   948       have "c i = 0" by (vector vec_eq_iff)}
```
```   949     hence ?rhs by blast }
```
```   950   moreover
```
```   951   { assume H: ?rhs
```
```   952     { fix x assume x: "A *v x = 0"
```
```   953       let ?c = "\<lambda>i. ((x\$i ):: real)"
```
```   954       from H[rule_format, of ?c, unfolded matrix_mult_sum[symmetric], OF x]
```
```   955       have "x = 0" by vector }
```
```   956   }
```
```   957   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
```
```   958 qed
```
```   959
```
```   960 lemma matrix_right_invertible_independent_rows:
```
```   961   fixes A :: "real^'n^'m"
```
```   962   shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
```
```   963     (\<forall>c. sum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
```
```   964   unfolding left_invertible_transpose[symmetric]
```
```   965     matrix_left_invertible_independent_columns
```
```   966   by (simp add: column_transpose)
```
```   967
```
```   968 lemma matrix_right_invertible_span_columns:
```
```   969   "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
```
```   970     span (columns A) = UNIV" (is "?lhs = ?rhs")
```
```   971 proof -
```
```   972   let ?U = "UNIV :: 'm set"
```
```   973   have fU: "finite ?U" by simp
```
```   974   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). sum (\<lambda>i. (x\$i) *s column i A) ?U = y)"
```
```   975     unfolding matrix_right_invertible_surjective matrix_mult_sum surj_def
```
```   976     apply (subst eq_commute)
```
```   977     apply rule
```
```   978     done
```
```   979   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
```
```   980   { assume h: ?lhs
```
```   981     { fix x:: "real ^'n"
```
```   982       from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
```
```   983         where y: "sum (\<lambda>i. (y\$i) *s column i A) ?U = x" by blast
```
```   984       have "x \<in> span (columns A)"
```
```   985         unfolding y[symmetric]
```
```   986         apply (rule span_sum)
```
```   987         unfolding scalar_mult_eq_scaleR
```
```   988         apply (rule span_mul)
```
```   989         apply (rule span_superset)
```
```   990         unfolding columns_def
```
```   991         apply blast
```
```   992         done
```
```   993     }
```
```   994     then have ?rhs unfolding rhseq by blast }
```
```   995   moreover
```
```   996   { assume h:?rhs
```
```   997     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). sum (\<lambda>i. (x\$i) *s column i A) ?U = y"
```
```   998     { fix y
```
```   999       have "?P y"
```
```  1000       proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
```
```  1001         show "\<exists>x::real ^ 'm. sum (\<lambda>i. (x\$i) *s column i A) ?U = 0"
```
```  1002           by (rule exI[where x=0], simp)
```
```  1003       next
```
```  1004         fix c y1 y2
```
```  1005         assume y1: "y1 \<in> columns A" and y2: "?P y2"
```
```  1006         from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
```
```  1007           unfolding columns_def by blast
```
```  1008         from y2 obtain x:: "real ^'m" where
```
```  1009           x: "sum (\<lambda>i. (x\$i) *s column i A) ?U = y2" by blast
```
```  1010         let ?x = "(\<chi> j. if j = i then c + (x\$i) else (x\$j))::real^'m"
```
```  1011         show "?P (c*s y1 + y2)"
```
```  1012         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)
```
```  1013           fix j
```
```  1014           have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```  1015               else (x\$xa) * ((column xa A\$j))) = (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))"
```
```  1016             using i(1) by (simp add: field_simps)
```
```  1017           have "sum (\<lambda>xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```  1018               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"
```
```  1019             apply (rule sum.cong[OF refl])
```
```  1020             using th apply blast
```
```  1021             done
```
```  1022           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"
```
```  1023             by (simp add: sum.distrib)
```
```  1024           also have "\<dots> = c * ((column i A)\$j) + sum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U"
```
```  1025             unfolding sum.delta[OF fU]
```
```  1026             using i(1) by simp
```
```  1027           finally show "sum (\<lambda>xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```  1028             else (x\$xa) * ((column xa A\$j))) ?U = c * ((column i A)\$j) + sum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U" .
```
```  1029         qed
```
```  1030       next
```
```  1031         show "y \<in> span (columns A)"
```
```  1032           unfolding h by blast
```
```  1033       qed
```
```  1034     }
```
```  1035     then have ?lhs unfolding lhseq ..
