src/HOL/Library/Euclidean_Space.thy
author huffman
Tue Jun 02 19:42:44 2009 -0700 (2009-06-02)
changeset 31398 b67a3ac4882d
parent 31389 3affcbc60c6d
child 31399 d9769f093160
permissions -rw-r--r--
generalize lemma norm_triangle_sub
     1 (* Title:      Library/Euclidean_Space
     2    Author:     Amine Chaieb, University of Cambridge
     3 *)
     4 
     5 header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
     6 
     7 theory Euclidean_Space
     8 imports
     9   Complex_Main "~~/src/HOL/Decision_Procs/Dense_Linear_Order"
    10   Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type
    11   Inner_Product
    12 uses "positivstellensatz.ML" ("normarith.ML")
    13 begin
    14 
    15 text{* Some common special cases.*}
    16 
    17 lemma forall_1: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
    18   by (metis num1_eq_iff)
    19 
    20 lemma exhaust_2:
    21   fixes x :: 2 shows "x = 1 \<or> x = 2"
    22 proof (induct x)
    23   case (of_int z)
    24   then have "0 <= z" and "z < 2" by simp_all
    25   then have "z = 0 | z = 1" by arith
    26   then show ?case by auto
    27 qed
    28 
    29 lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
    30   by (metis exhaust_2)
    31 
    32 lemma exhaust_3:
    33   fixes x :: 3 shows "x = 1 \<or> x = 2 \<or> x = 3"
    34 proof (induct x)
    35   case (of_int z)
    36   then have "0 <= z" and "z < 3" by simp_all
    37   then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
    38   then show ?case by auto
    39 qed
    40 
    41 lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
    42   by (metis exhaust_3)
    43 
    44 lemma UNIV_1: "UNIV = {1::1}"
    45   by (auto simp add: num1_eq_iff)
    46 
    47 lemma UNIV_2: "UNIV = {1::2, 2::2}"
    48   using exhaust_2 by auto
    49 
    50 lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
    51   using exhaust_3 by auto
    52 
    53 lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
    54   unfolding UNIV_1 by simp
    55 
    56 lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
    57   unfolding UNIV_2 by simp
    58 
    59 lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
    60   unfolding UNIV_3 by (simp add: add_ac)
    61 
    62 subsection{* Basic componentwise operations on vectors. *}
    63 
    64 instantiation "^" :: (plus,type) plus
    65 begin
    66 definition  vector_add_def : "op + \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) + (y$i)))"
    67 instance ..
    68 end
    69 
    70 instantiation "^" :: (times,type) times
    71 begin
    72   definition vector_mult_def : "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
    73   instance ..
    74 end
    75 
    76 instantiation "^" :: (minus,type) minus begin
    77   definition vector_minus_def : "op - \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) - (y$i)))"
    78 instance ..
    79 end
    80 
    81 instantiation "^" :: (uminus,type) uminus begin
    82   definition vector_uminus_def : "uminus \<equiv> (\<lambda> x.  (\<chi> i. - (x$i)))"
    83 instance ..
    84 end
    85 instantiation "^" :: (zero,type) zero begin
    86   definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)"
    87 instance ..
    88 end
    89 
    90 instantiation "^" :: (one,type) one begin
    91   definition vector_one_def : "1 \<equiv> (\<chi> i. 1)"
    92 instance ..
    93 end
    94 
    95 instantiation "^" :: (ord,type) ord
    96  begin
    97 definition vector_less_eq_def:
    98   "less_eq (x :: 'a ^'b) y = (ALL i. x$i <= y$i)"
    99 definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i. x$i < y$i)"
   100 
   101 instance by (intro_classes)
   102 end
   103 
   104 instantiation "^" :: (scaleR, type) scaleR
   105 begin
   106 definition vector_scaleR_def: "scaleR = (\<lambda> r x.  (\<chi> i. scaleR r (x$i)))"
   107 instance ..
   108 end
   109 
   110 text{* Also the scalar-vector multiplication. *}
   111 
   112 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
   113   where "c *s x = (\<chi> i. c * (x$i))"
   114 
   115 text{* Constant Vectors *} 
   116 
   117 definition "vec x = (\<chi> i. x)"
   118 
   119 text{* Dot products. *}
   120 
   121 definition dot :: "'a::{comm_monoid_add, times} ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" (infix "\<bullet>" 70) where
   122   "x \<bullet> y = setsum (\<lambda>i. x$i * y$i) UNIV"
   123 
   124 lemma dot_1[simp]: "(x::'a::{comm_monoid_add, times}^1) \<bullet> y = (x$1) * (y$1)"
   125   by (simp add: dot_def setsum_1)
   126 
   127 lemma dot_2[simp]: "(x::'a::{comm_monoid_add, times}^2) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2)"
   128   by (simp add: dot_def setsum_2)
   129 
   130 lemma dot_3[simp]: "(x::'a::{comm_monoid_add, times}^3) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2) + (x$3) * (y$3)"
   131   by (simp add: dot_def setsum_3)
   132 
   133 subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
   134 
   135 method_setup vector = {*
   136 let
   137   val ss1 = HOL_basic_ss addsimps [@{thm dot_def}, @{thm setsum_addf} RS sym,
   138   @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
   139   @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
   140   val ss2 = @{simpset} addsimps
   141              [@{thm vector_add_def}, @{thm vector_mult_def},
   142               @{thm vector_minus_def}, @{thm vector_uminus_def},
   143               @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def},
   144               @{thm vector_scaleR_def},
   145               @{thm Cart_lambda_beta}, @{thm vector_scalar_mult_def}]
   146  fun vector_arith_tac ths =
   147    simp_tac ss1
   148    THEN' (fn i => rtac @{thm setsum_cong2} i
   149          ORELSE rtac @{thm setsum_0'} i
   150          ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i)
   151    (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
   152    THEN' asm_full_simp_tac (ss2 addsimps ths)
   153  in
   154   Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths)))
   155  end
   156 *} "Lifts trivial vector statements to real arith statements"
   157 
   158 lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def)
   159 lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def)
   160 
   161 
   162 
   163 text{* Obvious "component-pushing". *}
   164 
   165 lemma vec_component [simp]: "(vec x :: 'a ^ 'n)$i = x"
   166   by (vector vec_def)
   167 
   168 lemma vector_add_component [simp]:
   169   fixes x y :: "'a::{plus} ^ 'n"
   170   shows "(x + y)$i = x$i + y$i"
   171   by vector
   172 
   173 lemma vector_minus_component [simp]:
   174   fixes x y :: "'a::{minus} ^ 'n"
   175   shows "(x - y)$i = x$i - y$i"
   176   by vector
   177 
   178 lemma vector_mult_component [simp]:
   179   fixes x y :: "'a::{times} ^ 'n"
   180   shows "(x * y)$i = x$i * y$i"
   181   by vector
   182 
   183 lemma vector_smult_component [simp]:
   184   fixes y :: "'a::{times} ^ 'n"
   185   shows "(c *s y)$i = c * (y$i)"
   186   by vector
   187 
   188 lemma vector_uminus_component [simp]:
   189   fixes x :: "'a::{uminus} ^ 'n"
   190   shows "(- x)$i = - (x$i)"
   191   by vector
   192 
   193 lemma vector_scaleR_component [simp]:
   194   fixes x :: "'a::scaleR ^ 'n"
   195   shows "(scaleR r x)$i = scaleR r (x$i)"
   196   by vector
   197 
   198 lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
   199 
   200 lemmas vector_component =
   201   vec_component vector_add_component vector_mult_component
   202   vector_smult_component vector_minus_component vector_uminus_component
   203   vector_scaleR_component cond_component
   204 
   205 subsection {* Some frequently useful arithmetic lemmas over vectors. *}
   206 
   207 instance "^" :: (semigroup_add,type) semigroup_add
   208   apply (intro_classes) by (vector add_assoc)
   209 
   210 
   211 instance "^" :: (monoid_add,type) monoid_add
   212   apply (intro_classes) by vector+
   213 
   214 instance "^" :: (group_add,type) group_add
   215   apply (intro_classes) by (vector algebra_simps)+
   216 
   217 instance "^" :: (ab_semigroup_add,type) ab_semigroup_add
   218   apply (intro_classes) by (vector add_commute)
   219 
   220 instance "^" :: (comm_monoid_add,type) comm_monoid_add
   221   apply (intro_classes) by vector
   222 
   223 instance "^" :: (ab_group_add,type) ab_group_add
   224   apply (intro_classes) by vector+
   225 
   226 instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add
   227   apply (intro_classes)
   228   by (vector Cart_eq)+
   229 
   230 instance "^" :: (cancel_ab_semigroup_add,type) cancel_ab_semigroup_add
   231   apply (intro_classes)
   232   by (vector Cart_eq)
   233 
   234 instance "^" :: (real_vector, type) real_vector
   235   by default (vector scaleR_left_distrib scaleR_right_distrib)+
   236 
   237 instance "^" :: (semigroup_mult,type) semigroup_mult
   238   apply (intro_classes) by (vector mult_assoc)
   239 
   240 instance "^" :: (monoid_mult,type) monoid_mult
   241   apply (intro_classes) by vector+
   242 
   243 instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult
   244   apply (intro_classes) by (vector mult_commute)
   245 
   246 instance "^" :: (ab_semigroup_idem_mult,type) ab_semigroup_idem_mult
   247   apply (intro_classes) by (vector mult_idem)
   248 
   249 instance "^" :: (comm_monoid_mult,type) comm_monoid_mult
   250   apply (intro_classes) by vector
   251 
   252 fun vector_power :: "('a::{one,times} ^'n) \<Rightarrow> nat \<Rightarrow> 'a^'n" where
   253   "vector_power x 0 = 1"
   254   | "vector_power x (Suc n) = x * vector_power x n"
   255 
   256 instance "^" :: (semiring,type) semiring
   257   apply (intro_classes) by (vector ring_simps)+
   258 
   259 instance "^" :: (semiring_0,type) semiring_0
   260   apply (intro_classes) by (vector ring_simps)+
   261 instance "^" :: (semiring_1,type) semiring_1
   262   apply (intro_classes) by vector
   263 instance "^" :: (comm_semiring,type) comm_semiring
   264   apply (intro_classes) by (vector ring_simps)+
   265 
   266 instance "^" :: (comm_semiring_0,type) comm_semiring_0 by (intro_classes)
   267 instance "^" :: (cancel_comm_monoid_add, type) cancel_comm_monoid_add ..
   268 instance "^" :: (semiring_0_cancel,type) semiring_0_cancel by (intro_classes)
   269 instance "^" :: (comm_semiring_0_cancel,type) comm_semiring_0_cancel by (intro_classes)
   270 instance "^" :: (ring,type) ring by (intro_classes)
   271 instance "^" :: (semiring_1_cancel,type) semiring_1_cancel by (intro_classes)
   272 instance "^" :: (comm_semiring_1,type) comm_semiring_1 by (intro_classes)
   273 
   274 instance "^" :: (ring_1,type) ring_1 ..
   275 
   276 instance "^" :: (real_algebra,type) real_algebra
   277   apply intro_classes
   278   apply (simp_all add: vector_scaleR_def ring_simps)
   279   apply vector
   280   apply vector
   281   done
   282 
   283 instance "^" :: (real_algebra_1,type) real_algebra_1 ..
