src/HOL/Analysis/Cartesian_Euclidean_Space.thy
 author immler Mon Feb 26 09:58:47 2018 +0100 (17 months ago) changeset 67728 d97a28a006f9 parent 67719 bffb7482faaa child 67731 184c293f0a33 permissions -rw-r--r--
generalized
```     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_sum:
```
```   143   assumes "finite S"
```
```   144   shows "vec(sum f S) = sum (vec \<circ> f) S"
```
```   145   using assms
```
```   146 proof induct
```
```   147   case empty
```
```   148   then show ?case by simp
```
```   149 next
```
```   150   case insert
```
```   151   then show ?case by (auto simp add: vec_add)
```
```   152 qed
```
```   153
```
```   154 text\<open>Obvious "component-pushing".\<close>
```
```   155
```
```   156 lemma vec_component [simp]: "vec x \$ i = x"
```
```   157   by vector
```
```   158
```
```   159 lemma vector_mult_component [simp]: "(x * y)\$i = x\$i * y\$i"
```
```   160   by vector
```
```   161
```
```   162 lemma vector_smult_component [simp]: "(c *s y)\$i = c * (y\$i)"
```
```   163   by vector
```
```   164
```
```   165 lemma cond_component: "(if b then x else y)\$i = (if b then x\$i else y\$i)" by vector
```
```   166
```
```   167 lemmas vector_component =
```
```   168   vec_component vector_add_component vector_mult_component
```
```   169   vector_smult_component vector_minus_component vector_uminus_component
```
```   170   vector_scaleR_component cond_component
```
```   171
```
```   172
```
```   173 subsection \<open>Some frequently useful arithmetic lemmas over vectors.\<close>
```
```   174
```
```   175 instance vec :: (semigroup_mult, finite) semigroup_mult
```
```   176   by standard (vector mult.assoc)
```
```   177
```
```   178 instance vec :: (monoid_mult, finite) monoid_mult
```
```   179   by standard vector+
```
```   180
```
```   181 instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
```
```   182   by standard (vector mult.commute)
```
```   183
```
```   184 instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
```
```   185   by standard vector
```
```   186
```
```   187 instance vec :: (semiring, finite) semiring
```
```   188   by standard (vector field_simps)+
```
```   189
```
```   190 instance vec :: (semiring_0, finite) semiring_0
```
```   191   by standard (vector field_simps)+
```
```   192 instance vec :: (semiring_1, finite) semiring_1
```
```   193   by standard vector
```
```   194 instance vec :: (comm_semiring, finite) comm_semiring
```
```   195   by standard (vector field_simps)+
```
```   196
```
```   197 instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
```
```   198 instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
```
```   199 instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
```
```   200 instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
```
```   201 instance vec :: (ring, finite) ring ..
```
```   202 instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
```
```   203 instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
```
```   204
```
```   205 instance vec :: (ring_1, finite) ring_1 ..
```
```   206
```
```   207 instance vec :: (real_algebra, finite) real_algebra
```
```   208   by standard (simp_all add: vec_eq_iff)
```
```   209
```
```   210 instance vec :: (real_algebra_1, finite) real_algebra_1 ..
```
```   211
```
```   212 lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)\$i = of_nat n"
```
```   213 proof (induct n)
```
```   214   case 0
```
```   215   then show ?case by vector
```
```   216 next
```
```   217   case Suc
```
```   218   then show ?case by vector
```
```   219 qed
```
```   220
```
```   221 lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) \$ i = 1"
```
```   222   by vector
```
```   223
```
```   224 lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) \$ i = - 1"
```
```   225   by vector
```
```   226
```
```   227 instance vec :: (semiring_char_0, finite) semiring_char_0
```
```   228 proof
```
```   229   fix m n :: nat
```
```   230   show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
```
```   231     by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
```
```   232 qed
```
```   233
```
```   234 instance vec :: (numeral, finite) numeral ..
```
```   235 instance vec :: (semiring_numeral, finite) semiring_numeral ..
```
```   236
```
```   237 lemma numeral_index [simp]: "numeral w \$ i = numeral w"
```
```   238   by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
```
```   239
```
```   240 lemma neg_numeral_index [simp]: "- numeral w \$ i = - numeral w"
```
```   241   by (simp only: vector_uminus_component numeral_index)
```
```   242
```
```   243 instance vec :: (comm_ring_1, finite) comm_ring_1 ..
```
```   244 instance vec :: (ring_char_0, finite) ring_char_0 ..
```
```   245
```
```   246 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
```
```   247   by (vector mult.assoc)
```
```   248 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
```
```   249   by (vector field_simps)
```
```   250 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
```
```   251   by (vector field_simps)
```
```   252 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
```
```   253 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
```
```   254 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
```
```   255   by (vector field_simps)
```
```   256 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
```
```   257 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
```
```   258 lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
```
```   259 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
```
```   260 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
```
```   261   by (vector field_simps)
```
```   262
```
```   263 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
```
```   264   by (simp add: vec_eq_iff)
```
```   265
```
```   266 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
```
```   267
```
```   268 lemma norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
```
```   269   by (simp add: inner_axis' norm_eq_1)
```
```   270
```
```   271 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
```
```   272   by vector
```
```   273
```
```   274 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
```
```   275   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
```
```   276
```
```   277 lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
```
```   278   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
```
```   279
```
```   280 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
```
```   281   by (metis vector_mul_lcancel)
```
```   282
```
```   283 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
```
```   284   by (metis vector_mul_rcancel)
```
```   285
```
```   286 lemma component_le_norm_cart: "\<bar>x\$i\<bar> <= norm x"
```
```   287   apply (simp add: norm_vec_def)
```
```   288   apply (rule member_le_L2_set, simp_all)
```
```   289   done
```
```   290
```
```   291 lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x\$i\<bar> <= e"
```
```   292   by (metis component_le_norm_cart order_trans)
```
```   293
```
```   294 lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x\$i\<bar> < e"
```
```   295   by (metis component_le_norm_cart le_less_trans)
```
```   296
```
```   297 lemma norm_le_l1_cart: "norm x <= sum(\<lambda>i. \<bar>x\$i\<bar>) UNIV"
```
```   298   by (simp add: norm_vec_def L2_set_le_sum)
```
```   299
```
```   300 lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
```
```   301   unfolding scaleR_vec_def vector_scalar_mult_def by simp
```
```   302
```
```   303 lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
```
```   304   unfolding dist_norm scalar_mult_eq_scaleR
```
```   305   unfolding scaleR_right_diff_distrib[symmetric] by simp
```
```   306
```
```   307 lemma sum_component [simp]:
```
```   308   fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
```
```   309   shows "(sum f S)\$i = sum (\<lambda>x. (f x)\$i) S"
```
```   310 proof (cases "finite S")
```
```   311   case True
```
```   312   then show ?thesis by induct simp_all
```
```   313 next
```
```   314   case False
```
```   315   then show ?thesis by simp
```
```   316 qed
```
```   317
```
```   318 lemma sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)\$i ) S)"
```
```   319   by (simp add: vec_eq_iff)
```
```   320
```
```   321 lemma sum_cmul:
```
```   322   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
```
```   323   shows "sum (\<lambda>x. c *s f x) S = c *s sum f S"
```
```   324   by (simp add: vec_eq_iff sum_distrib_left)
```
```   325
```
```   326 lemma sum_norm_allsubsets_bound_cart:
```
```   327   fixes f:: "'a \<Rightarrow> real ^'n"
```
```   328   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
```
```   329   shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
```
```   330   using sum_norm_allsubsets_bound[OF assms]
```
```   331   by simp
```
```   332
```
```   333 subsection\<open>Closures and interiors of halfspaces\<close>
```
```   334
```
```   335 lemma interior_halfspace_le [simp]:
```
```   336   assumes "a \<noteq> 0"
```
```   337     shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
```
```   338 proof -
```
```   339   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
```
```   340   proof -
```
```   341     obtain e where "e>0" and e: "cball x e \<subseteq> S"
```
```   342       using \<open>open S\<close> open_contains_cball x by blast
```
```   343     then have "x + (e / norm a) *\<^sub>R a \<in> cball x e"
```
```   344       by (simp add: dist_norm)
```
```   345     then have "x + (e / norm a) *\<^sub>R a \<in> S"
