src/HOL/Analysis/Cartesian_Euclidean_Space.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 63886 685fb01256af child 63918 6bf55e6e0b75 permissions -rw-r--r--
tuned proofs;
```     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 (* Henstock_Kurzweil_Integration *)
```
```     5 begin
```
```     6
```
```     7 lemma subspace_special_hyperplane: "subspace {x. x \$ k = 0}"
```
```     8   by (simp add: subspace_def)
```
```     9
```
```    10 lemma delta_mult_idempotent:
```
```    11   "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)"
```
```    12   by simp
```
```    13
```
```    14 lemma setsum_UNIV_sum:
```
```    15   fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
```
```    16   shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
```
```    17   apply (subst UNIV_Plus_UNIV [symmetric])
```
```    18   apply (subst setsum.Plus)
```
```    19   apply simp_all
```
```    20   done
```
```    21
```
```    22 lemma setsum_mult_product:
```
```    23   "setsum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
```
```    24   unfolding setsum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]
```
```    25 proof (rule setsum.cong, simp, rule setsum.reindex_cong)
```
```    26   fix i
```
```    27   show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
```
```    28   show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
```
```    29   proof safe
```
```    30     fix j assume "j \<in> {i * B..<i * B + B}"
```
```    31     then show "j \<in> (\<lambda>j. j + i * B) ` {..<B}"
```
```    32       by (auto intro!: image_eqI[of _ _ "j - i * B"])
```
```    33   qed simp
```
```    34 qed simp
```
```    35
```
```    36
```
```    37 subsection\<open>Basic componentwise operations on vectors.\<close>
```
```    38
```
```    39 instantiation vec :: (times, finite) times
```
```    40 begin
```
```    41
```
```    42 definition "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x\$i) * (y\$i)))"
```
```    43 instance ..
```
```    44
```
```    45 end
```
```    46
```
```    47 instantiation vec :: (one, finite) one
```
```    48 begin
```
```    49
```
```    50 definition "1 \<equiv> (\<chi> i. 1)"
```
```    51 instance ..
```
```    52
```
```    53 end
```
```    54
```
```    55 instantiation vec :: (ord, finite) ord
```
```    56 begin
```
```    57
```
```    58 definition "x \<le> y \<longleftrightarrow> (\<forall>i. x\$i \<le> y\$i)"
```
```    59 definition "x < (y::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
```
```    60 instance ..
```
```    61
```
```    62 end
```
```    63
```
```    64 text\<open>The ordering on one-dimensional vectors is linear.\<close>
```
```    65
```
```    66 class cart_one =
```
```    67   assumes UNIV_one: "card (UNIV :: 'a set) = Suc 0"
```
```    68 begin
```
```    69
```
```    70 subclass finite
```
```    71 proof
```
```    72   from UNIV_one show "finite (UNIV :: 'a set)"
```
```    73     by (auto intro!: card_ge_0_finite)
```
```    74 qed
```
```    75
```
```    76 end
```
```    77
```
```    78 instance vec:: (order, finite) order
```
```    79   by standard (auto simp: less_eq_vec_def less_vec_def vec_eq_iff
```
```    80       intro: order.trans order.antisym order.strict_implies_order)
```
```    81
```
```    82 instance vec :: (linorder, cart_one) linorder
```
```    83 proof
```
```    84   obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"
```
```    85   proof -
```
```    86     have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)
```
```    87     then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)
```
```    88     then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto
```
```    89     then show thesis by (auto intro: that)
```
```    90   qed
```
```    91   fix x y :: "'a^'b::cart_one"
```
```    92   note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
```
```    93   show "x \<le> y \<or> y \<le> x" by auto
```
```    94 qed
```
```    95
```
```    96 text\<open>Constant Vectors\<close>
```
```    97
```
```    98 definition "vec x = (\<chi> i. x)"
```
```    99
```
```   100 lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b"
```
```   101   by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis)
```
```   102
```
```   103 text\<open>Also the scalar-vector multiplication.\<close>
```
```   104
```
```   105 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
```
```   106   where "c *s x = (\<chi> i. c * (x\$i))"
```
```   107
```
```   108
```
```   109 subsection \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space.\<close>
```
```   110
```
```   111 lemma setsum_cong_aux:
```
```   112   "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
```
```   113   by (auto intro: setsum.cong)
```
```   114
```
```   115 hide_fact (open) setsum_cong_aux
```
```   116
```
```   117 method_setup vector = \<open>
```
```   118 let
```
```   119   val ss1 =
```
```   120     simpset_of (put_simpset HOL_basic_ss @{context}
```
```   121       addsimps [@{thm setsum.distrib} RS sym,
```
```   122       @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
```
```   123       @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym])
```
```   124   val ss2 =
```
```   125     simpset_of (@{context} addsimps
```
```   126              [@{thm plus_vec_def}, @{thm times_vec_def},
```
```   127               @{thm minus_vec_def}, @{thm uminus_vec_def},
```
```   128               @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
```
```   129               @{thm scaleR_vec_def},
```
```   130               @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}])
```
```   131   fun vector_arith_tac ctxt ths =
```
```   132     simp_tac (put_simpset ss1 ctxt)
```
```   133     THEN' (fn i => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.setsum_cong_aux} i
```
```   134          ORELSE resolve_tac ctxt @{thms setsum.neutral} i
```
```   135          ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i)
```
```   136     (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
```
```   137     THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths)
```
```   138 in
```
```   139   Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths))
```
```   140 end
```
```   141 \<close> "lift trivial vector statements to real arith statements"
```
```   142
```
```   143 lemma vec_0[simp]: "vec 0 = 0" by vector
```
```   144 lemma vec_1[simp]: "vec 1 = 1" by vector
```
```   145
```
```   146 lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
```
```   147
```
```   148 lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
```
```   149
```
```   150 lemma vec_add: "vec(x + y) = vec x + vec y"  by vector
```
```   151 lemma vec_sub: "vec(x - y) = vec x - vec y" by vector
```
```   152 lemma vec_cmul: "vec(c * x) = c *s vec x " by vector
```
```   153 lemma vec_neg: "vec(- x) = - vec x " by vector
```
```   154
```
```   155 lemma vec_setsum:
```
```   156   assumes "finite S"
```
```   157   shows "vec(setsum f S) = setsum (vec \<circ> f) S"
```
```   158   using assms
```
```   159 proof induct
```
```   160   case empty
```
```   161   then show ?case by simp
```
```   162 next
```
```   163   case insert
```
```   164   then show ?case by (auto simp add: vec_add)
```
```   165 qed
```
```   166
```
```   167 text\<open>Obvious "component-pushing".\<close>
```
```   168
```
```   169 lemma vec_component [simp]: "vec x \$ i = x"
```
```   170   by vector
```
```   171
```
```   172 lemma vector_mult_component [simp]: "(x * y)\$i = x\$i * y\$i"
```
```   173   by vector
```
```   174
```
```   175 lemma vector_smult_component [simp]: "(c *s y)\$i = c * (y\$i)"
```
```   176   by vector
```
```   177
```
```   178 lemma cond_component: "(if b then x else y)\$i = (if b then x\$i else y\$i)" by vector
```
```   179
```
```   180 lemmas vector_component =
```
```   181   vec_component vector_add_component vector_mult_component
```
```   182   vector_smult_component vector_minus_component vector_uminus_component
```
```   183   vector_scaleR_component cond_component
```
```   184
```
```   185
```
```   186 subsection \<open>Some frequently useful arithmetic lemmas over vectors.\<close>
```
```   187
```
```   188 instance vec :: (semigroup_mult, finite) semigroup_mult
```
```   189   by standard (vector mult.assoc)
```
```   190
```
```   191 instance vec :: (monoid_mult, finite) monoid_mult
```
```   192   by standard vector+
```
```   193
```
```   194 instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
```
```   195   by standard (vector mult.commute)
```
```   196
```
```   197 instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
```
```   198   by standard vector
```
```   199
```
```   200 instance vec :: (semiring, finite) semiring
```
```   201   by standard (vector field_simps)+
```
```   202
```
```   203 instance vec :: (semiring_0, finite) semiring_0
```
```   204   by standard (vector field_simps)+
```
```   205 instance vec :: (semiring_1, finite) semiring_1
```
```   206   by standard vector
```
```   207 instance vec :: (comm_semiring, finite) comm_semiring
```
```   208   by standard (vector field_simps)+
```
```   209
```
```   210 instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
```
```   211 instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
```
```   212 instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
```
```   213 instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
```
```   214 instance vec :: (ring, finite) ring ..