   284 
   285 lemma of_nat_index:
   286   "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
   287   apply (induct n)
   288   apply vector
   289   apply vector
   290   done
   291 lemma zero_index[simp]:
   292   "(0 :: 'a::zero ^'n)$i = 0" by vector
   293 
   294 lemma one_index[simp]:
   295   "(1 :: 'a::one ^'n)$i = 1" by vector
   296 
   297 lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \<noteq> 0"
   298 proof-
   299   have "(1::'a) + of_nat n = 0 \<longleftrightarrow> of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp
   300   also have "\<dots> \<longleftrightarrow> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff)
   301   finally show ?thesis by simp
   302 qed
   303 
   304 instance "^" :: (semiring_char_0,type) semiring_char_0
   305 proof (intro_classes)
   306   fix m n ::nat
   307   show "(of_nat m :: 'a^'b) = of_nat n \<longleftrightarrow> m = n"
   308     by (simp add: Cart_eq of_nat_index)
   309 qed
   310 
   311 instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes
   312 instance "^" :: (ring_char_0,type) ring_char_0 by intro_classes
   313 
   314 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
   315   by (vector mult_assoc)
   316 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
   317   by (vector ring_simps)
   318 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
   319   by (vector ring_simps)
   320 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
   321 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
   322 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
   323   by (vector ring_simps)
   324 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
   325 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
   326 lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
   327 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
   328 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
   329   by (vector ring_simps)
   330 
   331 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
   332   by (simp add: Cart_eq)
   333 
   334 subsection {* Square root of sum of squares *}
   335 
   336 definition
   337   "setL2 f A = sqrt (\<Sum>i\<in>A. (f i)\<twosuperior>)"
   338 
   339 lemma setL2_cong:
   340   "\<lbrakk>A = B; \<And>x. x \<in> B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
   341   unfolding setL2_def by simp
   342 
   343 lemma strong_setL2_cong:
   344   "\<lbrakk>A = B; \<And>x. x \<in> B =simp=> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
   345   unfolding setL2_def simp_implies_def by simp
   346 
   347 lemma setL2_infinite [simp]: "\<not> finite A \<Longrightarrow> setL2 f A = 0"
   348   unfolding setL2_def by simp
   349 
   350 lemma setL2_empty [simp]: "setL2 f {} = 0"
   351   unfolding setL2_def by simp
   352 
   353 lemma setL2_insert [simp]:
   354   "\<lbrakk>finite F; a \<notin> F\<rbrakk> \<Longrightarrow>
   355     setL2 f (insert a F) = sqrt ((f a)\<twosuperior> + (setL2 f F)\<twosuperior>)"
   356   unfolding setL2_def by (simp add: setsum_nonneg)
   357 
   358 lemma setL2_nonneg [simp]: "0 \<le> setL2 f A"
   359   unfolding setL2_def by (simp add: setsum_nonneg)
   360 
   361 lemma setL2_0': "\<forall>a\<in>A. f a = 0 \<Longrightarrow> setL2 f A = 0"
   362   unfolding setL2_def by simp
   363 
   364 lemma setL2_mono:
   365   assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i"
   366   assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"
   367   shows "setL2 f K \<le> setL2 g K"
   368   unfolding setL2_def
   369   by (simp add: setsum_nonneg setsum_mono power_mono prems)
   370 
   371 lemma setL2_right_distrib:
   372   "0 \<le> r \<Longrightarrow> r * setL2 f A = setL2 (\<lambda>x. r * f x) A"
   373   unfolding setL2_def
   374   apply (simp add: power_mult_distrib)
   375   apply (simp add: setsum_right_distrib [symmetric])
   376   apply (simp add: real_sqrt_mult setsum_nonneg)
   377   done
   378 
   379 lemma setL2_left_distrib:
   380   "0 \<le> r \<Longrightarrow> setL2 f A * r = setL2 (\<lambda>x. f x * r) A"
   381   unfolding setL2_def
   382   apply (simp add: power_mult_distrib)
   383   apply (simp add: setsum_left_distrib [symmetric])
   384   apply (simp add: real_sqrt_mult setsum_nonneg)
   385   done
   386 
   387 lemma setsum_nonneg_eq_0_iff:
   388   fixes f :: "'a \<Rightarrow> 'b::pordered_ab_group_add"
   389   shows "\<lbrakk>finite A; \<forall>x\<in>A. 0 \<le> f x\<rbrakk> \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
   390   apply (induct set: finite, simp)
   391   apply (simp add: add_nonneg_eq_0_iff setsum_nonneg)
   392   done
   393 
   394 lemma setL2_eq_0_iff: "finite A \<Longrightarrow> setL2 f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
   395   unfolding setL2_def
   396   by (simp add: setsum_nonneg setsum_nonneg_eq_0_iff)
   397 
   398 lemma setL2_triangle_ineq:
   399   shows "setL2 (\<lambda>i. f i + g i) A \<le> setL2 f A + setL2 g A"
   400 proof (cases "finite A")
   401   case False
   402   thus ?thesis by simp
   403 next
   404   case True
   405   thus ?thesis
   406   proof (induct set: finite)
   407     case empty
   408     show ?case by simp
   409   next
   410     case (insert x F)
   411     hence "sqrt ((f x + g x)\<twosuperior> + (setL2 (\<lambda>i. f i + g i) F)\<twosuperior>) \<le>
   412            sqrt ((f x + g x)\<twosuperior> + (setL2 f F + setL2 g F)\<twosuperior>)"
   413       by (intro real_sqrt_le_mono add_left_mono power_mono insert
   414                 setL2_nonneg add_increasing zero_le_power2)
   415     also have
   416       "\<dots> \<le> sqrt ((f x)\<twosuperior> + (setL2 f F)\<twosuperior>) + sqrt ((g x)\<twosuperior> + (setL2 g F)\<twosuperior>)"
   417       by (rule real_sqrt_sum_squares_triangle_ineq)
   418     finally show ?case
   419       using insert by simp
   420   qed
   421 qed
   422 
   423 lemma sqrt_sum_squares_le_sum:
   424   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) \<le> x + y"
   425   apply (rule power2_le_imp_le)
   426   apply (simp add: power2_sum)
   427   apply (simp add: mult_nonneg_nonneg)
   428   apply (simp add: add_nonneg_nonneg)
   429   done
   430 
   431 lemma setL2_le_setsum [rule_format]:
   432   "(\<forall>i\<in>A. 0 \<le> f i) \<longrightarrow> setL2 f A \<le> setsum f A"
   433   apply (cases "finite A")
   434   apply (induct set: finite)
   435   apply simp
   436   apply clarsimp
   437   apply (erule order_trans [OF sqrt_sum_squares_le_sum])
   438   apply simp
   439   apply simp
   440   apply simp
   441   done
   442 
   443 lemma sqrt_sum_squares_le_sum_abs: "sqrt (x\<twosuperior> + y\<twosuperior>) \<le> \<bar>x\<bar> + \<bar>y\<bar>"
   444   apply (rule power2_le_imp_le)
   445   apply (simp add: power2_sum)
   446   apply (simp add: mult_nonneg_nonneg)
   447   apply (simp add: add_nonneg_nonneg)
   448   done
   449 
   450 lemma setL2_le_setsum_abs: "setL2 f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
   451   apply (cases "finite A")
   452   apply (induct set: finite)
   453   apply simp
   454   apply simp
   455   apply (rule order_trans [OF sqrt_sum_squares_le_sum_abs])
   456   apply simp
   457   apply simp
   458   done
   459 
   460 lemma setL2_mult_ineq_lemma:
   461   fixes a b c d :: real
   462   shows "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
   463 proof -
   464   have "0 \<le> (a * d - b * c)\<twosuperior>" by simp
   465   also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * d) * (b * c)"
   466     by (simp only: power2_diff power_mult_distrib)
   467   also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * c) * (b * d)"
   468     by simp
   469   finally show "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
   470     by simp
   471 qed
   472 
   473 lemma setL2_mult_ineq: "(\<Sum>i\<in>A. \<bar>f i\<bar> * \<bar>g i\<bar>) \<le> setL2 f A * setL2 g A"
   474   apply (cases "finite A")
   475   apply (induct set: finite)
   476   apply simp
   477   apply (rule power2_le_imp_le, simp)
   478   apply (rule order_trans)
   479   apply (rule power_mono)
   480   apply (erule add_left_mono)
   481   apply (simp add: add_nonneg_nonneg mult_nonneg_nonneg setsum_nonneg)
   482   apply (simp add: power2_sum)
   483   apply (simp add: power_mult_distrib)
   484   apply (simp add: right_distrib left_distrib)
   485   apply (rule ord_le_eq_trans)
   486   apply (rule setL2_mult_ineq_lemma)
   487   apply simp
   488   apply (intro mult_nonneg_nonneg setL2_nonneg)
   489   apply simp
   490   done
   491 
   492 lemma member_le_setL2: "\<lbrakk>finite A; i \<in> A\<rbrakk> \<Longrightarrow> f i \<le> setL2 f A"
   493   apply (rule_tac s="insert i (A - {i})" and t="A" in subst)
   494   apply fast
   495   apply (subst setL2_insert)
   496   apply simp
   497   apply simp
   498   apply simp
   499   done
   500 
   501 subsection {* Metric *}
   502 
   503 instantiation "^" :: (metric_space, finite) metric_space
   504 begin
   505 
   506 definition dist_vector_def:
   507   "dist (x::'a^'b) (y::'a^'b) = setL2 (\<lambda>i. dist (x$i) (y$i)) UNIV"
   508 
   509 instance proof
   510   fix x y :: "'a ^ 'b"
   511   show "dist x y = 0 \<longleftrightarrow> x = y"
   512     unfolding dist_vector_def
   513     by (simp add: setL2_eq_0_iff Cart_eq)
   514 next
   515   fix x y z :: "'a ^ 'b"
   516   show "dist x y \<le> dist x z + dist y z"
   517     unfolding dist_vector_def
   518     apply (rule order_trans [OF _ setL2_triangle_ineq])
   519     apply (simp add: setL2_mono dist_triangle2)
   520     done
   521 qed
   522 
   523 end
   524 
   525 lemma dist_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
   526 unfolding dist_vector_def
   527 by (rule member_le_setL2) simp_all
   528 
   529 lemma tendsto_Cart_nth:
   530   fixes X :: "'a \<Rightarrow> 'b::metric_space ^ 'n::finite"
   531   assumes "tendsto (\<lambda>n. X n) a net"
   532   shows "tendsto (\<lambda>n. X n $ i) (a $ i) net"
   533 proof (rule tendstoI)
   534   fix e :: real assume "0 < e"
   535   with assms have "eventually (\<lambda>n. dist (X n) a < e) net"
   536     by (rule tendstoD)
   537   thus "eventually (\<lambda>n. dist (X n $ i) (a $ i) < e) net"
   538   proof (rule eventually_elim1)
   539     fix n :: 'a
   540     have "dist (X n $ i) (a $ i) \<le> dist (X n) a"
   541       by (rule dist_nth_le)
   542     also assume "dist (X n) a < e"
   543     finally show "dist (X n $ i) (a $ i) < e" .
   544   qed
   545 qed
   546 
   547 lemma LIMSEQ_Cart_nth:
   548   "(X ----> a) \<Longrightarrow> (\<lambda>n. X n $ i) ----> a $ i"
   549 unfolding LIMSEQ_conv_tendsto by (rule tendsto_Cart_nth)
   550 
   551 lemma LIM_Cart_nth:
   552   "(f -- x --> y) \<Longrightarrow> (\<lambda>x. f x $ i) -- x --> y $ i"
   553 unfolding LIM_conv_tendsto by (rule tendsto_Cart_nth)
   554 
   555 lemma Cauchy_Cart_nth:
   556   fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n::finite"
   557   assumes "Cauchy (\<lambda>n. X n)"
   558   shows "Cauchy (\<lambda>n. X n $ i)"
   559 proof (rule metric_CauchyI)
   560   fix e :: real assume "0 < e"
   561   obtain M where "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
   562     using metric_CauchyD [OF `Cauchy X` `0 < e`] by fast
   563   moreover
   564   {
   565     fix m n
   566     assume "M \<le> m" "M \<le> n"
   567     have "dist (X m $ i) (X n $ i) \<le> dist (X m) (X n)"
   568       by (rule dist_nth_le)
   569     also assume "dist (X m) (X n) < e"
   570     finally have "dist (X m $ i) (X n $ i) < e" .
   571   }
   572   ultimately
   573   have "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m $ i) (X n $ i) < e" by fast
   574   thus "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m $ i) (X n $ i) < e" ..
   575 qed
   576 
   577 lemma LIMSEQ_vector:
   578   fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n::finite"
   579   assumes X: "\<And>i. (\<lambda>n. X n $ i) ----> (a $ i)"
   580   shows "X ----> a"
   581 proof (rule metric_LIMSEQ_I)
   582   fix r :: real assume "0 < r"
   583   then have "0 < r / of_nat CARD('n)" (is "0 < ?s")
   584     by (simp add: divide_pos_pos)
   585   def N \<equiv> "\<lambda>i. LEAST N. \<forall>n\<ge>N. dist (X n $ i) (a $ i) < ?s"
   586   def M \<equiv> "Max (range N)"
   587   have "\<And>i. \<exists>N. \<forall>n\<ge>N. dist (X n $ i) (a $ i) < ?s"
   588     using X `0 < ?s` by (rule metric_LIMSEQ_D)
   589   hence "\<And>i. \<forall>n\<ge>N i. dist (X n $ i) (a $ i) < ?s"
   590     unfolding N_def by (rule LeastI_ex)
   591   hence M: "\<And>i. \<forall>n\<ge>M. dist (X n $ i) (a $ i) < ?s"
   592     unfolding M_def by simp
   593   {
   594     fix n :: nat assume "M \<le> n"
   595     have "dist (X n) a = setL2 (\<lambda>i. dist (X n $ i) (a $ i)) UNIV"
   596       unfolding dist_vector_def ..
   597     also have "\<dots> \<le> setsum (\<lambda>i. dist (X n $ i) (a $ i)) UNIV"
   598       by (rule setL2_le_setsum [OF zero_le_dist])
   599     also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
   600       by (rule setsum_strict_mono, simp_all add: M `M \<le> n`)
   601     also have "\<dots> = r"
   602       by simp
   603     finally have "dist (X n) a < r" .
   604   }
   605   hence "\<forall>n\<ge>M. dist (X n) a < r"
   606     by simp
   607   then show "\<exists>M. \<forall>n\<ge>M. dist (X n) a < r" ..
   608 qed
   609 
   610 lemma Cauchy_vector:
   611   fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n::finite"
   612   assumes X: "\<And>i. Cauchy (\<lambda>n. X n $ i)"
   613   shows "Cauchy (\<lambda>n. X n)"
   614 proof (rule metric_CauchyI)
   615   fix r :: real assume "0 < r"
   616   then have "0 < r / of_nat CARD('n)" (is "0 < ?s")
   617     by (simp add: divide_pos_pos)
   618   def N \<equiv> "\<lambda>i. LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"
   619   def M \<equiv> "Max (range N)"
   620   have "\<And>i. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"
   621     using X `0 < ?s` by (rule metric_CauchyD)
   622   hence "\<And>i. \<forall>m\<ge>N i. \<forall>n\<ge>N i. dist (X m $ i) (X n $ i) < ?s"
   623     unfolding N_def by (rule LeastI_ex)
   624   hence M: "\<And>i. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m $ i) (X n $ i) < ?s"
   625     unfolding M_def by simp
   626   {
   627     fix m n :: nat
   628     assume "M \<le> m" "M \<le> n"
   629     have "dist (X m) (X n) = setL2 (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
   630       unfolding dist_vector_def ..
   631     also have "\<dots> \<le> setsum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
   632       by (rule setL2_le_setsum [OF zero_le_dist])
   633     also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
   634       by (rule setsum_strict_mono, simp_all add: M `M \<le> m` `M \<le> n`)
   635     also have "\<dots> = r"
   636       by simp
   637     finally have "dist (X m) (X n) < r" .