```
```  1036   }
```
```  1037   ultimately show ?thesis by blast
```
```  1038 qed
```
```  1039
```
```  1040 lemma matrix_left_invertible_span_rows:
```
```  1041   "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
```
```  1042   unfolding right_invertible_transpose[symmetric]
```
```  1043   unfolding columns_transpose[symmetric]
```
```  1044   unfolding matrix_right_invertible_span_columns
```
```  1045   ..
```
```  1046
```
```  1047 text \<open>The same result in terms of square matrices.\<close>
```
```  1048
```
```  1049 lemma matrix_left_right_inverse:
```
```  1050   fixes A A' :: "real ^'n^'n"
```
```  1051   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
```
```  1052 proof -
```
```  1053   { fix A A' :: "real ^'n^'n"
```
```  1054     assume AA': "A ** A' = mat 1"
```
```  1055     have sA: "surj (( *v) A)"
```
```  1056       unfolding surj_def
```
```  1057       apply clarify
```
```  1058       apply (rule_tac x="(A' *v y)" in exI)
```
```  1059       apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
```
```  1060       done
```
```  1061     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
```
```  1062     obtain f' :: "real ^'n \<Rightarrow> real ^'n"
```
```  1063       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
```
```  1064     have th: "matrix f' ** A = mat 1"
```
```  1065       by (simp add: matrix_eq matrix_works[OF f'(1)]
```
```  1066           matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
```
```  1067     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
```
```  1068     hence "matrix f' = A'"
```
```  1069       by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
```
```  1070     hence "matrix f' ** A = A' ** A" by simp
```
```  1071     hence "A' ** A = mat 1" by (simp add: th)
```
```  1072   }
```
```  1073   then show ?thesis by blast
```
```  1074 qed
```
```  1075
```
```  1076 text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
```
```  1077
```
```  1078 definition "rowvector v = (\<chi> i j. (v\$j))"
```
```  1079
```
```  1080 definition "columnvector v = (\<chi> i j. (v\$i))"
```
```  1081
```
```  1082 lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
```
```  1083   by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
```
```  1084
```
```  1085 lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
```
```  1086   by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
```
```  1087
```
```  1088 lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
```
```  1089   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
```
```  1090
```
```  1091 lemma dot_matrix_product:
```
```  1092   "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))\$1)\$1"
```
```  1093   by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
```
```  1094
```
```  1095 lemma dot_matrix_vector_mul:
```
```  1096   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
```
```  1097   shows "(A *v x) \<bullet> (B *v y) =
```
```  1098       (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))\$1)\$1"
```
```  1099   unfolding dot_matrix_product transpose_columnvector[symmetric]
```
```  1100     dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
```
```  1101
```
```  1102 lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x\$i\<bar> |i. i\<in>UNIV}"
```
```  1103   by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
```
```  1104
```
```  1105 lemma component_le_infnorm_cart: "\<bar>x\$i\<bar> \<le> infnorm (x::real^'n)"
```
```  1106   using Basis_le_infnorm[of "axis i 1" x]
```
```  1107   by (simp add: Basis_vec_def axis_eq_axis inner_axis)
```
```  1108
```
```  1109 lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x \$ i)"
```
```  1110   unfolding continuous_def by (rule tendsto_vec_nth)
```
```  1111
```
```  1112 lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x \$ i)"
```
```  1113   unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
```
```  1114
```
```  1115 lemma continuous_on_vec_lambda[continuous_intros]:
```
```  1116   "(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)"
```
```  1117   unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)
```
```  1118
```
```  1119 lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x\$i}"
```
```  1120   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1121