```
```   215 instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
```
```   216 instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
```
```   217
```
```   218 instance vec :: (ring_1, finite) ring_1 ..
```
```   219
```
```   220 instance vec :: (real_algebra, finite) real_algebra
```
```   221   by standard (simp_all add: vec_eq_iff)
```
```   222
```
```   223 instance vec :: (real_algebra_1, finite) real_algebra_1 ..
```
```   224
```
```   225 lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)\$i = of_nat n"
```
```   226 proof (induct n)
```
```   227   case 0
```
```   228   then show ?case by vector
```
```   229 next
```
```   230   case Suc
```
```   231   then show ?case by vector
```
```   232 qed
```
```   233
```
```   234 lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) \$ i = 1"
```
```   235   by vector
```
```   236
```
```   237 lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) \$ i = - 1"
```
```   238   by vector
```
```   239
```
```   240 instance vec :: (semiring_char_0, finite) semiring_char_0
```
```   241 proof
```
```   242   fix m n :: nat
```
```   243   show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
```
```   244     by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
```
```   245 qed
```
```   246
```
```   247 instance vec :: (numeral, finite) numeral ..
```
```   248 instance vec :: (semiring_numeral, finite) semiring_numeral ..
```
```   249
```
```   250 lemma numeral_index [simp]: "numeral w \$ i = numeral w"
```
```   251   by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
```
```   252
```
```   253 lemma neg_numeral_index [simp]: "- numeral w \$ i = - numeral w"
```
```   254   by (simp only: vector_uminus_component numeral_index)
```
```   255
```
```   256 instance vec :: (comm_ring_1, finite) comm_ring_1 ..
```
```   257 instance vec :: (ring_char_0, finite) ring_char_0 ..
```
```   258
```
```   259 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
```
```   260   by (vector mult.assoc)
```
```   261 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
```
```   262   by (vector field_simps)
```
```   263 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
```
```   264   by (vector field_simps)
```
```   265 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
```
```   266 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
```
```   267 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
```
```   268   by (vector field_simps)
```
```   269 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
```
```   270 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
```
```   271 lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
```
```   272 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
```
```   273 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
```
```   274   by (vector field_simps)
```
```   275
```
```   276 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
```
```   277   by (simp add: vec_eq_iff)
```
```   278
```
```   279 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
```
```   280 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
```
```   281   by vector
```
```   282 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
```
```   283   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
```
```   284 lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
```
```   285   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
```
```   286 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
```
```   287   by (metis vector_mul_lcancel)
```
```   288 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
```
```   289   by (metis vector_mul_rcancel)
```
```   290
```
```   291 lemma component_le_norm_cart: "\<bar>x\$i\<bar> <= norm x"
```
```   292   apply (simp add: norm_vec_def)
```
```   293   apply (rule member_le_setL2, simp_all)
```
```   294   done
```
```   295
```
```   296 lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x\$i\<bar> <= e"
```
```   297   by (metis component_le_norm_cart order_trans)
```
```   298
```
```   299 lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x\$i\<bar> < e"
```
```   300   by (metis component_le_norm_cart le_less_trans)
```
```   301
```
```   302 lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x\$i\<bar>) UNIV"
```
```   303   by (simp add: norm_vec_def setL2_le_setsum)
```
```   304
```
```   305 lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
```
```   306   unfolding scaleR_vec_def vector_scalar_mult_def by simp
```
```   307
```
```   308 lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
```
```   309   unfolding dist_norm scalar_mult_eq_scaleR
```
```   310   unfolding scaleR_right_diff_distrib[symmetric] by simp
```
```   311
```
```   312 lemma setsum_component [simp]:
```
```   313   fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
```
```   314   shows "(setsum f S)\$i = setsum (\<lambda>x. (f x)\$i) S"
```
```   315 proof (cases "finite S")
```
```   316   case True
```
```   317   then show ?thesis by induct simp_all
```
```   318 next
```
```   319   case False
```
```   320   then show ?thesis by simp
```
```   321 qed
```
```   322
```
```   323 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)\$i ) S)"
```
```   324   by (simp add: vec_eq_iff)
```
```   325
```
```   326 lemma setsum_cmul:
```
```   327   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
```
```   328   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
```
```   329   by (simp add: vec_eq_iff setsum_right_distrib)
```
```   330
```
```   331 lemma setsum_norm_allsubsets_bound_cart:
```
```   332   fixes f:: "'a \<Rightarrow> real ^'n"
```
```   333   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
```
```   334   shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
```
```   335   using setsum_norm_allsubsets_bound[OF assms]
```
```   336   by simp
```
```   337
```
```   338 subsection\<open>Closures and interiors of halfspaces\<close>
```
```   339
```
```   340 lemma interior_halfspace_le [simp]:
```
```   341   assumes "a \<noteq> 0"
```
```   342     shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
```
```   343 proof -
```
```   344   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
```
```   345   proof -
```
```   346     obtain e where "e>0" and e: "cball x e \<subseteq> S"
```
```   347       using \<open>open S\<close> open_contains_cball x by blast
```
```   348     then have "x + (e / norm a) *\<^sub>R a \<in> cball x e"
```
```   349       by (simp add: dist_norm)
```
```   350     then have "x + (e / norm a) *\<^sub>R a \<in> S"
```
```   351       using e by blast
```
```   352     then have "x + (e / norm a) *\<^sub>R a \<in> {x. a \<bullet> x \<le> b}"
```
```   353       using S by blast
```
```   354     moreover have "e * (a \<bullet> a) / norm a > 0"
```
```   355       by (simp add: \<open>0 < e\<close> assms)
```
```   356     ultimately show ?thesis
```
```   357       by (simp add: algebra_simps)
```
```   358   qed
```
```   359   show ?thesis
```
```   360     by (rule interior_unique) (auto simp: open_halfspace_lt *)
```
```   361 qed
```
```   362
```
```   363 lemma interior_halfspace_ge [simp]:
```
```   364    "a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"
```
```   365 using interior_halfspace_le [of "-a" "-b"] by simp
```
```   366
```
```   367 lemma interior_halfspace_component_le [simp]:
```
```   368      "interior {x. x\$k \<le> a} = {x :: (real,'n::finite) vec. x\$k < a}" (is "?LE")
```
```   369   and interior_halfspace_component_ge [simp]:
```
```   370      "interior {x. x\$k \<ge> a} = {x :: (real,'n::finite) vec. x\$k > a}" (is "?GE")
```
```   371 proof -
```
```   372   have "axis k (1::real) \<noteq> 0"
```
```   373     by (simp add: axis_def vec_eq_iff)
```
```   374   moreover have "axis k (1::real) \<bullet> x = x\$k" for x
```
```   375     by (simp add: cart_eq_inner_axis inner_commute)
```
```   376   ultimately show ?LE ?GE
```
```   377     using interior_halfspace_le [of "axis k (1::real)" a]
```
```   378           interior_halfspace_ge [of "axis k (1::real)" a] by auto
```
```   379 qed
```
```   380
```
```   381 lemma closure_halfspace_lt [simp]:
```
```   382   assumes "a \<noteq> 0"
```
```   383     shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
```
```   384 proof -
```
```   385   have [simp]: "-{x. a \<bullet> x < b} = {x. a \<bullet> x \<ge> b}"
```
```   386     by (force simp:)
```
```   387   then show ?thesis
```
```   388     using interior_halfspace_ge [of a b] assms
```
```   389     by (force simp: closure_interior)
```
```   390 qed
```
```   391
```
```   392 lemma closure_halfspace_gt [simp]:
```
```   393    "a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"
```
```   394 using closure_halfspace_lt [of "-a" "-b"] by simp
```
```   395
```