   638   }
   639   hence "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r"
   640     by simp
   641   then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
   642 qed
   643 
   644 subsection {* Norms *}
   645 
   646 instantiation "^" :: (real_normed_vector, finite) real_normed_vector
   647 begin
   648 
   649 definition vector_norm_def:
   650   "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) UNIV"
   651 
   652 definition vector_sgn_def:
   653   "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
   654 
   655 instance proof
   656   fix a :: real and x y :: "'a ^ 'b"
   657   show "0 \<le> norm x"
   658     unfolding vector_norm_def
   659     by (rule setL2_nonneg)
   660   show "norm x = 0 \<longleftrightarrow> x = 0"
   661     unfolding vector_norm_def
   662     by (simp add: setL2_eq_0_iff Cart_eq)
   663   show "norm (x + y) \<le> norm x + norm y"
   664     unfolding vector_norm_def
   665     apply (rule order_trans [OF _ setL2_triangle_ineq])
   666     apply (simp add: setL2_mono norm_triangle_ineq)
   667     done
   668   show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
   669     unfolding vector_norm_def
   670     by (simp add: norm_scaleR setL2_right_distrib)
   671   show "sgn x = scaleR (inverse (norm x)) x"
   672     by (rule vector_sgn_def)
   673   show "dist x y = norm (x - y)"
   674     unfolding dist_vector_def vector_norm_def
   675     by (simp add: dist_norm)
   676 qed
   677 
   678 end
   679 
   680 lemma norm_nth_le: "norm (x $ i) \<le> norm x"
   681 unfolding vector_norm_def
   682 by (rule member_le_setL2) simp_all
   683 
   684 interpretation Cart_nth: bounded_linear "\<lambda>x. x $ i"
   685 apply default
   686 apply (rule vector_add_component)
   687 apply (rule vector_scaleR_component)
   688 apply (rule_tac x="1" in exI, simp add: norm_nth_le)
   689 done
   690 
   691 subsection {* Inner products *}
   692 
   693 instantiation "^" :: (real_inner, finite) real_inner
   694 begin
   695 
   696 definition vector_inner_def:
   697   "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
   698 
   699 instance proof
   700   fix r :: real and x y z :: "'a ^ 'b"
   701   show "inner x y = inner y x"
   702     unfolding vector_inner_def
   703     by (simp add: inner_commute)
   704   show "inner (x + y) z = inner x z + inner y z"
   705     unfolding vector_inner_def
   706     by (simp add: inner_left_distrib setsum_addf)
   707   show "inner (scaleR r x) y = r * inner x y"
   708     unfolding vector_inner_def
   709     by (simp add: inner_scaleR_left setsum_right_distrib)
   710   show "0 \<le> inner x x"
   711     unfolding vector_inner_def
   712     by (simp add: setsum_nonneg)
   713   show "inner x x = 0 \<longleftrightarrow> x = 0"
   714     unfolding vector_inner_def
   715     by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
   716   show "norm x = sqrt (inner x x)"
   717     unfolding vector_inner_def vector_norm_def setL2_def
   718     by (simp add: power2_norm_eq_inner)
   719 qed
   720 
   721 end
   722 
   723 subsection{* Properties of the dot product.  *}
   724 
   725 lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x"
   726   by (vector mult_commute)
   727 lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \<bullet> z = (x \<bullet> z) + (y \<bullet> z)"
   728   by (vector ring_simps)
   729 lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)"
   730   by (vector ring_simps)
   731 lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)"
   732   by (vector ring_simps)
   733 lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)"
   734   by (vector ring_simps)
   735 lemma dot_lmult: "(c *s x) \<bullet> y = (c::'a::ring) * (x \<bullet> y)" by (vector ring_simps)
   736 lemma dot_rmult: "x \<bullet> (c *s y) = (c::'a::comm_ring) * (x \<bullet> y)" by (vector ring_simps)
   737 lemma dot_lneg: "(-x) \<bullet> (y::'a::ring ^ 'n) = -(x \<bullet> y)" by vector
   738 lemma dot_rneg: "(x::'a::ring ^ 'n) \<bullet> (-y) = -(x \<bullet> y)" by vector
   739 lemma dot_lzero[simp]: "0 \<bullet> x = (0::'a::{comm_monoid_add, mult_zero})" by vector
   740 lemma dot_rzero[simp]: "x \<bullet> 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector
   741 lemma dot_pos_le[simp]: "(0::'a\<Colon>ordered_ring_strict) <= x \<bullet> x"
   742   by (simp add: dot_def setsum_nonneg)
   743 
   744 lemma setsum_squares_eq_0_iff: assumes fS: "finite F" and fp: "\<forall>x \<in> F. f x \<ge> (0 ::'a::pordered_ab_group_add)" shows "setsum f F = 0 \<longleftrightarrow> (ALL x:F. f x = 0)"
   745 using fS fp setsum_nonneg[OF fp]
   746 proof (induct set: finite)
   747   case empty thus ?case by simp
   748 next
   749   case (insert x F)
   750   from insert.prems have Fx: "f x \<ge> 0" and Fp: "\<forall> a \<in> F. f a \<ge> 0" by simp_all
   751   from insert.hyps Fp setsum_nonneg[OF Fp]
   752   have h: "setsum f F = 0 \<longleftrightarrow> (\<forall>a \<in>F. f a = 0)" by metis
   753   from add_nonneg_eq_0_iff[OF Fx  setsum_nonneg[OF Fp]] insert.hyps(1,2)
   754   show ?case by (simp add: h)
   755 qed
   756 
   757 lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n::finite) = 0"
   758   by (simp add: dot_def setsum_squares_eq_0_iff Cart_eq)
   759 
   760 lemma dot_pos_lt[simp]: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n::finite) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x]
   761   by (auto simp add: le_less)
   762 
   763 subsection{* The collapse of the general concepts to dimension one. *}
   764 
   765 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
   766   by (simp add: Cart_eq forall_1)
   767 
   768 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
   769   apply auto
   770   apply (erule_tac x= "x$1" in allE)
   771   apply (simp only: vector_one[symmetric])
   772   done
   773 
   774 lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
   775   by (simp add: vector_norm_def UNIV_1)
   776 
   777 lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
   778   by (simp add: norm_vector_1)
   779 
   780 lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
   781   by (auto simp add: norm_real dist_norm)
   782 
   783 subsection {* A connectedness or intermediate value lemma with several applications. *}
   784 
   785 lemma connected_real_lemma:
   786   fixes f :: "real \<Rightarrow> real ^ 'n::finite"
   787   assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
   788   and dst: "\<And>e x. a <= x \<Longrightarrow> x <= b \<Longrightarrow> 0 < e ==> \<exists>d > 0. \<forall>y. abs(y - x) < d \<longrightarrow> dist(f y) (f x) < e"
   789   and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
   790   and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
   791   and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
   792   shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
   793 proof-
   794   let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
   795   have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
   796   have Sub: "\<exists>y. isUb UNIV ?S y"
   797     apply (rule exI[where x= b])
   798     using ab fb e12 by (auto simp add: isUb_def setle_def)
   799   from reals_complete[OF Se Sub] obtain l where
   800     l: "isLub UNIV ?S l"by blast
   801   have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
   802     apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
   803     by (metis linorder_linear)
   804   have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
   805     apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
   806     by (metis linorder_linear not_le)
   807     have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
   808     have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
   809     have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo
   810     {assume le2: "f l \<in> e2"
   811       from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
   812       hence lap: "l - a > 0" using alb by arith
   813       from e2[rule_format, OF le2] obtain e where
   814 	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
   815       from dst[OF alb e(1)] obtain d where
   816 	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
   817       have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1)
   818 	apply ferrack by arith
   819       then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
   820       from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
   821       from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
   822       moreover
   823       have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
   824       ultimately have False using e12 alb d' by auto}
   825     moreover
   826     {assume le1: "f l \<in> e1"
   827     from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
   828       hence blp: "b - l > 0" using alb by arith
   829       from e1[rule_format, OF le1] obtain e where
   830 	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
   831       from dst[OF alb e(1)] obtain d where
   832 	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
   833       have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo
   834       then obtain d' where d': "d' > 0" "d' < d" by metis
   835       from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
   836       hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
   837       with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
   838       with l d' have False
   839 	by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
   840     ultimately show ?thesis using alb by metis
   841 qed
   842 
   843 text{* One immediately useful corollary is the existence of square roots! --- Should help to get rid of all the development of square-root for reals as a special case @{typ "real^1"} *}
   844 
   845 lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
   846 proof-
   847   have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
   848   thus ?thesis by (simp add: ring_simps power2_eq_square)
   849 qed
   850 
   851 lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
   852   using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_eq, rule_format, of e x] apply (auto simp add: power2_eq_square)
   853   apply (rule_tac x="s" in exI)
   854   apply auto
   855   apply (erule_tac x=y in allE)
   856   apply auto
   857   done
   858 
   859 lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
   860   using real_sqrt_le_iff[of x "y^2"] by simp
   861 
   862 lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
   863   using real_sqrt_le_mono[of "x^2" y] by simp
   864 
   865 lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
   866   using real_sqrt_less_mono[of "x^2" y] by simp
   867 
   868 lemma sqrt_even_pow2: assumes n: "even n"
   869   shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
   870 proof-
   871   from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2
   872     by (auto simp add: nat_number)
   873   from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
   874     by (simp only: power_mult[symmetric] mult_commute)
   875   then show ?thesis  using m by simp
   876 qed
   877 
   878 lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
   879   apply (cases "x = 0", simp_all)
   880   using sqrt_divide_self_eq[of x]
   881   apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps)
   882   done
   883 
   884 text{* Hence derive more interesting properties of the norm. *}
   885 
   886 text {*
   887   This type-specific version is only here
   888   to make @{text normarith.ML} happy.
   889 *}
   890 lemma norm_0: "norm (0::real ^ _) = 0"
   891   by (rule norm_zero)
   892 
   893 lemma norm_mul[simp]: "norm(a *s x) = abs(a) * norm x"
   894   by (simp add: vector_norm_def vector_component setL2_right_distrib
   895            abs_mult cong: strong_setL2_cong)
   896 lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
   897   by (simp add: vector_norm_def dot_def setL2_def power2_eq_square)
   898 lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"
   899   by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
   900 lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
   901   by (simp add: real_vector_norm_def)
   902 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n::finite)" by (metis norm_eq_zero)
   903 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
   904   by vector
   905 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
   906   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
   907 lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
   908   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
   909 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
   910   by (metis vector_mul_lcancel)
   911 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
   912   by (metis vector_mul_rcancel)
   913 lemma norm_cauchy_schwarz:
   914   fixes x y :: "real ^ 'n::finite"
   915   shows "x \<bullet> y <= norm x * norm y"
   916 proof-
   917   {assume "norm x = 0"
   918     hence ?thesis by (simp add: dot_lzero dot_rzero)}
   919   moreover
   920   {assume "norm y = 0"
   921     hence ?thesis by (simp add: dot_lzero dot_rzero)}
   922   moreover
   923   {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
   924     let ?z = "norm y *s x - norm x *s y"
   925     from h have p: "norm x * norm y > 0" by (metis norm_ge_zero le_less zero_compare_simps)
   926     from dot_pos_le[of ?z]
   927     have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"
   928       apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)
   929       by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym)
   930     hence "x\<bullet>y \<le> (norm x ^2 * norm y ^2) / (norm x * norm y)" using p
   931       by (simp add: field_simps)
   932     hence ?thesis using h by (simp add: power2_eq_square)}
   933   ultimately show ?thesis by metis
   934 qed
   935 
   936 lemma norm_cauchy_schwarz_abs:
   937   fixes x y :: "real ^ 'n::finite"
   938   shows "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
   939   using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
   940   by (simp add: real_abs_def dot_rneg)
   941 
   942 lemma norm_triangle_sub:
   943   fixes x y :: "'a::real_normed_vector"
   944   shows "norm x \<le> norm y  + norm (x - y)"
   945   using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
   946 
   947 lemma norm_triangle_le: "norm(x::real ^'n::finite) + norm y <= e ==> norm(x + y) <= e"
   948   by (metis order_trans norm_triangle_ineq)
   949 lemma norm_triangle_lt: "norm(x::real ^'n::finite) + norm(y) < e ==> norm(x + y) < e"
   950   by (metis basic_trans_rules(21) norm_triangle_ineq)
   951 
   952 lemma setsum_delta:
   953   assumes fS: "finite S"
   954   shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
   955 proof-
   956   let ?f = "(\<lambda>k. if k=a then b k else 0)"
   957   {assume a: "a \<notin> S"
   958     hence "\<forall> k\<in> S. ?f k = 0" by simp
   959     hence ?thesis  using a by simp}
   960   moreover
   961   {assume a: "a \<in> S"
   962     let ?A = "S - {a}"
   963     let ?B = "{a}"
   964     have eq: "S = ?A \<union> ?B" using a by blast
   965     have dj: "?A \<inter> ?B = {}" by simp
   966     from fS have fAB: "finite ?A" "finite ?B" by auto
   967     have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
   968       using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
   969       by simp
   970     then have ?thesis  using a by simp}
   971   ultimately show ?thesis by blast
   972 qed
   973 
   974 lemma component_le_norm: "\<bar>x$i\<bar> <= norm (x::real ^ 'n::finite)"
   975   apply (simp add: vector_norm_def)
   976   apply (rule member_le_setL2, simp_all)
   977   done
   978 
   979 lemma norm_bound_component_le: "norm(x::real ^ 'n::finite) <= e
   980                 ==> \<bar>x$i\<bar> <= e"
   981   by (metis component_le_norm order_trans)
   982 
   983 lemma norm_bound_component_lt: "norm(x::real ^ 'n::finite) < e
   984                 ==> \<bar>x$i\<bar> < e"
   985   by (metis component_le_norm basic_trans_rules(21))
   986 
   987 lemma norm_le_l1: "norm (x:: real ^'n::finite) <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
   988   by (simp add: vector_norm_def setL2_le_setsum)
   989 
   990 lemma real_abs_norm: "\<bar>norm x\<bar> = norm (x :: real ^ _)"
   991   by (rule abs_norm_cancel)
   992 lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n::finite) - norm y\<bar> <= norm(x - y)"
   993   by (rule norm_triangle_ineq3)
   994 lemma norm_le: "norm(x::real ^ _) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
   995   by (simp add: real_vector_norm_def)
   996 lemma norm_lt: "norm(x::real ^ _) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
   997   by (simp add: real_vector_norm_def)
   998 lemma norm_eq: "norm (x::real ^ _) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
   999   by (simp add: order_eq_iff norm_le)
  1000 lemma norm_eq_1: "norm(x::real ^ _) = 1 \<longleftrightarrow> x \<bullet> x = 1"
  1001   by (simp add: real_vector_norm_def)
  1002 
  1003 text{* Squaring equations and inequalities involving norms.  *}
  1004 
  1005 lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
  1006   by (simp add: real_vector_norm_def)
  1007 
  1008 lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
  1009   by (auto simp add: real_vector_norm_def)
  1010 
  1011 lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
  1012 proof-
  1013   have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: ring_simps power2_eq_square)
  1014   also have "\<dots> \<longleftrightarrow> \<bar>x\<bar> \<le> \<bar>y\<bar>" apply (simp add: zero_compare_simps real_abs_def not_less) by arith
  1015 finally show ?thesis ..
