src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
 author hoelzl Thu Jan 31 11:31:22 2013 +0100 (2013-01-31) changeset 50998 501200635659 parent 50526 899c9c4e4a4c child 51489 f738e6dbd844 permissions -rw-r--r--
simplify heine_borel type class
```     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 \<longleftrightarrow> (\<forall>i. x\$i < y\$i)"
```
```    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 instantiation vec :: (linorder, cart_one) linorder
```
```    81 begin
```
```    82
```
```    83 instance
```
```    84 proof
```
```    85   obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"
```
```    86   proof -
```
```    87     have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)
```
```    88     then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)
```
```    89     then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto
```
```    90     then show thesis by (auto intro: that)
```
```    91   qed
```
```    92
```
```    93   note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
```
```    94   fix x y z :: "'a^'b::cart_one"
```
```    95   show "x \<le> x" "(x < y) = (x \<le> y \<and> \<not> y \<le> x)" "x \<le> y \<or> y \<le> x" by auto
```
```    96   { assume "x\<le>y" "y\<le>z" then show "x\<le>z" by auto }
```
```    97   { assume "x\<le>y" "y\<le>x" then show "x=y" by auto }
```
```    98 qed
```
```    99
```
```   100 end
```
```   101
```
```   102 text{* Constant Vectors *}
```
```   103
```
```   104 definition "vec x = (\<chi> i. x)"
```
```   105
```
```   106 text{* Also the scalar-vector multiplication. *}
```
```   107
```
```   108 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
```
```   109   where "c *s x = (\<chi> i. c * (x\$i))"
```
```   110
```
```   111
```
```   112 subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
```
```   113
```
```   114 method_setup vector = {*
```
```   115 let
```
```   116   val ss1 = HOL_basic_ss addsimps [@{thm setsum_addf} RS sym,
```
```   117     @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
```
```   118     @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
```
```   119   val ss2 = @{simpset} addsimps
```
```   120              [@{thm plus_vec_def}, @{thm times_vec_def},
```
```   121               @{thm minus_vec_def}, @{thm uminus_vec_def},
```
```   122               @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
```
```   123               @{thm scaleR_vec_def},
```
```   124               @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}]
```
```   125   fun vector_arith_tac ths =
```
```   126     simp_tac ss1
```
```   127     THEN' (fn i => rtac @{thm setsum_cong2} i
```
```   128          ORELSE rtac @{thm setsum_0'} i
```
```   129          ORELSE simp_tac (HOL_basic_ss addsimps [@{thm vec_eq_iff}]) i)
```
```   130     (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
```
```   131     THEN' asm_full_simp_tac (ss2 addsimps ths)
```
```   132 in
```
```   133   Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths)))
```
```   134 end
```
```   135 *} "lift trivial vector statements to real arith statements"
```
```   136
```
```   137 lemma vec_0[simp]: "vec 0 = 0" by (vector zero_vec_def)
```
```   138 lemma vec_1[simp]: "vec 1 = 1" by (vector one_vec_def)
```
```   139
```
```   140 lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
```
```   141
```
```   142 lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
```
```   143
```
```   144 lemma vec_add: "vec(x + y) = vec x + vec y"  by (vector vec_def)
```
```   145 lemma vec_sub: "vec(x - y) = vec x - vec y" by (vector vec_def)
```
```   146 lemma vec_cmul: "vec(c * x) = c *s vec x " by (vector vec_def)
```
```   147 lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def)
```
```   148
```
```   149 lemma vec_setsum:
```
```   150   assumes "finite S"
```
```   151   shows "vec(setsum f S) = setsum (vec o f) S"
```
```   152   using assms
```
```   153 proof induct
```
```   154   case empty
```
```   155   then show ?case by simp
```
```   156 next
```
```   157   case insert
```
```   158   then show ?case by (auto simp add: vec_add)
```
```   159 qed
```
```   160
```
```   161 text{* Obvious "component-pushing". *}
```
```   162
```
```   163 lemma vec_component [simp]: "vec x \$ i = x"
```
```   164   by (vector vec_def)
```
```   165
```
```   166 lemma vector_mult_component [simp]: "(x * y)\$i = x\$i * y\$i"
```
```   167   by vector
```
```   168
```
```   169 lemma vector_smult_component [simp]: "(c *s y)\$i = c * (y\$i)"
```
```   170   by vector
```
```   171
```
```   172 lemma cond_component: "(if b then x else y)\$i = (if b then x\$i else y\$i)" by vector
```
```   173
```
```   174 lemmas vector_component =
```
```   175   vec_component vector_add_component vector_mult_component
```
```   176   vector_smult_component vector_minus_component vector_uminus_component
```
```   177   vector_scaleR_component cond_component
```
```   178
```
```   179
```
```   180 subsection {* Some frequently useful arithmetic lemmas over vectors. *}
```
```   181
```
```   182 instance vec :: (semigroup_mult, finite) semigroup_mult
```
```   183   by default (vector mult_assoc)
```
```   184
```
```   185 instance vec :: (monoid_mult, finite) monoid_mult
```
```   186   by default vector+
```
```   187
```
```   188 instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
```
```   189   by default (vector mult_commute)
```
```   190
```
```   191 instance vec :: (ab_semigroup_idem_mult, finite) ab_semigroup_idem_mult
```
```   192   by default (vector mult_idem)
```
```   193
```
```   194 instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
```
```   195   by default vector
```
```   196
```
```   197 instance vec :: (semiring, finite) semiring
```
```   198   by default (vector field_simps)+
```
```   199
```
```   200 instance vec :: (semiring_0, finite) semiring_0
```
```   201   by default (vector field_simps)+
```
```   202 instance vec :: (semiring_1, finite) semiring_1
```
```   203   by default vector
```
```   204 instance vec :: (comm_semiring, finite) comm_semiring
```
```   205   by default (vector field_simps)+
```
```   206
```
```   207 instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
```
```   208 instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
```
```   209 instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
```
```   210 instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
```
```   211 instance vec :: (ring, finite) ring ..
```
```   212 instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
```
```   213 instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
```
```   214
```
```   215 instance vec :: (ring_1, finite) ring_1 ..
```
```   216
```
```   217 instance vec :: (real_algebra, finite) real_algebra
```
```   218   by default (simp_all add: vec_eq_iff)
```
```   219
```
```   220 instance vec :: (real_algebra_1, finite) real_algebra_1 ..
```
```   221
```
```   222 lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)\$i = of_nat n"
```
```   223 proof (induct n)
```
```   224   case 0
```
```   225   then show ?case by vector
```
```   226 next
```
```   227   case Suc
```
```   228   then show ?case by vector
```
```   229 qed
```
```   230
```
```   231 lemma one_index[simp]: "(1 :: 'a::one ^'n)\$i = 1"
```
```   232   by vector
```
```   233
```
```   234 instance vec :: (semiring_char_0, finite) semiring_char_0
```
```   235 proof
```
```   236   fix m n :: nat
```
```   237   show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
```
```   238     by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
```
```   239 qed
```
```   240
```
```   241 instance vec :: (numeral, finite) numeral ..
