src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
 author immler Tue Mar 18 10:12:57 2014 +0100 (2014-03-18) changeset 56188 0268784f60da parent 55522 23d2cbac6dce child 56189 c4daa97ac57a permissions -rw-r--r--
use cbox to relax class constraints
```     1 header {*Instanciates the finite cartesian product of euclidean spaces as a euclidean space.*}
```
```     2
```
```     3 theory Cartesian_Euclidean_Space
```
```     4 imports Finite_Cartesian_Product Integration
```
```     5 begin
```
```     6
```
```     7 lemma delta_mult_idempotent:
```
```     8   "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)"
```
```     9   by (cases "k=a") auto
```
```    10
```
```    11 lemma setsum_Plus:
```
```    12   "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
```
```    13     (\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))"
```
```    14   unfolding Plus_def
```
```    15   by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)
```
```    16
```
```    17 lemma setsum_UNIV_sum:
```
```    18   fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
```
```    19   shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
```
```    20   apply (subst UNIV_Plus_UNIV [symmetric])
```
```    21   apply (rule setsum_Plus [OF finite finite])
```
```    22   done
```
```    23
```
```    24 lemma setsum_mult_product:
```
```    25   "setsum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
```
```    26   unfolding sumr_group[of h B A, unfolded atLeast0LessThan, symmetric]
```
```    27 proof (rule setsum_cong, simp, rule setsum_reindex_cong)
```
```    28   fix i
```
```    29   show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
```
```    30   show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
```
```    31   proof safe
```
```    32     fix j assume "j \<in> {i * B..<i * B + B}"
```
```    33     then show "j \<in> (\<lambda>j. j + i * B) ` {..<B}"
```
```    34       by (auto intro!: image_eqI[of _ _ "j - i * B"])
```
```    35   qed simp
```
```    36 qed simp
```
```    37
```
```    38
```
```    39 subsection{* Basic componentwise operations on vectors. *}
```
```    40
```
```    41 instantiation vec :: (times, finite) times
```
```    42 begin
```
```    43
```
```    44 definition "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x\$i) * (y\$i)))"
```
```    45 instance ..
```
```    46
```
```    47 end
```
```    48
```
```    49 instantiation vec :: (one, finite) one
```
```    50 begin
```
```    51
```
```    52 definition "1 \<equiv> (\<chi> i. 1)"
```
```    53 instance ..
```
```    54
```
```    55 end
```
```    56
```
```    57 instantiation vec :: (ord, finite) ord
```
```    58 begin
```
```    59
```
```    60 definition "x \<le> y \<longleftrightarrow> (\<forall>i. x\$i \<le> y\$i)"
```
```    61 definition "x < (y::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
```
```    62 instance ..
```
```    63
```
```    64 end
```
```    65
```
```    66 text{* The ordering on one-dimensional vectors is linear. *}
```
```    67
```
```    68 class cart_one =
```
```    69   assumes UNIV_one: "card (UNIV \<Colon> 'a set) = Suc 0"
```
```    70 begin
```
```    71
```
```    72 subclass finite
```
```    73 proof
```
```    74   from UNIV_one show "finite (UNIV :: 'a set)"
```
```    75     by (auto intro!: card_ge_0_finite)
```
```    76 qed
```
```    77
```
```    78 end
```
```    79
```
```    80 instance vec:: (order, finite) order
```
```    81   by default (auto simp: less_eq_vec_def less_vec_def vec_eq_iff
```
```    82       intro: order.trans order.antisym order.strict_implies_order)
```
```    83
```
```    84 instance vec :: (linorder, cart_one) linorder
```
```    85 proof
```
```    86   obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"
```
```    87   proof -
```
```    88     have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)
```
```    89     then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)
```
```    90     then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto
```
```    91     then show thesis by (auto intro: that)
```
```    92   qed
```
```    93   fix x y :: "'a^'b::cart_one"
```
```    94   note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
```
```    95   show "x \<le> y \<or> y \<le> x" by auto
```
```    96 qed
```
```    97
```
```    98 text{* Constant Vectors *}
```
```    99
```
```   100 definition "vec x = (\<chi> i. x)"
```
```   101
```
```   102 lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b"
```
```   103   by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis)
```
```   104
```
```   105 text{* Also the scalar-vector multiplication. *}
```
```   106
```
```   107 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
```
```   108   where "c *s x = (\<chi> i. c * (x\$i))"
```
```   109
```
```   110
```
```   111 subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
```
```   112
```
```   113 method_setup vector = {*
```
```   114 let
```
```   115   val ss1 =
```
```   116     simpset_of (put_simpset HOL_basic_ss @{context}
```
```   117       addsimps [@{thm setsum_addf} RS sym,
```
```   118       @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
```
```   119       @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym])
```
```   120   val ss2 =
```
```   121     simpset_of (@{context} addsimps
```
```   122              [@{thm plus_vec_def}, @{thm times_vec_def},
```
```   123               @{thm minus_vec_def}, @{thm uminus_vec_def},
```
```   124               @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
```
```   125               @{thm scaleR_vec_def},
```
```   126               @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}])
```
```   127   fun vector_arith_tac ctxt ths =
```
```   128     simp_tac (put_simpset ss1 ctxt)
```
```   129     THEN' (fn i => rtac @{thm setsum_cong2} i
```
```   130          ORELSE rtac @{thm setsum_0'} i
```
```   131          ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i)
```
```   132     (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
```
```   133     THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths)
```
```   134 in
```
```   135   Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths))
```
```   136 end
```
```   137 *} "lift trivial vector statements to real arith statements"
```
```   138
```
```   139 lemma vec_0[simp]: "vec 0 = 0" by (vector zero_vec_def)
```
```   140 lemma vec_1[simp]: "vec 1 = 1" by (vector one_vec_def)
```
```   141
```
```   142 lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
```
```   143
```
```   144 lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
```
```   145
```
```   146 lemma vec_add: "vec(x + y) = vec x + vec y"  by (vector vec_def)
```
```   147 lemma vec_sub: "vec(x - y) = vec x - vec y" by (vector vec_def)
```
```   148 lemma vec_cmul: "vec(c * x) = c *s vec x " by (vector vec_def)
```
```   149 lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def)
```
```   150
```
```   151 lemma vec_setsum:
```
```   152   assumes "finite S"
```
```   153   shows "vec(setsum f S) = setsum (vec o f) S"
```
```   154   using assms
```
```   155 proof induct
```
```   156   case empty
```
```   157   then show ?case by simp
```
```   158 next
```
```   159   case insert
```
```   160   then show ?case by (auto simp add: vec_add)
```
```   161 qed
```
```   162
```
```   163 text{* Obvious "component-pushing". *}
```
```   164
```
```   165 lemma vec_component [simp]: "vec x \$ i = x"
```
```   166   by (vector vec_def)
```
```   167
```
```   168 lemma vector_mult_component [simp]: "(x * y)\$i = x\$i * y\$i"
```
```   169   by vector
```
```   170
```
```   171 lemma vector_smult_component [simp]: "(c *s y)\$i = c * (y\$i)"
```
```   172   by vector
```
```   173
```
```   174 lemma cond_component: "(if b then x else y)\$i = (if b then x\$i else y\$i)" by vector
```
```   175
```
```   176 lemmas vector_component =
```
```   177   vec_component vector_add_component vector_mult_component
```
```   178   vector_smult_component vector_minus_component vector_uminus_component
```
```   179   vector_scaleR_component cond_component
```
```   180
```
```   181
```
```   182 subsection {* Some frequently useful arithmetic lemmas over vectors. *}
```
```   183
```
```   184 instance vec :: (semigroup_mult, finite) semigroup_mult
```
```   185   by default (vector mult_assoc)
```
```   186
```
```   187 instance vec :: (monoid_mult, finite) monoid_mult
```
```   188   by default vector+
```
```   189
```
```   190 instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
```
```   191   by default (vector mult_commute)
```
```   192
```
```   193 instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
```
```   194   by default vector
```
```   195
```
```   196 instance vec :: (semiring, finite) semiring
```
```   197   by default (vector field_simps)+
```
```   198
```
```   199 instance vec :: (semiring_0, finite) semiring_0
```
```   200   by default (vector field_simps)+
```
```   201 instance vec :: (semiring_1, finite) semiring_1
```
```   202   by default vector
```
```   203 instance vec :: (comm_semiring, finite) comm_semiring
```
```   204   by default (vector field_simps)+
```
```   205
```
```   206 instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
```
```   207 instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
```
```   208 instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
```
```   209 instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
```
```   210 instance vec :: (ring, finite) ring ..