```   396 lemma closure_halfspace_component_lt [simp]:
```
```   397      "closure {x. x\$k < a} = {x :: (real,'n::finite) vec. x\$k \<le> a}" (is "?LE")
```
```   398   and closure_halfspace_component_gt [simp]:
```
```   399      "closure {x. x\$k > a} = {x :: (real,'n::finite) vec. x\$k \<ge> a}" (is "?GE")
```
```   400 proof -
```
```   401   have "axis k (1::real) \<noteq> 0"
```
```   402     by (simp add: axis_def vec_eq_iff)
```
```   403   moreover have "axis k (1::real) \<bullet> x = x\$k" for x
```
```   404     by (simp add: cart_eq_inner_axis inner_commute)
```
```   405   ultimately show ?LE ?GE
```
```   406     using closure_halfspace_lt [of "axis k (1::real)" a]
```
```   407           closure_halfspace_gt [of "axis k (1::real)" a] by auto
```
```   408 qed
```
```   409
```
```   410 lemma interior_hyperplane [simp]:
```
```   411   assumes "a \<noteq> 0"
```
```   412     shows "interior {x. a \<bullet> x = b} = {}"
```
```   413 proof -
```
```   414   have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
```
```   415     by (force simp:)
```
```   416   then show ?thesis
```
```   417     by (auto simp: assms)
```
```   418 qed
```
```   419
```
```   420 lemma frontier_halfspace_le:
```
```   421   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
```
```   422     shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
```
```   423 proof (cases "a = 0")
```
```   424   case True with assms show ?thesis by simp
```
```   425 next
```
```   426   case False then show ?thesis
```
```   427     by (force simp: frontier_def closed_halfspace_le)
```
```   428 qed
```
```   429
```
```   430 lemma frontier_halfspace_ge:
```
```   431   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
```
```   432     shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"
```
```   433 proof (cases "a = 0")
```
```   434   case True with assms show ?thesis by simp
```
```   435 next
```
```   436   case False then show ?thesis
```
```   437     by (force simp: frontier_def closed_halfspace_ge)
```
```   438 qed
```
```   439
```
```   440 lemma frontier_halfspace_lt:
```
```   441   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
```
```   442     shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"
```
```   443 proof (cases "a = 0")
```
```   444   case True with assms show ?thesis by simp
```
```   445 next
```
```   446   case False then show ?thesis
```
```   447     by (force simp: frontier_def interior_open open_halfspace_lt)
```
```   448 qed
```
```   449
```
```   450 lemma frontier_halfspace_gt:
```
```   451   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
```
```   452     shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"
```
```   453 proof (cases "a = 0")
```
```   454   case True with assms show ?thesis by simp
```
```   455 next
```
```   456   case False then show ?thesis
```
```   457     by (force simp: frontier_def interior_open open_halfspace_gt)
```
```   458 qed
```
```   459
```
```   460 lemma interior_standard_hyperplane:
```
```   461    "interior {x :: (real,'n::finite) vec. x\$k = a} = {}"
```
```   462 proof -
```
```   463   have "axis k (1::real) \<noteq> 0"
```
```   464     by (simp add: axis_def vec_eq_iff)
```
```   465   moreover have "axis k (1::real) \<bullet> x = x\$k" for x
```
```   466     by (simp add: cart_eq_inner_axis inner_commute)
```
```   467   ultimately show ?thesis
```
```   468     using interior_hyperplane [of "axis k (1::real)" a]
```
```   469     by force
```
```   470 qed
```
```   471
```
```   472 subsection \<open>Matrix operations\<close>
```
```   473
```
```   474 text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close>
```
```   475
```
```   476 definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
```
```   477     (infixl "**" 70)
```
```   478   where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m\$i)\$k) * ((m'\$k)\$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
```
```   479
```
```   480 definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
```
```   481     (infixl "*v" 70)
```
```   482   where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m\$i)\$j) * (x\$j)) (UNIV ::'n set)) :: 'a^'m"
```
```   483
```
```   484 definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
```
```   485     (infixl "v*" 70)
```
```   486   where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m\$i)\$j) * (v\$i)) (UNIV :: 'm set)) :: 'a^'n"
```
```   487
```
```   488 definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
```
```   489 definition transpose where
```
```   490   "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A\$j)\$i))"
```
```   491 definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A\$i)\$j))"
```
```   492 definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A\$i)\$j))"
```
```   493 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
```
```   494 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
```
```   495
```
```   496 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
```
```   497 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
```
```   498   by (vector matrix_matrix_mult_def setsum.distrib[symmetric] field_simps)
```
```   499
```
```   500 lemma matrix_mul_lid:
```
```   501   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
```
```   502   shows "mat 1 ** A = A"
```
```   503   apply (simp add: matrix_matrix_mult_def mat_def)
```
```   504   apply vector
```
```   505   apply (auto simp only: if_distrib cond_application_beta setsum.delta'[OF finite]
```
```   506     mult_1_left mult_zero_left if_True UNIV_I)
```
```   507   done
```
```   508
```
```   509
```
```   510 lemma matrix_mul_rid:
```
```   511   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
```
```   512   shows "A ** mat 1 = A"
```
```   513   apply (simp add: matrix_matrix_mult_def mat_def)
```
```   514   apply vector
```
```   515   apply (auto simp only: if_distrib cond_application_beta setsum.delta[OF finite]
```
```   516     mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
```
```   517   done
```
```   518
```
```   519 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
```
```   520   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult.assoc)
```
```   521   apply (subst setsum.commute)
```
```   522   apply simp
```
```   523   done
```
```   524
```
```   525 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
```
```   526   apply (vector matrix_matrix_mult_def matrix_vector_mult_def
```
```   527     setsum_right_distrib setsum_left_distrib mult.assoc)
```
```   528   apply (subst setsum.commute)
```
```   529   apply simp
```
```   530   done
```
```   531
```
```   532 lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
```
```   533   apply (vector matrix_vector_mult_def mat_def)
```
```   534   apply (simp add: if_distrib cond_application_beta setsum.delta' cong del: if_weak_cong)
```
```   535   done
```
```   536
```
```   537 lemma matrix_transpose_mul:
```
```   538     "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
```
```   539   by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
```
```   540
```
```   541 lemma matrix_eq:
```
```   542   fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
```
```   543   shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   544   apply auto
```
```   545   apply (subst vec_eq_iff)
```
```   546   apply clarify
```
```   547   apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
```
```   548   apply (erule_tac x="axis ia 1" in allE)
```
```   549   apply (erule_tac x="i" in allE)
```
```   550   apply (auto simp add: if_distrib cond_application_beta axis_def
```
```   551     setsum.delta[OF finite] cong del: if_weak_cong)
```
```   552   done
```
```   553
```
```   554 lemma matrix_vector_mul_component: "((A::real^_^_) *v x)\$k = (A\$k) \<bullet> x"
```
```   555   by (simp add: matrix_vector_mult_def inner_vec_def)
```
```   556
```
```   557 lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
```
```   558   apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib ac_simps)
```
```   559   apply (subst setsum.commute)
```
```   560   apply simp
```
```   561   done
```
```   562
```
```   563 lemma transpose_mat: "transpose (mat n) = mat n"
```
```   564   by (vector transpose_def mat_def)
```
```   565
```
```   566 lemma transpose_transpose: "transpose(transpose A) = A"
```
```   567   by (vector transpose_def)
```
```   568
```
```   569 lemma row_transpose:
```
```   570   fixes A:: "'a::semiring_1^_^_"
```
```   571   shows "row i (transpose A) = column i A"
```
```   572   by (simp add: row_def column_def transpose_def vec_eq_iff)
```
```   573
```
```   574 lemma column_transpose:
```
```   575   fixes A:: "'a::semiring_1^_^_"
```
```   576   shows "column i (transpose A) = row i A"
```