```
```   242 instance vec :: (semiring_numeral, finite) semiring_numeral ..
```
```   243
```
```   244 lemma numeral_index [simp]: "numeral w \$ i = numeral w"
```
```   245   by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
```
```   246
```
```   247 lemma neg_numeral_index [simp]: "neg_numeral w \$ i = neg_numeral w"
```
```   248   by (simp only: neg_numeral_def vector_uminus_component numeral_index)
```
```   249
```
```   250 instance vec :: (comm_ring_1, finite) comm_ring_1 ..
```
```   251 instance vec :: (ring_char_0, finite) ring_char_0 ..
```
```   252
```
```   253 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
```
```   254   by (vector mult_assoc)
```
```   255 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
```
```   256   by (vector field_simps)
```
```   257 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
```
```   258   by (vector field_simps)
```
```   259 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
```
```   260 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
```
```   261 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
```
```   262   by (vector field_simps)
```
```   263 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
```
```   264 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
```
```   265 lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
```
```   266 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
```
```   267 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
```
```   268   by (vector field_simps)
```
```   269
```
```   270 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
```
```   271   by (simp add: vec_eq_iff)
```
```   272
```
```   273 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
```
```   274 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
```
```   275   by vector
```
```   276 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
```
```   277   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
```
```   278 lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
```
```   279   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
```
```   280 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
```
```   281   by (metis vector_mul_lcancel)
```
```   282 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
```
```   283   by (metis vector_mul_rcancel)
```
```   284
```
```   285 lemma component_le_norm_cart: "\<bar>x\$i\<bar> <= norm x"
```
```   286   apply (simp add: norm_vec_def)
```
```   287   apply (rule member_le_setL2, simp_all)
```
```   288   done
```
```   289
```
```   290 lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x\$i\<bar> <= e"
```
```   291   by (metis component_le_norm_cart order_trans)
```
```   292
```
```   293 lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x\$i\<bar> < e"
```
```   294   by (metis component_le_norm_cart basic_trans_rules(21))
```
```   295
```
```   296 lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x\$i\<bar>) UNIV"
```
```   297   by (simp add: norm_vec_def setL2_le_setsum)
```
```   298
```
```   299 lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
```
```   300   unfolding scaleR_vec_def vector_scalar_mult_def by simp
```
```   301
```
```   302 lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
```
```   303   unfolding dist_norm scalar_mult_eq_scaleR
```
```   304   unfolding scaleR_right_diff_distrib[symmetric] by simp
```
```   305
```
```   306 lemma setsum_component [simp]:
```
```   307   fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
```
```   308   shows "(setsum f S)\$i = setsum (\<lambda>x. (f x)\$i) S"
```
```   309 proof (cases "finite S")
```
```   310   case True
```
```   311   then show ?thesis by induct simp_all
```
```   312 next
```
```   313   case False
```
```   314   then show ?thesis by simp
```
```   315 qed
```
```   316
```
```   317 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)\$i ) S)"
```
```   318   by (simp add: vec_eq_iff)
```
```   319
```
```   320 lemma setsum_cmul:
```
```   321   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
```
```   322   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
```
```   323   by (simp add: vec_eq_iff setsum_right_distrib)
```
```   324
```
```   325 (* TODO: use setsum_norm_allsubsets_bound *)
```
```   326 lemma setsum_norm_allsubsets_bound_cart:
```
```   327   fixes f:: "'a \<Rightarrow> real ^'n"
```
```   328   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
```
```   329   shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
```
```   330   using setsum_norm_allsubsets_bound[OF assms]
```
```   331   by (simp add: DIM_cart Basis_real_def)
```
```   332
```
```   333 instance vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
```
```   334 proof
```
```   335   fix x y::"'a^'b"
```
```   336   show "(x \<le> y) = (\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i)"
```
```   337     unfolding less_eq_vec_def apply(subst eucl_le) by (simp add: Basis_vec_def inner_axis)
```
```   338   show"(x < y) = (\<forall>i\<in>Basis. x \<bullet> i < y \<bullet> i)"
```
```   339     unfolding less_vec_def apply(subst eucl_less) by (simp add: Basis_vec_def inner_axis)
```
```   340 qed
```
```   341
```
```   342 subsection {* Matrix operations *}
```
```   343
```
```   344 text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
```
```   345
```
```   346 definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
```
```   347     (infixl "**" 70)
```
```   348   where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m\$i)\$k) * ((m'\$k)\$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
```
```   349
```
```   350 definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
```
```   351     (infixl "*v" 70)
```
```   352   where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m\$i)\$j) * (x\$j)) (UNIV ::'n set)) :: 'a^'m"
```
```   353
```
```   354 definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
```
```   355     (infixl "v*" 70)
```
```   356   where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m\$i)\$j) * (v\$i)) (UNIV :: 'm set)) :: 'a^'n"
```
```   357
```
```   358 definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
```
```   359 definition transpose where
```
```   360   "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A\$j)\$i))"
```
```   361 definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A\$i)\$j))"
```
```   362 definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A\$i)\$j))"
```
```   363 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
```
```   364 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
```
```   365
```
```   366 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
```
```   367 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
```
```   368   by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps)
```
```   369
```
```   370 lemma matrix_mul_lid:
```
```   371   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
```
```   372   shows "mat 1 ** A = A"
```
```   373   apply (simp add: matrix_matrix_mult_def mat_def)
```
```   374   apply vector
```
```   375   apply (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite]
```
```   376     mult_1_left mult_zero_left if_True UNIV_I)
```
```   377   done
```
```   378
```
```   379
```
```   380 lemma matrix_mul_rid:
```
```   381   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
```
```   382   shows "A ** mat 1 = A"
```
```   383   apply (simp add: matrix_matrix_mult_def mat_def)
```
```   384   apply vector
```
```   385   apply (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite]
```
```   386     mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
```
```   387   done
```
```   388
```
```   389 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
```
```   390   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
```
```   391   apply (subst setsum_commute)
```
```   392   apply simp
```
```   393   done
```
```   394
```
```   395 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
```
```   396   apply (vector matrix_matrix_mult_def matrix_vector_mult_def
```
```   397     setsum_right_distrib setsum_left_distrib mult_assoc)
```
```   398   apply (subst setsum_commute)
```
```   399   apply simp
```
```   400   done
```
```   401
```
```   402 lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
```
```   403   apply (vector matrix_vector_mult_def mat_def)
```
```   404   apply (simp add: if_distrib cond_application_beta setsum_delta' cong del: if_weak_cong)
```
```   405   done
```
```   406
```
```   407 lemma matrix_transpose_mul:
```
```   408     "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
```
```   409   by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute)
```
```   410
```
```   411 lemma matrix_eq:
```
```   412   fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
```
```   413   shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   414   apply auto
```
```   415   apply (subst vec_eq_iff)
```
```   416   apply clarify
```
```   417   apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
```
```   418   apply (erule_tac x="axis ia 1" in allE)
```
```   419   apply (erule_tac x="i" in allE)
```
```   420   apply (auto simp add: if_distrib cond_application_beta axis_def
```
```   421     setsum_delta[OF finite] cong del: if_weak_cong)
```
```   422   done
```
```   423
```
```   424 lemma matrix_vector_mul_component: "((A::real^_^_) *v x)\$k = (A\$k) \<bullet> x"
```
```   425   by (simp add: matrix_vector_mult_def inner_vec_def)
```
```   426
```
```   427 lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
```