```
```   211 instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
```
```   212 instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
```
```   213
```
```   214 instance vec :: (ring_1, finite) ring_1 ..
```
```   215
```
```   216 instance vec :: (real_algebra, finite) real_algebra
```
```   217   by default (simp_all add: vec_eq_iff)
```
```   218
```
```   219 instance vec :: (real_algebra_1, finite) real_algebra_1 ..
```
```   220
```
```   221 lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)\$i = of_nat n"
```
```   222 proof (induct n)
```
```   223   case 0
```
```   224   then show ?case by vector
```
```   225 next
```
```   226   case Suc
```
```   227   then show ?case by vector
```
```   228 qed
```
```   229
```
```   230 lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) \$ i = 1"
```
```   231   by vector
```
```   232
```
```   233 lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) \$ i = - 1"
```
```   234   by vector
```
```   235
```
```   236 instance vec :: (semiring_char_0, finite) semiring_char_0
```
```   237 proof
```
```   238   fix m n :: nat
```
```   239   show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
```
```   240     by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
```
```   241 qed
```
```   242
```
```   243 instance vec :: (numeral, finite) numeral ..
```
```   244 instance vec :: (semiring_numeral, finite) semiring_numeral ..
```
```   245
```
```   246 lemma numeral_index [simp]: "numeral w \$ i = numeral w"
```
```   247   by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
```
```   248
```
```   249 lemma neg_numeral_index [simp]: "- numeral w \$ i = - numeral w"
```
```   250   by (simp only: vector_uminus_component numeral_index)
```
```   251
```
```   252 instance vec :: (comm_ring_1, finite) comm_ring_1 ..
```
```   253 instance vec :: (ring_char_0, finite) ring_char_0 ..
```
```   254
```
```   255 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
```
```   256   by (vector mult_assoc)
```
```   257 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
```
```   258   by (vector field_simps)
```
```   259 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
```
```   260   by (vector field_simps)
```
```   261 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
```
```   262 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
```
```   263 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
```
```   264   by (vector field_simps)
```
```   265 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
```
```   266 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
```
```   267 lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
```
```   268 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
```
```   269 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
```
```   270   by (vector field_simps)
```
```   271
```
```   272 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
```
```   273   by (simp add: vec_eq_iff)
```
```   274
```
```   275 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
```
```   276 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
```
```   277   by vector
```
```   278 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
```
```   279   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
```
```   280 lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
```
```   281   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
```
```   282 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
```
```   283   by (metis vector_mul_lcancel)
```
```   284 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
```
```   285   by (metis vector_mul_rcancel)
```
```   286
```
```   287 lemma component_le_norm_cart: "\<bar>x\$i\<bar> <= norm x"
```
```   288   apply (simp add: norm_vec_def)
```
```   289   apply (rule member_le_setL2, simp_all)
```
```   290   done
```
```   291
```
```   292 lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x\$i\<bar> <= e"
```
```   293   by (metis component_le_norm_cart order_trans)
```
```   294
```
```   295 lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x\$i\<bar> < e"
```
```   296   by (metis component_le_norm_cart le_less_trans)
```
```   297
```
```   298 lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x\$i\<bar>) UNIV"
```
```   299   by (simp add: norm_vec_def setL2_le_setsum)
```
```   300
```
```   301 lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
```
```   302   unfolding scaleR_vec_def vector_scalar_mult_def by simp
```
```   303
```
```   304 lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
```
```   305   unfolding dist_norm scalar_mult_eq_scaleR
```
```   306   unfolding scaleR_right_diff_distrib[symmetric] by simp
```
```   307
```
```   308 lemma setsum_component [simp]:
```
```   309   fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
```
```   310   shows "(setsum f S)\$i = setsum (\<lambda>x. (f x)\$i) S"
```
```   311 proof (cases "finite S")
```
```   312   case True
```
```   313   then show ?thesis by induct simp_all
```
```   314 next
```
```   315   case False
```
```   316   then show ?thesis by simp
```
```   317 qed
```
```   318
```
```   319 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)\$i ) S)"
```
```   320   by (simp add: vec_eq_iff)
```
```   321
```
```   322 lemma setsum_cmul:
```
```   323   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
```
```   324   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
```
```   325   by (simp add: vec_eq_iff setsum_right_distrib)
```
```   326
```
```   327 lemma setsum_norm_allsubsets_bound_cart:
```
```   328   fixes f:: "'a \<Rightarrow> real ^'n"
```
```   329   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
```
```   330   shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
```
```   331   using setsum_norm_allsubsets_bound[OF assms]
```
```   332   by (simp add: DIM_cart Basis_real_def)
```
```   333
```
```   334 instantiation vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
```
```   335 begin
```
```   336
```
```   337 definition "inf x y = (\<chi> i. inf (x \$ i) (y \$ i))"
```
```   338 definition "sup x y = (\<chi> i. sup (x \$ i) (y \$ i))"
```
```   339 definition "Inf X = (\<chi> i. (INF x:X. x \$ i))"
```
```   340 definition "Sup X = (\<chi> i. (SUP x:X. x \$ i))"
```
```   341 definition "abs x = (\<chi> i. abs (x \$ i))"
```
```   342
```
```   343 instance
```
```   344   apply default
```
```   345   unfolding euclidean_representation_setsum'
```
```   346   apply (auto simp: less_eq_vec_def inf_vec_def sup_vec_def Inf_vec_def Sup_vec_def inner_axis
```
```   347     Basis_vec_def inner_Basis_inf_left inner_Basis_sup_left inner_Basis_INF_left
```
```   348     inner_Basis_SUP_left eucl_le[where 'a='a] less_le_not_le abs_vec_def abs_inner)
```
```   349   done
```
```   350
```
```   351 end
```
```   352
```
```   353 subsection {* Matrix operations *}
```
```   354
```
```   355 text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
```
```   356
```
```   357 definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
```
```   358     (infixl "**" 70)
```
```   359   where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m\$i)\$k) * ((m'\$k)\$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
```
```   360
```
```   361 definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
```
```   362     (infixl "*v" 70)
```
```   363   where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m\$i)\$j) * (x\$j)) (UNIV ::'n set)) :: 'a^'m"
```
```   364
```
```   365 definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
```
```   366     (infixl "v*" 70)
```
```   367   where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m\$i)\$j) * (v\$i)) (UNIV :: 'm set)) :: 'a^'n"
```
```   368
```
```   369 definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
```
```   370 definition transpose where
```
```   371   "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A\$j)\$i))"
```
```   372 definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A\$i)\$j))"
```
```   373 definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A\$i)\$j))"
```
```   374 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
```
```   375 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
```
```   376
```
```   377 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
```
```   378 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
```
```   379   by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps)
```
```   380
```
```   381 lemma matrix_mul_lid:
```
```   382   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
```
```   383   shows "mat 1 ** A = A"
```
```   384   apply (simp add: matrix_matrix_mult_def mat_def)
```
```   385   apply vector
```
```   386   apply (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite]
```
```   387     mult_1_left mult_zero_left if_True UNIV_I)