  1016 qed
  1017 
  1018 lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
  1019   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
  1020   using norm_ge_zero[of x]
  1021   apply arith
  1022   done
  1023 
  1024 lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
  1025   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
  1026   using norm_ge_zero[of x]
  1027   apply arith
  1028   done
  1029 
  1030 lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
  1031   by (metis not_le norm_ge_square)
  1032 lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
  1033   by (metis norm_le_square not_less)
  1034 
  1035 text{* Dot product in terms of the norm rather than conversely. *}
  1036 
  1037 lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
  1038   by (simp add: norm_pow_2 dot_ladd dot_radd dot_sym)
  1039 
  1040 lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
  1041   by (simp add: norm_pow_2 dot_ladd dot_radd dot_lsub dot_rsub dot_sym)
  1042 
  1043 
  1044 text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
  1045 
  1046 lemma vector_eq: "(x:: real ^ 'n::finite) = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y\<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
  1047 proof
  1048   assume "?lhs" then show ?rhs by simp
  1049 next
  1050   assume ?rhs
  1051   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y\<bullet> y = 0" by simp
  1052   hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
  1053     by (simp add: dot_rsub dot_lsub dot_sym)
  1054   then have "(x - y) \<bullet> (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub)
  1055   then show "x = y" by (simp add: dot_eq_0)
  1056 qed
  1057 
  1058 
  1059 subsection{* General linear decision procedure for normed spaces. *}
  1060 
  1061 lemma norm_cmul_rule_thm: "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(c *s x)"
  1062   apply (clarsimp simp add: norm_mul)
  1063   apply (rule mult_mono1)
  1064   apply simp_all
  1065   done
  1066 
  1067   (* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
  1068 lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n::finite) \<Longrightarrow> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)"
  1069   apply (rule norm_triangle_le) by simp
  1070 
  1071 lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"
  1072   by (simp add: ring_simps)
  1073 
  1074 lemma pth_1: "(x::real^'n) == 1 *s x" by (simp only: vector_smult_lid)
  1075 lemma pth_2: "x - (y::real^'n) == x + -y" by (atomize (full)) simp
  1076 lemma pth_3: "(-x::real^'n) == -1 *s x" by vector
  1077 lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+
  1078 lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector
  1079 lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps)
  1080 lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all
  1081 lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps)
  1082 lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z"
  1083   "c *s x + (d *s x + z) == (c + d) *s x + z"
  1084   "(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+
  1085 lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector
  1086 lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y"
  1087   "(c *s x + z) + d *s y == c *s x + (z + d *s y)"
  1088   "c *s x + (d *s y + z) == c *s x + (d *s y + z)"
  1089   "(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))"
  1090   by ((atomize (full)), vector)+
  1091 lemma pth_c: "(c::real) *s x + d *s y == d *s y + c *s x"
  1092   "(c *s x + z) + d *s y == d *s y + (c *s x + z)"
  1093   "c *s x + (d *s y + z) == d *s y + (c *s x + z)"
  1094   "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+
  1095 lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector
  1096 
  1097 lemma norm_imp_pos_and_ge: "norm (x::real ^ _) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
  1098   by (atomize) (auto simp add: norm_ge_zero)
  1099 
  1100 lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
  1101 
  1102 lemma norm_pths:
  1103   "(x::real ^'n::finite) = y \<longleftrightarrow> norm (x - y) \<le> 0"
  1104   "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
  1105   using norm_ge_zero[of "x - y"] by auto
  1106 
  1107 lemma vector_dist_norm:
  1108   fixes x y :: "real ^ _"
  1109   shows "dist x y = norm (x - y)"
  1110   by (rule dist_norm)
  1111 
  1112 use "normarith.ML"
  1113 
  1114 method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
  1115 *} "Proves simple linear statements about vector norms"
  1116 
  1117 
  1118 
  1119 text{* Hence more metric properties. *}
  1120 
  1121 lemma dist_triangle_alt:
  1122   fixes x y z :: "'a::metric_space"
  1123   shows "dist y z <= dist x y + dist x z"
  1124 using dist_triangle [of y z x] by (simp add: dist_commute)
  1125 
  1126 lemma dist_pos_lt:
  1127   fixes x y :: "'a::metric_space"
  1128   shows "x \<noteq> y ==> 0 < dist x y"
  1129 by (simp add: zero_less_dist_iff)
  1130 
  1131 lemma dist_nz:
  1132   fixes x y :: "'a::metric_space"
  1133   shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
  1134 by (simp add: zero_less_dist_iff)
  1135 
  1136 lemma dist_triangle_le:
  1137   fixes x y z :: "'a::metric_space"
  1138   shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
  1139 by (rule order_trans [OF dist_triangle2])
  1140 
  1141 lemma dist_triangle_lt:
  1142   fixes x y z :: "'a::metric_space"
  1143   shows "dist x z + dist y z < e ==> dist x y < e"
  1144 by (rule le_less_trans [OF dist_triangle2])
  1145 
  1146 lemma dist_triangle_half_l:
  1147   fixes x1 x2 y :: "'a::metric_space"
  1148   shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
  1149 by (rule dist_triangle_lt [where z=y], simp)
  1150 
  1151 lemma dist_triangle_half_r:
  1152   fixes x1 x2 y :: "'a::metric_space"
  1153   shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
  1154 by (rule dist_triangle_half_l, simp_all add: dist_commute)
  1155 
  1156 lemma dist_triangle_add:
  1157   fixes x y x' y' :: "'a::real_normed_vector"
  1158   shows "dist (x + y) (x' + y') <= dist x x' + dist y y'"
  1159 unfolding dist_norm by (rule norm_diff_triangle_ineq)
  1160 
  1161 lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
  1162   unfolding dist_norm vector_ssub_ldistrib[symmetric] norm_mul ..
  1163 
  1164 lemma dist_triangle_add_half:
  1165   fixes x x' y y' :: "'a::real_normed_vector"
  1166   shows "dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 \<Longrightarrow> dist(x + y) (x' + y') < e"
  1167 by (rule le_less_trans [OF dist_triangle_add], simp)
  1168 
  1169 lemma setsum_component [simp]:
  1170   fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
  1171   shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
  1172   by (cases "finite S", induct S set: finite, simp_all)
  1173 
  1174 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
  1175   by (simp add: Cart_eq)
  1176 
  1177 lemma setsum_clauses:
  1178   shows "setsum f {} = 0"
  1179   and "finite S \<Longrightarrow> setsum f (insert x S) =
  1180                  (if x \<in> S then setsum f S else f x + setsum f S)"
  1181   by (auto simp add: insert_absorb)
  1182 
  1183 lemma setsum_cmul:
  1184   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
  1185   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
  1186   by (simp add: Cart_eq setsum_right_distrib)
  1187 
  1188 lemma setsum_norm:
  1189   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1190   assumes fS: "finite S"
  1191   shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
  1192 proof(induct rule: finite_induct[OF fS])
  1193   case 1 thus ?case by simp
  1194 next
  1195   case (2 x S)
  1196   from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
  1197   also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
  1198     using "2.hyps" by simp
  1199   finally  show ?case  using "2.hyps" by simp
  1200 qed
  1201 
  1202 lemma real_setsum_norm:
  1203   fixes f :: "'a \<Rightarrow> real ^'n::finite"
  1204   assumes fS: "finite S"
  1205   shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
  1206 proof(induct rule: finite_induct[OF fS])
  1207   case 1 thus ?case by simp
  1208 next
  1209   case (2 x S)
  1210   from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
  1211   also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
  1212     using "2.hyps" by simp
  1213   finally  show ?case  using "2.hyps" by simp
  1214 qed
  1215 
  1216 lemma setsum_norm_le:
  1217   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1218   assumes fS: "finite S"
  1219   and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
  1220   shows "norm (setsum f S) \<le> setsum g S"
  1221 proof-
  1222   from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
  1223     by - (rule setsum_mono, simp)
  1224   then show ?thesis using setsum_norm[OF fS, of f] fg
  1225     by arith
  1226 qed
  1227 
  1228 lemma real_setsum_norm_le:
  1229   fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
  1230   assumes fS: "finite S"
  1231   and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
  1232   shows "norm (setsum f S) \<le> setsum g S"
  1233 proof-
  1234   from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
  1235     by - (rule setsum_mono, simp)
  1236   then show ?thesis using real_setsum_norm[OF fS, of f] fg
  1237     by arith
  1238 qed
  1239 
  1240 lemma setsum_norm_bound:
  1241   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1242   assumes fS: "finite S"
  1243   and K: "\<forall>x \<in> S. norm (f x) \<le> K"
  1244   shows "norm (setsum f S) \<le> of_nat (card S) * K"
  1245   using setsum_norm_le[OF fS K] setsum_constant[symmetric]
  1246   by simp
  1247 
  1248 lemma real_setsum_norm_bound:
  1249   fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
  1250   assumes fS: "finite S"
  1251   and K: "\<forall>x \<in> S. norm (f x) \<le> K"
  1252   shows "norm (setsum f S) \<le> of_nat (card S) * K"
  1253   using real_setsum_norm_le[OF fS K] setsum_constant[symmetric]
  1254   by simp
  1255 
  1256 lemma setsum_vmul:
  1257   fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector,semiring, mult_zero}"
  1258   assumes fS: "finite S"
  1259   shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
  1260 proof(induct rule: finite_induct[OF fS])
  1261   case 1 then show ?case by (simp add: vector_smult_lzero)
  1262 next
  1263   case (2 x F)
  1264   from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v"
  1265     by simp
  1266   also have "\<dots> = f x *s v + setsum f F *s v"
  1267     by (simp add: vector_sadd_rdistrib)
  1268   also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp
  1269   finally show ?case .
  1270 qed
  1271 
  1272 (* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \<Rightarrow> real ^'n"]  ---
  1273  Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *)
  1274 
  1275 lemma setsum_add_split: assumes mn: "(m::nat) \<le> n + 1"
  1276   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
  1277 proof-
  1278   let ?A = "{m .. n}"
  1279   let ?B = "{n + 1 .. n + p}"
  1280   have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto
  1281   have d: "?A \<inter> ?B = {}" by auto
  1282   from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto
  1283 qed
  1284 
  1285 lemma setsum_natinterval_left:
  1286   assumes mn: "(m::nat) <= n"
  1287   shows "setsum f {m..n} = f m + setsum f {m + 1..n}"
  1288 proof-
  1289   from mn have "{m .. n} = insert m {m+1 .. n}" by auto
  1290   then show ?thesis by auto
  1291 qed
  1292 
  1293 lemma setsum_natinterval_difff:
  1294   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
  1295   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
  1296           (if m <= n then f m - f(n + 1) else 0)"
  1297 by (induct n, auto simp add: ring_simps not_le le_Suc_eq)
  1298 
  1299 lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
  1300 
  1301 lemma setsum_setsum_restrict:
  1302   "finite S \<Longrightarrow> finite T \<Longrightarrow> setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y\<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"
  1303   apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
  1304   by (rule setsum_commute)
  1305 
  1306 lemma setsum_image_gen: assumes fS: "finite S"
  1307   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
  1308 proof-
  1309   {fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto}
  1310   note th0 = this
  1311   have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
  1312     apply (rule setsum_cong2)
  1313     by (simp add: th0)
  1314   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
  1315     apply (rule setsum_setsum_restrict[OF fS])
  1316     by (rule finite_imageI[OF fS])
  1317   finally show ?thesis .
  1318 qed
  1319 
  1320     (* FIXME: Here too need stupid finiteness assumption on T!!! *)
  1321 lemma setsum_group:
  1322   assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
  1323   shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
  1324 
  1325 apply (subst setsum_image_gen[OF fS, of g f])
  1326 apply (rule setsum_mono_zero_right[OF fT fST])
  1327 by (auto intro: setsum_0')
  1328 
  1329 lemma vsum_norm_allsubsets_bound:
  1330   fixes f:: "'a \<Rightarrow> real ^'n::finite"
  1331   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
  1332   shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
  1333 proof-
  1334   let ?d = "real CARD('n)"
  1335   let ?nf = "\<lambda>x. norm (f x)"
  1336   let ?U = "UNIV :: 'n set"
  1337   have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x $ i\<bar>) P) ?U"
  1338     by (rule setsum_commute)
  1339   have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
  1340   have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
  1341     apply (rule setsum_mono)
  1342     by (rule norm_le_l1)
  1343   also have "\<dots> \<le> 2 * ?d * e"
  1344     unfolding th0 th1
  1345   proof(rule setsum_bounded)
  1346     fix i assume i: "i \<in> ?U"
  1347     let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
  1348     let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
  1349     have thp: "P = ?Pp \<union> ?Pn" by auto
  1350     have thp0: "?Pp \<inter> ?Pn ={}" by auto
  1351     have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
  1352     have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
  1353       using component_le_norm[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
  1354       by (auto intro: abs_le_D1)
  1355     have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
  1356       using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
  1357       by (auto simp add: setsum_negf intro: abs_le_D1)
  1358     have "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn"
  1359       apply (subst thp)
  1360       apply (rule setsum_Un_zero)
  1361       using fP thp0 by auto
  1362     also have "\<dots> \<le> 2*e" using Pne Ppe by arith
  1363     finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
  1364   qed
  1365   finally show ?thesis .