```   428   apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
```
```   429   apply (subst setsum_commute)
```
```   430   apply simp
```
```   431   done
```
```   432
```
```   433 lemma transpose_mat: "transpose (mat n) = mat n"
```
```   434   by (vector transpose_def mat_def)
```
```   435
```
```   436 lemma transpose_transpose: "transpose(transpose A) = A"
```
```   437   by (vector transpose_def)
```
```   438
```
```   439 lemma row_transpose:
```
```   440   fixes A:: "'a::semiring_1^_^_"
```
```   441   shows "row i (transpose A) = column i A"
```
```   442   by (simp add: row_def column_def transpose_def vec_eq_iff)
```
```   443
```
```   444 lemma column_transpose:
```
```   445   fixes A:: "'a::semiring_1^_^_"
```
```   446   shows "column i (transpose A) = row i A"
```
```   447   by (simp add: row_def column_def transpose_def vec_eq_iff)
```
```   448
```
```   449 lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
```
```   450   by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
```
```   451
```
```   452 lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
```
```   453   by (metis transpose_transpose rows_transpose)
```
```   454
```
```   455 text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
```
```   456
```
```   457 lemma matrix_mult_dot: "A *v x = (\<chi> i. A\$i \<bullet> x)"
```
```   458   by (simp add: matrix_vector_mult_def inner_vec_def)
```
```   459
```
```   460 lemma matrix_mult_vsum:
```
```   461   "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x\$i) *s column i A) (UNIV:: 'n set)"
```
```   462   by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
```
```   463
```
```   464 lemma vector_componentwise:
```
```   465   "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x\$i) * (axis i 1 :: 'a^'n) \$ j)"
```
```   466   by (simp add: axis_def if_distrib setsum_cases vec_eq_iff)
```
```   467
```
```   468 lemma basis_expansion: "setsum (\<lambda>i. (x\$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
```
```   469   by (auto simp add: axis_def vec_eq_iff if_distrib setsum_cases cong del: if_weak_cong)
```
```   470
```
```   471 lemma linear_componentwise:
```
```   472   fixes f:: "real ^'m \<Rightarrow> real ^ _"
```
```   473   assumes lf: "linear f"
```
```   474   shows "(f x)\$j = setsum (\<lambda>i. (x\$i) * (f (axis i 1)\$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
```
```   475 proof -
```
```   476   let ?M = "(UNIV :: 'm set)"
```
```   477   let ?N = "(UNIV :: 'n set)"
```
```   478   have fM: "finite ?M" by simp
```
```   479   have "?rhs = (setsum (\<lambda>i.(x\$i) *\<^sub>R f (axis i 1) ) ?M)\$j"
```
```   480     unfolding setsum_component by simp
```
```   481   then show ?thesis
```
```   482     unfolding linear_setsum_mul[OF lf fM, symmetric]
```
```   483     unfolding scalar_mult_eq_scaleR[symmetric]
```
```   484     unfolding basis_expansion
```
```   485     by simp
```
```   486 qed
```
```   487
```
```   488 text{* Inverse matrices  (not necessarily square) *}
```
```   489
```
```   490 definition
```
```   491   "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
```
```   492
```
```   493 definition
```
```   494   "matrix_inv(A:: 'a::semiring_1^'n^'m) =
```
```   495     (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
```
```   496
```
```   497 text{* Correspondence between matrices and linear operators. *}
```
```   498
```
```   499 definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
```
```   500   where "matrix f = (\<chi> i j. (f(axis j 1))\$i)"
```
```   501
```
```   502 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
```
```   503   by (simp add: linear_def matrix_vector_mult_def vec_eq_iff
```
```   504       field_simps setsum_right_distrib setsum_addf)
```
```   505
```
```   506 lemma matrix_works:
```
```   507   assumes lf: "linear f"
```
```   508   shows "matrix f *v x = f (x::real ^ 'n)"
```
```   509   apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
```
```   510   apply clarify
```
```   511   apply (rule linear_componentwise[OF lf, symmetric])
```
```   512   done
```
```   513
```
```   514 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
```
```   515   by (simp add: ext matrix_works)
```
```   516
```
```   517 lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
```
```   518   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
```
```   519
```
```   520 lemma matrix_compose:
```
```   521   assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
```
```   522     and lg: "linear (g::real^'m \<Rightarrow> real^_)"
```
```   523   shows "matrix (g o f) = matrix g ** matrix f"
```
```   524   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
```
```   525   by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
```
```   526
```
```   527 lemma matrix_vector_column:
```
```   528   "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x\$i) *s ((transpose A)\$i)) (UNIV:: 'n set)"
```
```   529   by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult_commute)
```
```   530
```
```   531 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
```
```   532   apply (rule adjoint_unique)
```
```   533   apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
```
```   534     setsum_left_distrib setsum_right_distrib)
```
```   535   apply (subst setsum_commute)
```
```   536   apply (auto simp add: mult_ac)
```
```   537   done
```
```   538
```
```   539 lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
```
```   540   shows "matrix(adjoint f) = transpose(matrix f)"
```
```   541   apply (subst matrix_vector_mul[OF lf])
```
```   542   unfolding adjoint_matrix matrix_of_matrix_vector_mul
```
```   543   apply rule
```
```   544   done
```
```   545
```
```   546
```
```   547 subsection {* lambda skolemization on cartesian products *}
```
```   548
```
```   549 (* FIXME: rename do choice_cart *)
```
```   550
```
```   551 lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
```
```   552    (\<exists>x::'a ^ 'n. \<forall>i. P i (x \$ i))" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   553 proof -
```
```   554   let ?S = "(UNIV :: 'n set)"
```
```   555   { assume H: "?rhs"
```
```   556     then have ?lhs by auto }
```
```   557   moreover
```
```   558   { assume H: "?lhs"
```
```   559     then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
```
```   560     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
```
```   561     { fix i
```
```   562       from f have "P i (f i)" by metis
```
```   563       then have "P i (?x \$ i)" by auto
```
```   564     }
```
```   565     hence "\<forall>i. P i (?x\$i)" by metis
```
```   566     hence ?rhs by metis }
```
```   567   ultimately show ?thesis by metis
```
```   568 qed
```
```   569
```
```   570 lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
```
```   571   unfolding inner_simps scalar_mult_eq_scaleR by auto
```
```   572
```
```   573 lemma left_invertible_transpose:
```
```   574   "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
```
```   575   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
```
```   576
```
```   577 lemma right_invertible_transpose:
```
```   578   "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
```
```   579   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
```
```   580
```
```   581 lemma matrix_left_invertible_injective:
```
```   582   "(\<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)"
```
```   583 proof -
```
```   584   { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
```
```   585     from xy have "B*v (A *v x) = B *v (A*v y)" by simp
```
```   586     hence "x = y"
```
```   587       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
```
```   588   moreover
```
```   589   { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
```
```   590     hence i: "inj (op *v A)" unfolding inj_on_def by auto
```
```   591     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
```
```   592     obtain g where g: "linear g" "g o op *v A = id" by blast
```
```   593     have "matrix g ** A = mat 1"
```
```   594       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
```
```   595       using g(2) by (simp add: fun_eq_iff)
```
```   596     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
```
```   597   ultimately show ?thesis by blast
```
```   598 qed
```
```   599
```
```   600 lemma matrix_left_invertible_ker:
```
```   601   "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
```
```   602   unfolding matrix_left_invertible_injective
```
```   603   using linear_injective_0[OF matrix_vector_mul_linear, of A]
```
```   604   by (simp add: inj_on_def)
```
```   605
```
```   606 lemma matrix_right_invertible_surjective:
```
```   607   "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
```
```   608 proof -
```
```   609   { fix B :: "real ^'m^'n"
```
```   610     assume AB: "A ** B = mat 1"
```
```   611     { fix x :: "real ^ 'm"
```
```   612       have "A *v (B *v x) = x"
```
```   613         by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
```
```   614     hence "surj (op *v A)" unfolding surj_def by metis }
```
```   615   moreover
```
```   616   { assume sf: "surj (op *v A)"
```
```   617     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
```
```   618     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
```
```   619       by blast
```
```   620
```
```   621     have "A ** (matrix g) = mat 1"
```
```   622       unfolding matrix_eq  matrix_vector_mul_lid
```
```   623         matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
```
```   624       using g(2) unfolding o_def fun_eq_iff id_def
```
```   625       .