```
```   346       using e by blast
```
```   347     then have "x + (e / norm a) *\<^sub>R a \<in> {x. a \<bullet> x \<le> b}"
```
```   348       using S by blast
```
```   349     moreover have "e * (a \<bullet> a) / norm a > 0"
```
```   350       by (simp add: \<open>0 < e\<close> assms)
```
```   351     ultimately show ?thesis
```
```   352       by (simp add: algebra_simps)
```
```   353   qed
```
```   354   show ?thesis
```
```   355     by (rule interior_unique) (auto simp: open_halfspace_lt *)
```
```   356 qed
```
```   357
```
```   358 lemma interior_halfspace_ge [simp]:
```
```   359    "a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"
```
```   360 using interior_halfspace_le [of "-a" "-b"] by simp
```
```   361
```
```   362 lemma interior_halfspace_component_le [simp]:
```
```   363      "interior {x. x\$k \<le> a} = {x :: (real,'n::finite) vec. x\$k < a}" (is "?LE")
```
```   364   and interior_halfspace_component_ge [simp]:
```
```   365      "interior {x. x\$k \<ge> a} = {x :: (real,'n::finite) vec. x\$k > a}" (is "?GE")
```
```   366 proof -
```
```   367   have "axis k (1::real) \<noteq> 0"
```
```   368     by (simp add: axis_def vec_eq_iff)
```
```   369   moreover have "axis k (1::real) \<bullet> x = x\$k" for x
```
```   370     by (simp add: cart_eq_inner_axis inner_commute)
```
```   371   ultimately show ?LE ?GE
```
```   372     using interior_halfspace_le [of "axis k (1::real)" a]
```
```   373           interior_halfspace_ge [of "axis k (1::real)" a] by auto
```
```   374 qed
```
```   375
```
```   376 lemma closure_halfspace_lt [simp]:
```
```   377   assumes "a \<noteq> 0"
```
```   378     shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
```
```   379 proof -
```
```   380   have [simp]: "-{x. a \<bullet> x < b} = {x. a \<bullet> x \<ge> b}"
```
```   381     by (force simp:)
```
```   382   then show ?thesis
```
```   383     using interior_halfspace_ge [of a b] assms
```
```   384     by (force simp: closure_interior)
```
```   385 qed
```
```   386
```
```   387 lemma closure_halfspace_gt [simp]:
```
```   388    "a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"
```
```   389 using closure_halfspace_lt [of "-a" "-b"] by simp
```
```   390
```
```   391 lemma closure_halfspace_component_lt [simp]:
```
```   392      "closure {x. x\$k < a} = {x :: (real,'n::finite) vec. x\$k \<le> a}" (is "?LE")
```
```   393   and closure_halfspace_component_gt [simp]:
```
```   394      "closure {x. x\$k > a} = {x :: (real,'n::finite) vec. x\$k \<ge> a}" (is "?GE")
```
```   395 proof -
```
```   396   have "axis k (1::real) \<noteq> 0"
```
```   397     by (simp add: axis_def vec_eq_iff)
```
```   398   moreover have "axis k (1::real) \<bullet> x = x\$k" for x
```
```   399     by (simp add: cart_eq_inner_axis inner_commute)
```
```   400   ultimately show ?LE ?GE
```
```   401     using closure_halfspace_lt [of "axis k (1::real)" a]
```
```   402           closure_halfspace_gt [of "axis k (1::real)" a] by auto
```
```   403 qed
```
```   404
```
```   405 lemma interior_hyperplane [simp]:
```
```   406   assumes "a \<noteq> 0"
```
```   407     shows "interior {x. a \<bullet> x = b} = {}"
```
```   408 proof -
```
```   409   have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
```
```   410     by (force simp:)
```
```   411   then show ?thesis
```
```   412     by (auto simp: assms)
```
```   413 qed
```
```   414
```
```   415 lemma frontier_halfspace_le:
```
```   416   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
```
```   417     shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
```
```   418 proof (cases "a = 0")
```
```   419   case True with assms show ?thesis by simp
```
```   420 next
```
```   421   case False then show ?thesis
```
```   422     by (force simp: frontier_def closed_halfspace_le)
```
```   423 qed
```
```   424
```
```   425 lemma frontier_halfspace_ge:
```
```   426   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
```
```   427     shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"
```
```   428 proof (cases "a = 0")
```
```   429   case True with assms show ?thesis by simp
```
```   430 next
```
```   431   case False then show ?thesis
```
```   432     by (force simp: frontier_def closed_halfspace_ge)
```
```   433 qed
```
```   434
```
```   435 lemma frontier_halfspace_lt:
```
```   436   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
```
```   437     shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"
```
```   438 proof (cases "a = 0")
```
```   439   case True with assms show ?thesis by simp
```
```   440 next
```
```   441   case False then show ?thesis
```
```   442     by (force simp: frontier_def interior_open open_halfspace_lt)
```
```   443 qed
```
```   444
```
```   445 lemma frontier_halfspace_gt:
```
```   446   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
```
```   447     shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"
```
```   448 proof (cases "a = 0")
```
```   449   case True with assms show ?thesis by simp
```
```   450 next
```
```   451   case False then show ?thesis
```
```   452     by (force simp: frontier_def interior_open open_halfspace_gt)
```
```   453 qed
```
```   454
```
```   455 lemma interior_standard_hyperplane:
```
```   456    "interior {x :: (real,'n::finite) vec. x\$k = a} = {}"
```
```   457 proof -
```
```   458   have "axis k (1::real) \<noteq> 0"
```
```   459     by (simp add: axis_def vec_eq_iff)
```
```   460   moreover have "axis k (1::real) \<bullet> x = x\$k" for x
```
```   461     by (simp add: cart_eq_inner_axis inner_commute)
```
```   462   ultimately show ?thesis
```
```   463     using interior_hyperplane [of "axis k (1::real)" a]
```
```   464     by force
```
```   465 qed
```
```   466
```
```   467 subsection \<open>Matrix operations\<close>
```
```   468
```
```   469 text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close>
```
```   470
```
```   471 definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
```
```   472     (infixl "**" 70)
```
```   473   where "m ** m' == (\<chi> i j. sum (\<lambda>k. ((m\$i)\$k) * ((m'\$k)\$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
```
```   474
```
```   475 definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
```
```   476     (infixl "*v" 70)
```
```   477   where "m *v x \<equiv> (\<chi> i. sum (\<lambda>j. ((m\$i)\$j) * (x\$j)) (UNIV ::'n set)) :: 'a^'m"
```
```   478
```
```   479 definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
```
```   480     (infixl "v*" 70)
```
```   481   where "v v* m == (\<chi> j. sum (\<lambda>i. ((m\$i)\$j) * (v\$i)) (UNIV :: 'm set)) :: 'a^'n"
```
```   482
```
```   483 definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
```
```   484 definition transpose where
```
```   485   "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A\$j)\$i))"
```
```   486 definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A\$i)\$j))"
```
```   487 definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A\$i)\$j))"
```
```   488 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
```
```   489 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
```
```   490
```
```   491 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
```
```   492 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
```
```   493   by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
```
```   494
```
```   495 lemma matrix_mul_lid [simp]:
```
```   496   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
```
```   497   shows "mat 1 ** A = A"
```
```   498   apply (simp add: matrix_matrix_mult_def mat_def)
```
```   499   apply vector
```
```   500   apply (auto simp only: if_distrib cond_application_beta sum.delta'[OF finite]
```
```   501     mult_1_left mult_zero_left if_True UNIV_I)
```
```   502   done
```
```   503
```
```   504
```
```   505 lemma matrix_mul_rid [simp]:
```
```   506   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
```
```   507   shows "A ** mat 1 = A"
```
```   508   apply (simp add: matrix_matrix_mult_def mat_def)
```
```   509   apply vector
```
```   510   apply (auto simp only: if_distrib cond_application_beta sum.delta[OF finite]
```
```   511     mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
```
```   512   done
```
```   513
```
```   514 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
```
```   515   apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc)
```
```   516   apply (subst sum.swap)
```
```   517   apply simp
```
```   518   done
```
```   519
```
```   520 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
```
```   521   apply (vector matrix_matrix_mult_def matrix_vector_mult_def
```
```   522     sum_distrib_left sum_distrib_right mult.assoc)
```
```   523   apply (subst sum.swap)
```
```   524   apply simp
```
```   525   done
```
```   526
```
```   527 lemma matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
```
```   528   apply (vector matrix_vector_mult_def mat_def)
```
```   529   apply (simp add: if_distrib cond_application_beta sum.delta' cong del: if_weak_cong)
```
```   530   done
```
```   531
```
```   532 lemma matrix_transpose_mul:
```
```   533     "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
```
```   534   by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
```
```   535
```
```   536 lemma matrix_eq:
```
```   537   fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
```
```   538   shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   539   apply auto
```
```   540   apply (subst vec_eq_iff)
```
```   541   apply clarify