  1366 qed
  1367 
  1368 lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> (y::'a::{comm_ring}^'n) = setsum (\<lambda>x. f x \<bullet> y) S "
  1369   by (induct rule: finite_induct, auto simp add: dot_lzero dot_ladd dot_radd)
  1370 
  1371 lemma dot_rsum: "finite S \<Longrightarrow> (y::'a::{comm_ring}^'n) \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
  1372   by (induct rule: finite_induct, auto simp add: dot_rzero dot_radd)
  1373 
  1374 subsection{* Basis vectors in coordinate directions. *}
  1375 
  1376 
  1377 definition "basis k = (\<chi> i. if i = k then 1 else 0)"
  1378 
  1379 lemma basis_component [simp]: "basis k $ i = (if k=i then 1 else 0)"
  1380   unfolding basis_def by simp
  1381 
  1382 lemma delta_mult_idempotent:
  1383   "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" by (cases "k=a", auto)
  1384 
  1385 lemma norm_basis:
  1386   shows "norm (basis k :: real ^'n::finite) = 1"
  1387   apply (simp add: basis_def real_vector_norm_def dot_def)
  1388   apply (vector delta_mult_idempotent)
  1389   using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"]
  1390   apply auto
  1391   done
  1392 
  1393 lemma norm_basis_1: "norm(basis 1 :: real ^'n::{finite,one}) = 1"
  1394   by (rule norm_basis)
  1395 
  1396 lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n::finite). norm x = c"
  1397   apply (rule exI[where x="c *s basis arbitrary"])
  1398   by (simp only: norm_mul norm_basis)
  1399 
  1400 lemma vector_choose_dist: assumes e: "0 <= e"
  1401   shows "\<exists>(y::real^'n::finite). dist x y = e"
  1402 proof-
  1403   from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"
  1404     by blast
  1405   then have "dist x (x - c) = e" by (simp add: dist_norm)
  1406   then show ?thesis by blast
  1407 qed
  1408 
  1409 lemma basis_inj: "inj (basis :: 'n \<Rightarrow> real ^'n::finite)"
  1410   by (simp add: inj_on_def Cart_eq)
  1411 
  1412 lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
  1413   by auto
  1414 
  1415 lemma basis_expansion:
  1416   "setsum (\<lambda>i. (x$i) *s basis i) UNIV = (x::('a::ring_1) ^'n::finite)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
  1417   by (auto simp add: Cart_eq cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
  1418 
  1419 lemma basis_expansion_unique:
  1420   "setsum (\<lambda>i. f i *s basis i) UNIV = (x::('a::comm_ring_1) ^'n::finite) \<longleftrightarrow> (\<forall>i. f i = x$i)"
  1421   by (simp add: Cart_eq setsum_delta cond_value_iff cong del: if_weak_cong)
  1422 
  1423 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
  1424   by auto
  1425 
  1426 lemma dot_basis:
  1427   shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i :: 'a^'n::finite) = (x$i :: 'a::semiring_1)"
  1428   by (auto simp add: dot_def basis_def cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)
  1429 
  1430 lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
  1431   by (auto simp add: Cart_eq)
  1432 
  1433 lemma basis_nonzero:
  1434   shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
  1435   by (simp add: basis_eq_0)
  1436 
  1437 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = (z::'a::semiring_1^'n::finite)"
  1438   apply (auto simp add: Cart_eq dot_basis)
  1439   apply (erule_tac x="basis i" in allE)
  1440   apply (simp add: dot_basis)
  1441   apply (subgoal_tac "y = z")
  1442   apply simp
  1443   apply (simp add: Cart_eq)
  1444   done
  1445 
  1446 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = (y::'a::semiring_1^'n::finite)"
  1447   apply (auto simp add: Cart_eq dot_basis)
  1448   apply (erule_tac x="basis i" in allE)
  1449   apply (simp add: dot_basis)
  1450   apply (subgoal_tac "x = y")
  1451   apply simp
  1452   apply (simp add: Cart_eq)
  1453   done
  1454 
  1455 subsection{* Orthogonality. *}
  1456 
  1457 definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
  1458 
  1459 lemma orthogonal_basis:
  1460   shows "orthogonal (basis i :: 'a^'n::finite) x \<longleftrightarrow> x$i = (0::'a::ring_1)"
  1461   by (auto simp add: orthogonal_def dot_def basis_def cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
  1462 
  1463 lemma orthogonal_basis_basis:
  1464   shows "orthogonal (basis i :: 'a::ring_1^'n::finite) (basis j) \<longleftrightarrow> i \<noteq> j"
  1465   unfolding orthogonal_basis[of i] basis_component[of j] by simp
  1466 
  1467   (* FIXME : Maybe some of these require less than comm_ring, but not all*)
  1468 lemma orthogonal_clauses:
  1469   "orthogonal a (0::'a::comm_ring ^'n)"
  1470   "orthogonal a x ==> orthogonal a (c *s x)"
  1471   "orthogonal a x ==> orthogonal a (-x)"
  1472   "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x + y)"
  1473   "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x - y)"
  1474   "orthogonal 0 a"
  1475   "orthogonal x a ==> orthogonal (c *s x) a"
  1476   "orthogonal x a ==> orthogonal (-x) a"
  1477   "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x + y) a"
  1478   "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x - y) a"
  1479   unfolding orthogonal_def dot_rneg dot_rmult dot_radd dot_rsub
  1480   dot_lzero dot_rzero dot_lneg dot_lmult dot_ladd dot_lsub
  1481   by simp_all
  1482 
  1483 lemma orthogonal_commute: "orthogonal (x::'a::{ab_semigroup_mult,comm_monoid_add} ^'n)y \<longleftrightarrow> orthogonal y x"
  1484   by (simp add: orthogonal_def dot_sym)
  1485 
  1486 subsection{* Explicit vector construction from lists. *}
  1487 
  1488 primrec from_nat :: "nat \<Rightarrow> 'a::{monoid_add,one}"
  1489 where "from_nat 0 = 0" | "from_nat (Suc n) = 1 + from_nat n"
  1490 
  1491 lemma from_nat [simp]: "from_nat = of_nat"
  1492 by (rule ext, induct_tac x, simp_all)
  1493 
  1494 primrec
  1495   list_fun :: "nat \<Rightarrow> _ list \<Rightarrow> _ \<Rightarrow> _"
  1496 where
  1497   "list_fun n [] = (\<lambda>x. 0)"
  1498 | "list_fun n (x # xs) = fun_upd (list_fun (Suc n) xs) (from_nat n) x"
  1499 
  1500 definition "vector l = (\<chi> i. list_fun 1 l i)"
  1501 (*definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"*)
  1502 
  1503 lemma vector_1: "(vector[x]) $1 = x"
  1504   unfolding vector_def by simp
  1505 
  1506 lemma vector_2:
  1507  "(vector[x,y]) $1 = x"
  1508  "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
  1509   unfolding vector_def by simp_all
  1510 
  1511 lemma vector_3:
  1512  "(vector [x,y,z] ::('a::zero)^3)$1 = x"
  1513  "(vector [x,y,z] ::('a::zero)^3)$2 = y"
  1514  "(vector [x,y,z] ::('a::zero)^3)$3 = z"
  1515   unfolding vector_def by simp_all
  1516 
  1517 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
  1518   apply auto
  1519   apply (erule_tac x="v$1" in allE)
  1520   apply (subgoal_tac "vector [v$1] = v")
  1521   apply simp
  1522   apply (vector vector_def)
  1523   apply (simp add: forall_1)
  1524   done
  1525 
  1526 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
  1527   apply auto
  1528   apply (erule_tac x="v$1" in allE)
  1529   apply (erule_tac x="v$2" in allE)
  1530   apply (subgoal_tac "vector [v$1, v$2] = v")
  1531   apply simp
  1532   apply (vector vector_def)
  1533   apply (simp add: forall_2)
  1534   done
  1535 
  1536 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
  1537   apply auto
  1538   apply (erule_tac x="v$1" in allE)
  1539   apply (erule_tac x="v$2" in allE)
  1540   apply (erule_tac x="v$3" in allE)
  1541   apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
  1542   apply simp
  1543   apply (vector vector_def)
  1544   apply (simp add: forall_3)
  1545   done
  1546 
  1547 subsection{* Linear functions. *}
  1548 
  1549 definition "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *s x) = c *s f x)"
  1550 
  1551 lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. (c::'a::comm_semiring) *s f x)"
  1552   by (vector linear_def Cart_eq ring_simps)
  1553 
  1554 lemma linear_compose_neg: "linear (f :: 'a ^'n \<Rightarrow> 'a::comm_ring ^'m) ==> linear (\<lambda>x. -(f(x)))" by (vector linear_def Cart_eq)
  1555 
  1556 lemma linear_compose_add: "linear (f :: 'a ^'n \<Rightarrow> 'a::semiring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
  1557   by (vector linear_def Cart_eq ring_simps)
  1558 
  1559 lemma linear_compose_sub: "linear (f :: 'a ^'n \<Rightarrow> 'a::ring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
  1560   by (vector linear_def Cart_eq ring_simps)
  1561 
  1562 lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
  1563   by (simp add: linear_def)
  1564 
  1565 lemma linear_id: "linear id" by (simp add: linear_def id_def)
  1566 
  1567 lemma linear_zero: "linear (\<lambda>x. 0::'a::semiring_1 ^ 'n)" by (simp add: linear_def)
  1568 
  1569 lemma linear_compose_setsum:
  1570   assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a :: 'a::semiring_1 ^ 'n \<Rightarrow> 'a ^ 'm)"
  1571   shows "linear(\<lambda>x. setsum (\<lambda>a. f a x :: 'a::semiring_1 ^'m) S)"
  1572   using lS
  1573   apply (induct rule: finite_induct[OF fS])
  1574   by (auto simp add: linear_zero intro: linear_compose_add)
  1575 
  1576 lemma linear_vmul_component:
  1577   fixes f:: "'a::semiring_1^'m \<Rightarrow> 'a^'n"
  1578   assumes lf: "linear f"
  1579   shows "linear (\<lambda>x. f x $ k *s v)"
  1580   using lf
  1581   apply (auto simp add: linear_def )
  1582   by (vector ring_simps)+
  1583 
  1584 lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)"
  1585   unfolding linear_def
  1586   apply clarsimp
  1587   apply (erule allE[where x="0::'a"])
  1588   apply simp
  1589   done
  1590 
  1591 lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def)
  1592 
  1593 lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \<Rightarrow> _) ==> f (-x) = - f x"
  1594   unfolding vector_sneg_minus1
  1595   using linear_cmul[of f] by auto
  1596 
  1597 lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
  1598 
  1599 lemma linear_sub: "linear (f::'a::ring_1 ^'n \<Rightarrow> _) ==> f(x - y) = f x - f y"
  1600   by (simp add: diff_def linear_add linear_neg)
  1601 
  1602 lemma linear_setsum:
  1603   fixes f:: "'a::semiring_1^'n \<Rightarrow> _"
  1604   assumes lf: "linear f" and fS: "finite S"
  1605   shows "f (setsum g S) = setsum (f o g) S"
  1606 proof (induct rule: finite_induct[OF fS])
  1607   case 1 thus ?case by (simp add: linear_0[OF lf])
  1608 next
  1609   case (2 x F)
  1610   have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
  1611     by simp
  1612   also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
  1613   also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
  1614   finally show ?case .
  1615 qed
  1616 
  1617 lemma linear_setsum_mul:
  1618   fixes f:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m"
  1619   assumes lf: "linear f" and fS: "finite S"
  1620   shows "f (setsum (\<lambda>i. c i *s v i) S) = setsum (\<lambda>i. c i *s f (v i)) S"
  1621   using linear_setsum[OF lf fS, of "\<lambda>i. c i *s v i" , unfolded o_def]
  1622   linear_cmul[OF lf] by simp
  1623 
  1624 lemma linear_injective_0:
  1625   assumes lf: "linear (f:: 'a::ring_1 ^ 'n \<Rightarrow> _)"
  1626   shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
  1627 proof-
  1628   have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
  1629   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
  1630   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
  1631     by (simp add: linear_sub[OF lf])
  1632   also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
  1633   finally show ?thesis .
  1634 qed
  1635 
  1636 lemma linear_bounded:
  1637   fixes f:: "real ^'m::finite \<Rightarrow> real ^'n::finite"
  1638   assumes lf: "linear f"
  1639   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
  1640 proof-
  1641   let ?S = "UNIV:: 'm set"
  1642   let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
  1643   have fS: "finite ?S" by simp
  1644   {fix x:: "real ^ 'm"
  1645     let ?g = "(\<lambda>i. (x$i) *s (basis i) :: real ^ 'm)"
  1646     have "norm (f x) = norm (f (setsum (\<lambda>i. (x$i) *s (basis i)) ?S))"
  1647       by (simp only:  basis_expansion)
  1648     also have "\<dots> = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)"
  1649       using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf]
  1650       by auto
  1651     finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)" .
  1652     {fix i assume i: "i \<in> ?S"
  1653       from component_le_norm[of x i]
  1654       have "norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x"
  1655       unfolding norm_mul
  1656       apply (simp only: mult_commute)
  1657       apply (rule mult_mono)
  1658       by (auto simp add: ring_simps norm_ge_zero) }
  1659     then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
  1660     from real_setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (f (basis i))", OF th]
  1661     have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
  1662   then show ?thesis by blast
  1663 qed
  1664 
  1665 lemma linear_bounded_pos:
  1666   fixes f:: "real ^'n::finite \<Rightarrow> real ^ 'm::finite"
  1667   assumes lf: "linear f"
  1668   shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
  1669 proof-
  1670   from linear_bounded[OF lf] obtain B where
  1671     B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
  1672   let ?K = "\<bar>B\<bar> + 1"
  1673   have Kp: "?K > 0" by arith
  1674     {assume C: "B < 0"
  1675       have "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff)
  1676       with C have "B * norm (1:: real ^ 'n) < 0"
  1677 	by (simp add: zero_compare_simps)
  1678       with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp
  1679     }
  1680     then have Bp: "B \<ge> 0" by ferrack
  1681     {fix x::"real ^ 'n"
  1682       have "norm (f x) \<le> ?K *  norm x"
  1683       using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
  1684       apply (auto simp add: ring_simps split add: abs_split)
  1685       apply (erule order_trans, simp)
  1686       done
  1687   }
  1688   then show ?thesis using Kp by blast
  1689 qed
  1690 
  1691 subsection{* Bilinear functions. *}
  1692 
  1693 definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
  1694 
  1695 lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
  1696   by (simp add: bilinear_def linear_def)
  1697 lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
  1698   by (simp add: bilinear_def linear_def)
  1699 
  1700 lemma bilinear_lmul: "bilinear h ==> h (c *s x) y = c *s (h x y)"
  1701   by (simp add: bilinear_def linear_def)
  1702 
  1703 lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)"
  1704   by (simp add: bilinear_def linear_def)
  1705 
  1706 lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)"
  1707   by (simp only: vector_sneg_minus1 bilinear_lmul)
  1708 
  1709 lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y"
  1710   by (simp only: vector_sneg_minus1 bilinear_rmul)
  1711 
  1712 lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
  1713   using add_imp_eq[of x y 0] by auto
  1714 
  1715 lemma bilinear_lzero:
  1716   fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h 0 x = 0"
  1717   using bilinear_ladd[OF bh, of 0 0 x]
  1718     by (simp add: eq_add_iff ring_simps)
  1719 
  1720 lemma bilinear_rzero:
  1721   fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h x 0 = 0"
  1722   using bilinear_radd[OF bh, of x 0 0 ]
  1723     by (simp add: eq_add_iff ring_simps)
  1724 
  1725 lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ 'n)) z = h x z - h y z"
  1726   by (simp  add: diff_def bilinear_ladd bilinear_lneg)
  1727 
  1728 lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ 'n)) = h z x - h z y"
  1729   by (simp  add: diff_def bilinear_radd bilinear_rneg)
  1730 
  1731 lemma bilinear_setsum:
  1732   fixes h:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m \<Rightarrow> 'a ^ 'k"
  1733   assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
  1734   shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
  1735 proof-
  1736   have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
  1737     apply (rule linear_setsum[unfolded o_def])
  1738     using bh fS by (auto simp add: bilinear_def)
  1739   also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
  1740     apply (rule setsum_cong, simp)
  1741     apply (rule linear_setsum[unfolded o_def])
  1742     using bh fT by (auto simp add: bilinear_def)
  1743   finally show ?thesis unfolding setsum_cartesian_product .