```   577   by (simp add: row_def column_def transpose_def vec_eq_iff)
```
```   578
```
```   579 lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
```
```   580   by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
```
```   581
```
```   582 lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
```
```   583   by (metis transpose_transpose rows_transpose)
```
```   584
```
```   585 text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
```
```   586
```
```   587 lemma matrix_mult_dot: "A *v x = (\<chi> i. A\$i \<bullet> x)"
```
```   588   by (simp add: matrix_vector_mult_def inner_vec_def)
```
```   589
```
```   590 lemma matrix_mult_vsum:
```
```   591   "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x\$i) *s column i A) (UNIV:: 'n set)"
```
```   592   by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
```
```   593
```
```   594 lemma vector_componentwise:
```
```   595   "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x\$i) * (axis i 1 :: 'a^'n) \$ j)"
```
```   596   by (simp add: axis_def if_distrib setsum.If_cases vec_eq_iff)
```
```   597
```
```   598 lemma basis_expansion: "setsum (\<lambda>i. (x\$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
```
```   599   by (auto simp add: axis_def vec_eq_iff if_distrib setsum.If_cases cong del: if_weak_cong)
```
```   600
```
```   601 lemma linear_componentwise:
```
```   602   fixes f:: "real ^'m \<Rightarrow> real ^ _"
```
```   603   assumes lf: "linear f"
```
```   604   shows "(f x)\$j = setsum (\<lambda>i. (x\$i) * (f (axis i 1)\$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
```
```   605 proof -
```
```   606   let ?M = "(UNIV :: 'm set)"
```
```   607   let ?N = "(UNIV :: 'n set)"
```
```   608   have "?rhs = (setsum (\<lambda>i.(x\$i) *\<^sub>R f (axis i 1) ) ?M)\$j"
```
```   609     unfolding setsum_component by simp
```
```   610   then show ?thesis
```
```   611     unfolding linear_setsum_mul[OF lf, symmetric]
```
```   612     unfolding scalar_mult_eq_scaleR[symmetric]
```
```   613     unfolding basis_expansion
```
```   614     by simp
```
```   615 qed
```
```   616
```
```   617 text\<open>Inverse matrices  (not necessarily square)\<close>
```
```   618
```
```   619 definition
```
```   620   "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
```
```   621
```
```   622 definition
```
```   623   "matrix_inv(A:: 'a::semiring_1^'n^'m) =
```
```   624     (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
```
```   625
```
```   626 text\<open>Correspondence between matrices and linear operators.\<close>
```
```   627
```
```   628 definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
```
```   629   where "matrix f = (\<chi> i j. (f(axis j 1))\$i)"
```
```   630
```
```   631 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
```
```   632   by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
```
```   633       field_simps setsum_right_distrib setsum.distrib)
```
```   634
```
```   635 lemma matrix_works:
```
```   636   assumes lf: "linear f"
```
```   637   shows "matrix f *v x = f (x::real ^ 'n)"
```
```   638   apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)
```
```   639   apply clarify
```
```   640   apply (rule linear_componentwise[OF lf, symmetric])
```
```   641   done
```
```   642
```
```   643 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
```
```   644   by (simp add: ext matrix_works)
```
```   645
```
```   646 lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
```
```   647   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
```
```   648
```
```   649 lemma matrix_compose:
```
```   650   assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
```
```   651     and lg: "linear (g::real^'m \<Rightarrow> real^_)"
```
```   652   shows "matrix (g \<circ> f) = matrix g ** matrix f"
```
```   653   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
```
```   654   by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
```
```   655
```
```   656 lemma matrix_vector_column:
```
```   657   "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x\$i) *s ((transpose A)\$i)) (UNIV:: 'n set)"
```
```   658   by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
```
```   659
```
```   660 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
```
```   661   apply (rule adjoint_unique)
```
```   662   apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
```
```   663     setsum_left_distrib setsum_right_distrib)
```
```   664   apply (subst setsum.commute)
```
```   665   apply (auto simp add: ac_simps)
```
```   666   done
```
```   667
```
```   668 lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
```
```   669   shows "matrix(adjoint f) = transpose(matrix f)"
```
```   670   apply (subst matrix_vector_mul[OF lf])
```
```   671   unfolding adjoint_matrix matrix_of_matrix_vector_mul
```
```   672   apply rule
```
```   673   done
```
```   674
```
```   675
```
```   676 subsection \<open>lambda skolemization on cartesian products\<close>
```
```   677
```
```   678 (* FIXME: rename do choice_cart *)
```
```   679
```
```   680 lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
```
```   681    (\<exists>x::'a ^ 'n. \<forall>i. P i (x \$ i))" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   682 proof -
```
```   683   let ?S = "(UNIV :: 'n set)"
```
```   684   { assume H: "?rhs"
```
```   685     then have ?lhs by auto }
```
```   686   moreover
```
```   687   { assume H: "?lhs"
```
```   688     then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
```
```   689     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
```
```   690     { fix i
```
```   691       from f have "P i (f i)" by metis
```
```   692       then have "P i (?x \$ i)" by auto
```
```   693     }
```
```   694     hence "\<forall>i. P i (?x\$i)" by metis
```
```   695     hence ?rhs by metis }
```
```   696   ultimately show ?thesis by metis
```
```   697 qed
```
```   698
```
```   699 lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
```
```   700   unfolding inner_simps scalar_mult_eq_scaleR by auto
```
```   701
```
```   702 lemma left_invertible_transpose:
```
```   703   "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
```
```   704   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
```
```   705
```
```   706 lemma right_invertible_transpose:
```
```   707   "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
```
```   708   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
```
```   709
```
```   710 lemma matrix_left_invertible_injective:
```
```   711   "(\<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)"
```
```   712 proof -
```
```   713   { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
```
```   714     from xy have "B*v (A *v x) = B *v (A*v y)" by simp
```
```   715     hence "x = y"
```
```   716       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
```
```   717   moreover
```
```   718   { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
```
```   719     hence i: "inj (op *v A)" unfolding inj_on_def by auto
```
```   720     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
```
```   721     obtain g where g: "linear g" "g \<circ> op *v A = id" by blast
```
```   722     have "matrix g ** A = mat 1"
```
```   723       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
```
```   724       using g(2) by (simp add: fun_eq_iff)
```
```   725     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
```
```   726   ultimately show ?thesis by blast
```
```   727 qed
```
```   728
```
```   729 lemma matrix_left_invertible_ker:
```
```   730   "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
```
```   731   unfolding matrix_left_invertible_injective
```
```   732   using linear_injective_0[OF matrix_vector_mul_linear, of A]
```
```   733   by (simp add: inj_on_def)
```
```   734
```
```   735 lemma matrix_right_invertible_surjective:
```
```   736   "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
```
```   737 proof -
```
```   738   { fix B :: "real ^'m^'n"
```
```   739     assume AB: "A ** B = mat 1"
```
```   740     { fix x :: "real ^ 'm"
```
```   741       have "A *v (B *v x) = x"
```
```   742         by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
```
```   743     hence "surj (op *v A)" unfolding surj_def by metis }
```
```   744   moreover
```
```   745   { assume sf: "surj (op *v A)"
```
```   746     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
```
```   747     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A \<circ> g = id"
```
```   748       by blast
```
```   749
```
```   750     have "A ** (matrix g) = mat 1"
```
```   751       unfolding matrix_eq  matrix_vector_mul_lid
```
```   752         matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
```
```   753       using g(2) unfolding o_def fun_eq_iff id_def
```
```   754       .