```
```   542   apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
```
```   543   apply (erule_tac x="axis ia 1" in allE)
```
```   544   apply (erule_tac x="i" in allE)
```
```   545   apply (auto simp add: if_distrib cond_application_beta axis_def
```
```   546     sum.delta[OF finite] cong del: if_weak_cong)
```
```   547   done
```
```   548
```
```   549 lemma matrix_vector_mul_component: "((A::real^_^_) *v x)\$k = (A\$k) \<bullet> x"
```
```   550   by (simp add: matrix_vector_mult_def inner_vec_def)
```
```   551
```
```   552 lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
```
```   553   apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps)
```
```   554   apply (subst sum.swap)
```
```   555   apply simp
```
```   556   done
```
```   557
```
```   558 lemma transpose_mat [simp]: "transpose (mat n) = mat n"
```
```   559   by (vector transpose_def mat_def)
```
```   560
```
```   561 lemma transpose_transpose [simp]: "transpose(transpose A) = A"
```
```   562   by (vector transpose_def)
```
```   563
```
```   564 lemma row_transpose [simp]:
```
```   565   fixes A:: "'a::semiring_1^_^_"
```
```   566   shows "row i (transpose A) = column i A"
```
```   567   by (simp add: row_def column_def transpose_def vec_eq_iff)
```
```   568
```
```   569 lemma column_transpose [simp]:
```
```   570   fixes A:: "'a::semiring_1^_^_"
```
```   571   shows "column i (transpose A) = row i A"
```
```   572   by (simp add: row_def column_def transpose_def vec_eq_iff)
```
```   573
```
```   574 lemma rows_transpose [simp]: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
```
```   575   by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
```
```   576
```
```   577 lemma columns_transpose [simp]: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
```
```   578   by (metis transpose_transpose rows_transpose)
```
```   579
```
```   580 lemma matrix_mult_transpose_dot_column:
```
```   581   fixes A :: "real^'n^'n"
```
```   582   shows "transpose A ** A = (\<chi> i j. (column i A) \<bullet> (column j A))"
```
```   583   by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)
```
```   584
```
```   585 lemma matrix_mult_transpose_dot_row:
```
```   586   fixes A :: "real^'n^'n"
```
```   587   shows "A ** transpose A = (\<chi> i j. (row i A) \<bullet> (row j A))"
```
```   588   by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)
```
```   589
```
```   590 text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
```
```   591
```
```   592 lemma matrix_mult_dot: "A *v x = (\<chi> i. A\$i \<bullet> x)"
```
```   593   by (simp add: matrix_vector_mult_def inner_vec_def)
```
```   594
```
```   595 lemma matrix_mult_sum:
```
```   596   "(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x\$i) *s column i A) (UNIV:: 'n set)"
```
```   597   by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
```
```   598
```
```   599 lemma vector_componentwise:
```
```   600   "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x\$i) * (axis i 1 :: 'a^'n) \$ j)"
```
```   601   by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)
```
```   602
```
```   603 lemma basis_expansion: "sum (\<lambda>i. (x\$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
```
```   604   by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
```
```   605
```
```   606 lemma linear_componentwise_expansion:
```
```   607   fixes f:: "real ^'m \<Rightarrow> real ^ _"
```
```   608   assumes lf: "linear f"
```
```   609   shows "(f x)\$j = sum (\<lambda>i. (x\$i) * (f (axis i 1)\$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
```
```   610 proof -
```
```   611   let ?M = "(UNIV :: 'm set)"
```
```   612   let ?N = "(UNIV :: 'n set)"
```
```   613   have "?rhs = (sum (\<lambda>i.(x\$i) *\<^sub>R f (axis i 1) ) ?M)\$j"
```
```   614     unfolding sum_component by simp
```
```   615   then show ?thesis
```
```   616     unfolding linear_sum_mul[OF lf, symmetric]
```
```   617     unfolding scalar_mult_eq_scaleR[symmetric]
```
```   618     unfolding basis_expansion
```
```   619     by simp
```
```   620 qed
```
```   621
```
```   622 subsection\<open>Inverse matrices  (not necessarily square)\<close>
```
```   623
```
```   624 definition
```
```   625   "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
```
```   626
```
```   627 definition
```
```   628   "matrix_inv(A:: 'a::semiring_1^'n^'m) =
```
```   629     (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
```
```   630
```
```   631 text\<open>Correspondence between matrices and linear operators.\<close>
```
```   632
```
```   633 definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
```
```   634   where "matrix f = (\<chi> i j. (f(axis j 1))\$i)"
```
```   635
```
```   636 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
```
```   637   by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
```
```   638       field_simps sum_distrib_left sum.distrib)
```
```   639
```
```   640 lemma
```
```   641   fixes A :: "real^'n^'m"
```
```   642   shows matrix_vector_mult_linear_continuous_at [continuous_intros]: "isCont (( *v) A) z"
```
```   643     and matrix_vector_mult_linear_continuous_on [continuous_intros]: "continuous_on S (( *v) A)"
```
```   644   by (simp_all add: linear_linear linear_continuous_at linear_continuous_on matrix_vector_mul_linear)
```
```   645
```
```   646 lemma matrix_vector_mult_add_distrib [algebra_simps]:
```
```   647   "A *v (x + y) = A *v x + A *v y"
```
```   648   by (vector matrix_vector_mult_def sum.distrib distrib_left)
```
```   649
```
```   650 lemma matrix_vector_mult_diff_distrib [algebra_simps]:
```
```   651   fixes A :: "'a::ring_1^'n^'m"
```
```   652   shows "A *v (x - y) = A *v x - A *v y"
```
```   653   by (vector matrix_vector_mult_def sum_subtractf right_diff_distrib)
```
```   654
```
```   655 lemma matrix_vector_mult_scaleR[algebra_simps]:
```
```   656   fixes A :: "real^'n^'m"
```
```   657   shows "A *v (c *\<^sub>R x) = c *\<^sub>R (A *v x)"
```
```   658   using linear_iff matrix_vector_mul_linear by blast
```
```   659
```
```   660 lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
```
```   661   by (simp add: matrix_vector_mult_def vec_eq_iff)
```
```   662
```
```   663 lemma matrix_vector_mult_0 [simp]: "0 *v w = 0"
```
```   664   by (simp add: matrix_vector_mult_def vec_eq_iff)
```
```   665
```
```   666 lemma matrix_vector_mult_add_rdistrib [algebra_simps]:
```
```   667   "(A + B) *v x = (A *v x) + (B *v x)"
```
```   668   by (vector matrix_vector_mult_def sum.distrib distrib_right)
```
```   669
```
```   670 lemma matrix_vector_mult_diff_rdistrib [algebra_simps]:
```
```   671   fixes A :: "'a :: ring_1^'n^'m"
```
```   672   shows "(A - B) *v x = (A *v x) - (B *v x)"
```
```   673   by (vector matrix_vector_mult_def sum_subtractf left_diff_distrib)
```
```   674
```
```   675 lemma matrix_works:
```
```   676   assumes lf: "linear f"
```
```   677   shows "matrix f *v x = f (x::real ^ 'n)"
```
```   678   apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)
```
```   679   by (simp add: linear_componentwise_expansion lf)
```
```   680
```
```   681 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
```
```   682   by (simp add: ext matrix_works)
```
```   683
```
```   684 declare matrix_vector_mul [symmetric, simp]
```
```   685
```
```   686 lemma matrix_of_matrix_vector_mul [simp]: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
```
```   687   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
```
```   688
```
```   689 lemma matrix_compose:
```
```   690   assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
```
```   691     and lg: "linear (g::real^'m \<Rightarrow> real^_)"
```
```   692   shows "matrix (g \<circ> f) = matrix g ** matrix f"
```
```   693   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
```
```   694   by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
```
```   695
```
```   696 lemma matrix_vector_column:
```
```   697   "(A::'a::comm_semiring_1^'n^_) *v x = sum (\<lambda>i. (x\$i) *s ((transpose A)\$i)) (UNIV:: 'n set)"
```
```   698   by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
```
```   699
```
```   700 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
```
```   701   apply (rule adjoint_unique)
```
```   702   apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
```
```   703     sum_distrib_right sum_distrib_left)
```
```   704   apply (subst sum.swap)
```
```   705   apply (auto simp add: ac_simps)
```
```   706   done
```
```   707
```
```   708 lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
```
```   709   shows "matrix(adjoint f) = transpose(matrix f)"
```
```   710   apply (subst matrix_vector_mul[OF lf])
```
```   711   unfolding adjoint_matrix matrix_of_matrix_vector_mul
```
```   712   apply rule
```
```   713   done
```
```   714
```
```   715
```
```   716 subsection\<open>Some bounds on components etc. relative to operator norm.\<close>
```
```   717
```
```   718 lemma norm_column_le_onorm:
```
```   719   fixes A :: "real^'n^'m"
```
```   720   shows "norm(column i A) \<le> onorm(( *v) A)"
```
```   721 proof -
```
```   722   have bl: "bounded_linear (( *v) A)"
```
```   723     by (simp add: linear_linear matrix_vector_mul_linear)
```
```   724   have "norm (\<chi> j. A \$ j \$ i) \<le> norm (A *v axis i 1)"
```
```   725     by (simp add: matrix_mult_dot cart_eq_inner_axis)
```
```   726   also have "\<dots> \<le> onorm (( *v) A)"
```
```   727     using onorm [OF bl, of "axis i 1"] by (auto simp: axis_in_Basis)
```
```   728   finally have "norm (\<chi> j. A \$ j \$ i) \<le> onorm (( *v) A)" .