  1744 qed
  1745 
  1746 lemma bilinear_bounded:
  1747   fixes h:: "real ^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real ^ 'k::finite"
  1748   assumes bh: "bilinear h"
  1749   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
  1750 proof-
  1751   let ?M = "UNIV :: 'm set"
  1752   let ?N = "UNIV :: 'n set"
  1753   let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
  1754   have fM: "finite ?M" and fN: "finite ?N" by simp_all
  1755   {fix x:: "real ^ 'm" and  y :: "real^'n"
  1756     have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$i) *s basis i) ?M) (setsum (\<lambda>i. (y$i) *s basis i) ?N))" unfolding basis_expansion ..
  1757     also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$i) *s basis i) ((y$j) *s basis j)) (?M \<times> ?N))"  unfolding bilinear_setsum[OF bh fM fN] ..
  1758     finally have th: "norm (h x y) = \<dots>" .
  1759     have "norm (h x y) \<le> ?B * norm x * norm y"
  1760       apply (simp add: setsum_left_distrib th)
  1761       apply (rule real_setsum_norm_le)
  1762       using fN fM
  1763       apply simp
  1764       apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)
  1765       apply (rule mult_mono)
  1766       apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
  1767       apply (rule mult_mono)
  1768       apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
  1769       done}
  1770   then show ?thesis by metis
  1771 qed
  1772 
  1773 lemma bilinear_bounded_pos:
  1774   fixes h:: "real ^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real ^ 'k::finite"
  1775   assumes bh: "bilinear h"
  1776   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
  1777 proof-
  1778   from bilinear_bounded[OF bh] obtain B where
  1779     B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
  1780   let ?K = "\<bar>B\<bar> + 1"
  1781   have Kp: "?K > 0" by arith
  1782   have KB: "B < ?K" by arith
  1783   {fix x::"real ^'m" and y :: "real ^'n"
  1784     from KB Kp
  1785     have "B * norm x * norm y \<le> ?K * norm x * norm y"
  1786       apply -
  1787       apply (rule mult_right_mono, rule mult_right_mono)
  1788       by (auto simp add: norm_ge_zero)
  1789     then have "norm (h x y) \<le> ?K * norm x * norm y"
  1790       using B[rule_format, of x y] by simp}
  1791   with Kp show ?thesis by blast
  1792 qed
  1793 
  1794 subsection{* Adjoints. *}
  1795 
  1796 definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
  1797 
  1798 lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
  1799 
  1800 lemma adjoint_works_lemma:
  1801   fixes f:: "'a::ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
  1802   assumes lf: "linear f"
  1803   shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
  1804 proof-
  1805   let ?N = "UNIV :: 'n set"
  1806   let ?M = "UNIV :: 'm set"
  1807   have fN: "finite ?N" by simp
  1808   have fM: "finite ?M" by simp
  1809   {fix y:: "'a ^ 'm"
  1810     let ?w = "(\<chi> i. (f (basis i) \<bullet> y)) :: 'a ^ 'n"
  1811     {fix x
  1812       have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *s basis i) ?N) \<bullet> y"
  1813 	by (simp only: basis_expansion)
  1814       also have "\<dots> = (setsum (\<lambda>i. (x$i) *s f (basis i)) ?N) \<bullet> y"
  1815 	unfolding linear_setsum[OF lf fN]
  1816 	by (simp add: linear_cmul[OF lf])
  1817       finally have "f x \<bullet> y = x \<bullet> ?w"
  1818 	apply (simp only: )
  1819 	apply (simp add: dot_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] ring_simps)
  1820 	done}
  1821   }
  1822   then show ?thesis unfolding adjoint_def
  1823     some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
  1824     using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
  1825     by metis
  1826 qed
  1827 
  1828 lemma adjoint_works:
  1829   fixes f:: "'a::ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
  1830   assumes lf: "linear f"
  1831   shows "x \<bullet> adjoint f y = f x \<bullet> y"
  1832   using adjoint_works_lemma[OF lf] by metis
  1833 
  1834 
  1835 lemma adjoint_linear:
  1836   fixes f :: "'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
  1837   assumes lf: "linear f"
  1838   shows "linear (adjoint f)"
  1839   by (simp add: linear_def vector_eq_ldot[symmetric] dot_radd dot_rmult adjoint_works[OF lf])
  1840 
  1841 lemma adjoint_clauses:
  1842   fixes f:: "'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
  1843   assumes lf: "linear f"
  1844   shows "x \<bullet> adjoint f y = f x \<bullet> y"
  1845   and "adjoint f y \<bullet> x = y \<bullet> f x"
  1846   by (simp_all add: adjoint_works[OF lf] dot_sym )
  1847 
  1848 lemma adjoint_adjoint:
  1849   fixes f:: "'a::comm_ring_1 ^ 'n::finite \<Rightarrow> 'a ^ 'm::finite"
  1850   assumes lf: "linear f"
  1851   shows "adjoint (adjoint f) = f"
  1852   apply (rule ext)
  1853   by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf])
  1854 
  1855 lemma adjoint_unique:
  1856   fixes f:: "'a::comm_ring_1 ^ 'n::finite \<Rightarrow> 'a ^ 'm::finite"
  1857   assumes lf: "linear f" and u: "\<forall>x y. f' x \<bullet> y = x \<bullet> f y"
  1858   shows "f' = adjoint f"
  1859   apply (rule ext)
  1860   using u
  1861   by (simp add: vector_eq_rdot[symmetric] adjoint_clauses[OF lf])
  1862 
  1863 text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
  1864 
  1865 consts generic_mult :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" (infixr "\<star>" 75)
  1866 
  1867 defs (overloaded)
  1868 matrix_matrix_mult_def: "(m:: ('a::semiring_1) ^'n^'m) \<star> (m' :: 'a ^'p^'n) \<equiv> (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
  1869 
  1870 abbreviation
  1871   matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)
  1872   where "m ** m' == m\<star> m'"
  1873 
  1874 defs (overloaded)
  1875   matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
  1876 
  1877 abbreviation
  1878   matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)
  1879   where
  1880   "m *v v == m \<star> v"
  1881 
  1882 defs (overloaded)
  1883   vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (x$i)) (UNIV :: 'm set)) :: 'a^'n"
  1884 
  1885 abbreviation
  1886   vactor_matrix_mult' :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)
  1887   where
  1888   "v v* m == v \<star> m"
  1889 
  1890 definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
  1891 definition "(transp::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
  1892 definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
  1893 definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
  1894 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
  1895 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
  1896 
  1897 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
  1898 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \<star> B) + (A \<star> C)"
  1899   by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps)
  1900 
  1901 lemma setsum_delta':
  1902   assumes fS: "finite S" shows
  1903   "setsum (\<lambda>k. if a = k then b k else 0) S =
  1904      (if a\<in> S then b a else 0)"
  1905   using setsum_delta[OF fS, of a b, symmetric]
  1906   by (auto intro: setsum_cong)
  1907 
  1908 lemma matrix_mul_lid:
  1909   fixes A :: "'a::semiring_1 ^ 'm ^ 'n::finite"
  1910   shows "mat 1 ** A = A"
  1911   apply (simp add: matrix_matrix_mult_def mat_def)
  1912   apply vector
  1913   by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite]  mult_1_left mult_zero_left if_True UNIV_I)
  1914 
  1915 
  1916 lemma matrix_mul_rid:
  1917   fixes A :: "'a::semiring_1 ^ 'm::finite ^ 'n"
  1918   shows "A ** mat 1 = A"
  1919   apply (simp add: matrix_matrix_mult_def mat_def)
  1920   apply vector
  1921   by (auto simp only: cond_value_iff cond_application_beta setsum_delta[OF finite]  mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
  1922 
  1923 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
  1924   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
  1925   apply (subst setsum_commute)
  1926   apply simp
  1927   done
  1928 
  1929 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
  1930   apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
  1931   apply (subst setsum_commute)
  1932   apply simp
  1933   done
  1934 
  1935 lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n::finite)"
  1936   apply (vector matrix_vector_mult_def mat_def)
  1937   by (simp add: cond_value_iff cond_application_beta
  1938     setsum_delta' cong del: if_weak_cong)
  1939 
  1940 lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)"
  1941   by (simp add: matrix_matrix_mult_def transp_def Cart_eq mult_commute)
  1942 
  1943 lemma matrix_eq:
  1944   fixes A B :: "'a::semiring_1 ^ 'n::finite ^ 'm"
  1945   shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
  1946   apply auto
  1947   apply (subst Cart_eq)
  1948   apply clarify
  1949   apply (clarsimp simp add: matrix_vector_mult_def basis_def cond_value_iff cond_application_beta Cart_eq cong del: if_weak_cong)
  1950   apply (erule_tac x="basis ia" in allE)
  1951   apply (erule_tac x="i" in allE)
  1952   by (auto simp add: basis_def cond_value_iff cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong)
  1953 
  1954 lemma matrix_vector_mul_component:
  1955   shows "((A::'a::semiring_1^'n'^'m) *v x)$k = (A$k) \<bullet> x"
  1956   by (simp add: matrix_vector_mult_def dot_def)
  1957 
  1958 lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) v* A) \<bullet> y = x \<bullet> (A *v y)"
  1959   apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
  1960   apply (subst setsum_commute)
  1961   by simp
  1962 
  1963 lemma transp_mat: "transp (mat n) = mat n"
  1964   by (vector transp_def mat_def)
  1965 
  1966 lemma transp_transp: "transp(transp A) = A"
  1967   by (vector transp_def)
  1968 
  1969 lemma row_transp:
  1970   fixes A:: "'a::semiring_1^'n^'m"
  1971   shows "row i (transp A) = column i A"
  1972   by (simp add: row_def column_def transp_def Cart_eq)
  1973 
  1974 lemma column_transp:
  1975   fixes A:: "'a::semiring_1^'n^'m"
  1976   shows "column i (transp A) = row i A"
  1977   by (simp add: row_def column_def transp_def Cart_eq)
  1978 
  1979 lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A"
  1980 by (auto simp add: rows_def columns_def row_transp intro: set_ext)
  1981 
  1982 lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp)
  1983 
  1984 text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
  1985 
  1986 lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
  1987   by (simp add: matrix_vector_mult_def dot_def)
  1988 
  1989 lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
  1990   by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute)
  1991 
  1992 lemma vector_componentwise:
  1993   "(x::'a::ring_1^'n::finite) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) (UNIV :: 'n set))"
  1994   apply (subst basis_expansion[symmetric])
  1995   by (vector Cart_eq setsum_component)
  1996 
  1997 lemma linear_componentwise:
  1998   fixes f:: "'a::ring_1 ^ 'm::finite \<Rightarrow> 'a ^ 'n"
  1999   assumes lf: "linear f"
  2000   shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
  2001 proof-
  2002   let ?M = "(UNIV :: 'm set)"
  2003   let ?N = "(UNIV :: 'n set)"
  2004   have fM: "finite ?M" by simp
  2005   have "?rhs = (setsum (\<lambda>i.(x$i) *s f (basis i) ) ?M)$j"
  2006     unfolding vector_smult_component[symmetric]
  2007     unfolding setsum_component[of "(\<lambda>i.(x$i) *s f (basis i :: 'a^'m))" ?M]
  2008     ..
  2009   then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion ..
  2010 qed
  2011 
  2012 text{* Inverse matrices  (not necessarily square) *}
  2013 
  2014 definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
  2015 
  2016 definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
  2017         (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
  2018 
  2019 text{* Correspondence between matrices and linear operators. *}
  2020 
  2021 definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
  2022 where "matrix f = (\<chi> i j. (f(basis j))$i)"
  2023 
  2024 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::'a::comm_semiring_1 ^ 'n))"
  2025   by (simp add: linear_def matrix_vector_mult_def Cart_eq ring_simps setsum_right_distrib setsum_addf)
  2026 
  2027 lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n::finite)"
  2028 apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute)
  2029 apply clarify
  2030 apply (rule linear_componentwise[OF lf, symmetric])
  2031 done
  2032 
  2033 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::'a::comm_ring_1 ^ 'n::finite))" by (simp add: ext matrix_works)
  2034 
  2035 lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n::finite)) = A"
  2036   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
  2037 
  2038 lemma matrix_compose:
  2039   assumes lf: "linear (f::'a::comm_ring_1^'n::finite \<Rightarrow> 'a^'m::finite)"
  2040   and lg: "linear (g::'a::comm_ring_1^'m::finite \<Rightarrow> 'a^'k)"
  2041   shows "matrix (g o f) = matrix g ** matrix f"
  2042   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
  2043   by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
  2044 
  2045 lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s ((transp A)$i)) (UNIV:: 'n set)"
  2046   by (simp add: matrix_vector_mult_def transp_def Cart_eq mult_commute)
  2047 
  2048 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::'a::comm_ring_1^'n::finite^'m::finite) *v x) = (\<lambda>x. transp A *v x)"
  2049   apply (rule adjoint_unique[symmetric])
  2050   apply (rule matrix_vector_mul_linear)
  2051   apply (simp add: transp_def dot_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
  2052   apply (subst setsum_commute)
  2053   apply (auto simp add: mult_ac)
  2054   done
  2055 
  2056 lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n::finite \<Rightarrow> 'a ^ 'm::finite)"
  2057   shows "matrix(adjoint f) = transp(matrix f)"
  2058   apply (subst matrix_vector_mul[OF lf])
  2059   unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
  2060 
  2061 subsection{* Interlude: Some properties of real sets *}
  2062 
  2063 lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
  2064   shows "\<forall>n \<ge> m. d n < e m"
  2065   using prems apply auto
  2066   apply (erule_tac x="n" in allE)
  2067   apply (erule_tac x="n" in allE)
  2068   apply auto
  2069   done
  2070 
  2071 
  2072 lemma real_convex_bound_lt:
  2073   assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"
  2074   and uv: "u + v = 1"
  2075   shows "u * x + v * y < a"
  2076 proof-
  2077   have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
  2078   have "a = a * (u + v)" unfolding uv  by simp
  2079   hence th: "u * a + v * a = a" by (simp add: ring_simps)
  2080   from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_compare_simps)
  2081   from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_compare_simps)
  2082   from xa ya u v have "u * x + v * y < u * a + v * a"
  2083     apply (cases "u = 0", simp_all add: uv')
  2084     apply(rule mult_strict_left_mono)
  2085     using uv' apply simp_all
  2086 
  2087     apply (rule add_less_le_mono)
  2088     apply(rule mult_strict_left_mono)
  2089     apply simp_all
  2090     apply (rule mult_left_mono)
  2091     apply simp_all
  2092     done
  2093   thus ?thesis unfolding th .