```
```   755     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
```
```   756   }
```
```   757   ultimately show ?thesis unfolding surj_def by blast
```
```   758 qed
```
```   759
```
```   760 lemma matrix_left_invertible_independent_columns:
```
```   761   fixes A :: "real^'n^'m"
```
```   762   shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
```
```   763       (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
```
```   764     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   765 proof -
```
```   766   let ?U = "UNIV :: 'n set"
```
```   767   { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
```
```   768     { fix c i
```
```   769       assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
```
```   770       let ?x = "\<chi> i. c i"
```
```   771       have th0:"A *v ?x = 0"
```
```   772         using c
```
```   773         unfolding matrix_mult_vsum vec_eq_iff
```
```   774         by auto
```
```   775       from k[rule_format, OF th0] i
```
```   776       have "c i = 0" by (vector vec_eq_iff)}
```
```   777     hence ?rhs by blast }
```
```   778   moreover
```
```   779   { assume H: ?rhs
```
```   780     { fix x assume x: "A *v x = 0"
```
```   781       let ?c = "\<lambda>i. ((x\$i ):: real)"
```
```   782       from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
```
```   783       have "x = 0" by vector }
```
```   784   }
```
```   785   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
```
```   786 qed
```
```   787
```
```   788 lemma matrix_right_invertible_independent_rows:
```
```   789   fixes A :: "real^'n^'m"
```
```   790   shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
```
```   791     (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
```
```   792   unfolding left_invertible_transpose[symmetric]
```
```   793     matrix_left_invertible_independent_columns
```
```   794   by (simp add: column_transpose)
```
```   795
```
```   796 lemma matrix_right_invertible_span_columns:
```
```   797   "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
```
```   798     span (columns A) = UNIV" (is "?lhs = ?rhs")
```
```   799 proof -
```
```   800   let ?U = "UNIV :: 'm set"
```
```   801   have fU: "finite ?U" by simp
```
```   802   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x\$i) *s column i A) ?U = y)"
```
```   803     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
```
```   804     apply (subst eq_commute)
```
```   805     apply rule
```
```   806     done
```
```   807   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
```
```   808   { assume h: ?lhs
```
```   809     { fix x:: "real ^'n"
```
```   810       from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
```
```   811         where y: "setsum (\<lambda>i. (y\$i) *s column i A) ?U = x" by blast
```
```   812       have "x \<in> span (columns A)"
```
```   813         unfolding y[symmetric]
```
```   814         apply (rule span_setsum)
```
```   815         unfolding scalar_mult_eq_scaleR
```
```   816         apply (rule span_mul)
```
```   817         apply (rule span_superset)
```
```   818         unfolding columns_def
```
```   819         apply blast
```
```   820         done
```
```   821     }
```
```   822     then have ?rhs unfolding rhseq by blast }
```
```   823   moreover
```
```   824   { assume h:?rhs
```
```   825     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x\$i) *s column i A) ?U = y"
```
```   826     { fix y
```
```   827       have "?P y"
```
```   828       proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
```
```   829         show "\<exists>x::real ^ 'm. setsum (\<lambda>i. (x\$i) *s column i A) ?U = 0"
```
```   830           by (rule exI[where x=0], simp)
```
```   831       next
```
```   832         fix c y1 y2
```
```   833         assume y1: "y1 \<in> columns A" and y2: "?P y2"
```
```   834         from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
```
```   835           unfolding columns_def by blast
```
```   836         from y2 obtain x:: "real ^'m" where
```
```   837           x: "setsum (\<lambda>i. (x\$i) *s column i A) ?U = y2" by blast
```
```   838         let ?x = "(\<chi> j. if j = i then c + (x\$i) else (x\$j))::real^'m"
```
```   839         show "?P (c*s y1 + y2)"
```
```   840         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)
```
```   841           fix j
```
```   842           have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```   843               else (x\$xa) * ((column xa A\$j))) = (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))"
```
```   844             using i(1) by (simp add: field_simps)
```
```   845           have "setsum (\<lambda>xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```   846               else (x\$xa) * ((column xa A\$j))) ?U = setsum (\<lambda>xa. (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))) ?U"
```
```   847             apply (rule setsum.cong[OF refl])
```
```   848             using th apply blast
```
```   849             done
```
```   850           also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)\$j) else 0) ?U + setsum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U"
```
```   851             by (simp add: setsum.distrib)
```
```   852           also have "\<dots> = c * ((column i A)\$j) + setsum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U"
```
```   853             unfolding setsum.delta[OF fU]
```
```   854             using i(1) by simp
```
```   855           finally show "setsum (\<lambda>xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```   856             else (x\$xa) * ((column xa A\$j))) ?U = c * ((column i A)\$j) + setsum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U" .
```
```   857         qed
```
```   858       next
```
```   859         show "y \<in> span (columns A)"
```
```   860           unfolding h by blast
```
```   861       qed
```
```   862     }
```
```   863     then have ?lhs unfolding lhseq ..
```
```   864   }
```
```   865   ultimately show ?thesis by blast
```
```   866 qed
```
```   867
```
```   868 lemma matrix_left_invertible_span_rows:
```
```   869   "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
```
```   870   unfolding right_invertible_transpose[symmetric]
```
```   871   unfolding columns_transpose[symmetric]
```
```   872   unfolding matrix_right_invertible_span_columns
```
```   873   ..