```
```   729   then show ?thesis
```
```   730     unfolding column_def .
```
```   731 qed
```
```   732
```
```   733 lemma matrix_component_le_onorm:
```
```   734   fixes A :: "real^'n^'m"
```
```   735   shows "\<bar>A \$ i \$ j\<bar> \<le> onorm(( *v) A)"
```
```   736 proof -
```
```   737   have "\<bar>A \$ i \$ j\<bar> \<le> norm (\<chi> n. (A \$ n \$ j))"
```
```   738     by (metis (full_types, lifting) component_le_norm_cart vec_lambda_beta)
```
```   739   also have "\<dots> \<le> onorm (( *v) A)"
```
```   740     by (metis (no_types) column_def norm_column_le_onorm)
```
```   741   finally show ?thesis .
```
```   742 qed
```
```   743
```
```   744 lemma component_le_onorm:
```
```   745   fixes f :: "real^'m \<Rightarrow> real^'n"
```
```   746   shows "linear f \<Longrightarrow> \<bar>matrix f \$ i \$ j\<bar> \<le> onorm f"
```
```   747   by (metis matrix_component_le_onorm matrix_vector_mul)
```
```   748
```
```   749 lemma onorm_le_matrix_component_sum:
```
```   750   fixes A :: "real^'n^'m"
```
```   751   shows "onorm(( *v) A) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A \$ i \$ j\<bar>)"
```
```   752 proof (rule onorm_le)
```
```   753   fix x
```
```   754   have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) \$ i\<bar>)"
```
```   755     by (rule norm_le_l1_cart)
```
```   756   also have "\<dots> \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A \$ i \$ j\<bar> * norm x)"
```
```   757   proof (rule sum_mono)
```
```   758     fix i
```
```   759     have "\<bar>(A *v x) \$ i\<bar> \<le> \<bar>\<Sum>j\<in>UNIV. A \$ i \$ j * x \$ j\<bar>"
```
```   760       by (simp add: matrix_vector_mult_def)
```
```   761     also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A \$ i \$ j * x \$ j\<bar>)"
```
```   762       by (rule sum_abs)
```
```   763     also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A \$ i \$ j\<bar> * norm x)"
```
```   764       by (rule sum_mono) (simp add: abs_mult component_le_norm_cart mult_left_mono)
```
```   765     finally show "\<bar>(A *v x) \$ i\<bar> \<le> (\<Sum>j\<in>UNIV. \<bar>A \$ i \$ j\<bar> * norm x)" .
```
```   766   qed
```
```   767   finally show "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A \$ i \$ j\<bar>) * norm x"
```
```   768     by (simp add: sum_distrib_right)
```
```   769 qed
```
```   770
```
```   771 lemma onorm_le_matrix_component:
```
```   772   fixes A :: "real^'n^'m"
```
```   773   assumes "\<And>i j. abs(A\$i\$j) \<le> B"
```
```   774   shows "onorm(( *v) A) \<le> real (CARD('m)) * real (CARD('n)) * B"
```
```   775 proof (rule onorm_le)
```
```   776   fix x :: "(real, 'n) vec"
```
```   777   have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) \$ i\<bar>)"
```
```   778     by (rule norm_le_l1_cart)
```
```   779   also have "\<dots> \<le> (\<Sum>i::'m \<in>UNIV. real (CARD('n)) * B * norm x)"
```
```   780   proof (rule sum_mono)
```
```   781     fix i
```
```   782     have "\<bar>(A *v x) \$ i\<bar> \<le> norm(A \$ i) * norm x"
```
```   783       by (simp add: matrix_mult_dot Cauchy_Schwarz_ineq2)
```
```   784     also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A \$ i \$ j\<bar>) * norm x"
```
```   785       by (simp add: mult_right_mono norm_le_l1_cart)
```
```   786     also have "\<dots> \<le> real (CARD('n)) * B * norm x"
```
```   787       by (simp add: assms sum_bounded_above mult_right_mono)
```
```   788     finally show "\<bar>(A *v x) \$ i\<bar> \<le> real (CARD('n)) * B * norm x" .
```
```   789   qed
```
```   790   also have "\<dots> \<le> CARD('m) * real (CARD('n)) * B * norm x"
```
```   791     by simp
```
```   792   finally show "norm (A *v x) \<le> CARD('m) * real (CARD('n)) * B * norm x" .
```
```   793 qed
```
```   794
```
```   795 subsection \<open>lambda skolemization on cartesian products\<close>
```
```   796
```
```   797 lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
```
```   798    (\<exists>x::'a ^ 'n. \<forall>i. P i (x \$ i))" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   799 proof -
```
```   800   let ?S = "(UNIV :: 'n set)"
```
```   801   { assume H: "?rhs"
```
```   802     then have ?lhs by auto }
```
```   803   moreover
```
```   804   { assume H: "?lhs"
```
```   805     then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
```
```   806     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
```
```   807     { fix i
```
```   808       from f have "P i (f i)" by metis
```
```   809       then have "P i (?x \$ i)" by auto
```
```   810     }
```
```   811     hence "\<forall>i. P i (?x\$i)" by metis
```
```   812     hence ?rhs by metis }
```
```   813   ultimately show ?thesis by metis
```
```   814 qed
```
```   815
```
```   816 lemma rational_approximation:
```
```   817   assumes "e > 0"
```
```   818   obtains r::real where "r \<in> \<rat>" "\<bar>r - x\<bar> < e"
```
```   819   using Rats_dense_in_real [of "x - e/2" "x + e/2"] assms by auto
```
```   820
```
```   821 lemma matrix_rational_approximation:
```
```   822   fixes A :: "real^'n^'m"
```
```   823   assumes "e > 0"
```
```   824   obtains B where "\<And>i j. B\$i\$j \<in> \<rat>" "onorm(\<lambda>x. (A - B) *v x) < e"
```
```   825 proof -
```
```   826   have "\<forall>i j. \<exists>q \<in> \<rat>. \<bar>q - A \$ i \$ j\<bar> < e / (2 * CARD('m) * CARD('n))"
```
```   827     using assms by (force intro: rational_approximation [of "e / (2 * CARD('m) * CARD('n))"])
```
```   828   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))"
```
```   829     by (auto simp: lambda_skolem Bex_def)
```
```   830   show ?thesis
```
```   831   proof
```
```   832     have "onorm (( *v) (A - B)) \<le> real CARD('m) * real CARD('n) *
```
```   833     (e / (2 * real CARD('m) * real CARD('n)))"
```
```   834       apply (rule onorm_le_matrix_component)
```
```   835       using Bclo by (simp add: abs_minus_commute less_imp_le)
```
```   836     also have "\<dots> < e"
```
```   837       using \<open>0 < e\<close> by (simp add: divide_simps)
```
```   838     finally show "onorm (( *v) (A - B)) < e" .