```
```   626     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
```
```   627   }
```
```   628   ultimately show ?thesis unfolding surj_def by blast
```
```   629 qed
```
```   630
```
```   631 lemma matrix_left_invertible_independent_columns:
```
```   632   fixes A :: "real^'n^'m"
```
```   633   shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
```
```   634       (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
```
```   635     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   636 proof -
```
```   637   let ?U = "UNIV :: 'n set"
```
```   638   { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
```
```   639     { fix c i
```
```   640       assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
```
```   641       let ?x = "\<chi> i. c i"
```
```   642       have th0:"A *v ?x = 0"
```
```   643         using c
```
```   644         unfolding matrix_mult_vsum vec_eq_iff
```
```   645         by auto
```
```   646       from k[rule_format, OF th0] i
```
```   647       have "c i = 0" by (vector vec_eq_iff)}
```
```   648     hence ?rhs by blast }
```
```   649   moreover
```
```   650   { assume H: ?rhs
```
```   651     { fix x assume x: "A *v x = 0"
```
```   652       let ?c = "\<lambda>i. ((x\$i ):: real)"
```
```   653       from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
```
```   654       have "x = 0" by vector }
```
```   655   }
```
```   656   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
```
```   657 qed
```
```   658
```
```   659 lemma matrix_right_invertible_independent_rows:
```
```   660   fixes A :: "real^'n^'m"
```
```   661   shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
```
```   662     (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
```
```   663   unfolding left_invertible_transpose[symmetric]
```
```   664     matrix_left_invertible_independent_columns
```
```   665   by (simp add: column_transpose)
```
```   666
```
```   667 lemma matrix_right_invertible_span_columns:
```
```   668   "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
```
```   669     span (columns A) = UNIV" (is "?lhs = ?rhs")
```
```   670 proof -
```
```   671   let ?U = "UNIV :: 'm set"
```
```   672   have fU: "finite ?U" by simp
```
```   673   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x\$i) *s column i A) ?U = y)"
```
```   674     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
```
```   675     apply (subst eq_commute)
```
```   676     apply rule
```
```   677     done
```
```   678   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
```
```   679   { assume h: ?lhs
```
```   680     { fix x:: "real ^'n"
```
```   681       from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
```
```   682         where y: "setsum (\<lambda>i. (y\$i) *s column i A) ?U = x" by blast
```
```   683       have "x \<in> span (columns A)"
```
```   684         unfolding y[symmetric]
```
```   685         apply (rule span_setsum[OF fU])
```
```   686         apply clarify
```
```   687         unfolding scalar_mult_eq_scaleR
```
```   688         apply (rule span_mul)
```
```   689         apply (rule span_superset)
```
```   690         unfolding columns_def
```
```   691         apply blast
```
```   692         done
```
```   693     }
```
```   694     then have ?rhs unfolding rhseq by blast }
```
```   695   moreover
```
```   696   { assume h:?rhs
```
```   697     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x\$i) *s column i A) ?U = y"
```
```   698     { fix y
```
```   699       have "?P y"
```
```   700       proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
```
```   701         show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x\$i) *s column i A) ?U = 0"
```
```   702           by (rule exI[where x=0], simp)
```
```   703       next
```
```   704         fix c y1 y2
```
```   705         assume y1: "y1 \<in> columns A" and y2: "?P y2"
```
```   706         from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
```
```   707           unfolding columns_def by blast
```
```   708         from y2 obtain x:: "real ^'m" where
```
```   709           x: "setsum (\<lambda>i. (x\$i) *s column i A) ?U = y2" by blast
```
```   710         let ?x = "(\<chi> j. if j = i then c + (x\$i) else (x\$j))::real^'m"
```
```   711         show "?P (c*s y1 + y2)"
```
```   712         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)
```
```   713           fix j
```
```   714           have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```   715               else (x\$xa) * ((column xa A\$j))) = (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))"
```
```   716             using i(1) by (simp add: field_simps)
```
```   717           have "setsum (\<lambda>xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```   718               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"
```
```   719             apply (rule setsum_cong[OF refl])
```
```   720             using th apply blast
```
```   721             done
```
```   722           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"
```
```   723             by (simp add: setsum_addf)
```
```   724           also have "\<dots> = c * ((column i A)\$j) + setsum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U"
```
```   725             unfolding setsum_delta[OF fU]
```
```   726             using i(1) by simp
```
```   727           finally show "setsum (\<lambda>xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
```
```   728             else (x\$xa) * ((column xa A\$j))) ?U = c * ((column i A)\$j) + setsum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U" .
```
```   729         qed
```
```   730       next
```
```   731         show "y \<in> span (columns A)"
```
```   732           unfolding h by blast
```
```   733       qed
```
```   734     }
```
```   735     then have ?lhs unfolding lhseq ..
```
```   736   }
```
```   737   ultimately show ?thesis by blast
```
```   738 qed
```
```   739
```
```   740 lemma matrix_left_invertible_span_rows:
```
```   741   "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
```
```   742   unfolding right_invertible_transpose[symmetric]
```
```   743   unfolding columns_transpose[symmetric]
```
```   744   unfolding matrix_right_invertible_span_columns
```
```   745   ..