```
```   388   done
```
```   389
```
```   390
```
```   391 lemma matrix_mul_rid:
```
```   392   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
```
```   393   shows "A ** mat 1 = A"
```
```   394   apply (simp add: matrix_matrix_mult_def mat_def)
```
```   395   apply vector
```
```   396   apply (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite]
```
```   397     mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
```
```   398   done
```
```   399
```
```   400 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
```
```   401   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
```
```   402   apply (subst setsum_commute)
```
```   403   apply simp
```
```   404   done
```
```   405
```
```   406 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
```
```   407   apply (vector matrix_matrix_mult_def matrix_vector_mult_def
```
```   408     setsum_right_distrib setsum_left_distrib mult_assoc)
```
```   409   apply (subst setsum_commute)
```
```   410   apply simp
```
```   411   done
```
```   412
```
```   413 lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
```
```   414   apply (vector matrix_vector_mult_def mat_def)
```
```   415   apply (simp add: if_distrib cond_application_beta setsum_delta' cong del: if_weak_cong)
```
```   416   done
```
```   417
```
```   418 lemma matrix_transpose_mul:
```
```   419     "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
```
```   420   by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute)
```
```   421
```
```   422 lemma matrix_eq:
```
```   423   fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
```
```   424   shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   425   apply auto
```
```   426   apply (subst vec_eq_iff)
```
```   427   apply clarify
```
```   428   apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
```
```   429   apply (erule_tac x="axis ia 1" in allE)
```
```   430   apply (erule_tac x="i" in allE)
```
```   431   apply (auto simp add: if_distrib cond_application_beta axis_def
```
```   432     setsum_delta[OF finite] cong del: if_weak_cong)
```
```   433   done
```
```   434
```
```   435 lemma matrix_vector_mul_component: "((A::real^_^_) *v x)\$k = (A\$k) \<bullet> x"
```
```   436   by (simp add: matrix_vector_mult_def inner_vec_def)
```
```   437
```
```   438 lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
```
```   439   apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
```
```   440   apply (subst setsum_commute)
```
```   441   apply simp
```
```   442   done
```
```   443
```
```   444 lemma transpose_mat: "transpose (mat n) = mat n"
```
```   445   by (vector transpose_def mat_def)
```
```   446
```
```   447 lemma transpose_transpose: "transpose(transpose A) = A"
```
```   448   by (vector transpose_def)
```
```   449
```
```   450 lemma row_transpose:
```
```   451   fixes A:: "'a::semiring_1^_^_"
```
```   452   shows "row i (transpose A) = column i A"
```
```   453   by (simp add: row_def column_def transpose_def vec_eq_iff)
```
```   454
```
```   455 lemma column_transpose:
```
```   456   fixes A:: "'a::semiring_1^_^_"
```
```   457   shows "column i (transpose A) = row i A"
```
```   458   by (simp add: row_def column_def transpose_def vec_eq_iff)
```
```   459
```
```   460 lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
```
```   461   by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
```
```   462
```
```   463 lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
```
```   464   by (metis transpose_transpose rows_transpose)
```
```   465
```
```   466 text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
```
```   467
```
```   468 lemma matrix_mult_dot: "A *v x = (\<chi> i. A\$i \<bullet> x)"
```
```   469   by (simp add: matrix_vector_mult_def inner_vec_def)
```
```   470
```
```   471 lemma matrix_mult_vsum:
```
```   472   "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x\$i) *s column i A) (UNIV:: 'n set)"
```
```   473   by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
```
```   474
```
```   475 lemma vector_componentwise:
```
```   476   "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x\$i) * (axis i 1 :: 'a^'n) \$ j)"
```
```   477   by (simp add: axis_def if_distrib setsum_cases vec_eq_iff)
```
```   478
```
```   479 lemma basis_expansion: "setsum (\<lambda>i. (x\$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
```
```   480   by (auto simp add: axis_def vec_eq_iff if_distrib setsum_cases cong del: if_weak_cong)
```
```   481
```
```   482 lemma linear_componentwise:
```
```   483   fixes f:: "real ^'m \<Rightarrow> real ^ _"
```
```   484   assumes lf: "linear f"
```
```   485   shows "(f x)\$j = setsum (\<lambda>i. (x\$i) * (f (axis i 1)\$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
```
```   486 proof -
```
```   487   let ?M = "(UNIV :: 'm set)"
```
```   488   let ?N = "(UNIV :: 'n set)"
```
```   489   have fM: "finite ?M" by simp
```
```   490   have "?rhs = (setsum (\<lambda>i.(x\$i) *\<^sub>R f (axis i 1) ) ?M)\$j"
```
```   491     unfolding setsum_component by simp
```
```   492   then show ?thesis
```
```   493     unfolding linear_setsum_mul[OF lf fM, symmetric]
```
```   494     unfolding scalar_mult_eq_scaleR[symmetric]
```
```   495     unfolding basis_expansion
```
```   496     by simp
```
```   497 qed
```
```   498
```
```   499 text{* Inverse matrices  (not necessarily square) *}
```
```   500
```
```   501 definition
```
```   502   "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
```
```   503
```
```   504 definition
```
```   505   "matrix_inv(A:: 'a::semiring_1^'n^'m) =
```
```   506     (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
```
```   507
```
```   508 text{* Correspondence between matrices and linear operators. *}
```
```   509
```
```   510 definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
```
```   511   where "matrix f = (\<chi> i j. (f(axis j 1))\$i)"
```
```   512
```
```   513 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
```
```   514   by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
```
```   515       field_simps setsum_right_distrib setsum_addf)
```
```   516
```
```   517 lemma matrix_works:
```
```   518   assumes lf: "linear f"
```
```   519   shows "matrix f *v x = f (x::real ^ 'n)"
```
```   520   apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
```
```   521   apply clarify
```
```   522   apply (rule linear_componentwise[OF lf, symmetric])
```
```   523   done
```
```   524
```
```   525 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
```
```   526   by (simp add: ext matrix_works)
```
```   527
```
```   528 lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
```
```   529   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
```
```   530
```
```   531 lemma matrix_compose:
```
```   532   assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
```
```   533     and lg: "linear (g::real^'m \<Rightarrow> real^_)"
```
```   534   shows "matrix (g o f) = matrix g ** matrix f"
```
```   535   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
```
```   536   by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
```
```   537
```
```   538 lemma matrix_vector_column:
```
```   539   "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x\$i) *s ((transpose A)\$i)) (UNIV:: 'n set)"
```
```   540   by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult_commute)
```
```   541
```
```   542 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
```
```   543   apply (rule adjoint_unique)
```
```   544   apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
```
```   545     setsum_left_distrib setsum_right_distrib)
```
```   546   apply (subst setsum_commute)
```
```   547   apply (auto simp add: mult_ac)
```
```   548   done
```
```   549
```
```   550 lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
```
```   551   shows "matrix(adjoint f) = transpose(matrix f)"
```
```   552   apply (subst matrix_vector_mul[OF lf])
```
```   553   unfolding adjoint_matrix matrix_of_matrix_vector_mul
```
```   554   apply rule
```
```   555   done
```
```   556
```
```   557
```
```   558 subsection {* lambda skolemization on cartesian products *}
```
```   559
```
```   560 (* FIXME: rename do choice_cart *)
```
```   561
```
```   562 lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
```
```   563    (\<exists>x::'a ^ 'n. \<forall>i. P i (x \$ i))" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   564 proof -
```
```   565   let ?S = "(UNIV :: 'n set)"
```
```   566   { assume H: "?rhs"
```
```   567     then have ?lhs by auto }
```
```   568   moreover
```
```   569   { assume H: "?lhs"
```
```   570     then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
```
```   571     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
```
```   572     { fix i