```
```   839   qed (use B in auto)
```
```   840 qed
```
```   841
```
```   842 lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
```
```   843   unfolding inner_simps scalar_mult_eq_scaleR by auto
```
```   844
```
```   845 lemma left_invertible_transpose:
```
```   846   "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
```
```   847   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
```
```   848
```
```   849 lemma right_invertible_transpose:
```
```   850   "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
```
```   851   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
```
```   852
```
```   853 lemma matrix_left_invertible_injective:
```
```   854   "(\<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)"
```
```   855 proof -
```
```   856   { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
```
```   857     from xy have "B*v (A *v x) = B *v (A*v y)" by simp
```
```   858     hence "x = y"
```
```   859       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
```
```   860   moreover
```
```   861   { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
```
```   862     hence i: "inj (( *v) A)" unfolding inj_on_def by auto
```
```   863     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
```
```   864     obtain g where g: "linear g" "g \<circ> ( *v) A = id" by blast
```
```   865     have "matrix g ** A = mat 1"
```
```   866       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
```
```   867       using g(2) by (simp add: fun_eq_iff)
```
```   868     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
```
```   869   ultimately show ?thesis by blast
```
```   870 qed
```
```   871
```
```   872 lemma matrix_left_invertible_ker:
```
```   873   "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
```
```   874   unfolding matrix_left_invertible_injective
```
```   875   using linear_injective_0[OF matrix_vector_mul_linear, of A]
```
```   876   by (simp add: inj_on_def)
```
```   877
```
```   878 lemma matrix_right_invertible_surjective:
```
```   879   "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
```
```   880 proof -
```
```   881   { fix B :: "real ^'m^'n"
```
```   882     assume AB: "A ** B = mat 1"
```
```   883     { fix x :: "real ^ 'm"
```
```   884       have "A *v (B *v x) = x"
```
```   885         by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
```
```   886     hence "surj (( *v) A)" unfolding surj_def by metis }
```
```   887   moreover
```
```   888   { assume sf: "surj (( *v) A)"
```
```   889     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
```
```   890     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "( *v) A \<circ> g = id"
```
```   891       by blast
```
```   892
```
```   893     have "A ** (matrix g) = mat 1"
```
```   894       unfolding matrix_eq  matrix_vector_mul_lid
```
```   895         matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
```
```   896       using g(2) unfolding o_def fun_eq_iff id_def
```
```   897       .
```
```   898     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
```
```   899   }
```
```   900   ultimately show ?thesis unfolding surj_def by blast
```
```   901 qed
```
```   902
```
```   903 lemma matrix_left_invertible_independent_columns:
```
```   904   fixes A :: "real^'n^'m"
```
```   905   shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
```
```   906       (\<forall>c. sum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
```
```   907     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   908 proof -
```
```   909   let ?U = "UNIV :: 'n set"
```
```   910   { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
```
```   911     { fix c i
```
```   912       assume c: "sum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
```
```   913       let ?x = "\<chi> i. c i"
```
```   914       have th0:"A *v ?x = 0"
```
```   915         using c
```
```   916         unfolding matrix_mult_sum vec_eq_iff
```
```   917         by auto
```
```   918       from k[rule_format, OF th0] i
```
```   919       have "c i = 0" by (vector vec_eq_iff)}
```
```   920     hence ?rhs by blast }
```
```   921   moreover
```
```   922   { assume H: ?rhs
```
```   923     { fix x assume x: "A *v x = 0"
```
```   924       let ?c = "\<lambda>i. ((x\$i ):: real)"
```
```   925       from H[rule_format, of ?c, unfolded matrix_mult_sum[symmetric], OF x]
```
```   926       have "x = 0" by vector }
```
```   927   }
```
```   928   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
```
```   929 qed
```
```   930
```
```   931 lemma matrix_right_invertible_independent_rows:
```
```   932   fixes A :: "real^'n^'m"
```
```   933   shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
```
```   934     (\<forall>c. sum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
```
```   935   unfolding left_invertible_transpose[symmetric]
```
```   936     matrix_left_invertible_independent_columns
```
```   937   by (simp add: column_transpose)
```
```   938
```
```   939 lemma matrix_right_invertible_span_columns:
```
```   940   "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
```
```   941     span (columns A) = UNIV" (is "?lhs = ?rhs")
```
```   942 proof -
```
```   943   let ?U = "UNIV :: 'm set"
```
```   944   have fU: "finite ?U" by simp
```
```   945   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). sum (\<lambda>i. (x\$i) *s column i A) ?U = y)"
```
```   946     unfolding matrix_right_invertible_surjective matrix_mult_sum surj_def
```
```   947     apply (subst eq_commute)
```
```   948     apply rule
```
```   949     done
```
```   950   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
```
```   951   { assume h: ?lhs
```
```   952     { fix x:: "real ^'n"
```
```   953       from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
```
```   954         where y: "sum (\<lambda>i. (y\$i) *s column i A) ?U = x" by blast
```
```   955       have "x \<in> span (columns A)"
```
```   956         unfolding y[symmetric]
```
```   957         apply (rule span_sum)
```
```   958         unfolding scalar_mult_eq_scaleR
```
```   959         apply (rule span_mul)
```
```   960         apply (rule span_superset)
```
```   961         unfolding columns_def
```
```   962         apply blast
```
```   963         done
```
```   964     }
```
```   965     then have ?rhs unfolding rhseq by blast }
```
```   966   moreover
```
```   967   { assume h:?rhs
```
```   968     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). sum (\<lambda>i. (x\$i) *s column i A) ?U = y"
```
```   969     { fix y
```
```   970       have "?P y"
```
```   971       proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
```
```   972         show "\<exists>x::real ^ 'm. sum (\<lambda>i. (x\$i) *s column i A) ?U = 0"
```
```   973           by (rule exI[where x=0], simp)
```
```   974       next
```
```   975         fix c y1 y2
```
```   976         assume y1: "y1 \<in> columns A" and y2: "?P y2"
```
```   977         from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
```
```   978           unfolding columns_def by blast
```
```   979         from y2 obtain x:: "real ^'m" where
```
```   980           x: "sum (\<lambda>i. (x\$i) *s column i A) ?U = y2" by blast
```
```   981         let ?x = "(\<chi> j. if j = i then c + (x\$i) else (x\$j))::real^'m"
```
```   982         show "?P (c*s y1 + y2)"
```
```   983         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)
```
```   984           fix j
```
```   985           have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```   986               else (x\$xa) * ((column xa A\$j))) = (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))"
```
```   987             using i(1) by (simp add: field_simps)
```
```   988           have "sum (\<lambda>xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```   989               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"
```
```   990             apply (rule sum.cong[OF refl])
```
```   991             using th apply blast
```
```   992             done
```
```   993           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"
```
```   994             by (simp add: sum.distrib)
```
```   995           also have "\<dots> = c * ((column i A)\$j) + sum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U"
```
```   996             unfolding sum.delta[OF fU]
```
```   997             using i(1) by simp
```
```   998           finally show "sum (\<lambda>xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```   999             else (x\$xa) * ((column xa A\$j))) ?U = c * ((column i A)\$j) + sum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U" .
```
```  1000         qed
```
```  1001       next
```
```  1002         show "y \<in> span (columns A)"
```
```  1003           unfolding h by blast
```
```  1004       qed
```
```  1005     }
```
```  1006     then have ?lhs unfolding lhseq ..
```
```  1007   }
```
```  1008   ultimately show ?thesis by blast
```
```  1009 qed
```
```  1010
```
```  1011 lemma matrix_left_invertible_span_rows:
```
```  1012   "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
```
```  1013   unfolding right_invertible_transpose[symmetric]
```
```  1014   unfolding columns_transpose[symmetric]
```
```  1015   unfolding matrix_right_invertible_span_columns
```
```  1016   ..