```
```  1122 lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x \$ i) ` s)"
```
```  1123   unfolding bounded_def
```
```  1124   apply clarify
```
```  1125   apply (rule_tac x="x \$ i" in exI)
```
```  1126   apply (rule_tac x="e" in exI)
```
```  1127   apply clarify
```
```  1128   apply (rule order_trans [OF dist_vec_nth_le], simp)
```
```  1129   done
```
```  1130
```
```  1131 lemma compact_lemma_cart:
```
```  1132   fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
```
```  1133   assumes f: "bounded (range f)"
```
```  1134   shows "\<exists>l r. strict_mono r \<and>
```
```  1135         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
```
```  1136     (is "?th d")
```
```  1137 proof -
```
```  1138   have "\<forall>d' \<subseteq> d. ?th d'"
```
```  1139     by (rule compact_lemma_general[where unproj=vec_lambda])
```
```  1140       (auto intro!: f bounded_component_cart simp: vec_lambda_eta)
```
```  1141   then show "?th d" by simp
```
```  1142 qed
```
```  1143
```
```  1144 instance vec :: (heine_borel, finite) heine_borel
```
```  1145 proof
```
```  1146   fix f :: "nat \<Rightarrow> 'a ^ 'b"
```
```  1147   assume f: "bounded (range f)"
```
```  1148   then obtain l r where r: "strict_mono r"
```
```  1149       and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) \$ i) (l \$ i) < e) sequentially"
```
```  1150     using compact_lemma_cart [OF f] by blast
```
```  1151   let ?d = "UNIV::'b set"
```
```  1152   { fix e::real assume "e>0"
```
```  1153     hence "0 < e / (real_of_nat (card ?d))"
```
```  1154       using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
```
```  1155     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially"
```
```  1156       by simp
```
```  1157     moreover
```
```  1158     { fix n
```
```  1159       assume n: "\<forall>i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))"
```
```  1160       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) \$ i) (l \$ i))"
```
```  1161         unfolding dist_vec_def using zero_le_dist by (rule L2_set_le_sum)
```
```  1162       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
```
```  1163         by (rule sum_strict_mono) (simp_all add: n)
```
```  1164       finally have "dist (f (r n)) l < e" by simp
```
```  1165     }
```
```  1166     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
```
```  1167       by (rule eventually_mono)
```
```  1168   }
```
```  1169   hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp
```
```  1170   with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
```
```  1171 qed
```
```  1172
```
```  1173 lemma interval_cart:
```
```  1174   fixes a :: "real^'n"
```
```  1175   shows "box a b = {x::real^'n. \<forall>i. a\$i < x\$i \<and> x\$i < b\$i}"
```
```  1176     and "cbox a b = {x::real^'n. \<forall>i. a\$i \<le> x\$i \<and> x\$i \<le> b\$i}"
```
```  1177   by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
```
```  1178
```
```  1179 lemma mem_box_cart:
```
```  1180   fixes a :: "real^'n"
```
```  1181   shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a\$i < x\$i \<and> x\$i < b\$i)"
```
```  1182     and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a\$i \<le> x\$i \<and> x\$i \<le> b\$i)"
```
```  1183   using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
```
```  1184
```
```  1185 lemma interval_eq_empty_cart:
```
```  1186   fixes a :: "real^'n"
```
```  1187   shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b\$i \<le> a\$i))" (is ?th1)
```
```  1188     and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b\$i < a\$i))" (is ?th2)
```
```  1189 proof -
```
```  1190   { fix i x assume as:"b\$i \<le> a\$i" and x:"x\<in>box a b"
```
```  1191     hence "a \$ i < x \$ i \<and> x \$ i < b \$ i" unfolding mem_box_cart by auto
```
```  1192     hence "a\$i < b\$i" by auto
```
```  1193     hence False using as by auto }
```
```  1194   moreover
```
```  1195   { assume as:"\<forall>i. \<not> (b\$i \<le> a\$i)"
```
```  1196     let ?x = "(1/2) *\<^sub>R (a + b)"
```
```  1197     { fix i
```
```  1198       have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto
```
```  1199       hence "a\$i < ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i < b\$i"
```