```
```   874
```
```   875 text \<open>The same result in terms of square matrices.\<close>
```
```   876
```
```   877 lemma matrix_left_right_inverse:
```
```   878   fixes A A' :: "real ^'n^'n"
```
```   879   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
```
```   880 proof -
```
```   881   { fix A A' :: "real ^'n^'n"
```
```   882     assume AA': "A ** A' = mat 1"
```
```   883     have sA: "surj (op *v A)"
```
```   884       unfolding surj_def
```
```   885       apply clarify
```
```   886       apply (rule_tac x="(A' *v y)" in exI)
```
```   887       apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
```
```   888       done
```
```   889     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
```
```   890     obtain f' :: "real ^'n \<Rightarrow> real ^'n"
```
```   891       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
```
```   892     have th: "matrix f' ** A = mat 1"
```
```   893       by (simp add: matrix_eq matrix_works[OF f'(1)]
```
```   894           matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
```
```   895     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
```
```   896     hence "matrix f' = A'"
```
```   897       by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
```
```   898     hence "matrix f' ** A = A' ** A" by simp
```
```   899     hence "A' ** A = mat 1" by (simp add: th)
```
```   900   }
```
```   901   then show ?thesis by blast
```
```   902 qed
```
```   903
```
```   904 text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
```
```   905
```
```   906 definition "rowvector v = (\<chi> i j. (v\$j))"
```
```   907
```
```   908 definition "columnvector v = (\<chi> i j. (v\$i))"
```
```   909
```
```   910 lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
```
```   911   by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
```
```   912
```
```   913 lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
```
```   914   by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
```
```   915
```
```   916 lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
```
```   917   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
```
```   918
```
```   919 lemma dot_matrix_product:
```
```   920   "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))\$1)\$1"
```
```   921   by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
```
```   922
```
```   923 lemma dot_matrix_vector_mul:
```
```   924   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
```
```   925   shows "(A *v x) \<bullet> (B *v y) =
```
```   926       (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))\$1)\$1"
```
```   927   unfolding dot_matrix_product transpose_columnvector[symmetric]
```
```   928     dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
```
```   929
```
```   930 lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x\$i\<bar> |i. i\<in>UNIV}"
```
```   931   by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
```
```   932
```
```   933 lemma component_le_infnorm_cart: "\<bar>x\$i\<bar> \<le> infnorm (x::real^'n)"
```
```   934   using Basis_le_infnorm[of "axis i 1" x]
```
```   935   by (simp add: Basis_vec_def axis_eq_axis inner_axis)
```
```   936
```
```   937 lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x \$ i)"
```
```   938   unfolding continuous_def by (rule tendsto_vec_nth)
```
```   939
```
```   940 lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x \$ i)"
```
```   941   unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
```
```   942
```
```   943 lemma continuous_on_vec_lambda[continuous_intros]:
```
```   944   "(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)"
```
```   945   unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)
```
```   946
```
```   947 lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x\$i}"
```
```   948   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```   949
```
```   950 lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x \$ i) ` s)"
```
```   951   unfolding bounded_def
```
```   952   apply clarify
```
```   953   apply (rule_tac x="x \$ i" in exI)
```
```   954   apply (rule_tac x="e" in exI)
```
```   955   apply clarify
```
```   956   apply (rule order_trans [OF dist_vec_nth_le], simp)
```
```   957   done
```
```   958
```
```   959 lemma compact_lemma_cart:
```
```   960   fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
```
```   961   assumes f: "bounded (range f)"
```
```   962   shows "\<exists>l r. subseq r \<and>
```
```   963         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
```
```   964     (is "?th d")
```
```   965 proof -
```
```   966   have "\<forall>d' \<subseteq> d. ?th d'"
```
```   967     by (rule compact_lemma_general[where unproj=vec_lambda])
```
```   968       (auto intro!: f bounded_component_cart simp: vec_lambda_eta)
```
```   969   then show "?th d" by simp
```
```   970 qed
```
```   971
```
```   972 instance vec :: (heine_borel, finite) heine_borel
```
```   973 proof
```
```   974   fix f :: "nat \<Rightarrow> 'a ^ 'b"
```
```   975   assume f: "bounded (range f)"
```
```   976   then obtain l r where r: "subseq r"
```
```   977       and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) \$ i) (l \$ i) < e) sequentially"
```
```   978     using compact_lemma_cart [OF f] by blast
```
```   979   let ?d = "UNIV::'b set"
```
```   980   { fix e::real assume "e>0"
```
```   981     hence "0 < e / (real_of_nat (card ?d))"
```
```   982       using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
```
```   983     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially"
```
```   984       by simp
```
```   985     moreover
```
```   986     { fix n
```
```   987       assume n: "\<forall>i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))"
```
```   988       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) \$ i) (l \$ i))"
```
```   989         unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
```
```   990       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
```
```   991         by (rule setsum_strict_mono) (simp_all add: n)
```
```   992       finally have "dist (f (r n)) l < e" by simp
```
```   993     }
```
```   994     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
```
```   995       by (rule eventually_mono)
```
```   996   }
```
```   997   hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp
```
```   998   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
```
```   999 qed
```
```  1000
```
```  1001 lemma interval_cart:
```
```  1002   fixes a :: "real^'n"
```
```  1003   shows "box a b = {x::real^'n. \<forall>i. a\$i < x\$i \<and> x\$i < b\$i}"
```
```  1004     and "cbox a b = {x::real^'n. \<forall>i. a\$i \<le> x\$i \<and> x\$i \<le> b\$i}"
```
```  1005   by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
```
```  1006
```
```  1007 lemma mem_interval_cart:
```
```  1008   fixes a :: "real^'n"
```
```  1009   shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a\$i < x\$i \<and> x\$i < b\$i)"
```
```  1010     and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a\$i \<le> x\$i \<and> x\$i \<le> b\$i)"
```
```  1011   using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
```
```  1012
```
```  1013 lemma interval_eq_empty_cart:
```
```  1014   fixes a :: "real^'n"
```
```  1015   shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b\$i \<le> a\$i))" (is ?th1)
```
```  1016     and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b\$i < a\$i))" (is ?th2)
```
```  1017 proof -
```
```  1018   { fix i x assume as:"b\$i \<le> a\$i" and x:"x\<in>box a b"
```
```  1019     hence "a \$ i < x \$ i \<and> x \$ i < b \$ i" unfolding mem_interval_cart by auto
```
```  1020     hence "a\$i < b\$i" by auto
```
```  1021     hence False using as by auto }
```
```  1022   moreover
```
```  1023   { assume as:"\<forall>i. \<not> (b\$i \<le> a\$i)"
```
```  1024     let ?x = "(1/2) *\<^sub>R (a + b)"
```
```  1025     { fix i
```
```  1026       have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto
```
```  1027       hence "a\$i < ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i < b\$i"
```
```  1028         unfolding vector_smult_component and vector_add_component
```
```  1029         by auto }
```
```  1030     hence "box a b \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
```
```  1031   ultimately show ?th1 by blast
```
```  1032
```
```  1033   { fix i x assume as:"b\$i < a\$i" and x:"x\<in>cbox a b"
```
```  1034     hence "a \$ i \<le> x \$ i \<and> x \$ i \<le> b \$ i" unfolding mem_interval_cart by auto
```
```  1035     hence "a\$i \<le> b\$i" by auto
```
```  1036     hence False using as by auto }
```
```  1037   moreover
```
```  1038   { assume as:"\<forall>i. \<not> (b\$i < a\$i)"
```
```  1039     let ?x = "(1/2) *\<^sub>R (a + b)"
```
```  1040     { fix i
```
```  1041       have "a\$i \<le> b\$i" using as[THEN spec[where x=i]] by auto