```
```  1017
```
```  1018 text \<open>The same result in terms of square matrices.\<close>
```
```  1019
```
```  1020 lemma matrix_left_right_inverse:
```
```  1021   fixes A A' :: "real ^'n^'n"
```
```  1022   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
```
```  1023 proof -
```
```  1024   { fix A A' :: "real ^'n^'n"
```
```  1025     assume AA': "A ** A' = mat 1"
```
```  1026     have sA: "surj (( *v) A)"
```
```  1027       unfolding surj_def
```
```  1028       apply clarify
```
```  1029       apply (rule_tac x="(A' *v y)" in exI)
```
```  1030       apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
```
```  1031       done
```
```  1032     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
```
```  1033     obtain f' :: "real ^'n \<Rightarrow> real ^'n"
```
```  1034       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
```
```  1035     have th: "matrix f' ** A = mat 1"
```
```  1036       by (simp add: matrix_eq matrix_works[OF f'(1)]
```
```  1037           matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
```
```  1038     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
```
```  1039     hence "matrix f' = A'"
```
```  1040       by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
```
```  1041     hence "matrix f' ** A = A' ** A" by simp
```
```  1042     hence "A' ** A = mat 1" by (simp add: th)
```
```  1043   }
```
```  1044   then show ?thesis by blast
```
```  1045 qed
```
```  1046
```
```  1047 text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
```
```  1048
```
```  1049 definition "rowvector v = (\<chi> i j. (v\$j))"
```
```  1050
```
```  1051 definition "columnvector v = (\<chi> i j. (v\$i))"
```
```  1052
```
```  1053 lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
```
```  1054   by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
```
```  1055
```
```  1056 lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
```
```  1057   by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
```
```  1058
```
```  1059 lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
```
```  1060   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
```
```  1061
```
```  1062 lemma dot_matrix_product:
```
```  1063   "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))\$1)\$1"
```
```  1064   by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
```
```  1065
```
```  1066 lemma dot_matrix_vector_mul:
```
```  1067   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
```
```  1068   shows "(A *v x) \<bullet> (B *v y) =
```
```  1069       (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))\$1)\$1"
```
```  1070   unfolding dot_matrix_product transpose_columnvector[symmetric]
```
```  1071     dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
```
```  1072
```
```  1073 lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x\$i\<bar> |i. i\<in>UNIV}"
```
```  1074   by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
```
```  1075
```
```  1076 lemma component_le_infnorm_cart: "\<bar>x\$i\<bar> \<le> infnorm (x::real^'n)"
```
```  1077   using Basis_le_infnorm[of "axis i 1" x]
```
```  1078   by (simp add: Basis_vec_def axis_eq_axis inner_axis)
```
```  1079
```
```  1080 lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x \$ i)"
```
```  1081   unfolding continuous_def by (rule tendsto_vec_nth)
```
```  1082
```
```  1083 lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x \$ i)"
```
```  1084   unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
```
```  1085
```
```  1086 lemma continuous_on_vec_lambda[continuous_intros]:
```
```  1087   "(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)"
```
```  1088   unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)
```
```  1089
```
```  1090 lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x\$i}"
```
```  1091   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1092
```
```  1093 lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x \$ i) ` s)"
```
```  1094   unfolding bounded_def
```
```  1095   apply clarify
```
```  1096   apply (rule_tac x="x \$ i" in exI)
```
```  1097   apply (rule_tac x="e" in exI)
```
```  1098   apply clarify
```
```  1099   apply (rule order_trans [OF dist_vec_nth_le], simp)
```
```  1100   done
```
```  1101
```
```  1102 lemma compact_lemma_cart:
```
```  1103   fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
```
```  1104   assumes f: "bounded (range f)"
```
```  1105   shows "\<exists>l r. strict_mono r \<and>
```
```  1106         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
```
```  1107     (is "?th d")
```
```  1108 proof -
```
```  1109   have "\<forall>d' \<subseteq> d. ?th d'"
```
```  1110     by (rule compact_lemma_general[where unproj=vec_lambda])
```
```  1111       (auto intro!: f bounded_component_cart simp: vec_lambda_eta)
```
```  1112   then show "?th d" by simp
```
```  1113 qed
```
```  1114
```
```  1115 instance vec :: (heine_borel, finite) heine_borel
```
```  1116 proof
```
```  1117   fix f :: "nat \<Rightarrow> 'a ^ 'b"
```
```  1118   assume f: "bounded (range f)"
```
```  1119   then obtain l r where r: "strict_mono r"
```
```  1120       and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) \$ i) (l \$ i) < e) sequentially"
```
```  1121     using compact_lemma_cart [OF f] by blast
```
```  1122   let ?d = "UNIV::'b set"
```
```  1123   { fix e::real assume "e>0"
```
```  1124     hence "0 < e / (real_of_nat (card ?d))"
```
```  1125       using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
```
```  1126     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially"
```
```  1127       by simp
```
```  1128     moreover
```
```  1129     { fix n
```
```  1130       assume n: "\<forall>i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))"
```
```  1131       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) \$ i) (l \$ i))"
```
```  1132         unfolding dist_vec_def using zero_le_dist by (rule L2_set_le_sum)
```
```  1133       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
```
```  1134         by (rule sum_strict_mono) (simp_all add: n)
```
```  1135       finally have "dist (f (r n)) l < e" by simp
```
```  1136     }
```
```  1137     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
```
```  1138       by (rule eventually_mono)
```
```  1139   }
```
```  1140   hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp
```
```  1141   with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
```
```  1142 qed
```
```  1143
```
```  1144 lemma interval_cart:
```
```  1145   fixes a :: "real^'n"
```
```  1146   shows "box a b = {x::real^'n. \<forall>i. a\$i < x\$i \<and> x\$i < b\$i}"
```
```  1147     and "cbox a b = {x::real^'n. \<forall>i. a\$i \<le> x\$i \<and> x\$i \<le> b\$i}"
```
```  1148   by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
```
```  1149
```
```  1150 lemma mem_box_cart:
```
```  1151   fixes a :: "real^'n"
```
```  1152   shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a\$i < x\$i \<and> x\$i < b\$i)"
```
```  1153     and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a\$i \<le> x\$i \<and> x\$i \<le> b\$i)"
```
```  1154   using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
```
```  1155
```
```  1156 lemma interval_eq_empty_cart:
```
```  1157   fixes a :: "real^'n"
```
```  1158   shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b\$i \<le> a\$i))" (is ?th1)
```
```  1159     and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b\$i < a\$i))" (is ?th2)
```
```  1160 proof -
```
```  1161   { fix i x assume as:"b\$i \<le> a\$i" and x:"x\<in>box a b"
```
```  1162     hence "a \$ i < x \$ i \<and> x \$ i < b \$ i" unfolding mem_box_cart by auto
```
```  1163     hence "a\$i < b\$i" by auto
```
```  1164     hence False using as by auto }
```
```  1165   moreover
```
```  1166   { assume as:"\<forall>i. \<not> (b\$i \<le> a\$i)"
```
```  1167     let ?x = "(1/2) *\<^sub>R (a + b)"
```
```  1168     { fix i
```
```  1169       have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto
```
```  1170       hence "a\$i < ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i < b\$i"
```
```  1171         unfolding vector_smult_component and vector_add_component
```
```  1172         by auto }
```
```  1173     hence "box a b \<noteq> {}" using mem_box_cart(1)[of "?x" a b] by auto }
```
```  1174   ultimately show ?th1 by blast
```
```  1175
```
```  1176   { fix i x assume as:"b\$i < a\$i" and x:"x\<in>cbox a b"
```
```  1177     hence "a \$ i \<le> x \$ i \<and> x \$ i \<le> b \$ i" unfolding mem_box_cart by auto
```
```  1178     hence "a\$i \<le> b\$i" by auto
```
```  1179     hence False using as by auto }
```
```  1180   moreover
```
```  1181   { assume as:"\<forall>i. \<not> (b\$i < a\$i)"
```
```  1182     let ?x = "(1/2) *\<^sub>R (a + b)"
```
```  1183     { fix i
```
```  1184       have "a\$i \<le> b\$i" using as[THEN spec[where x=i]] by auto
```
```  1185       hence "a\$i \<le> ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i \<le> b\$i"