```
```   573       from f have "P i (f i)" by metis
```
```   574       then have "P i (?x \$ i)" by auto
```
```   575     }
```
```   576     hence "\<forall>i. P i (?x\$i)" by metis
```
```   577     hence ?rhs by metis }
```
```   578   ultimately show ?thesis by metis
```
```   579 qed
```
```   580
```
```   581 lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
```
```   582   unfolding inner_simps scalar_mult_eq_scaleR by auto
```
```   583
```
```   584 lemma left_invertible_transpose:
```
```   585   "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
```
```   586   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
```
```   587
```
```   588 lemma right_invertible_transpose:
```
```   589   "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
```
```   590   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
```
```   591
```
```   592 lemma matrix_left_invertible_injective:
```
```   593   "(\<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)"
```
```   594 proof -
```
```   595   { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
```
```   596     from xy have "B*v (A *v x) = B *v (A*v y)" by simp
```
```   597     hence "x = y"
```
```   598       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
```
```   599   moreover
```
```   600   { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
```
```   601     hence i: "inj (op *v A)" unfolding inj_on_def by auto
```
```   602     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
```
```   603     obtain g where g: "linear g" "g o op *v A = id" by blast
```
```   604     have "matrix g ** A = mat 1"
```
```   605       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
```
```   606       using g(2) by (simp add: fun_eq_iff)
```
```   607     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
```
```   608   ultimately show ?thesis by blast
```
```   609 qed
```
```   610
```
```   611 lemma matrix_left_invertible_ker:
```
```   612   "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
```
```   613   unfolding matrix_left_invertible_injective
```
```   614   using linear_injective_0[OF matrix_vector_mul_linear, of A]
```
```   615   by (simp add: inj_on_def)
```
```   616
```
```   617 lemma matrix_right_invertible_surjective:
```
```   618   "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
```
```   619 proof -
```
```   620   { fix B :: "real ^'m^'n"
```
```   621     assume AB: "A ** B = mat 1"
```
```   622     { fix x :: "real ^ 'm"
```
```   623       have "A *v (B *v x) = x"
```
```   624         by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
```
```   625     hence "surj (op *v A)" unfolding surj_def by metis }
```
```   626   moreover
```
```   627   { assume sf: "surj (op *v A)"
```
```   628     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
```
```   629     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
```
```   630       by blast
```
```   631
```
```   632     have "A ** (matrix g) = mat 1"
```
```   633       unfolding matrix_eq  matrix_vector_mul_lid
```
```   634         matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
```
```   635       using g(2) unfolding o_def fun_eq_iff id_def
```
```   636       .
```
```   637     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
```
```   638   }
```
```   639   ultimately show ?thesis unfolding surj_def by blast
```
```   640 qed
```
```   641
```
```   642 lemma matrix_left_invertible_independent_columns:
```
```   643   fixes A :: "real^'n^'m"
```
```   644   shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
```
```   645       (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
```
```   646     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   647 proof -
```
```   648   let ?U = "UNIV :: 'n set"
```
```   649   { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
```
```   650     { fix c i
```
```   651       assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
```
```   652       let ?x = "\<chi> i. c i"
```
```   653       have th0:"A *v ?x = 0"
```
```   654         using c
```
```   655         unfolding matrix_mult_vsum vec_eq_iff
```
```   656         by auto
```
```   657       from k[rule_format, OF th0] i
```
```   658       have "c i = 0" by (vector vec_eq_iff)}
```
```   659     hence ?rhs by blast }
```
```   660   moreover
```
```   661   { assume H: ?rhs
```
```   662     { fix x assume x: "A *v x = 0"
```
```   663       let ?c = "\<lambda>i. ((x\$i ):: real)"
```
```   664       from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
```
```   665       have "x = 0" by vector }
```
```   666   }
```
```   667   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
```
```   668 qed
```
```   669
```
```   670 lemma matrix_right_invertible_independent_rows:
```
```   671   fixes A :: "real^'n^'m"
```
```   672   shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
```
```   673     (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
```
```   674   unfolding left_invertible_transpose[symmetric]
```
```   675     matrix_left_invertible_independent_columns
```
```   676   by (simp add: column_transpose)
```
```   677
```
```   678 lemma matrix_right_invertible_span_columns:
```
```   679   "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
```
```   680     span (columns A) = UNIV" (is "?lhs = ?rhs")
```
```   681 proof -
```
```   682   let ?U = "UNIV :: 'm set"
```
```   683   have fU: "finite ?U" by simp
```
```   684   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x\$i) *s column i A) ?U = y)"
```
```   685     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
```
```   686     apply (subst eq_commute)
```
```   687     apply rule
```
```   688     done
```
```   689   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
```
```   690   { assume h: ?lhs
```
```   691     { fix x:: "real ^'n"
```
```   692       from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
```
```   693         where y: "setsum (\<lambda>i. (y\$i) *s column i A) ?U = x" by blast
```
```   694       have "x \<in> span (columns A)"
```
```   695         unfolding y[symmetric]
```
```   696         apply (rule span_setsum[OF fU])
```
```   697         apply clarify
```
```   698         unfolding scalar_mult_eq_scaleR
```
```   699         apply (rule span_mul)
```
```   700         apply (rule span_superset)
```
```   701         unfolding columns_def
```
```   702         apply blast
```
```   703         done
```
```   704     }
```
```   705     then have ?rhs unfolding rhseq by blast }
```
```   706   moreover
```
```   707   { assume h:?rhs
```
```   708     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x\$i) *s column i A) ?U = y"
```
```   709     { fix y
```
```   710       have "?P y"
```
```   711       proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
```
```   712         show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x\$i) *s column i A) ?U = 0"
```
```   713           by (rule exI[where x=0], simp)
```
```   714       next
```
```   715         fix c y1 y2
```
```   716         assume y1: "y1 \<in> columns A" and y2: "?P y2"
```
```   717         from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
```
```   718           unfolding columns_def by blast
```
```   719         from y2 obtain x:: "real ^'m" where
```
```   720           x: "setsum (\<lambda>i. (x\$i) *s column i A) ?U = y2" by blast
```
```   721         let ?x = "(\<chi> j. if j = i then c + (x\$i) else (x\$j))::real^'m"
```
```   722         show "?P (c*s y1 + y2)"
```
```   723         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)
```
```   724           fix j
```
```   725           have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```   726               else (x\$xa) * ((column xa A\$j))) = (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))"
```
```   727             using i(1) by (simp add: field_simps)
```
```   728           have "setsum (\<lambda>xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```   729               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"
```
```   730             apply (rule setsum_cong[OF refl])
```
```   731             using th apply blast
```
```   732             done
```
```   733           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"
```
```   734             by (simp add: setsum_addf)
```
```   735           also have "\<dots> = c * ((column i A)\$j) + setsum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U"
```
```   736             unfolding setsum_delta[OF fU]
```
```   737             using i(1) by simp
```
```   738           finally show "setsum (\<lambda>xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```   739             else (x\$xa) * ((column xa A\$j))) ?U = c * ((column i A)\$j) + setsum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U" .