  2094 qed
  2095 
  2096 lemma real_convex_bound_le:
  2097   assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"
  2098   and uv: "u + v = 1"
  2099   shows "u * x + v * y \<le> a"
  2100 proof-
  2101   from xa ya u v have "u * x + v * y \<le> u * a + v * a" by (simp add: add_mono mult_left_mono)
  2102   also have "\<dots> \<le> (u + v) * a" by (simp add: ring_simps)
  2103   finally show ?thesis unfolding uv by simp
  2104 qed
  2105 
  2106 lemma infinite_enumerate: assumes fS: "infinite S"
  2107   shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
  2108 unfolding subseq_def
  2109 using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
  2110 
  2111 lemma approachable_lt_le: "(\<exists>(d::real)>0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
  2112 apply auto
  2113 apply (rule_tac x="d/2" in exI)
  2114 apply auto
  2115 done
  2116 
  2117 
  2118 lemma triangle_lemma:
  2119   assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
  2120   shows "x <= y + z"
  2121 proof-
  2122   have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y  by (simp add: zero_compare_simps)
  2123   with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square ring_simps)
  2124   from y z have yz: "y + z \<ge> 0" by arith
  2125   from power2_le_imp_le[OF th yz] show ?thesis .
  2126 qed
  2127 
  2128 
  2129 lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
  2130    (\<exists>x::'a ^ 'n. \<forall>i. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")
  2131 proof-
  2132   let ?S = "(UNIV :: 'n set)"
  2133   {assume H: "?rhs"
  2134     then have ?lhs by auto}
  2135   moreover
  2136   {assume H: "?lhs"
  2137     then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
  2138     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
  2139     {fix i
  2140       from f have "P i (f i)" by metis
  2141       then have "P i (?x$i)" by auto
  2142     }
  2143     hence "\<forall>i. P i (?x$i)" by metis
  2144     hence ?rhs by metis }
  2145   ultimately show ?thesis by metis
  2146 qed
  2147 
  2148 (* Supremum and infimum of real sets *)
  2149 
  2150 
  2151 definition rsup:: "real set \<Rightarrow> real" where
  2152   "rsup S = (SOME a. isLub UNIV S a)"
  2153 
  2154 lemma rsup_alt: "rsup S = (SOME a. (\<forall>x \<in> S. x \<le> a) \<and> (\<forall>b. (\<forall>x \<in> S. x \<le> b) \<longrightarrow> a \<le> b))"  by (auto simp  add: isLub_def rsup_def leastP_def isUb_def setle_def setge_def)
  2155 
  2156 lemma rsup: assumes Se: "S \<noteq> {}" and b: "\<exists>b. S *<= b"
  2157   shows "isLub UNIV S (rsup S)"
  2158 using Se b
  2159 unfolding rsup_def
  2160 apply clarify
  2161 apply (rule someI_ex)
  2162 apply (rule reals_complete)
  2163 by (auto simp add: isUb_def setle_def)
  2164 
  2165 lemma rsup_le: assumes Se: "S \<noteq> {}" and Sb: "S *<= b" shows "rsup S \<le> b"
  2166 proof-
  2167   from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def)
  2168   from rsup[OF Se] Sb have "isLub UNIV S (rsup S)"  by blast
  2169   then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def)
  2170 qed
  2171 
  2172 lemma rsup_finite_Max: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2173   shows "rsup S = Max S"
  2174 using fS Se
  2175 proof-
  2176   let ?m = "Max S"
  2177   from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis
  2178   with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def)
  2179   from Max_in[OF fS Se] lub have mrS: "?m \<le> rsup S"
  2180     by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)
  2181   moreover
  2182   have "rsup S \<le> ?m" using Sm lub
  2183     by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
  2184   ultimately  show ?thesis by arith
  2185 qed
  2186 
  2187 lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2188   shows "rsup S \<in> S"
  2189   using rsup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis
  2190 
  2191 lemma rsup_finite_Ub: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2192   shows "isUb S S (rsup S)"
  2193   using rsup_finite_Max[OF fS Se] rsup_finite_in[OF fS Se] Max_ge[OF fS]
  2194   unfolding isUb_def setle_def by metis
  2195 
  2196 lemma rsup_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2197   shows "a \<le> rsup S \<longleftrightarrow> (\<exists> x \<in> S. a \<le> x)"
  2198 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2199 
  2200 lemma rsup_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2201   shows "a \<ge> rsup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)"
  2202 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2203 
  2204 lemma rsup_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2205   shows "a < rsup S \<longleftrightarrow> (\<exists> x \<in> S. a < x)"
  2206 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2207 
  2208 lemma rsup_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2209   shows "a > rsup S \<longleftrightarrow> (\<forall> x \<in> S. a > x)"
  2210 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2211 
  2212 lemma rsup_unique: assumes b: "S *<= b" and S: "\<forall>b' < b. \<exists>x \<in> S. b' < x"
  2213   shows "rsup S = b"
  2214 using b S
  2215 unfolding setle_def rsup_alt
  2216 apply -
  2217 apply (rule some_equality)
  2218 apply (metis  linorder_not_le order_eq_iff[symmetric])+
  2219 done
  2220 
  2221 lemma rsup_le_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. T *<= b) \<Longrightarrow> rsup S \<le> rsup T"
  2222   apply (rule rsup_le)
  2223   apply simp
  2224   using rsup[of T] by (auto simp add: isLub_def leastP_def setge_def setle_def isUb_def)
  2225 
  2226 lemma isUb_def': "isUb R S = (\<lambda>x. S *<= x \<and> x \<in> R)"
  2227   apply (rule ext)
  2228   by (metis isUb_def)
  2229 
  2230 lemma UNIV_trivial: "UNIV x" using UNIV_I[of x] by (metis mem_def)
  2231 lemma rsup_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
  2232   shows "a \<le> rsup S \<and> rsup S \<le> b"
  2233 proof-
  2234   from rsup[OF Se] u have lub: "isLub UNIV S (rsup S)" by blast
  2235   hence b: "rsup S \<le> b" using u by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
  2236   from Se obtain y where y: "y \<in> S" by blast
  2237   from lub l have "a \<le> rsup S" apply (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
  2238     apply (erule ballE[where x=y])
  2239     apply (erule ballE[where x=y])
  2240     apply arith
  2241     using y apply auto
  2242     done
  2243   with b show ?thesis by blast
  2244 qed
  2245 
  2246 lemma rsup_abs_le: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rsup S\<bar> \<le> a"
  2247   unfolding abs_le_interval_iff  using rsup_bounds[of S "-a" a]
  2248   by (auto simp add: setge_def setle_def)
  2249 
  2250 lemma rsup_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rsup S - l\<bar> \<le> e"
  2251 proof-
  2252   have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
  2253   show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th
  2254     by  (auto simp add: setge_def setle_def)
  2255 qed
  2256 
  2257 definition rinf:: "real set \<Rightarrow> real" where
  2258   "rinf S = (SOME a. isGlb UNIV S a)"
  2259 
  2260 lemma rinf_alt: "rinf S = (SOME a. (\<forall>x \<in> S. x \<ge> a) \<and> (\<forall>b. (\<forall>x \<in> S. x \<ge> b) \<longrightarrow> a \<ge> b))"  by (auto simp  add: isGlb_def rinf_def greatestP_def isLb_def setle_def setge_def)
  2261 
  2262 lemma reals_complete_Glb: assumes Se: "\<exists>x. x \<in> S" and lb: "\<exists> y. isLb UNIV S y"
  2263   shows "\<exists>(t::real). isGlb UNIV S t"
  2264 proof-
  2265   let ?M = "uminus ` S"
  2266   from lb have th: "\<exists>y. isUb UNIV ?M y" apply (auto simp add: isUb_def isLb_def setle_def setge_def)
  2267     by (rule_tac x="-y" in exI, auto)
  2268   from Se have Me: "\<exists>x. x \<in> ?M" by blast
  2269   from reals_complete[OF Me th] obtain t where t: "isLub UNIV ?M t" by blast
  2270   have "isGlb UNIV S (- t)" using t
  2271     apply (auto simp add: isLub_def isGlb_def leastP_def greatestP_def setle_def setge_def isUb_def isLb_def)
  2272     apply (erule_tac x="-y" in allE)
  2273     apply auto
  2274     done
  2275   then show ?thesis by metis
  2276 qed
  2277 
  2278 lemma rinf: assumes Se: "S \<noteq> {}" and b: "\<exists>b. b <=* S"
  2279   shows "isGlb UNIV S (rinf S)"
  2280 using Se b
  2281 unfolding rinf_def
  2282 apply clarify
  2283 apply (rule someI_ex)
  2284 apply (rule reals_complete_Glb)
  2285 apply (auto simp add: isLb_def setle_def setge_def)
  2286 done
  2287 
  2288 lemma rinf_ge: assumes Se: "S \<noteq> {}" and Sb: "b <=* S" shows "rinf S \<ge> b"
  2289 proof-
  2290   from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def)
  2291   from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)"  by blast
  2292   then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def)
  2293 qed
  2294 
  2295 lemma rinf_finite_Min: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2296   shows "rinf S = Min S"
  2297 using fS Se
  2298 proof-
  2299   let ?m = "Min S"
  2300   from Min_le[OF fS] have Sm: "\<forall> x\<in> S. x \<ge> ?m" by metis
  2301   with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def)
  2302   from Min_in[OF fS Se] glb have mrS: "?m \<ge> rinf S"
  2303     by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def)
  2304   moreover
  2305   have "rinf S \<ge> ?m" using Sm glb
  2306     by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def)
  2307   ultimately  show ?thesis by arith
  2308 qed
  2309 
  2310 lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2311   shows "rinf S \<in> S"
  2312   using rinf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis
  2313 
  2314 lemma rinf_finite_Lb: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2315   shows "isLb S S (rinf S)"
  2316   using rinf_finite_Min[OF fS Se] rinf_finite_in[OF fS Se] Min_le[OF fS]
  2317   unfolding isLb_def setge_def by metis
  2318 
  2319 lemma rinf_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2320   shows "a \<le> rinf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)"
  2321 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2322 
  2323 lemma rinf_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2324   shows "a \<ge> rinf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)"
  2325 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2326 
  2327 lemma rinf_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2328   shows "a < rinf S \<longleftrightarrow> (\<forall> x \<in> S. a < x)"
  2329 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2330 
  2331 lemma rinf_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2332   shows "a > rinf S \<longleftrightarrow> (\<exists> x \<in> S. a > x)"
  2333 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2334 
  2335 lemma rinf_unique: assumes b: "b <=* S" and S: "\<forall>b' > b. \<exists>x \<in> S. b' > x"
  2336   shows "rinf S = b"
  2337 using b S
  2338 unfolding setge_def rinf_alt
  2339 apply -
  2340 apply (rule some_equality)
  2341 apply (metis  linorder_not_le order_eq_iff[symmetric])+
  2342 done
  2343 
  2344 lemma rinf_ge_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. b <=* T) \<Longrightarrow> rinf S >= rinf T"
  2345   apply (rule rinf_ge)
  2346   apply simp
  2347   using rinf[of T] by (auto simp add: isGlb_def greatestP_def setge_def setle_def isLb_def)
  2348 
  2349 lemma isLb_def': "isLb R S = (\<lambda>x. x <=* S \<and> x \<in> R)"
  2350   apply (rule ext)
  2351   by (metis isLb_def)
  2352 
  2353 lemma rinf_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
  2354   shows "a \<le> rinf S \<and> rinf S \<le> b"
  2355 proof-
  2356   from rinf[OF Se] l have lub: "isGlb UNIV S (rinf S)" by blast
  2357   hence b: "a \<le> rinf S" using l by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
  2358   from Se obtain y where y: "y \<in> S" by blast
  2359   from lub u have "b \<ge> rinf S" apply (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
  2360     apply (erule ballE[where x=y])
  2361     apply (erule ballE[where x=y])
  2362     apply arith
  2363     using y apply auto
  2364     done
  2365   with b show ?thesis by blast
  2366 qed
  2367 
  2368 lemma rinf_abs_ge: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rinf S\<bar> \<le> a"
  2369   unfolding abs_le_interval_iff  using rinf_bounds[of S "-a" a]
  2370   by (auto simp add: setge_def setle_def)
  2371 
  2372 lemma rinf_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rinf S - l\<bar> \<le> e"
  2373 proof-
  2374   have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
  2375   show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th
  2376     by  (auto simp add: setge_def setle_def)
  2377 qed
  2378 
  2379 
  2380 
  2381 subsection{* Operator norm. *}
  2382 
  2383 definition "onorm f = rsup {norm (f x)| x. norm x = 1}"
  2384 
  2385 lemma norm_bound_generalize:
  2386   fixes f:: "real ^'n::finite \<Rightarrow> real^'m::finite"
  2387   assumes lf: "linear f"
  2388   shows "(\<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> b) \<longleftrightarrow> (\<forall>x. norm (f x) \<le> b * norm x)" (is "?lhs \<longleftrightarrow> ?rhs")
  2389 proof-
  2390   {assume H: ?rhs
  2391     {fix x :: "real^'n" assume x: "norm x = 1"