```
```  1186         unfolding vector_smult_component and vector_add_component
```
```  1187         by auto }
```
```  1188     hence "cbox a b \<noteq> {}" using mem_box_cart(2)[of "?x" a b] by auto  }
```
```  1189   ultimately show ?th2 by blast
```
```  1190 qed
```
```  1191
```
```  1192 lemma interval_ne_empty_cart:
```
```  1193   fixes a :: "real^'n"
```
```  1194   shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a\$i \<le> b\$i)"
```
```  1195     and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a\$i < b\$i)"
```
```  1196   unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
```
```  1197     (* BH: Why doesn't just "auto" work here? *)
```
```  1198
```
```  1199 lemma subset_interval_imp_cart:
```
```  1200   fixes a :: "real^'n"
```
```  1201   shows "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
```
```  1202     and "(\<forall>i. a\$i < c\$i \<and> d\$i < b\$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
```
```  1203     and "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
```
```  1204     and "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> box c d \<subseteq> box a b"
```
```  1205   unfolding subset_eq[unfolded Ball_def] unfolding mem_box_cart
```
```  1206   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
```
```  1207
```
```  1208 lemma interval_sing:
```
```  1209   fixes a :: "'a::linorder^'n"
```
```  1210   shows "{a .. a} = {a} \<and> {a<..<a} = {}"
```
```  1211   apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
```
```  1212   done
```
```  1213
```
```  1214 lemma subset_interval_cart:
```
```  1215   fixes a :: "real^'n"
```
```  1216   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)
```
```  1217     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)
```
```  1218     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)
```
```  1219     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)
```
```  1220   using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
```
```  1221
```
```  1222 lemma disjoint_interval_cart:
```
```  1223   fixes a::"real^'n"
```
```  1224   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)
```
```  1225     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)
```
```  1226     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)
```
```  1227     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)
```
```  1228   using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
```
```  1229
```
```  1230 lemma Int_interval_cart:
```
```  1231   fixes a :: "real^'n"
```
```  1232   shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a\$i) (c\$i)) .. (\<chi> i. min (b\$i) (d\$i))}"
```
```  1233   unfolding Int_interval
```
```  1234   by (auto simp: mem_box less_eq_vec_def)
```
```  1235     (auto simp: Basis_vec_def inner_axis)
```
```  1236
```
```  1237 lemma closed_interval_left_cart:
```
```  1238   fixes b :: "real^'n"
```
```  1239   shows "closed {x::real^'n. \<forall>i. x\$i \<le> b\$i}"
```
```  1240   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1241
```
```  1242 lemma closed_interval_right_cart:
```
```  1243   fixes a::"real^'n"
```
```  1244   shows "closed {x::real^'n. \<forall>i. a\$i \<le> x\$i}"
```
```  1245   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1246
```
```  1247 lemma is_interval_cart:
```
```  1248   "is_interval (s::(real^'n) set) \<longleftrightarrow>
```
```  1249     (\<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)"
```
```  1250   by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
```
```  1251
```
```  1252 lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x\$i \<le> a}"
```
```  1253   by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1254
```
```  1255 lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x\$i \<ge> a}"
```
```  1256   by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1257
```
```  1258 lemma open_halfspace_component_lt_cart: "open {x::real^'n. x\$i < a}"
```
```  1259   by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
```
```  1260
```
```  1261 lemma open_halfspace_component_gt_cart: "open {x::real^'n. x\$i  > a}"
```
```  1262   by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
```
```  1263
```
```  1264 lemma Lim_component_le_cart:
```
```  1265   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1266   assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x \$i \<le> b) net"
```
```  1267   shows "l\$i \<le> b"
```
```  1268   by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
```
```  1269
```
```  1270 lemma Lim_component_ge_cart:
```
```  1271   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1272   assumes "(f \<longlongrightarrow> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)\$i) net"
```
```  1273   shows "b \<le> l\$i"
```
```  1274   by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
```
```  1275
```
```  1276 lemma Lim_component_eq_cart:
```
```  1277   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1278   assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\$i = b) net"
```
```  1279   shows "l\$i = b"
```
```  1280   using ev[unfolded order_eq_iff eventually_conj_iff] and
```
```  1281     Lim_component_ge_cart[OF net, of b i] and
```
```  1282     Lim_component_le_cart[OF net, of i b] by auto
```
```  1283
```
```  1284 lemma connected_ivt_component_cart:
```
```  1285   fixes x :: "real^'n"
```
```  1286   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)"
```
```  1287   using connected_ivt_hyperplane[of s x y "axis k 1" a]
```
```  1288   by (auto simp add: inner_axis inner_commute)
```
```  1289
```
```  1290 lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x\$i = 0)}"
```
```  1291   unfolding subspace_def by auto
```
```  1292
```
```  1293 lemma closed_substandard_cart:
```
```  1294   "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x\$i = 0}"
```
```  1295 proof -
```
```  1296   { fix i::'n
```
```  1297     have "closed {x::'a ^ 'n. P i \<longrightarrow> x\$i = 0}"
```
```  1298       by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) }
```
```  1299   thus ?thesis
```
```  1300     unfolding Collect_all_eq by (simp add: closed_INT)
```
```  1301 qed
```
```  1302
```
```  1303 lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x\$i = 0} = card d"
```
```  1304   (is "dim ?A = _")
```
```  1305 proof -
```
```  1306   let ?a = "\<lambda>x. axis x 1 :: real^'n"
```
```  1307   have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
```
```  1308     by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
```
```  1309   have "?a ` d \<subseteq> Basis"
```
```  1310     by (auto simp: Basis_vec_def)
```
```  1311   thus ?thesis
```
```  1312     using dim_substandard[of "?a ` d"] card_image[of ?a d]
```
```  1313     by (auto simp: axis_eq_axis inj_on_def *)
```
```  1314 qed
```
```  1315
```
```  1316 lemma dim_subset_UNIV_cart:
```
```  1317   fixes S :: "(real^'n) set"
```
```  1318   shows "dim S \<le> CARD('n)"
```
```  1319   by (metis dim_subset_UNIV DIM_cart DIM_real mult.right_neutral)
```
```  1320
```
```  1321 lemma affinity_inverses:
```
```  1322   assumes m0: "m \<noteq> (0::'a::field)"
```
```  1323   shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
```
```  1324   "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
```
```  1325   using m0
```
```  1326   apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
```
```  1327   apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
```
```  1328   done
```
```  1329
```
```  1330 lemma vector_affinity_eq:
```
```  1331   assumes m0: "(m::'a::field) \<noteq> 0"
```
```  1332   shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
```
```  1333 proof
```
```  1334   assume h: "m *s x + c = y"
```
```  1335   hence "m *s x = y - c" by (simp add: field_simps)
```
```  1336   hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
```
```  1337   then show "x = inverse m *s y + - (inverse m *s c)"
```
```  1338     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
```
```  1339 next
```
```  1340   assume h: "x = inverse m *s y + - (inverse m *s c)"
```
```  1341   show "m *s x + c = y" unfolding h
```
```  1342     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
```
```  1343 qed
```
```  1344
```
```  1345 lemma vector_eq_affinity:
```
```  1346     "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
```
```  1347   using vector_affinity_eq[where m=m and x=x and y=y and c=c]
```
```  1348   by metis
```
```  1349
```
```  1350 lemma vector_cart:
```
```  1351   fixes f :: "real^'n \<Rightarrow> real"
```
```  1352   shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
```
```  1353   unfolding euclidean_eq_iff[where 'a="real^'n"]
```
```  1354   by simp (simp add: Basis_vec_def inner_axis)
```
```  1355
```
```  1356 lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
```
```  1357   by (rule vector_cart)
```
```  1358
```
```  1359 subsection "Convex Euclidean Space"
```
```  1360