```
```   740         qed
```
```   741       next
```
```   742         show "y \<in> span (columns A)"
```
```   743           unfolding h by blast
```
```   744       qed
```
```   745     }
```
```   746     then have ?lhs unfolding lhseq ..
```
```   747   }
```
```   748   ultimately show ?thesis by blast
```
```   749 qed
```
```   750
```
```   751 lemma matrix_left_invertible_span_rows:
```
```   752   "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
```
```   753   unfolding right_invertible_transpose[symmetric]
```
```   754   unfolding columns_transpose[symmetric]
```
```   755   unfolding matrix_right_invertible_span_columns
```
```   756   ..
```
```   757
```
```   758 text {* The same result in terms of square matrices. *}
```
```   759
```
```   760 lemma matrix_left_right_inverse:
```
```   761   fixes A A' :: "real ^'n^'n"
```
```   762   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
```
```   763 proof -
```
```   764   { fix A A' :: "real ^'n^'n"
```
```   765     assume AA': "A ** A' = mat 1"
```
```   766     have sA: "surj (op *v A)"
```
```   767       unfolding surj_def
```
```   768       apply clarify
```
```   769       apply (rule_tac x="(A' *v y)" in exI)
```
```   770       apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
```
```   771       done
```
```   772     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
```
```   773     obtain f' :: "real ^'n \<Rightarrow> real ^'n"
```
```   774       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
```
```   775     have th: "matrix f' ** A = mat 1"
```
```   776       by (simp add: matrix_eq matrix_works[OF f'(1)]
```
```   777           matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
```
```   778     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
```
```   779     hence "matrix f' = A'"
```
```   780       by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
```
```   781     hence "matrix f' ** A = A' ** A" by simp
```
```   782     hence "A' ** A = mat 1" by (simp add: th)
```
```   783   }
```
```   784   then show ?thesis by blast
```
```   785 qed
```
```   786
```
```   787 text {* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *}
```
```   788
```
```   789 definition "rowvector v = (\<chi> i j. (v\$j))"
```
```   790
```
```   791 definition "columnvector v = (\<chi> i j. (v\$i))"
```
```   792
```
```   793 lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
```
```   794   by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
```
```   795
```
```   796 lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
```
```   797   by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
```
```   798
```
```   799 lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
```
```   800   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
```
```   801
```
```   802 lemma dot_matrix_product:
```
```   803   "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))\$1)\$1"
```
```   804   by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
```
```   805
```
```   806 lemma dot_matrix_vector_mul:
```
```   807   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
```
```   808   shows "(A *v x) \<bullet> (B *v y) =
```
```   809       (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))\$1)\$1"
```
```   810   unfolding dot_matrix_product transpose_columnvector[symmetric]
```
```   811     dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
```
```   812
```
```   813
```
```   814 lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x\$i) |i. i\<in>UNIV}"
```
```   815   by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
```
```   816
```
```   817 lemma component_le_infnorm_cart: "\<bar>x\$i\<bar> \<le> infnorm (x::real^'n)"
```
```   818   using Basis_le_infnorm[of "axis i 1" x]
```
```   819   by (simp add: Basis_vec_def axis_eq_axis inner_axis)
```
```   820
```
```   821 lemma continuous_component: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x \$ i)"
```
```   822   unfolding continuous_def by (rule tendsto_vec_nth)
```
```   823
```
```   824 lemma continuous_on_component: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x \$ i)"
```
```   825   unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
```
```   826
```
```   827 lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x\$i}"
```
```   828   by (simp add: Collect_all_eq closed_INT closed_Collect_le)
```
```   829
```
```   830 lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x \$ i) ` s)"
```
```   831   unfolding bounded_def
```
```   832   apply clarify
```
```   833   apply (rule_tac x="x \$ i" in exI)
```
```   834   apply (rule_tac x="e" in exI)
```
```   835   apply clarify
```
```   836   apply (rule order_trans [OF dist_vec_nth_le], simp)
```
```   837   done
```
```   838
```
```   839 lemma compact_lemma_cart:
```
```   840   fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
```
```   841   assumes f: "bounded (range f)"
```
```   842   shows "\<forall>d.