```  1200         unfolding vector_smult_component and vector_add_component
```
```  1201         by auto }
```
```  1202     hence "box a b \<noteq> {}" using mem_box_cart(1)[of "?x" a b] by auto }
```
```  1203   ultimately show ?th1 by blast
```
```  1204
```
```  1205   { fix i x assume as:"b\$i < a\$i" and x:"x\<in>cbox a b"
```
```  1206     hence "a \$ i \<le> x \$ i \<and> x \$ i \<le> b \$ i" unfolding mem_box_cart by auto
```
```  1207     hence "a\$i \<le> b\$i" by auto
```
```  1208     hence False using as by auto }
```
```  1209   moreover
```
```  1210   { assume as:"\<forall>i. \<not> (b\$i < a\$i)"
```
```  1211     let ?x = "(1/2) *\<^sub>R (a + b)"
```
```  1212     { fix i
```
```  1213       have "a\$i \<le> b\$i" using as[THEN spec[where x=i]] by auto
```
```  1214       hence "a\$i \<le> ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i \<le> b\$i"
```
```  1215         unfolding vector_smult_component and vector_add_component
```
```  1216         by auto }
```
```  1217     hence "cbox a b \<noteq> {}" using mem_box_cart(2)[of "?x" a b] by auto  }
```
```  1218   ultimately show ?th2 by blast
```
```  1219 qed
```
```  1220
```
```  1221 lemma interval_ne_empty_cart:
```
```  1222   fixes a :: "real^'n"
```
```  1223   shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a\$i \<le> b\$i)"
```
```  1224     and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a\$i < b\$i)"
```
```  1225   unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
```
```  1226     (* BH: Why doesn't just "auto" work here? *)
```
```  1227
```
```  1228 lemma subset_interval_imp_cart:
```
```  1229   fixes a :: "real^'n"
```
```  1230   shows "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
```
```  1231     and "(\<forall>i. a\$i < c\$i \<and> d\$i < b\$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
```
```  1232     and "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
```
```  1233     and "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> box c d \<subseteq> box a b"
```
```  1234   unfolding subset_eq[unfolded Ball_def] unfolding mem_box_cart
```
```  1235   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
```
```  1236
```
```  1237 lemma interval_sing:
```
```  1238   fixes a :: "'a::linorder^'n"
```
```  1239   shows "{a .. a} = {a} \<and> {a<..<a} = {}"
```
```  1240   apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
```
```  1241   done
```
```  1242
```
```  1243 lemma subset_interval_cart:
```
```  1244   fixes a :: "real^'n"
```
```  1245   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)
```
```  1246     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)
```
```  1247     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)
```
```  1248     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)
```
```  1249   using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
```
```  1250
```
```  1251 lemma disjoint_interval_cart:
```
```  1252   fixes a::"real^'n"
```
```  1253   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)
```
```  1254     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)
```
```  1255     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)
```
```  1256     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)
```
```  1257   using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
```
```  1258
```
```  1259 lemma Int_interval_cart:
```
```  1260   fixes a :: "real^'n"
```
```  1261   shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a\$i) (c\$i)) .. (\<chi> i. min (b\$i) (d\$i))}"
```
```  1262   unfolding Int_interval
```
```  1263   by (auto simp: mem_box less_eq_vec_def)
```
```  1264     (auto simp: Basis_vec_def inner_axis)
```
```  1265
```
```  1266 lemma closed_interval_left_cart:
```
```  1267   fixes b :: "real^'n"
```
```  1268   shows "closed {x::real^'n. \<forall>i. x\$i \<le> b\$i}"
```
```  1269   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1270
```
```  1271 lemma closed_interval_right_cart:
```
```  1272   fixes a::"real^'n"
```
```  1273   shows "closed {x::real^'n. \<forall>i. a\$i \<le> x\$i}"
```
```  1274   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1275
```
```  1276 lemma is_interval_cart:
```
```  1277   "is_interval (s::(real^'n) set) \<longleftrightarrow>
```
```  1278     (\<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)"