```
```  1042       hence "a\$i \<le> ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i \<le> b\$i"
```
```  1043         unfolding vector_smult_component and vector_add_component
```
```  1044         by auto }
```
```  1045     hence "cbox a b \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
```
```  1046   ultimately show ?th2 by blast
```
```  1047 qed
```
```  1048
```
```  1049 lemma interval_ne_empty_cart:
```
```  1050   fixes a :: "real^'n"
```
```  1051   shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a\$i \<le> b\$i)"
```
```  1052     and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a\$i < b\$i)"
```
```  1053   unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
```
```  1054     (* BH: Why doesn't just "auto" work here? *)
```
```  1055
```
```  1056 lemma subset_interval_imp_cart:
```
```  1057   fixes a :: "real^'n"
```
```  1058   shows "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
```
```  1059     and "(\<forall>i. a\$i < c\$i \<and> d\$i < b\$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
```
```  1060     and "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
```
```  1061     and "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> box c d \<subseteq> box a b"
```
```  1062   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
```
```  1063   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
```
```  1064
```
```  1065 lemma interval_sing:
```
```  1066   fixes a :: "'a::linorder^'n"
```
```  1067   shows "{a .. a} = {a} \<and> {a<..<a} = {}"
```
```  1068   apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
```
```  1069   done
```
```  1070
```
```  1071 lemma subset_interval_cart:
```
```  1072   fixes a :: "real^'n"
```
```  1073   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)
```
```  1074     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)
```
```  1075     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)
```
```  1076     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)
```
```  1077   using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
```
```  1078
```
```  1079 lemma disjoint_interval_cart:
```
```  1080   fixes a::"real^'n"
```
```  1081   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)
```
```  1082     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)
```
```  1083     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)
```
```  1084     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)
```
```  1085   using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
```
```  1086
```
```  1087 lemma inter_interval_cart:
```
```  1088   fixes a :: "real^'n"
```
```  1089   shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a\$i) (c\$i)) .. (\<chi> i. min (b\$i) (d\$i))}"
```
```  1090   unfolding inter_interval
```
```  1091   by (auto simp: mem_box less_eq_vec_def)
```
```  1092     (auto simp: Basis_vec_def inner_axis)
```
```  1093
```
```  1094 lemma closed_interval_left_cart:
```
```  1095   fixes b :: "real^'n"
```
```  1096   shows "closed {x::real^'n. \<forall>i. x\$i \<le> b\$i}"
```
```  1097   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1098
```
```  1099 lemma closed_interval_right_cart:
```
```  1100   fixes a::"real^'n"
```
```  1101   shows "closed {x::real^'n. \<forall>i. a\$i \<le> x\$i}"
```
```  1102   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1103
```
```  1104 lemma is_interval_cart:
```
```  1105   "is_interval (s::(real^'n) set) \<longleftrightarrow>
```
```  1106     (\<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)"
```
```  1107   by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
```
```  1108
```
```  1109 lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x\$i \<le> a}"
```
```  1110   by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1111
```
```  1112 lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x\$i \<ge> a}"
```
```  1113   by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
```
```  1114
```
```  1115 lemma open_halfspace_component_lt_cart: "open {x::real^'n. x\$i < a}"
```
```  1116   by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
```
```  1117
```
```  1118 lemma open_halfspace_component_gt_cart: "open {x::real^'n. x\$i  > a}"
```
```  1119   by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
```
```  1120
```
```  1121 lemma Lim_component_le_cart:
```
```  1122   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1123   assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x \$i \<le> b) net"
```
```  1124   shows "l\$i \<le> b"
```
```  1125   by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
```
```  1126
```
```  1127 lemma Lim_component_ge_cart:
```
```  1128   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1129   assumes "(f \<longlongrightarrow> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)\$i) net"
```
```  1130   shows "b \<le> l\$i"
```
```  1131   by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
```
```  1132
```
```  1133 lemma Lim_component_eq_cart:
```
```  1134   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1135   assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\$i = b) net"
```
```  1136   shows "l\$i = b"
```
```  1137   using ev[unfolded order_eq_iff eventually_conj_iff] and
```
```  1138     Lim_component_ge_cart[OF net, of b i] and
```
```  1139     Lim_component_le_cart[OF net, of i b] by auto
```
```  1140
```
```  1141 lemma connected_ivt_component_cart:
```
```  1142   fixes x :: "real^'n"
```
```  1143   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)"
```
```  1144   using connected_ivt_hyperplane[of s x y "axis k 1" a]
```
```  1145   by (auto simp add: inner_axis inner_commute)
```
```  1146
```
```  1147 lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x\$i = 0)}"
```
```  1148   unfolding subspace_def by auto
```
```  1149
```
```  1150 lemma closed_substandard_cart:
```
```  1151   "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x\$i = 0}"
```
```  1152 proof -
```
```  1153   { fix i::'n
```
```  1154     have "closed {x::'a ^ 'n. P i \<longrightarrow> x\$i = 0}"
```
```  1155       by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) }
```
```  1156   thus ?thesis
```
```  1157     unfolding Collect_all_eq by (simp add: closed_INT)
```
```  1158 qed
```
```  1159
```
```  1160 lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x\$i = 0} = card d"
```
```  1161   (is "dim ?A = _")
```
```  1162 proof -
```
```  1163   let ?a = "\<lambda>x. axis x 1 :: real^'n"
```
```  1164   have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
```
```  1165     by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
```
```  1166   have "?a ` d \<subseteq> Basis"
```
```  1167     by (auto simp: Basis_vec_def)
```
```  1168   thus ?thesis
```
```  1169     using dim_substandard[of "?a ` d"] card_image[of ?a d]
```
```  1170     by (auto simp: axis_eq_axis inj_on_def *)
```
```  1171 qed
```
```  1172
```
```  1173 lemma affinity_inverses:
```
```  1174   assumes m0: "m \<noteq> (0::'a::field)"
```
```  1175   shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
```
```  1176   "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
```
```  1177   using m0
```
```  1178   apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
```
```  1179   apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
```
```  1180   done
```
```  1181
```
```  1182 lemma vector_affinity_eq:
```
```  1183   assumes m0: "(m::'a::field) \<noteq> 0"
```
```  1184   shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
```
```  1185 proof
```
```  1186   assume h: "m *s x + c = y"
```
```  1187   hence "m *s x = y - c" by (simp add: field_simps)
```
```  1188   hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
```
```  1189   then show "x = inverse m *s y + - (inverse m *s c)"
```
```  1190     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
```
```  1191 next
```
```  1192   assume h: "x = inverse m *s y + - (inverse m *s c)"
```
```  1193   show "m *s x + c = y" unfolding h
```
```  1194     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
```
```  1195 qed
```
```  1196
```
```  1197 lemma vector_eq_affinity:
```
```  1198     "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
```
```  1199   using vector_affinity_eq[where m=m and x=x and y=y and c=c]
```
```  1200   by metis
```
```  1201
```
```  1202 lemma vector_cart:
```
```  1203   fixes f :: "real^'n \<Rightarrow> real"
```
```  1204   shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
```
```  1205   unfolding euclidean_eq_iff[where 'a="real^'n"]
```
```  1206   by simp (simp add: Basis_vec_def inner_axis)
```
```  1207
```
```  1208 lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
```
```  1209   by (rule vector_cart)
```
```  1210
```
```  1211 subsection "Convex Euclidean Space"
```
```  1212
```