```
```   843         \<exists>l r. subseq r \<and>
```
```   844         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
```
```   845 proof
```
```   846   fix d :: "'n set"
```
```   847   have "finite d" by simp
```
```   848   thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
```
```   849       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
```
```   850   proof (induct d)
```
```   851     case empty
```
```   852     thus ?case unfolding subseq_def by auto
```
```   853   next
```
```   854     case (insert k d)
```
```   855     obtain l1::"'a^'n" and r1 where r1:"subseq r1"
```
```   856       and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially"
```
```   857       using insert(3) by auto
```
```   858     have s': "bounded ((\<lambda>x. x \$ k) ` range f)" using `bounded (range f)`
```
```   859       by (auto intro!: bounded_component_cart)
```
```   860     have f': "\<forall>n. f (r1 n) \$ k \<in> (\<lambda>x. x \$ k) ` range f" by simp
```
```   861     have "bounded (range (\<lambda>i. f (r1 i) \$ k))"
```
```   862       by (metis (lifting) bounded_subset image_subsetI f' s')
```
```   863     then obtain l2 r2 where r2: "subseq r2"
```
```   864       and lr2: "((\<lambda>i. f (r1 (r2 i)) \$ k) ---> l2) sequentially"
```
```   865       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \$ k"] by (auto simp: o_def)
```
```   866     def r \<equiv> "r1 \<circ> r2"
```
```   867     have r: "subseq r"
```
```   868       using r1 and r2 unfolding r_def o_def subseq_def by auto
```
```   869     moreover
```
```   870     def l \<equiv> "(\<chi> i. if i = k then l2 else l1\$i)::'a^'n"
```
```   871     { fix e :: real assume "e > 0"
```
```   872       from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially"
```
```   873         by blast
```
```   874       from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \$ k) l2 < e) sequentially"
```
```   875         by (rule tendstoD)
```
```   876       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \$ i) (l1 \$ i) < e) sequentially"
```
```   877         by (rule eventually_subseq)
```
```   878       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \$ i) (l \$ i) < e) sequentially"
```
```   879         using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
```
```   880     }
```
```   881     ultimately show ?case by auto
```
```   882   qed
```
```   883 qed
```
```   884
```
```   885 instance vec :: (heine_borel, finite) heine_borel
```
```   886 proof
```
```   887   fix f :: "nat \<Rightarrow> 'a ^ 'b"
```
```   888   assume f: "bounded (range f)"
```
```   889   then obtain l r where r: "subseq r"
```
```   890       and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) \$ i) (l \$ i) < e) sequentially"
```
```   891     using compact_lemma_cart [OF f] by blast
```
```   892   let ?d = "UNIV::'b set"
```
```   893   { fix e::real assume "e>0"
```
```   894     hence "0 < e / (real_of_nat (card ?d))"
```
```   895       using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
```
```   896     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially"
```
```   897       by simp
```
```   898     moreover
```
```   899     { fix n
```
```   900       assume n: "\<forall>i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))"
```
```   901       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) \$ i) (l \$ i))"
```
```   902         unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
```
```   903       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
```
```   904         by (rule setsum_strict_mono) (simp_all add: n)
```
```   905       finally have "dist (f (r n)) l < e" by simp
```
```   906     }
```
```   907     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
```
```   908       by (rule eventually_elim1)
```
```   909   }
```
```   910   hence "((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
```
```   911   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
```
```   912 qed
```
```   913
```
```   914 lemma interval_cart:
```
```   915   fixes a :: "real^'n"
```
```   916   shows "box a b = {x::real^'n. \<forall>i. a\$i < x\$i \<and> x\$i < b\$i}"
```
```   917     and "cbox a b = {x::real^'n. \<forall>i. a\$i \<le> x\$i \<and> x\$i \<le> b\$i}"
```
```   918   by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
```
```   919
```
```   920 lemma mem_interval_cart:
```
```   921   fixes a :: "real^'n"
```
```   922   shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a\$i < x\$i \<and> x\$i < b\$i)"
```
```   923     and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a\$i \<le> x\$i \<and> x\$i \<le> b\$i)"
```
```   924   using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
```
```   925
```
```   926 lemma interval_eq_empty_cart:
```
```   927   fixes a :: "real^'n"
```
```   928   shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b\$i \<le> a\$i))" (is ?th1)
```
```   929     and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b\$i < a\$i))" (is ?th2)
```
```   930 proof -
```
```   931   { fix i x assume as:"b\$i \<le> a\$i" and x:"x\<in>box a b"
```
```   932     hence "a \$ i < x \$ i \<and> x \$ i < b \$ i" unfolding mem_interval_cart by auto
```
```   933     hence "a\$i < b\$i" by auto
```
```   934     hence False using as by auto }
```
```   935   moreover
```
```   936   { assume as:"\<forall>i. \<not> (b\$i \<le> a\$i)"
```
```   937     let ?x = "(1/2) *\<^sub>R (a + b)"
```
```   938     { fix i
```
```   939       have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto
```
```   940       hence "a\$i < ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i < b\$i"
```
```   941         unfolding vector_smult_component and vector_add_component
```
```   942         by auto }
```
```   943     hence "box a b \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
```
```   944   ultimately show ?th1 by blast
```
```   945
```
```   946   { fix i x assume as:"b\$i < a\$i" and x:"x\<in>cbox a b"
```
```   947     hence "a \$ i \<le> x \$ i \<and> x \$ i \<le> b \$ i" unfolding mem_interval_cart by auto
```
```   948     hence "a\$i \<le> b\$i" by auto
```
```   949     hence False using as by auto }
```
```   950   moreover
```
```   951   { assume as:"\<forall>i. \<not> (b\$i < a\$i)"
```
```   952     let ?x = "(1/2) *\<^sub>R (a + b)"
```
```   953     { fix i
```
```   954       have "a\$i \<le> b\$i" using as[THEN spec[where x=i]] by auto
```
```   955       hence "a\$i \<le> ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i \<le> b\$i"
```
```   956         unfolding vector_smult_component and vector_add_component
```
```   957         by auto }
```
```   958     hence "cbox a b \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
```
```   959   ultimately show ?th2 by blast
```
```   960 qed
```
```   961
```
```   962 lemma interval_ne_empty_cart:
```
```   963   fixes a :: "real^'n"
```
```   964   shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a\$i \<le> b\$i)"
```
```   965     and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a\$i < b\$i)"
```
```   966   unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
```
```   967     (* BH: Why doesn't just "auto" work here? *)
```
```   968
```
```   969 lemma subset_interval_imp_cart:
```
```   970   fixes a :: "real^'n"
```
```   971   shows "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
```
```   972     and "(\<forall>i. a\$i < c\$i \<and> d\$i < b\$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
```
```   973     and "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
```
```   974     and "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> box c d \<subseteq> box a b"
```
```   975   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
```
```   976   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
```
```   977
```
```   978 lemma interval_sing:
```
```   979   fixes a :: "'a::linorder^'n"
```
```   980   shows "{a .. a} = {a} \<and> {a<..<a} = {}"
```
```   981   apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
```
```   982   done
```
```   983
```
```   984 lemma subset_interval_cart:
```
```   985   fixes a :: "real^'n"
```
```   986   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)
```
```   987     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)