```
```  1279   by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
```
```  1280
```
```  1281 lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x\$i \<le> a}"
```
```  1282   by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1283
```
```  1284 lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x\$i \<ge> a}"
```
```  1285   by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1286
```
```  1287 lemma open_halfspace_component_lt_cart: "open {x::real^'n. x\$i < a}"
```
```  1288   by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
```
```  1289
```
```  1290 lemma open_halfspace_component_gt_cart: "open {x::real^'n. x\$i  > a}"
```
```  1291   by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
```
```  1292
```
```  1293 lemma Lim_component_le_cart:
```
```  1294   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1295   assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x \$i \<le> b) net"
```
```  1296   shows "l\$i \<le> b"
```
```  1297   by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
```
```  1298
```
```  1299 lemma Lim_component_ge_cart:
```
```  1300   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1301   assumes "(f \<longlongrightarrow> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)\$i) net"
```
```  1302   shows "b \<le> l\$i"
```
```  1303   by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
```
```  1304
```
```  1305 lemma Lim_component_eq_cart:
```
```  1306   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1307   assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\$i = b) net"
```
```  1308   shows "l\$i = b"
```
```  1309   using ev[unfolded order_eq_iff eventually_conj_iff] and
```
```  1310     Lim_component_ge_cart[OF net, of b i] and
```
```  1311     Lim_component_le_cart[OF net, of i b] by auto
```
```  1312
```
```  1313 lemma connected_ivt_component_cart:
```
```  1314   fixes x :: "real^'n"
```
```  1315   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)"
```
```  1316   using connected_ivt_hyperplane[of s x y "axis k 1" a]
```
```  1317   by (auto simp add: inner_axis inner_commute)
```
```  1318
```
```  1319 lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x\$i = 0)}"
```
```  1320   unfolding subspace_def by auto
```
```  1321
```
```  1322 lemma closed_substandard_cart:
```
```  1323   "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x\$i = 0}"
```
```  1324 proof -
```
```  1325   { fix i::'n
```
```  1326     have "closed {x::'a ^ 'n. P i \<longrightarrow> x\$i = 0}"
```
```  1327       by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) }
```
```  1328   thus ?thesis
```
```  1329     unfolding Collect_all_eq by (simp add: closed_INT)
```
```  1330 qed
```
```  1331
```
```  1332 lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x\$i = 0} = card d"
```
```  1333   (is "dim ?A = _")
```
```  1334 proof -
```
```  1335   let ?a = "\<lambda>x. axis x 1 :: real^'n"
```
```  1336   have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
```
```  1337     by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
```
```  1338   have "?a ` d \<subseteq> Basis"
```
```  1339     by (auto simp: Basis_vec_def)
```
```  1340   thus ?thesis
```
```  1341     using dim_substandard[of "?a ` d"] card_image[of ?a d]
```
```  1342     by (auto simp: axis_eq_axis inj_on_def *)
```
```  1343 qed
```
```  1344
```
```  1345 lemma dim_subset_UNIV_cart:
```
```  1346   fixes S :: "(real^'n) set"
```
```  1347   shows "dim S \<le> CARD('n)"
```
```  1348   by (metis dim_subset_UNIV DIM_cart DIM_real mult.right_neutral)
```
```  1349
```
```  1350 lemma affinity_inverses:
```
```  1351   assumes m0: "m \<noteq> (0::'a::field)"
```
```  1352   shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
```
```  1353   "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
```
```  1354   using m0
```
```  1355   apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
```
```  1356   apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
```
```  1357   done
```
```  1358
```
```  1359 lemma vector_affinity_eq:
```
```  1360   assumes m0: "(m::'a::field) \<noteq> 0"
```
```  1361   shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
```
```  1362 proof
```
```  1363   assume h: "m *s x + c = y"
```