```  1213 lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
```
```  1214   using const_vector_cart[of 1] by (simp add: one_vec_def)
```
```  1215
```
```  1216 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
```
```  1217 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
```
```  1218
```
```  1219 lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
```
```  1220
```
```  1221 lemma convex_box_cart:
```
```  1222   assumes "\<And>i. convex {x. P i x}"
```
```  1223   shows "convex {x. \<forall>i. P i (x\$i)}"
```
```  1224   using assms unfolding convex_def by auto
```
```  1225
```
```  1226 lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x\$i)}"
```
```  1227   by (rule convex_box_cart) (simp add: atLeast_def[symmetric])
```
```  1228
```
```  1229 lemma unit_interval_convex_hull_cart:
```
```  1230   "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x\$i = 0) \<or> (x\$i = 1)}"
```
```  1231   unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
```
```  1232   by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
```
```  1233
```
```  1234 lemma cube_convex_hull_cart:
```
```  1235   assumes "0 < d"
```
```  1236   obtains s::"(real^'n) set"
```
```  1237     where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
```
```  1238 proof -
```
```  1239   from assms obtain s where "finite s"
```
```  1240     and "cbox (x - setsum (op *\<^sub>R d) Basis) (x + setsum (op *\<^sub>R d) Basis) = convex hull s"
```
```  1241     by (rule cube_convex_hull)
```
```  1242   with that[of s] show thesis
```
```  1243     by (simp add: const_vector_cart)
```
```  1244 qed
```
```  1245
```
```  1246
```
```  1247 subsection "Derivative"
```
```  1248
```
```  1249 definition "jacobian f net = matrix(frechet_derivative f net)"
```
```  1250
```
```  1251 lemma jacobian_works:
```
```  1252   "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
```
```  1253     (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
```
```  1254   apply rule
```
```  1255   unfolding jacobian_def
```
```  1256   apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
```
```  1257   apply (rule differentiableI)
```
```  1258   apply assumption
```
```  1259   unfolding frechet_derivative_works
```
```  1260   apply assumption
```
```  1261   done
```
```  1262
```
```  1263
```
```  1264 subsection \<open>Component of the differential must be zero if it exists at a local
```
```  1265   maximum or minimum for that corresponding component.\<close>
```
```  1266
```
```  1267 lemma differential_zero_maxmin_cart:
```
```  1268   fixes f::"real^'a \<Rightarrow> real^'b"
```
```  1269   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))"
```
```  1270     "f differentiable (at x)"
```
```  1271   shows "jacobian f (at x) \$ k = 0"
```
```  1272   using differential_zero_maxmin_component[of "axis k 1" e x f] assms
```
```  1273     vector_cart[of "\<lambda>j. frechet_derivative f (at x) j \$ k"]
```
```  1274   by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
```
```  1275
```
```  1276 subsection \<open>Lemmas for working on @{typ "real^1"}\<close>
```
```  1277
```
```  1278 lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
```
```  1279   by (metis (full_types) num1_eq_iff)
```
```  1280
```
```  1281 lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
```
```  1282   by auto (metis (full_types) num1_eq_iff)
```
```  1283
```
```  1284 lemma exhaust_2:
```
```  1285   fixes x :: 2
```
```  1286   shows "x = 1 \<or> x = 2"
```
```  1287 proof (induct x)
```
```  1288   case (of_int z)
```
```  1289   then have "0 <= z" and "z < 2" by simp_all
```
```  1290   then have "z = 0 | z = 1" by arith
```
```  1291   then show ?case by auto
```
```  1292 qed
```
```  1293
```
```  1294 lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
```
```  1295   by (metis exhaust_2)
```
```  1296
```
```  1297 lemma exhaust_3:
```
```  1298   fixes x :: 3
```
```  1299   shows "x = 1 \<or> x = 2 \<or> x = 3"
```
```  1300 proof (induct x)
```
```  1301   case (of_int z)
```
```  1302   then have "0 <= z" and "z < 3" by simp_all
```
```  1303   then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
```
```  1304   then show ?case by auto
```
```  1305 qed
```
```  1306
```
```  1307 lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
```
```  1308   by (metis exhaust_3)
```
```  1309
```
```  1310 lemma UNIV_1 [simp]: "UNIV = {1::1}"
```
```  1311   by (auto simp add: num1_eq_iff)
```
```  1312
```
```  1313 lemma UNIV_2: "UNIV = {1::2, 2::2}"
```
```  1314   using exhaust_2 by auto
```
```  1315
```
```  1316 lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
```
```  1317   using exhaust_3 by auto
```
```  1318
```
```  1319 lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
```
```  1320   unfolding UNIV_1 by simp
```
```  1321
```
```  1322 lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
```
```  1323   unfolding UNIV_2 by simp
```
```  1324
```
```  1325 lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
```
```  1326   unfolding UNIV_3 by (simp add: ac_simps)
```
```  1327
```
```  1328 instantiation num1 :: cart_one
```
```  1329 begin
```
```  1330
```
```  1331 instance
```
```  1332 proof
```
```  1333   show "CARD(1) = Suc 0" by auto
```
```  1334 qed
```
```  1335
```
```  1336 end
```
```  1337
```
```  1338 subsection\<open>The collapse of the general concepts to dimension one.\<close>
```
```  1339
```
```  1340 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x\$1))"
```
```  1341   by (simp add: vec_eq_iff)
```
```  1342
```
```  1343 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
```
```  1344   apply auto
```
```  1345   apply (erule_tac x= "x\$1" in allE)
```
```  1346   apply (simp only: vector_one[symmetric])
```
```  1347   done
```
```  1348
```
```  1349 lemma norm_vector_1: "norm (x :: _^1) = norm (x\$1)"
```
```  1350   by (simp add: norm_vec_def)
```
```  1351
```
```  1352 lemma norm_real: "norm(x::real ^ 1) = \<bar>x\$1\<bar>"
```
```  1353   by (simp add: norm_vector_1)
```
```  1354
```
```  1355 lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x\$1) - (y\$1)\<bar>"
```
```  1356   by (auto simp add: norm_real dist_norm)
```
```  1357
```
```  1358
```
```  1359 subsection\<open>Explicit vector construction from lists.\<close>
```
```  1360
```
```  1361 definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
```
```  1362
```
```  1363 lemma vector_1: "(vector[x]) \$1 = x"
```
```  1364   unfolding vector_def by simp
```
```  1365
```
```  1366 lemma vector_2:
```
```  1367  "(vector[x,y]) \$1 = x"
```
```  1368  "(vector[x,y] :: 'a^2)\$2 = (y::'a::zero)"
```
```  1369   unfolding vector_def by simp_all
```
```  1370
```
```  1371 lemma vector_3:
```
```  1372  "(vector [x,y,z] ::('a::zero)^3)\$1 = x"
```
```  1373  "(vector [x,y,z] ::('a::zero)^3)\$2 = y"
```
```  1374  "(vector [x,y,z] ::('a::zero)^3)\$3 = z"
```
```  1375   unfolding vector_def by simp_all
```
```  1376
```
```  1377 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
```
```  1378   apply auto
```
```  1379   apply (erule_tac x="v\$1" in allE)
```
```  1380   apply (subgoal_tac "vector [v\$1] = v")
```
```  1381   apply simp
```
```  1382   apply (vector vector_def)
```
```  1383   apply simp
```
```  1384   done
```
```  1385
```
```  1386 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
```
```  1387   apply auto
```
```  1388   apply (erule_tac x="v\$1" in allE)
```
```  1389   apply (erule_tac x="v\$2" in allE)
```
```  1390   apply (subgoal_tac "vector [v\$1, v\$2] = v")
```
```  1391   apply simp
```
```  1392   apply (vector vector_def)
```
```  1393   apply (simp add: forall_2)
```
```  1394   done
```
```  1395
```
```  1396 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
```
```  1397   apply auto
```
```  1398   apply (erule_tac x="v\$1" in allE)
```
```  1399   apply (erule_tac x="v\$2" in allE)
```
```  1400   apply (erule_tac x="v\$3" in allE)
```
```  1401   apply (subgoal_tac "vector [v\$1, v\$2, v\$3] = v")
```
```  1402   apply simp
```
```  1403   apply (vector vector_def)
```
```  1404   apply (simp add: forall_3)
```
```  1405   done
```
```  1406
```
```  1407 lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x \$ k)"
```
```  1408   apply (rule bounded_linearI[where K=1])
```
```  1409   using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
```
```  1410
```
```  1411 lemma interval_split_cart:
```
```  1412   "{a..b::real^'n} \<inter> {x. x\$k \<le> c} = {a .. (\<chi> i. if i = k then min (b\$k) c else b\$i)}"
```
```  1413   "cbox a b \<inter> {x. x\$k \<ge> c} = {(\<chi> i. if i = k then max (a\$k) c else a\$i) .. b}"
```
```  1414   apply (rule_tac[!] set_eqI)
```
```  1415   unfolding Int_iff mem_interval_cart mem_Collect_eq interval_cbox_cart
```
```  1416   unfolding vec_lambda_beta
```
```  1417   by auto
```
```  1418
```
```  1419 end
```