```
```   988     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)
```
```   989     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)
```
```   990   using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
```
```   991
```
```   992 lemma disjoint_interval_cart:
```
```   993   fixes a::"real^'n"
```
```   994   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)
```
```   995     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)
```
```   996     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)
```
```   997     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)
```
```   998   using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
```
```   999
```
```  1000 lemma inter_interval_cart:
```
```  1001   fixes a :: "real^'n"
```
```  1002   shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a\$i) (c\$i)) .. (\<chi> i. min (b\$i) (d\$i))}"
```
```  1003   unfolding inter_interval
```
```  1004   by (auto simp: mem_box less_eq_vec_def)
```
```  1005     (auto simp: Basis_vec_def inner_axis)
```
```  1006
```
```  1007 lemma closed_interval_left_cart:
```
```  1008   fixes b :: "real^'n"
```
```  1009   shows "closed {x::real^'n. \<forall>i. x\$i \<le> b\$i}"
```
```  1010   by (simp add: Collect_all_eq closed_INT closed_Collect_le)
```
```  1011
```
```  1012 lemma closed_interval_right_cart:
```
```  1013   fixes a::"real^'n"
```
```  1014   shows "closed {x::real^'n. \<forall>i. a\$i \<le> x\$i}"
```
```  1015   by (simp add: Collect_all_eq closed_INT closed_Collect_le)
```
```  1016
```
```  1017 lemma is_interval_cart:
```
```  1018   "is_interval (s::(real^'n) set) \<longleftrightarrow>
```
```  1019     (\<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)"
```
```  1020   by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
```
```  1021
```
```  1022 lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x\$i \<le> a}"
```
```  1023   by (simp add: closed_Collect_le)
```
```  1024
```
```  1025 lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x\$i \<ge> a}"
```
```  1026   by (simp add: closed_Collect_le)
```
```  1027
```
```  1028 lemma open_halfspace_component_lt_cart: "open {x::real^'n. x\$i < a}"
```
```  1029   by (simp add: open_Collect_less)
```
```  1030
```
```  1031 lemma open_halfspace_component_gt_cart: "open {x::real^'n. x\$i  > a}"
```
```  1032   by (simp add: open_Collect_less)
```
```  1033
```
```  1034 lemma Lim_component_le_cart:
```
```  1035   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1036   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x \$i \<le> b) net"
```
```  1037   shows "l\$i \<le> b"
```
```  1038   by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
```
```  1039
```
```  1040 lemma Lim_component_ge_cart:
```
```  1041   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1042   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)\$i) net"
```
```  1043   shows "b \<le> l\$i"
```
```  1044   by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
```
```  1045
```
```  1046 lemma Lim_component_eq_cart:
```
```  1047   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1048   assumes net: "(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\$i = b) net"
```
```  1049   shows "l\$i = b"
```
```  1050   using ev[unfolded order_eq_iff eventually_conj_iff] and
```
```  1051     Lim_component_ge_cart[OF net, of b i] and
```
```  1052     Lim_component_le_cart[OF net, of i b] by auto
```
```  1053
```
```  1054 lemma connected_ivt_component_cart:
```
```  1055   fixes x :: "real^'n"
```
```  1056   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)"
```
```  1057   using connected_ivt_hyperplane[of s x y "axis k 1" a]
```
```  1058   by (auto simp add: inner_axis inner_commute)
```
```  1059
```
```  1060 lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x\$i = 0)}"
```
```  1061   unfolding subspace_def by auto
```
```  1062
```
```  1063 lemma closed_substandard_cart:
```
```  1064   "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x\$i = 0}"
```
```  1065 proof -
```
```  1066   { fix i::'n
```
```  1067     have "closed {x::'a ^ 'n. P i \<longrightarrow> x\$i = 0}"
```
```  1068       by (cases "P i") (simp_all add: closed_Collect_eq) }
```
```  1069   thus ?thesis
```
```  1070     unfolding Collect_all_eq by (simp add: closed_INT)
```
```  1071 qed
```
```  1072
```
```  1073 lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x\$i = 0} = card d"
```
```  1074   (is "dim ?A = _")
```
```  1075 proof -
```
```  1076   let ?a = "\<lambda>x. axis x 1 :: real^'n"
```
```  1077   have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
```
```  1078     by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
```
```  1079   have "?a ` d \<subseteq> Basis"
```
```  1080     by (auto simp: Basis_vec_def)
```
```  1081   thus ?thesis
```
```  1082     using dim_substandard[of "?a ` d"] card_image[of ?a d]
```
```  1083     by (auto simp: axis_eq_axis inj_on_def *)
```
```  1084 qed
```
```  1085
```
```  1086 lemma affinity_inverses:
```
```  1087   assumes m0: "m \<noteq> (0::'a::field)"
```
```  1088   shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
```
```  1089   "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
```
```  1090   using m0
```
```  1091   apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
```
```  1092   apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
```
```  1093   done
```
```  1094
```
```  1095 lemma vector_affinity_eq:
```
```  1096   assumes m0: "(m::'a::field) \<noteq> 0"
```
```  1097   shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
```
```  1098 proof
```
```  1099   assume h: "m *s x + c = y"
```
```  1100   hence "m *s x = y - c" by (simp add: field_simps)
```
```  1101   hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
```
```  1102   then show "x = inverse m *s y + - (inverse m *s c)"
```
```  1103     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
```
```  1104 next
```
```  1105   assume h: "x = inverse m *s y + - (inverse m *s c)"
```
```  1106   show "m *s x + c = y" unfolding h
```
```  1107     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
```
```  1108 qed
```
```  1109
```
```  1110 lemma vector_eq_affinity:
```
```  1111     "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
```
```  1112   using vector_affinity_eq[where m=m and x=x and y=y and c=c]
```
```  1113   by metis
```
```  1114
```
```  1115 lemma vector_cart:
```
```  1116   fixes f :: "real^'n \<Rightarrow> real"
```
```  1117   shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
```
```  1118   unfolding euclidean_eq_iff[where 'a="real^'n"]
```
```  1119   by simp (simp add: Basis_vec_def inner_axis)
```
```  1120
```
```  1121 lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
```
```  1122   by (rule vector_cart)
```
```  1123
```
```  1124 subsection "Convex Euclidean Space"
```
```  1125
```
```  1126 lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
```
```  1127   using const_vector_cart[of 1] by (simp add: one_vec_def)
```
```  1128
```
```  1129 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
```
```  1130 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
```
```  1131
```
```  1132 lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
```
```  1133
```
```  1134 lemma convex_box_cart:
```
```  1135   assumes "\<And>i. convex {x. P i x}"
```
```  1136   shows "convex {x. \<forall>i. P i (x\$i)}"
```
```  1137   using assms unfolding convex_def by auto
```
```  1138
```
```  1139 lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x\$i)}"
```
```  1140   by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)
```
```  1141
```
```  1142 lemma unit_interval_convex_hull_cart:
```
```  1143   "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x\$i = 0) \<or> (x\$i = 1)}"
```
```  1144   unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
```
```  1145   by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
```
```  1146
```
```  1147 lemma cube_convex_hull_cart:
```
```  1148   assumes "0 < d"
```
```  1149   obtains s::"(real^'n) set"
```
```  1150     where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
```
```  1151 proof -
```
```  1152   from assms obtain s where "finite s"
```