```
```   746
```
```   747 text {* The same result in terms of square matrices. *}
```
```   748
```
```   749 lemma matrix_left_right_inverse:
```
```   750   fixes A A' :: "real ^'n^'n"
```
```   751   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
```
```   752 proof -
```
```   753   { fix A A' :: "real ^'n^'n"
```
```   754     assume AA': "A ** A' = mat 1"
```
```   755     have sA: "surj (op *v A)"
```
```   756       unfolding surj_def
```
```   757       apply clarify
```
```   758       apply (rule_tac x="(A' *v y)" in exI)
```
```   759       apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
```
```   760       done
```
```   761     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
```
```   762     obtain f' :: "real ^'n \<Rightarrow> real ^'n"
```
```   763       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
```
```   764     have th: "matrix f' ** A = mat 1"
```
```   765       by (simp add: matrix_eq matrix_works[OF f'(1)]
```
```   766           matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
```
```   767     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
```
```   768     hence "matrix f' = A'"
```
```   769       by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
```
```   770     hence "matrix f' ** A = A' ** A" by simp
```
```   771     hence "A' ** A = mat 1" by (simp add: th)
```
```   772   }
```
```   773   then show ?thesis by blast
```
```   774 qed
```
```   775
```
```   776 text {* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *}
```
```   777
```
```   778 definition "rowvector v = (\<chi> i j. (v\$j))"
```
```   779
```
```   780 definition "columnvector v = (\<chi> i j. (v\$i))"
```
```   781
```
```   782 lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
```
```   783   by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
```
```   784
```
```   785 lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
```
```   786   by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
```
```   787
```
```   788 lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
```
```   789   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
```
```   790
```
```   791 lemma dot_matrix_product:
```
```   792   "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))\$1)\$1"
```
```   793   by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
```
```   794
```
```   795 lemma dot_matrix_vector_mul:
```
```   796   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
```
```   797   shows "(A *v x) \<bullet> (B *v y) =
```
```   798       (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))\$1)\$1"
```
```   799   unfolding dot_matrix_product transpose_columnvector[symmetric]
```
```   800     dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
```
```   801
```
```   802
```
```   803 lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x\$i) |i. i\<in>UNIV}"
```
```   804   by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
```
```   805
```
```   806 lemma component_le_infnorm_cart: "\<bar>x\$i\<bar> \<le> infnorm (x::real^'n)"
```
```   807   using Basis_le_infnorm[of "axis i 1" x]
```
```   808   by (simp add: Basis_vec_def axis_eq_axis inner_axis)
```
```   809
```
```   810 lemma continuous_component: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x \$ i)"
```
```   811   unfolding continuous_def by (rule tendsto_vec_nth)
```
```   812
```
```   813 lemma continuous_on_component: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x \$ i)"
```
```   814   unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
```
```   815
```
```   816 lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x\$i}"
```
```   817   by (simp add: Collect_all_eq closed_INT closed_Collect_le)
```
```   818
```
```   819 lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x \$ i) ` s)"
```
```   820   unfolding bounded_def
```
```   821   apply clarify
```
```   822   apply (rule_tac x="x \$ i" in exI)
```
```   823   apply (rule_tac x="e" in exI)
```
```   824   apply clarify
```
```   825   apply (rule order_trans [OF dist_vec_nth_le], simp)
```
```   826   done
```
```   827
```
```   828 lemma compact_lemma_cart:
```
```   829   fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
```
```   830   assumes f: "bounded (range f)"
```
```   831   shows "\<forall>d.
```
```   832         \<exists>l r. subseq r \<and>
```
```   833         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
```
```   834 proof
```
```   835   fix d :: "'n set"
```
```   836   have "finite d" by simp
```
```   837   thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
```
```   838       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
```
```   839   proof (induct d)
```
```   840     case empty
```
```   841     thus ?case unfolding subseq_def by auto
```
```   842   next
```
```   843     case (insert k d)
```
```   844     obtain l1::"'a^'n" and r1 where r1:"subseq r1"
```
```   845       and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially"
```
```   846       using insert(3) by auto
```
```   847     have s': "bounded ((\<lambda>x. x \$ k) ` range f)" using `bounded (range f)`
```
```   848       by (auto intro!: bounded_component_cart)
```
```   849     have f': "\<forall>n. f (r1 n) \$ k \<in> (\<lambda>x. x \$ k) ` range f" by simp
```
```   850     have "bounded (range (\<lambda>i. f (r1 i) \$ k))"
```
```   851       by (metis (lifting) bounded_subset image_subsetI f' s')
```
```   852     then obtain l2 r2 where r2: "subseq r2"
```
```   853       and lr2: "((\<lambda>i. f (r1 (r2 i)) \$ k) ---> l2) sequentially"
```
```   854       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \$ k"] by (auto simp: o_def)
```
```   855     def r \<equiv> "r1 \<circ> r2"
```
```   856     have r: "subseq r"
```
```   857       using r1 and r2 unfolding r_def o_def subseq_def by auto
```
```   858     moreover
```
```   859     def l \<equiv> "(\<chi> i. if i = k then l2 else l1\$i)::'a^'n"
```
```   860     { fix e :: real assume "e > 0"
```
```   861       from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially"
```
```   862         by blast
```
```   863       from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \$ k) l2 < e) sequentially"
```
```   864         by (rule tendstoD)
```
```   865       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \$ i) (l1 \$ i) < e) sequentially"
```
```   866         by (rule eventually_subseq)
```
```   867       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \$ i) (l \$ i) < e) sequentially"
```
```   868         using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
```
```   869     }
```
```   870     ultimately show ?case by auto
```
```   871   qed
```
```   872 qed
```
```   873
```
```   874 instance vec :: (heine_borel, finite) heine_borel
```
```   875 proof
```
```   876   fix f :: "nat \<Rightarrow> 'a ^ 'b"
```
```   877   assume f: "bounded (range f)"
```
```   878   then obtain l r where r: "subseq r"
```
```   879       and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) \$ i) (l \$ i) < e) sequentially"
```
```   880     using compact_lemma_cart [OF f] by blast
```
```   881   let ?d = "UNIV::'b set"
```
```   882   { fix e::real assume "e>0"
```
```   883     hence "0 < e / (real_of_nat (card ?d))"
```
```   884       using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
```
```   885     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially"
```