```
```  1361 lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
```
```  1362   using const_vector_cart[of 1] by (simp add: one_vec_def)
```
```  1363
```
```  1364 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
```
```  1365 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
```
```  1366
```
```  1367 lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
```
```  1368
```
```  1369 lemma convex_box_cart:
```
```  1370   assumes "\<And>i. convex {x. P i x}"
```
```  1371   shows "convex {x. \<forall>i. P i (x\$i)}"
```
```  1372   using assms unfolding convex_def by auto
```
```  1373
```
```  1374 lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x\$i)}"
```
```  1375   by (rule convex_box_cart) (simp add: atLeast_def[symmetric])
```
```  1376
```
```  1377 lemma unit_interval_convex_hull_cart:
```
```  1378   "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x\$i = 0) \<or> (x\$i = 1)}"
```
```  1379   unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
```
```  1380   by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
```
```  1381
```
```  1382 lemma cube_convex_hull_cart:
```
```  1383   assumes "0 < d"
```
```  1384   obtains s::"(real^'n) set"
```
```  1385     where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
```
```  1386 proof -
```
```  1387   from assms obtain s where "finite s"
```
```  1388     and "cbox (x - sum (( *\<^sub>R) d) Basis) (x + sum (( *\<^sub>R) d) Basis) = convex hull s"
```
```  1389     by (rule cube_convex_hull)
```
```  1390   with that[of s] show thesis
```
```  1391     by (simp add: const_vector_cart)
```
```  1392 qed
```
```  1393
```
```  1394
```
```  1395 subsection "Derivative"
```
```  1396
```
```  1397 definition "jacobian f net = matrix(frechet_derivative f net)"
```
```  1398
```
```  1399 lemma jacobian_works:
```
```  1400   "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
```
```  1401     (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
```
```  1402   apply rule
```
```  1403   unfolding jacobian_def
```
```  1404   apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
```
```  1405   apply (rule differentiableI)
```
```  1406   apply assumption
```
```  1407   unfolding frechet_derivative_works
```
```  1408   apply assumption
```
```  1409   done
```
```  1410
```
```  1411
```
```  1412 subsection \<open>Component of the differential must be zero if it exists at a local
```
```  1413   maximum or minimum for that corresponding component.\<close>
```
```  1414
```
```  1415 lemma differential_zero_maxmin_cart:
```
```  1416   fixes f::"real^'a \<Rightarrow> real^'b"
```
```  1417   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))"
```
```  1418     "f differentiable (at x)"
```
```  1419   shows "jacobian f (at x) \$ k = 0"
```
```  1420   using differential_zero_maxmin_component[of "axis k 1" e x f] assms
```
```  1421     vector_cart[of "\<lambda>j. frechet_derivative f (at x) j \$ k"]
```
```  1422   by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
```
```  1423
```
```  1424 subsection \<open>Lemmas for working on @{typ "real^1"}\<close>
```
```  1425
```
```  1426 lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
```
```  1427   by (metis (full_types) num1_eq_iff)
```
```  1428
```
```  1429 lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
```
```  1430   by auto (metis (full_types) num1_eq_iff)
```
```  1431
```
```  1432 lemma exhaust_2:
```
```  1433   fixes x :: 2
```
```  1434   shows "x = 1 \<or> x = 2"
```
```  1435 proof (induct x)
```
```  1436   case (of_int z)
```
```  1437   then have "0 <= z" and "z < 2" by simp_all
```
```  1438   then have "z = 0 | z = 1" by arith
```
```  1439   then show ?case by auto
```
```  1440 qed
```
```  1441
```
```  1442 lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
```
```  1443   by (metis exhaust_2)
```
```  1444
```
```  1445 lemma exhaust_3:
```
```  1446   fixes x :: 3
```
```  1447   shows "x = 1 \<or> x = 2 \<or> x = 3"
```
```  1448 proof (induct x)
```
```  1449   case (of_int z)
```
```  1450   then have "0 <= z" and "z < 3" by simp_all
```
```  1451   then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
```
```  1452   then show ?case by auto
```
```  1453 qed
```
```  1454
```
```  1455 lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
```
```  1456   by (metis exhaust_3)
```
```  1457
```
```  1458 lemma UNIV_1 [simp]: "UNIV = {1::1}"
```
```  1459   by (auto simp add: num1_eq_iff)
```
```  1460
```
```  1461 lemma UNIV_2: "UNIV = {1::2, 2::2}"
```
```  1462   using exhaust_2 by auto
```
```  1463
```
```  1464 lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
```
```  1465   using exhaust_3 by auto
```
```  1466
```
```  1467 lemma sum_1: "sum f (UNIV::1 set) = f 1"
```
```  1468   unfolding UNIV_1 by simp
```
```  1469
```
```  1470 lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2"
```
```  1471   unfolding UNIV_2 by simp
```
```  1472
```
```  1473 lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3"
```
```  1474   unfolding UNIV_3 by (simp add: ac_simps)
```
```  1475
```
```  1476 instantiation num1 :: cart_one
```
```  1477 begin
```
```  1478
```
```  1479 instance
```
```  1480 proof
```
```  1481   show "CARD(1) = Suc 0" by auto
```
```  1482 qed
```
```  1483
```
```  1484 end
```
```  1485
```
```  1486 subsection\<open>The collapse of the general concepts to dimension one.\<close>
```
```  1487
```
```  1488 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x\$1))"
```
```  1489   by (simp add: vec_eq_iff)
```
```  1490
```
```  1491 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
```
```  1492   apply auto
```
```  1493   apply (erule_tac x= "x\$1" in allE)
```
```  1494   apply (simp only: vector_one[symmetric])
```
```  1495   done
```
```  1496
```
```  1497 lemma norm_vector_1: "norm (x :: _^1) = norm (x\$1)"
```
```  1498   by (simp add: norm_vec_def)
```
```  1499
```
```  1500 lemma norm_real: "norm(x::real ^ 1) = \<bar>x\$1\<bar>"
```
```  1501   by (simp add: norm_vector_1)
```
```  1502
```
```  1503 lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x\$1) - (y\$1)\<bar>"
```
```  1504   by (auto simp add: norm_real dist_norm)
```
```  1505
```
```  1506
```
```  1507 subsection\<open>Explicit vector construction from lists.\<close>
```
```  1508
```
```  1509 definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
```
```  1510
```
```  1511 lemma vector_1: "(vector[x]) \$1 = x"
```
```  1512   unfolding vector_def by simp
```
```  1513
```
```  1514 lemma vector_2:
```
```  1515  "(vector[x,y]) \$1 = x"
```
```  1516  "(vector[x,y] :: 'a^2)\$2 = (y::'a::zero)"
```
```  1517   unfolding vector_def by simp_all
```
```  1518
```
```  1519 lemma vector_3:
```
```  1520  "(vector [x,y,z] ::('a::zero)^3)\$1 = x"
```
```  1521  "(vector [x,y,z] ::('a::zero)^3)\$2 = y"
```
```  1522  "(vector [x,y,z] ::('a::zero)^3)\$3 = z"
```
```  1523   unfolding vector_def by simp_all
```
```  1524
```
```  1525 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
```
```  1526   by (metis vector_1 vector_one)
```
```  1527
```
```  1528 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
```
```  1529   apply auto
```
```  1530   apply (erule_tac x="v\$1" in allE)
```
```  1531   apply (erule_tac x="v\$2" in allE)
```
```  1532   apply (subgoal_tac "vector [v\$1, v\$2] = v")
```
```  1533   apply simp
```
```  1534   apply (vector vector_def)
```
```  1535   apply (simp add: forall_2)
```
```  1536   done
```
```  1537
```
```  1538 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
```
```  1539   apply auto
```
```  1540   apply (erule_tac x="v\$1" in allE)
```
```  1541   apply (erule_tac x="v\$2" in allE)
```
```  1542   apply (erule_tac x="v\$3" in allE)
```
```  1543   apply (subgoal_tac "vector [v\$1, v\$2, v\$3] = v")
```
```  1544   apply simp
```
```  1545   apply (vector vector_def)
```
```  1546   apply (simp add: forall_3)
```
```  1547   done
```
```  1548
```
```  1549 lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x \$ k)"
```
```  1550   apply (rule bounded_linearI[where K=1])
```
```  1551   using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
```
```  1552
```
```  1553 lemma interval_split_cart:
```
```  1554   "{a..b::real^'n} \<inter> {x. x\$k \<le> c} = {a .. (\<chi> i. if i = k then min (b\$k) c else b\$i)}"
```
```  1555   "cbox a b \<inter> {x. x\$k \<ge> c} = {(\<chi> i. if i = k then max (a\$k) c else a\$i) .. b}"
```
```  1556   apply (rule_tac[!] set_eqI)
```
```  1557   unfolding Int_iff mem_box_cart mem_Collect_eq interval_cbox_cart
```
```  1558   unfolding vec_lambda_beta
```
```  1559   by auto
```
```  1560
```
```  1561 lemmas cartesian_euclidean_space_uniform_limit_intros[uniform_limit_intros] =
```
```  1562   bounded_linear.uniform_limit[OF blinfun.bounded_linear_right]
```
```  1563   bounded_linear.uniform_limit[OF bounded_linear_vec_nth]
```
```  1564   bounded_linear.uniform_limit[OF bounded_linear_component_cart]
```
```  1565
```
```  1566 end
```