```  1153     and "cbox (x - setsum (op *\<^sub>R d) Basis) (x + setsum (op *\<^sub>R d) Basis) = convex hull s"
```
```  1154     by (rule cube_convex_hull)
```
```  1155   with that[of s] show thesis
```
```  1156     by (simp add: const_vector_cart)
```
```  1157 qed
```
```  1158
```
```  1159
```
```  1160 subsection "Derivative"
```
```  1161
```
```  1162 lemma differentiable_at_imp_differentiable_on:
```
```  1163   "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
```
```  1164   by (metis differentiable_at_withinI differentiable_on_def)
```
```  1165
```
```  1166 definition "jacobian f net = matrix(frechet_derivative f net)"
```
```  1167
```
```  1168 lemma jacobian_works:
```
```  1169   "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
```
```  1170     (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
```
```  1171   apply rule
```
```  1172   unfolding jacobian_def
```
```  1173   apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
```
```  1174   apply (rule differentiableI)
```
```  1175   apply assumption
```
```  1176   unfolding frechet_derivative_works
```
```  1177   apply assumption
```
```  1178   done
```
```  1179
```
```  1180
```
```  1181 subsection {* Component of the differential must be zero if it exists at a local
```
```  1182   maximum or minimum for that corresponding component. *}
```
```  1183
```
```  1184 lemma differential_zero_maxmin_cart:
```
```  1185   fixes f::"real^'a \<Rightarrow> real^'b"
```
```  1186   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))"
```
```  1187     "f differentiable (at x)"
```
```  1188   shows "jacobian f (at x) \$ k = 0"
```
```  1189   using differential_zero_maxmin_component[of "axis k 1" e x f] assms
```
```  1190     vector_cart[of "\<lambda>j. frechet_derivative f (at x) j \$ k"]
```
```  1191   by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
```
```  1192
```
```  1193 subsection {* Lemmas for working on @{typ "real^1"} *}
```
```  1194
```
```  1195 lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
```
```  1196   by (metis (full_types) num1_eq_iff)
```
```  1197
```
```  1198 lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
```
```  1199   by auto (metis (full_types) num1_eq_iff)
```
```  1200
```
```  1201 lemma exhaust_2:
```
```  1202   fixes x :: 2
```
```  1203   shows "x = 1 \<or> x = 2"
```
```  1204 proof (induct x)
```
```  1205   case (of_int z)
```
```  1206   then have "0 <= z" and "z < 2" by simp_all
```
```  1207   then have "z = 0 | z = 1" by arith
```
```  1208   then show ?case by auto
```
```  1209 qed
```
```  1210
```
```  1211 lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
```
```  1212   by (metis exhaust_2)
```
```  1213
```
```  1214 lemma exhaust_3:
```
```  1215   fixes x :: 3
```
```  1216   shows "x = 1 \<or> x = 2 \<or> x = 3"
```
```  1217 proof (induct x)
```
```  1218   case (of_int z)
```
```  1219   then have "0 <= z" and "z < 3" by simp_all
```
```  1220   then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
```
```  1221   then show ?case by auto
```
```  1222 qed
```
```  1223
```
```  1224 lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
```
```  1225   by (metis exhaust_3)
```
```  1226
```
```  1227 lemma UNIV_1 [simp]: "UNIV = {1::1}"
```
```  1228   by (auto simp add: num1_eq_iff)
```
```  1229
```
```  1230 lemma UNIV_2: "UNIV = {1::2, 2::2}"
```
```  1231   using exhaust_2 by auto
```
```  1232
```
```  1233 lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
```
```  1234   using exhaust_3 by auto
```
```  1235
```
```  1236 lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
```
```  1237   unfolding UNIV_1 by simp
```
```  1238
```
```  1239 lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
```
```  1240   unfolding UNIV_2 by simp
```
```  1241
```
```  1242 lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
```
```  1243   unfolding UNIV_3 by (simp add: add_ac)
```
```  1244
```
```  1245 instantiation num1 :: cart_one
```
```  1246 begin
```
```  1247
```
```  1248 instance
```
```  1249 proof
```
```  1250   show "CARD(1) = Suc 0" by auto
```
```  1251 qed
```
```  1252
```
```  1253 end
```
```  1254
```
```  1255 subsection{* The collapse of the general concepts to dimension one. *}
```
```  1256
```
```  1257 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x\$1))"
```
```  1258   by (simp add: vec_eq_iff)
```
```  1259
```
```  1260 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
```
```  1261   apply auto
```
```  1262   apply (erule_tac x= "x\$1" in allE)
```
```  1263   apply (simp only: vector_one[symmetric])
```
```  1264   done
```
```  1265
```
```  1266 lemma norm_vector_1: "norm (x :: _^1) = norm (x\$1)"
```
```  1267   by (simp add: norm_vec_def)
```
```  1268
```
```  1269 lemma norm_real: "norm(x::real ^ 1) = abs(x\$1)"
```
```  1270   by (simp add: norm_vector_1)
```
```  1271
```
```  1272 lemma dist_real: "dist(x::real ^ 1) y = abs((x\$1) - (y\$1))"
```
```  1273   by (auto simp add: norm_real dist_norm)
```
```  1274
```
```  1275
```
```  1276 subsection{* Explicit vector construction from lists. *}
```
```  1277
```
```  1278 definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
```
```  1279
```
```  1280 lemma vector_1: "(vector[x]) \$1 = x"
```
```  1281   unfolding vector_def by simp
```
```  1282
```
```  1283 lemma vector_2:
```
```  1284  "(vector[x,y]) \$1 = x"
```
```  1285  "(vector[x,y] :: 'a^2)\$2 = (y::'a::zero)"
```
```  1286   unfolding vector_def by simp_all
```
```  1287
```
```  1288 lemma vector_3:
```
```  1289  "(vector [x,y,z] ::('a::zero)^3)\$1 = x"
```
```  1290  "(vector [x,y,z] ::('a::zero)^3)\$2 = y"
```
```  1291  "(vector [x,y,z] ::('a::zero)^3)\$3 = z"
```
```  1292   unfolding vector_def by simp_all
```
```  1293
```
```  1294 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
```
```  1295   apply auto
```
```  1296   apply (erule_tac x="v\$1" in allE)
```
```  1297   apply (subgoal_tac "vector [v\$1] = v")
```
```  1298   apply simp
```
```  1299   apply (vector vector_def)
```
```  1300   apply simp
```
```  1301   done
```
```  1302
```
```  1303 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
```
```  1304   apply auto
```
```  1305   apply (erule_tac x="v\$1" in allE)
```
```  1306   apply (erule_tac x="v\$2" in allE)
```
```  1307   apply (subgoal_tac "vector [v\$1, v\$2] = v")
```
```  1308   apply simp
```
```  1309   apply (vector vector_def)
```
```  1310   apply (simp add: forall_2)
```
```  1311   done
```
```  1312
```
```  1313 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
```
```  1314   apply auto
```
```  1315   apply (erule_tac x="v\$1" in allE)
```
```  1316   apply (erule_tac x="v\$2" in allE)
```
```  1317   apply (erule_tac x="v\$3" in allE)
```
```  1318   apply (subgoal_tac "vector [v\$1, v\$2, v\$3] = v")
```
```  1319   apply simp
```
```  1320   apply (vector vector_def)
```
```  1321   apply (simp add: forall_3)
```
```  1322   done
```
```  1323
```
```  1324 lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x \$ k)"
```
```  1325   apply (rule bounded_linearI[where K=1])
```
```  1326   using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
```
```  1327
```
```  1328 lemma integral_component_eq_cart[simp]:
```
```  1329   fixes f :: "'n::euclidean_space \<Rightarrow> real^'m"
```
```  1330   assumes "f integrable_on s"
```
```  1331   shows "integral s (\<lambda>x. f x \$ k) = integral s f \$ k"
```
```  1332   using integral_linear[OF assms(1) bounded_linear_component_cart,unfolded o_def] .
```
```  1333
```
```  1334 lemma interval_split_cart:
```
```  1335   "{a..b::real^'n} \<inter> {x. x\$k \<le> c} = {a .. (\<chi> i. if i = k then min (b\$k) c else b\$i)}"
```
```  1336   "cbox a b \<inter> {x. x\$k \<ge> c} = {(\<chi> i. if i = k then max (a\$k) c else a\$i) .. b}"
```
```  1337   apply (rule_tac[!] set_eqI)
```
```  1338   unfolding Int_iff mem_interval_cart mem_Collect_eq interval_cbox_cart
```
```  1339   unfolding vec_lambda_beta
```
```  1340   by auto
```
```  1341
```
```  1342 lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "\<forall>i. a\$i < b\$i \<and> u\$i < v\$i"
```
```  1343   shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
```
```  1344   using assms by (intro interval_bij_bij) (auto simp: Basis_vec_def inner_axis)
```
```  1345
```
```  1346 end
```