```   886       by simp
```
```   887     moreover
```
```   888     { fix n
```
```   889       assume n: "\<forall>i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))"
```
```   890       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) \$ i) (l \$ i))"
```
```   891         unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
```
```   892       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
```
```   893         by (rule setsum_strict_mono) (simp_all add: n)
```
```   894       finally have "dist (f (r n)) l < e" by simp
```
```   895     }
```
```   896     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
```
```   897       by (rule eventually_elim1)
```
```   898   }
```
```   899   hence "((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
```
```   900   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
```
```   901 qed
```
```   902
```
```   903 lemma interval_cart:
```
```   904   fixes a :: "'a::ord^'n"
```
```   905   shows "{a <..< b} = {x::'a^'n. \<forall>i. a\$i < x\$i \<and> x\$i < b\$i}"
```
```   906     and "{a .. b} = {x::'a^'n. \<forall>i. a\$i \<le> x\$i \<and> x\$i \<le> b\$i}"
```
```   907   by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
```
```   908
```
```   909 lemma mem_interval_cart:
```
```   910   fixes a :: "'a::ord^'n"
```
```   911   shows "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a\$i < x\$i \<and> x\$i < b\$i)"
```
```   912     and "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a\$i \<le> x\$i \<and> x\$i \<le> b\$i)"
```
```   913   using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
```
```   914
```
```   915 lemma interval_eq_empty_cart:
```
```   916   fixes a :: "real^'n"
```
```   917   shows "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b\$i \<le> a\$i))" (is ?th1)
```
```   918     and "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i. b\$i < a\$i))" (is ?th2)
```
```   919 proof -
```
```   920   { fix i x assume as:"b\$i \<le> a\$i" and x:"x\<in>{a <..< b}"
```
```   921     hence "a \$ i < x \$ i \<and> x \$ i < b \$ i" unfolding mem_interval_cart by auto
```
```   922     hence "a\$i < b\$i" by auto
```
```   923     hence False using as by auto }
```
```   924   moreover
```
```   925   { assume as:"\<forall>i. \<not> (b\$i \<le> a\$i)"
```
```   926     let ?x = "(1/2) *\<^sub>R (a + b)"
```
```   927     { fix i
```
```   928       have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto
```
```   929       hence "a\$i < ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i < b\$i"
```
```   930         unfolding vector_smult_component and vector_add_component
```
```   931         by auto }
```
```   932     hence "{a <..< b} \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
```
```   933   ultimately show ?th1 by blast
```
```   934
```
```   935   { fix i x assume as:"b\$i < a\$i" and x:"x\<in>{a .. b}"
```
```   936     hence "a \$ i \<le> x \$ i \<and> x \$ i \<le> b \$ i" unfolding mem_interval_cart by auto
```
```   937     hence "a\$i \<le> b\$i" by auto
```
```   938     hence False using as by auto }
```
```   939   moreover
```
```   940   { assume as:"\<forall>i. \<not> (b\$i < a\$i)"
```
```   941     let ?x = "(1/2) *\<^sub>R (a + b)"
```
```   942     { fix i
```
```   943       have "a\$i \<le> b\$i" using as[THEN spec[where x=i]] by auto
```
```   944       hence "a\$i \<le> ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i \<le> b\$i"
```
```   945         unfolding vector_smult_component and vector_add_component
```
```   946         by auto }
```
```   947     hence "{a .. b} \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
```
```   948   ultimately show ?th2 by blast
```
```   949 qed
```
```   950
```
```   951 lemma interval_ne_empty_cart:
```
```   952   fixes a :: "real^'n"
```
```   953   shows "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a\$i \<le> b\$i)"
```
```   954     and "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a\$i < b\$i)"
```
```   955   unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
```
```   956     (* BH: Why doesn't just "auto" work here? *)
```
```   957
```
```   958 lemma subset_interval_imp_cart:
```
```   959   fixes a :: "real^'n"
```
```   960   shows "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}"
```
```   961     and "(\<forall>i. a\$i < c\$i \<and> d\$i < b\$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}"
```
```   962     and "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}"
```
```   963     and "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
```
```   964   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
```
```   965   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
```
```   966
```
```   967 lemma interval_sing:
```
```   968   fixes a :: "'a::linorder^'n"
```
```   969   shows "{a .. a} = {a} \<and> {a<..<a} = {}"
```
```   970   apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
```
```   971   apply (simp add: order_eq_iff)
```
```   972   apply (auto simp add: not_less less_imp_le)
```
```   973   done
```
```   974
```
```   975 lemma interval_open_subset_closed_cart:
```
```   976   fixes a :: "'a::preorder^'n"
```
```   977   shows "{a<..<b} \<subseteq> {a .. b}"
```
```   978 proof (simp add: subset_eq, rule)
```
```   979   fix x
```
```   980   assume x: "x \<in>{a<..<b}"
```
```   981   { fix i
```
```   982     have "a \$ i \<le> x \$ i"
```
```   983       using x order_less_imp_le[of "a\$i" "x\$i"]
```
```   984       by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
```
```   985   }
```
```   986   moreover
```
```   987   { fix i
```
```   988     have "x \$ i \<le> b \$ i"
```
```   989       using x order_less_imp_le[of "x\$i" "b\$i"]
```
```   990       by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
```
```   991   }
```
```   992   ultimately
```
```   993   show "a \<le> x \<and> x \<le> b"
```
```   994     by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
```
```   995 qed
```
```   996
```
```   997 lemma subset_interval_cart:
```
```   998   fixes a :: "real^'n"
```
```   999   shows "{c .. d} \<subseteq> {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)
```
```  1000     and "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c\$i \<le> d\$i) --> (\<forall>i. a\$i < c\$i \<and> d\$i < b\$i)" (is ?th2)
```
```  1001     and "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c\$i < d\$i) --> (\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i)" (is ?th3)
```
```  1002     and "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c\$i < d\$i) --> (\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i)" (is ?th4)
```
```  1003   using subset_interval[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
```
```  1004
```
```  1005 lemma disjoint_interval_cart:
```
```  1006   fixes a::"real^'n"
```
```  1007   shows "{a .. b} \<inter> {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)
```
```  1008     and "{a .. b} \<inter> {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)
```
```  1009     and "{a<..<b} \<inter> {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)
```
```  1010     and "{a<..<b} \<inter> {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)
```
```  1011   using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
```
```  1012
```
```  1013 lemma inter_interval_cart:
```
```  1014   fixes a :: "'a::linorder^'n"
```