```  1364   hence "m *s x = y - c" by (simp add: field_simps)
```
```  1365   hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
```
```  1366   then show "x = inverse m *s y + - (inverse m *s c)"
```
```  1367     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
```
```  1368 next
```
```  1369   assume h: "x = inverse m *s y + - (inverse m *s c)"
```
```  1370   show "m *s x + c = y" unfolding h
```
```  1371     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
```
```  1372 qed
```
```  1373
```
```  1374 lemma vector_eq_affinity:
```
```  1375     "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
```
```  1376   using vector_affinity_eq[where m=m and x=x and y=y and c=c]
```
```  1377   by metis
```
```  1378
```
```  1379 lemma vector_cart:
```
```  1380   fixes f :: "real^'n \<Rightarrow> real"
```
```  1381   shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
```
```  1382   unfolding euclidean_eq_iff[where 'a="real^'n"]
```
```  1383   by simp (simp add: Basis_vec_def inner_axis)
```
```  1384
```
```  1385 lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
```
```  1386   by (rule vector_cart)
```
```  1387
```
```  1388 subsection "Convex Euclidean Space"
```
```  1389
```
```  1390 lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
```
```  1391   using const_vector_cart[of 1] by (simp add: one_vec_def)
```
```  1392
```
```  1393 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
```
```  1394 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
```
```  1395
```
```  1396 lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
```
```  1397
```
```  1398 lemma convex_box_cart:
```
```  1399   assumes "\<And>i. convex {x. P i x}"
```
```  1400   shows "convex {x. \<forall>i. P i (x\$i)}"
```
```  1401   using assms unfolding convex_def by auto
```
```  1402
```
```  1403 lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x\$i)}"
```
```  1404   by (rule convex_box_cart) (simp add: atLeast_def[symmetric])
```
```  1405
```
```  1406 lemma unit_interval_convex_hull_cart:
```
```  1407   "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x\$i = 0) \<or> (x\$i = 1)}"
```
```  1408   unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
```
```  1409   by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
```
```  1410
```
```  1411 lemma cube_convex_hull_cart:
```
```  1412   assumes "0 < d"
```
```  1413   obtains s::"(real^'n) set"
```
```  1414     where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
```
```  1415 proof -
```
```  1416   from assms obtain s where "finite s"
```
```  1417     and "cbox (x - sum (( *\<^sub>R) d) Basis) (x + sum (( *\<^sub>R) d) Basis) = convex hull s"
```
```  1418     by (rule cube_convex_hull)
```
```  1419   with that[of s] show thesis
```
```  1420     by (simp add: const_vector_cart)
```
```  1421 qed
```
```  1422
```
```  1423
```
```  1424 subsection "Derivative"
```
```  1425
```
```  1426 definition "jacobian f net = matrix(frechet_derivative f net)"
```
```  1427
```
```  1428 lemma jacobian_works:
```
```  1429   "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
```
```  1430     (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
```
```  1431   apply rule
```
```  1432   unfolding jacobian_def
```
```  1433   apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
```
```  1434   apply (rule differentiableI)
```
```  1435   apply assumption
```
```  1436   unfolding frechet_derivative_works
```
```  1437   apply assumption
```
```  1438   done
```
```  1439
```
```  1440
```
```  1441 subsection \<open>Component of the differential must be zero if it exists at a local
```
```  1442   maximum or minimum for that corresponding component\<close>
```
```  1443
```
```  1444 lemma differential_zero_maxmin_cart:
```
```  1445   fixes f::"real^'a \<Rightarrow> real^'b"
```
```  1446   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))"
```
```  1447     "f differentiable (at x)"
```
```  1448   shows "jacobian f (at x) \$ k = 0"
```
```  1449   using differential_zero_maxmin_component[of "axis k 1" e x f] assms
```
```  1450     vector_cart[of "\<lambda>j. frechet_derivative f (at x) j \$ k"]
```
```  1451   by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
```
```  1452
```
```  1453 subsection \<open>Lemmas for working on @{typ "real^1"}\<close>