```  1015   shows "{a .. b} \<inter> {c .. d} =  {(\<chi> i. max (a\$i) (c\$i)) .. (\<chi> i. min (b\$i) (d\$i))}"
```
```  1016   unfolding set_eq_iff and Int_iff and mem_interval_cart
```
```  1017   by auto
```
```  1018
```
```  1019 lemma closed_interval_left_cart:
```
```  1020   fixes b :: "real^'n"
```
```  1021   shows "closed {x::real^'n. \<forall>i. x\$i \<le> b\$i}"
```
```  1022   by (simp add: Collect_all_eq closed_INT closed_Collect_le)
```
```  1023
```
```  1024 lemma closed_interval_right_cart:
```
```  1025   fixes a::"real^'n"
```
```  1026   shows "closed {x::real^'n. \<forall>i. a\$i \<le> x\$i}"
```
```  1027   by (simp add: Collect_all_eq closed_INT closed_Collect_le)
```
```  1028
```
```  1029 lemma is_interval_cart:
```
```  1030   "is_interval (s::(real^'n) set) \<longleftrightarrow>
```
```  1031     (\<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)"
```
```  1032   by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
```
```  1033
```
```  1034 lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x\$i \<le> a}"
```
```  1035   by (simp add: closed_Collect_le)
```
```  1036
```
```  1037 lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x\$i \<ge> a}"
```
```  1038   by (simp add: closed_Collect_le)
```
```  1039
```
```  1040 lemma open_halfspace_component_lt_cart: "open {x::real^'n. x\$i < a}"
```
```  1041   by (simp add: open_Collect_less)
```
```  1042
```
```  1043 lemma open_halfspace_component_gt_cart: "open {x::real^'n. x\$i  > a}"
```
```  1044   by (simp add: open_Collect_less)
```
```  1045
```
```  1046 lemma Lim_component_le_cart:
```
```  1047   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1048   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x \$i \<le> b) net"
```
```  1049   shows "l\$i \<le> b"
```
```  1050   by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
```
```  1051
```
```  1052 lemma Lim_component_ge_cart:
```
```  1053   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1054   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)\$i) net"
```
```  1055   shows "b \<le> l\$i"
```
```  1056   by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
```
```  1057
```
```  1058 lemma Lim_component_eq_cart:
```
```  1059   fixes f :: "'a \<Rightarrow> real^'n"
```
```  1060   assumes net: "(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\$i = b) net"
```
```  1061   shows "l\$i = b"
```
```  1062   using ev[unfolded order_eq_iff eventually_conj_iff] and
```
```  1063     Lim_component_ge_cart[OF net, of b i] and
```
```  1064     Lim_component_le_cart[OF net, of i b] by auto
```
```  1065
```
```  1066 lemma connected_ivt_component_cart:
```
```  1067   fixes x :: "real^'n"
```
```  1068   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)"
```
```  1069   using connected_ivt_hyperplane[of s x y "axis k 1" a]
```
```  1070   by (auto simp add: inner_axis inner_commute)
```
```  1071
```
```  1072 lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x\$i = 0)}"
```
```  1073   unfolding subspace_def by auto
```
```  1074
```
```  1075 lemma closed_substandard_cart:
```
```  1076   "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x\$i = 0}"
```
```  1077 proof -
```
```  1078   { fix i::'n
```
```  1079     have "closed {x::'a ^ 'n. P i \<longrightarrow> x\$i = 0}"
```
```  1080       by (cases "P i") (simp_all add: closed_Collect_eq) }
```
```  1081   thus ?thesis
```
```  1082     unfolding Collect_all_eq by (simp add: closed_INT)
```
```  1083 qed
```
```  1084
```
```  1085 lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x\$i = 0} = card d"
```
```  1086   (is "dim ?A = _")
```
```  1087 proof -
```
```  1088   let ?a = "\<lambda>x. axis x 1 :: real^'n"
```
```  1089   have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
```
```  1090     by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
```
```  1091   have "?a ` d \<subseteq> Basis"
```
```  1092     by (auto simp: Basis_vec_def)
```
```  1093   thus ?thesis
```
```  1094     using dim_substandard[of "?a ` d"] card_image[of ?a d]
```
```  1095     by (auto simp: axis_eq_axis inj_on_def *)
```
```  1096 qed
```
```  1097
```
```  1098 lemma affinity_inverses:
```
```  1099   assumes m0: "m \<noteq> (0::'a::field)"
```
```  1100   shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
```
```  1101   "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
```
```  1102   using m0
```
```  1103   apply (auto simp add: fun_eq_iff vector_add_ldistrib)
```
```  1104   apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1[symmetric])
```
```  1105   done
```
```  1106
```
```  1107 lemma vector_affinity_eq:
```
```  1108   assumes m0: "(m::'a::field) \<noteq> 0"
```
```  1109   shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
```
```  1110 proof
```
```  1111   assume h: "m *s x + c = y"
```
```  1112   hence "m *s x = y - c" by (simp add: field_simps)
```
```  1113   hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
```
```  1114   then show "x = inverse m *s y + - (inverse m *s c)"
```
```  1115     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
```
```  1116 next
```
```  1117   assume h: "x = inverse m *s y + - (inverse m *s c)"
```
```  1118   show "m *s x + c = y" unfolding h diff_minus[symmetric]
```
```  1119     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
```
```  1120 qed
```
```  1121
```
```  1122 lemma vector_eq_affinity:
```
```  1123     "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
```
```  1124   using vector_affinity_eq[where m=m and x=x and y=y and c=c]
```
```  1125   by metis
```
```  1126
```
```  1127 lemma vector_cart:
```
```  1128   fixes f :: "real^'n \<Rightarrow> real"
```
```  1129   shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
```
```  1130   unfolding euclidean_eq_iff[where 'a="real^'n"]
```
```  1131   by simp (simp add: Basis_vec_def inner_axis)
```
```  1132
```
```  1133 lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
```
```  1134   by (rule vector_cart)
```
```  1135
```
```  1136 subsection "Convex Euclidean Space"
```
```  1137
```
```  1138 lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
```
```  1139   using const_vector_cart[of 1] by (simp add: one_vec_def)
```
```  1140
```
```  1141 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
```
```  1142 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
```
```  1143
```
```  1144 lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
```
```  1145
```
```  1146 lemma convex_box_cart:
```
```  1147   assumes "\<And>i. convex {x. P i x}"
```
```  1148   shows "convex {x. \<forall>i. P i (x\$i)}"
```
```  1149   using assms unfolding convex_def by auto
```
```  1150
```
```  1151 lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x\$i)}"
```
```  1152   by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)
```
```  1153
```
```  1154 lemma unit_interval_convex_hull_cart:
```
```  1155   "{0::real^'n .. 1} = convex hull {x. \<forall>i. (x\$i = 0) \<or> (x\$i = 1)}" (is "?int = convex hull ?points")
```
```  1156   unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"]
```
```  1157   by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
```
```  1158
```
```  1159 lemma cube_convex_hull_cart:
```
```  1160   assumes "0 < d"
```
```  1161   obtains s::"(real^'n) set"
```
```  1162     where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s"
```
```  1163 proof -
```
```  1164   from cube_convex_hull [OF assms, of x] guess s .