```
```  1454
```
```  1455 lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
```
```  1456   by (metis (full_types) num1_eq_iff)
```
```  1457
```
```  1458 lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
```
```  1459   by auto (metis (full_types) num1_eq_iff)
```
```  1460
```
```  1461 lemma exhaust_2:
```
```  1462   fixes x :: 2
```
```  1463   shows "x = 1 \<or> x = 2"
```
```  1464 proof (induct x)
```
```  1465   case (of_int z)
```
```  1466   then have "0 \<le> z" and "z < 2" by simp_all
```
```  1467   then have "z = 0 | z = 1" by arith
```
```  1468   then show ?case by auto
```
```  1469 qed
```
```  1470
```
```  1471 lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
```
```  1472   by (metis exhaust_2)
```
```  1473
```
```  1474 lemma exhaust_3:
```
```  1475   fixes x :: 3
```
```  1476   shows "x = 1 \<or> x = 2 \<or> x = 3"
```
```  1477 proof (induct x)
```
```  1478   case (of_int z)
```
```  1479   then have "0 \<le> z" and "z < 3" by simp_all
```
```  1480   then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
```
```  1481   then show ?case by auto
```
```  1482 qed
```
```  1483
```
```  1484 lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
```
```  1485   by (metis exhaust_3)
```
```  1486
```
```  1487 lemma UNIV_1 [simp]: "UNIV = {1::1}"
```
```  1488   by (auto simp add: num1_eq_iff)
```
```  1489
```
```  1490 lemma UNIV_2: "UNIV = {1::2, 2::2}"
```
```  1491   using exhaust_2 by auto
```
```  1492
```
```  1493 lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
```
```  1494   using exhaust_3 by auto
```
```  1495
```
```  1496 lemma sum_1: "sum f (UNIV::1 set) = f 1"
```
```  1497   unfolding UNIV_1 by simp
```
```  1498
```
```  1499 lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2"
```
```  1500   unfolding UNIV_2 by simp
```
```  1501
```
```  1502 lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3"
```
```  1503   unfolding UNIV_3 by (simp add: ac_simps)
```
```  1504
```
```  1505 lemma num1_eqI:
```
```  1506   fixes a::num1 shows "a = b"
```
```  1507   by (metis (full_types) UNIV_1 UNIV_I empty_iff insert_iff)
```
```  1508
```
```  1509 lemma num1_eq1 [simp]:
```
```  1510   fixes a::num1 shows "a = 1"
```
```  1511   by (rule num1_eqI)
```
```  1512
```
```  1513 instantiation num1 :: cart_one
```
```  1514 begin
```
```  1515
```
```  1516 instance
```
```  1517 proof
```
```  1518   show "CARD(1) = Suc 0" by auto
```
```  1519 qed
```
```  1520
```
```  1521 end
```
```  1522
```
```  1523 instantiation num1 :: linorder begin
```
```  1524 definition "a < b \<longleftrightarrow> Rep_num1 a < Rep_num1 b"
```
```  1525 definition "a \<le> b \<longleftrightarrow> Rep_num1 a \<le> Rep_num1 b"
```
```  1526 instance
```
```  1527   by intro_classes (auto simp: less_eq_num1_def less_num1_def intro: num1_eqI)
```
```  1528 end
```
```  1529
```
```  1530 instance num1 :: wellorder
```
```  1531   by intro_classes (auto simp: less_eq_num1_def less_num1_def)
```
```  1532
```
```  1533 subsection\<open>The collapse of the general concepts to dimension one\<close>
```
```  1534
```
```  1535 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x\$1))"
```
```  1536   by (simp add: vec_eq_iff)
```
```  1537
```
```  1538 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
```
```  1539   apply auto
```
```  1540   apply (erule_tac x= "x\$1" in allE)
```
```  1541   apply (simp only: vector_one[symmetric])
```
```  1542   done
```
```  1543
```
```  1544 lemma norm_vector_1: "norm (x :: _^1) = norm (x\$1)"
```
```  1545   by (simp add: norm_vec_def)
```
```  1546
```
```  1547 lemma dist_vector_1:
```
```  1548   fixes x :: "'a::real_normed_vector^1"
```
```  1549   shows "dist x y = dist (x\$1) (y\$1)"
```
```  1550   by (simp add: dist_norm norm_vector_1)
```
```  1551
```
```  1552 lemma norm_real: "norm(x::real ^ 1) = \<bar>x\$1\<bar>"
```
```  1553   by (simp add: norm_vector_1)
```
```  1554
```
```  1555 lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x\$1) - (y\$1)\<bar>"
```
```  1556   by (auto simp add: norm_real dist_norm)
```
```  1557
```
```  1558
```
```  1559 lemma tendsto_at_within_vector_1:
```
```  1560   fixes S :: "'a :: metric_space set"
```
```  1561   assumes "(f \<longlongrightarrow> fx) (at x within S)"
```