```
```  1165   with that[of s] show thesis by (simp add: const_vector_cart)
```
```  1166 qed
```
```  1167
```
```  1168
```
```  1169 subsection "Derivative"
```
```  1170
```
```  1171 lemma differentiable_at_imp_differentiable_on:
```
```  1172   "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
```
```  1173   unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI)
```
```  1174
```
```  1175 definition "jacobian f net = matrix(frechet_derivative f net)"
```
```  1176
```
```  1177 lemma jacobian_works:
```
```  1178   "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
```
```  1179     (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
```
```  1180   apply rule
```
```  1181   unfolding jacobian_def
```
```  1182   apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
```
```  1183   apply (rule differentiableI)
```
```  1184   apply assumption
```
```  1185   unfolding frechet_derivative_works
```
```  1186   apply assumption
```
```  1187   done
```
```  1188
```
```  1189
```
```  1190 subsection {* Component of the differential must be zero if it exists at a local
```
```  1191   maximum or minimum for that corresponding component. *}
```
```  1192
```
```  1193 lemma differential_zero_maxmin_cart:
```
```  1194   fixes f::"real^'a \<Rightarrow> real^'b"
```
```  1195   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))"
```
```  1196     "f differentiable (at x)"
```
```  1197   shows "jacobian f (at x) \$ k = 0"
```
```  1198   using differential_zero_maxmin_component[of "axis k 1" e x f] assms
```
```  1199     vector_cart[of "\<lambda>j. frechet_derivative f (at x) j \$ k"]
```
```  1200   by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
```
```  1201
```
```  1202 subsection {* Lemmas for working on @{typ "real^1"} *}
```
```  1203
```
```  1204 lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
```
```  1205   by (metis (full_types) num1_eq_iff)
```
```  1206
```
```  1207 lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
```
```  1208   by auto (metis (full_types) num1_eq_iff)
```
```  1209
```
```  1210 lemma exhaust_2:
```
```  1211   fixes x :: 2
```
```  1212   shows "x = 1 \<or> x = 2"
```
```  1213 proof (induct x)
```
```  1214   case (of_int z)
```
```  1215   then have "0 <= z" and "z < 2" by simp_all
```
```  1216   then have "z = 0 | z = 1" by arith
```
```  1217   then show ?case by auto
```
```  1218 qed
```
```  1219
```
```  1220 lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
```
```  1221   by (metis exhaust_2)
```
```  1222
```
```  1223 lemma exhaust_3:
```
```  1224   fixes x :: 3
```
```  1225   shows "x = 1 \<or> x = 2 \<or> x = 3"
```
```  1226 proof (induct x)
```
```  1227   case (of_int z)
```
```  1228   then have "0 <= z" and "z < 3" by simp_all
```
```  1229   then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
```
```  1230   then show ?case by auto
```
```  1231 qed
```
```  1232
```
```  1233 lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
```
```  1234   by (metis exhaust_3)
```
```  1235
```
```  1236 lemma UNIV_1 [simp]: "UNIV = {1::1}"
```
```  1237   by (auto simp add: num1_eq_iff)
```
```  1238
```
```  1239 lemma UNIV_2: "UNIV = {1::2, 2::2}"
```
```  1240   using exhaust_2 by auto
```
```  1241
```
```  1242 lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
```
```  1243   using exhaust_3 by auto
```
```  1244
```
```  1245 lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
```
```  1246   unfolding UNIV_1 by simp
```
```  1247
```
```  1248 lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
```
```  1249   unfolding UNIV_2 by simp
```
```  1250
```
```  1251 lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
```
```  1252   unfolding UNIV_3 by (simp add: add_ac)
```
```  1253
```
```  1254 instantiation num1 :: cart_one
```
```  1255 begin
```
```  1256
```
```  1257 instance
```
```  1258 proof
```
```  1259   show "CARD(1) = Suc 0" by auto
```
```  1260 qed
```
```  1261
```
```  1262 end
```
```  1263
```
```  1264 subsection{* The collapse of the general concepts to dimension one. *}
```
```  1265
```
```  1266 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x\$1))"
```
```  1267   by (simp add: vec_eq_iff)
```
```  1268
```
```  1269 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
```
```  1270   apply auto
```
```  1271   apply (erule_tac x= "x\$1" in allE)
```
```  1272   apply (simp only: vector_one[symmetric])
```
```  1273   done
```
```  1274
```
```  1275 lemma norm_vector_1: "norm (x :: _^1) = norm (x\$1)"
```
```  1276   by (simp add: norm_vec_def)
```
```  1277
```
```  1278 lemma norm_real: "norm(x::real ^ 1) = abs(x\$1)"
```
```  1279   by (simp add: norm_vector_1)
```
```  1280
```
```  1281 lemma dist_real: "dist(x::real ^ 1) y = abs((x\$1) - (y\$1))"
```
```  1282   by (auto simp add: norm_real dist_norm)
```
```  1283
```
```  1284
```
```  1285 subsection{* Explicit vector construction from lists. *}
```
```  1286
```
```  1287 definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
```
```  1288
```
```  1289 lemma vector_1: "(vector[x]) \$1 = x"
```
```  1290   unfolding vector_def by simp
```
```  1291
```
```  1292 lemma vector_2:
```
```  1293  "(vector[x,y]) \$1 = x"
```
```  1294  "(vector[x,y] :: 'a^2)\$2 = (y::'a::zero)"
```
```  1295   unfolding vector_def by simp_all
```
```  1296
```
```  1297 lemma vector_3:
```
```  1298  "(vector [x,y,z] ::('a::zero)^3)\$1 = x"
```
```  1299  "(vector [x,y,z] ::('a::zero)^3)\$2 = y"
```
```  1300  "(vector [x,y,z] ::('a::zero)^3)\$3 = z"
```
```  1301   unfolding vector_def by simp_all
```
```  1302
```
```  1303 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
```
```  1304   apply auto
```
```  1305   apply (erule_tac x="v\$1" in allE)
```
```  1306   apply (subgoal_tac "vector [v\$1] = v")
```
```  1307   apply simp
```
```  1308   apply (vector vector_def)
```
```  1309   apply simp
```
```  1310   done
```
```  1311
```
```  1312 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
```
```  1313   apply auto
```
```  1314   apply (erule_tac x="v\$1" in allE)
```
```  1315   apply (erule_tac x="v\$2" in allE)
```
```  1316   apply (subgoal_tac "vector [v\$1, v\$2] = v")
```
```  1317   apply simp
```
```  1318   apply (vector vector_def)
```
```  1319   apply (simp add: forall_2)
```
```  1320   done
```
```  1321
```
```  1322 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
```
```  1323   apply auto
```
```  1324   apply (erule_tac x="v\$1" in allE)
```
```  1325   apply (erule_tac x="v\$2" in allE)
```
```  1326   apply (erule_tac x="v\$3" in allE)
```
```  1327   apply (subgoal_tac "vector [v\$1, v\$2, v\$3] = v")
```
```  1328   apply simp
```
```  1329   apply (vector vector_def)
```
```  1330   apply (simp add: forall_3)
```
```  1331   done
```
```  1332
```
```  1333 lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x \$ k)"
```
```  1334   apply (rule bounded_linearI[where K=1])
```
```  1335   using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
```
```  1336
```
```  1337 lemma integral_component_eq_cart[simp]:
```
```  1338   fixes f :: "'n::ordered_euclidean_space \<Rightarrow> real^'m"
```
```  1339   assumes "f integrable_on s"
```
```  1340   shows "integral s (\<lambda>x. f x \$ k) = integral s f \$ k"
```
```  1341   using integral_linear[OF assms(1) bounded_linear_component_cart,unfolded o_def] .
```
```  1342
```
```  1343 lemma interval_split_cart:
```
```  1344   "{a..b::real^'n} \<inter> {x. x\$k \<le> c} = {a .. (\<chi> i. if i = k then min (b\$k) c else b\$i)}"
```
```  1345   "{a..b} \<inter> {x. x\$k \<ge> c} = {(\<chi> i. if i = k then max (a\$k) c else a\$i) .. b}"
```
```  1346   apply (rule_tac[!] set_eqI)
```
```  1347   unfolding Int_iff mem_interval_cart mem_Collect_eq
```
```  1348   unfolding vec_lambda_beta
```
```  1349   by auto
```
```  1350
```
```  1351 lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "\<forall>i. a\$i < b\$i \<and> u\$i < v\$i"
```
```  1352   shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
```
```  1353   using assms by (intro interval_bij_bij) (auto simp: Basis_vec_def inner_axis)
```
```  1354
```
```  1355 end
```