```  1562   shows "((\<lambda>y::'a^1. \<chi> i. f (y \$ 1)) \<longlongrightarrow> (vec fx::'a^1)) (at (vec x) within vec ` S)"
```
```  1563 proof (rule topological_tendstoI)
```
```  1564   fix T :: "('a^1) set"
```
```  1565   assume "open T" "vec fx \<in> T"
```
```  1566   have "\<forall>\<^sub>F x in at x within S. f x \<in> (\<lambda>x. x \$ 1) ` T"
```
```  1567     using \<open>open T\<close> \<open>vec fx \<in> T\<close> assms open_image_vec_nth tendsto_def by fastforce
```
```  1568   then show "\<forall>\<^sub>F x::'a^1 in at (vec x) within vec ` S. (\<chi> i. f (x \$ 1)) \<in> T"
```
```  1569     unfolding eventually_at dist_norm [symmetric]
```
```  1570     by (rule ex_forward)
```
```  1571        (use \<open>open T\<close> in
```
```  1572          \<open>fastforce simp: dist_norm dist_vec_def L2_set_def image_iff vector_one open_vec_def\<close>)
```
```  1573 qed
```
```  1574
```
```  1575 lemma has_derivative_vector_1:
```
```  1576   assumes der_g: "(g has_derivative (\<lambda>x. x * g' a)) (at a within S)"
```
```  1577   shows "((\<lambda>x. vec (g (x \$ 1))) has_derivative ( *\<^sub>R) (g' a))
```
```  1578          (at ((vec a)::real^1) within vec ` S)"
```
```  1579     using der_g
```
```  1580     apply (auto simp: Deriv.has_derivative_within bounded_linear_scaleR_right norm_vector_1)
```
```  1581     apply (drule tendsto_at_within_vector_1, vector)
```
```  1582     apply (auto simp: algebra_simps eventually_at tendsto_def)
```
```  1583     done
```
```  1584
```
```  1585
```
```  1586 subsection\<open>Explicit vector construction from lists\<close>
```
```  1587
```
```  1588 definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
```
```  1589
```
```  1590 lemma vector_1: "(vector[x]) \$1 = x"
```
```  1591   unfolding vector_def by simp
```
```  1592
```
```  1593 lemma vector_2:
```
```  1594  "(vector[x,y]) \$1 = x"
```
```  1595  "(vector[x,y] :: 'a^2)\$2 = (y::'a::zero)"
```
```  1596   unfolding vector_def by simp_all
```
```  1597
```
```  1598 lemma vector_3:
```
```  1599  "(vector [x,y,z] ::('a::zero)^3)\$1 = x"
```
```  1600  "(vector [x,y,z] ::('a::zero)^3)\$2 = y"
```
```  1601  "(vector [x,y,z] ::('a::zero)^3)\$3 = z"
```
```  1602   unfolding vector_def by simp_all
```
```  1603
```
```  1604 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
```
```  1605   by (metis vector_1 vector_one)
```
```  1606
```
```  1607 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
```
```  1608   apply auto
```
```  1609   apply (erule_tac x="v\$1" in allE)
```
```  1610   apply (erule_tac x="v\$2" in allE)
```
```  1611   apply (subgoal_tac "vector [v\$1, v\$2] = v")
```
```  1612   apply simp
```
```  1613   apply (vector vector_def)
```
```  1614   apply (simp add: forall_2)
```
```  1615   done
```
```  1616
```
```  1617 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
```
```  1618   apply auto
```
```  1619   apply (erule_tac x="v\$1" in allE)
```
```  1620   apply (erule_tac x="v\$2" in allE)
```
```  1621   apply (erule_tac x="v\$3" in allE)
```
```  1622   apply (subgoal_tac "vector [v\$1, v\$2, v\$3] = v")
```
```  1623   apply simp
```
```  1624   apply (vector vector_def)
```
```  1625   apply (simp add: forall_3)
```
```  1626   done
```
```  1627
```
```  1628 lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x \$ k)"
```
```  1629   apply (rule bounded_linearI[where K=1])
```
```  1630   using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
```
```  1631
```
```  1632 lemma interval_split_cart:
```
```  1633   "{a..b::real^'n} \<inter> {x. x\$k \<le> c} = {a .. (\<chi> i. if i = k then min (b\$k) c else b\$i)}"
```
```  1634   "cbox a b \<inter> {x. x\$k \<ge> c} = {(\<chi> i. if i = k then max (a\$k) c else a\$i) .. b}"
```
```  1635   apply (rule_tac[!] set_eqI)
```
```  1636   unfolding Int_iff mem_box_cart mem_Collect_eq interval_cbox_cart
```
```  1637   unfolding vec_lambda_beta
```
```  1638   by auto
```
```  1639
```
```  1640 lemmas cartesian_euclidean_space_uniform_limit_intros[uniform_limit_intros] =
```
```  1641   bounded_linear.uniform_limit[OF blinfun.bounded_linear_right]
```
```  1642   bounded_linear.uniform_limit[OF bounded_linear_vec_nth]
```
```  1643   bounded_linear.uniform_limit[OF bounded_linear_component_cart]
```
```  1644
```
```  1645 end
```