  2392       from H[rule_format, of x] x have "norm (f x) \<le> b" by simp}
  2393     then have ?lhs by blast }
  2394 
  2395   moreover
  2396   {assume H: ?lhs
  2397     from H[rule_format, of "basis arbitrary"]
  2398     have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis arbitrary)"]
  2399       by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])
  2400     {fix x :: "real ^'n"
  2401       {assume "x = 0"
  2402 	then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}
  2403       moreover
  2404       {assume x0: "x \<noteq> 0"
  2405 	hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
  2406 	let ?c = "1/ norm x"
  2407 	have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul)
  2408 	with H have "norm (f(?c*s x)) \<le> b" by blast
  2409 	hence "?c * norm (f x) \<le> b"
  2410 	  by (simp add: linear_cmul[OF lf] norm_mul)
  2411 	hence "norm (f x) \<le> b * norm x"
  2412 	  using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
  2413       ultimately have "norm (f x) \<le> b * norm x" by blast}
  2414     then have ?rhs by blast}
  2415   ultimately show ?thesis by blast
  2416 qed
  2417 
  2418 lemma onorm:
  2419   fixes f:: "real ^'n::finite \<Rightarrow> real ^'m::finite"
  2420   assumes lf: "linear f"
  2421   shows "norm (f x) <= onorm f * norm x"
  2422   and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
  2423 proof-
  2424   {
  2425     let ?S = "{norm (f x) |x. norm x = 1}"
  2426     have Se: "?S \<noteq> {}" using  norm_basis by auto
  2427     from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
  2428       unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
  2429     {from rsup[OF Se b, unfolded onorm_def[symmetric]]
  2430       show "norm (f x) <= onorm f * norm x"
  2431 	apply -
  2432 	apply (rule spec[where x = x])
  2433 	unfolding norm_bound_generalize[OF lf, symmetric]
  2434 	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
  2435     {
  2436       show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
  2437 	using rsup[OF Se b, unfolded onorm_def[symmetric]]
  2438 	unfolding norm_bound_generalize[OF lf, symmetric]
  2439 	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
  2440   }
  2441 qed
  2442 
  2443 lemma onorm_pos_le: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" shows "0 <= onorm f"
  2444   using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis arbitrary"], unfolded norm_basis] by simp
  2445 
  2446 lemma onorm_eq_0: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)"
  2447   shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
  2448   using onorm[OF lf]
  2449   apply (auto simp add: onorm_pos_le)
  2450   apply atomize
  2451   apply (erule allE[where x="0::real"])
  2452   using onorm_pos_le[OF lf]
  2453   apply arith
  2454   done
  2455 
  2456 lemma onorm_const: "onorm(\<lambda>x::real^'n::finite. (y::real ^ 'm::finite)) = norm y"
  2457 proof-
  2458   let ?f = "\<lambda>x::real^'n. (y::real ^ 'm)"
  2459   have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
  2460     by(auto intro: vector_choose_size set_ext)
  2461   show ?thesis
  2462     unfolding onorm_def th
  2463     apply (rule rsup_unique) by (simp_all  add: setle_def)
  2464 qed
  2465 
  2466 lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n::finite \<Rightarrow> real ^'m::finite)"
  2467   shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"
  2468   unfolding onorm_eq_0[OF lf, symmetric]
  2469   using onorm_pos_le[OF lf] by arith
  2470 
  2471 lemma onorm_compose:
  2472   assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)"
  2473   and lg: "linear (g::real^'k::finite \<Rightarrow> real^'n::finite)"
  2474   shows "onorm (f o g) <= onorm f * onorm g"
  2475   apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
  2476   unfolding o_def
  2477   apply (subst mult_assoc)
  2478   apply (rule order_trans)
  2479   apply (rule onorm(1)[OF lf])
  2480   apply (rule mult_mono1)
  2481   apply (rule onorm(1)[OF lg])
  2482   apply (rule onorm_pos_le[OF lf])
  2483   done
  2484 
  2485 lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real^'m::finite)"
  2486   shows "onorm (\<lambda>x. - f x) \<le> onorm f"
  2487   using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
  2488   unfolding norm_minus_cancel by metis
  2489 
  2490 lemma onorm_neg: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real^'m::finite)"
  2491   shows "onorm (\<lambda>x. - f x) = onorm f"
  2492   using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]]
  2493   by simp
  2494 
  2495 lemma onorm_triangle:
  2496   assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" and lg: "linear g"
  2497   shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
  2498   apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
  2499   apply (rule order_trans)
  2500   apply (rule norm_triangle_ineq)
  2501   apply (simp add: distrib)
  2502   apply (rule add_mono)
  2503   apply (rule onorm(1)[OF lf])
  2504   apply (rule onorm(1)[OF lg])
  2505   done
  2506 
  2507 lemma onorm_triangle_le: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
  2508   \<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"
  2509   apply (rule order_trans)
  2510   apply (rule onorm_triangle)
  2511   apply assumption+
  2512   done
  2513 
  2514 lemma onorm_triangle_lt: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
  2515   ==> onorm(\<lambda>x. f x + g x) < e"
  2516   apply (rule order_le_less_trans)
  2517   apply (rule onorm_triangle)
  2518   by assumption+
  2519 
  2520 (* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)
  2521 
  2522 definition vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x = (\<chi> i. x)"
  2523 
  2524 definition dest_vec1:: "'a ^1 \<Rightarrow> 'a"
  2525   where "dest_vec1 x = (x$1)"
  2526 
  2527 lemma vec1_component[simp]: "(vec1 x)$1 = x"
  2528   by (simp add: vec1_def)
  2529 
  2530 lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
  2531   by (simp_all add: vec1_def dest_vec1_def Cart_eq forall_1)
  2532 
  2533 lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))" by (metis vec1_dest_vec1)
  2534 
  2535 lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))" by (metis vec1_dest_vec1)
  2536 
  2537 lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))"  by (metis vec1_dest_vec1)
  2538 
  2539 lemma exists_dest_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(dest_vec1 x))"by (metis vec1_dest_vec1)
  2540 
  2541 lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
  2542 
  2543 lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
  2544 
  2545 lemma vec1_in_image_vec1: "vec1 x \<in> (vec1 ` S) \<longleftrightarrow> x \<in> S" by auto
  2546 
  2547 lemma vec1_vec: "vec1 x = vec x" by (vector vec1_def)
  2548 
  2549 lemma vec1_add: "vec1(x + y) = vec1 x + vec1 y" by (vector vec1_def)
  2550 lemma vec1_sub: "vec1(x - y) = vec1 x - vec1 y" by (vector vec1_def)
  2551 lemma vec1_cmul: "vec1(c* x) = c *s vec1 x " by (vector vec1_def)
  2552 lemma vec1_neg: "vec1(- x) = - vec1 x " by (vector vec1_def)
  2553 
  2554 lemma vec1_setsum: assumes fS: "finite S"
  2555   shows "vec1(setsum f S) = setsum (vec1 o f) S"
  2556   apply (induct rule: finite_induct[OF fS])
  2557   apply (simp add: vec1_vec)
  2558   apply (auto simp add: vec1_add)
  2559   done
  2560 
  2561 lemma dest_vec1_lambda: "dest_vec1(\<chi> i. x i) = x 1"
  2562   by (simp add: dest_vec1_def)
  2563 
  2564 lemma dest_vec1_vec: "dest_vec1(vec x) = x"
  2565   by (simp add: vec1_vec[symmetric])
  2566 
  2567 lemma dest_vec1_add: "dest_vec1(x + y) = dest_vec1 x + dest_vec1 y"
  2568  by (metis vec1_dest_vec1 vec1_add)
  2569 
  2570 lemma dest_vec1_sub: "dest_vec1(x - y) = dest_vec1 x - dest_vec1 y"
  2571  by (metis vec1_dest_vec1 vec1_sub)
  2572 
  2573 lemma dest_vec1_cmul: "dest_vec1(c*sx) = c * dest_vec1 x"
  2574  by (metis vec1_dest_vec1 vec1_cmul)
  2575 
  2576 lemma dest_vec1_neg: "dest_vec1(- x) = - dest_vec1 x"
  2577  by (metis vec1_dest_vec1 vec1_neg)
  2578 
  2579 lemma dest_vec1_0[simp]: "dest_vec1 0 = 0" by (metis vec_0 dest_vec1_vec)
  2580 
  2581 lemma dest_vec1_sum: assumes fS: "finite S"
  2582   shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"
  2583   apply (induct rule: finite_induct[OF fS])
  2584   apply (simp add: dest_vec1_vec)
  2585   apply (auto simp add: dest_vec1_add)
  2586   done
  2587 
  2588 lemma norm_vec1: "norm(vec1 x) = abs(x)"
  2589   by (simp add: vec1_def norm_real)
  2590 
  2591 lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)"
  2592   by (simp only: dist_real vec1_component)
  2593 lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
  2594   by (metis vec1_dest_vec1 norm_vec1)
  2595 
  2596 lemma linear_vmul_dest_vec1:
  2597   fixes f:: "'a::semiring_1^'n \<Rightarrow> 'a^1"
  2598   shows "linear f \<Longrightarrow> linear (\<lambda>x. dest_vec1(f x) *s v)"
  2599   unfolding dest_vec1_def
  2600   apply (rule linear_vmul_component)
  2601   by auto
  2602 
  2603 lemma linear_from_scalars:
  2604   assumes lf: "linear (f::'a::comm_ring_1 ^1 \<Rightarrow> 'a^'n)"
  2605   shows "f = (\<lambda>x. dest_vec1 x *s column 1 (matrix f))"
  2606   apply (rule ext)
  2607   apply (subst matrix_works[OF lf, symmetric])
  2608   apply (auto simp add: Cart_eq matrix_vector_mult_def dest_vec1_def column_def  mult_commute UNIV_1)
  2609   done
  2610 
  2611 lemma linear_to_scalars: assumes lf: "linear (f::'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a^1)"
  2612   shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
  2613   apply (rule ext)
  2614   apply (subst matrix_works[OF lf, symmetric])
  2615   apply (simp add: Cart_eq matrix_vector_mult_def vec1_def row_def dot_def mult_commute forall_1)
  2616   done
  2617 
  2618 lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
  2619   by (simp add: dest_vec1_eq[symmetric])
  2620 
  2621 lemma setsum_scalars: assumes fS: "finite S"
  2622   shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)"
  2623   unfolding vec1_setsum[OF fS] by simp
  2624 
  2625 lemma dest_vec1_wlog_le: "(\<And>(x::'a::linorder ^ 1) y. P x y \<longleftrightarrow> P y x)  \<Longrightarrow> (\<And>x y. dest_vec1 x <= dest_vec1 y ==> P x y) \<Longrightarrow> P x y"
  2626   apply (cases "dest_vec1 x \<le> dest_vec1 y")
  2627   apply simp
  2628   apply (subgoal_tac "dest_vec1 y \<le> dest_vec1 x")
  2629   apply (auto)
  2630   done
  2631 
  2632 text{* Pasting vectors. *}
  2633 
  2634 lemma linear_fstcart: "linear fstcart"
  2635   by (auto simp add: linear_def Cart_eq)
  2636 
  2637 lemma linear_sndcart: "linear sndcart"
  2638   by (auto simp add: linear_def Cart_eq)
  2639 
  2640 lemma fstcart_vec[simp]: "fstcart(vec x) = vec x"
  2641   by (simp add: Cart_eq)
  2642 
  2643 lemma fstcart_add[simp]:"fstcart(x + y) = fstcart (x::'a::{plus,times}^('b + 'c)) + fstcart y"
  2644   by (simp add: Cart_eq)
  2645 
  2646 lemma fstcart_cmul[simp]:"fstcart(c*s x) = c*s fstcart (x::'a::{plus,times}^('b + 'c))"
  2647   by (simp add: Cart_eq)
  2648 
  2649 lemma fstcart_neg[simp]:"fstcart(- x) = - fstcart (x::'a::ring_1^('b + 'c))"
  2650   by (simp add: Cart_eq)
  2651 
  2652 lemma fstcart_sub[simp]:"fstcart(x - y) = fstcart (x::'a::ring_1^('b + 'c)) - fstcart y"
  2653   by (simp add: Cart_eq)
  2654 
  2655 lemma fstcart_setsum:
  2656   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
  2657   assumes fS: "finite S"
  2658   shows "fstcart (setsum f S) = setsum (\<lambda>i. fstcart (f i)) S"
  2659   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
  2660 
  2661 lemma sndcart_vec[simp]: "sndcart(vec x) = vec x"
  2662   by (simp add: Cart_eq)
  2663 
  2664 lemma sndcart_add[simp]:"sndcart(x + y) = sndcart (x::'a::{plus,times}^('b + 'c)) + sndcart y"
  2665   by (simp add: Cart_eq)
  2666 
  2667 lemma sndcart_cmul[simp]:"sndcart(c*s x) = c*s sndcart (x::'a::{plus,times}^('b + 'c))"
  2668   by (simp add: Cart_eq)
  2669 
  2670 lemma sndcart_neg[simp]:"sndcart(- x) = - sndcart (x::'a::ring_1^('b + 'c))"
  2671   by (simp add: Cart_eq)
  2672 
  2673 lemma sndcart_sub[simp]:"sndcart(x - y) = sndcart (x::'a::ring_1^('b + 'c)) - sndcart y"
  2674   by (simp add: Cart_eq)
  2675 
  2676 lemma sndcart_setsum:
  2677   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
  2678   assumes fS: "finite S"
  2679   shows "sndcart (setsum f S) = setsum (\<lambda>i. sndcart (f i)) S"
  2680   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
  2681 
  2682 lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x"
  2683   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
  2684 
  2685 lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)"
  2686   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
  2687 
  2688 lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1"
  2689   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
  2690 
  2691 lemma pastecart_neg[simp]: "pastecart (- (x::'a::ring_1^_)) (- y) = - pastecart x y"
  2692   unfolding vector_sneg_minus1 pastecart_cmul ..
  2693 
  2694 lemma pastecart_sub: "pastecart (x1::'a::ring_1^_) y1 - pastecart x2 y2 = pastecart (x1 - x2) (y1 - y2)"
  2695   by (simp add: diff_def pastecart_neg[symmetric] del: pastecart_neg)
  2696 
  2697 lemma pastecart_setsum:
  2698   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
  2699   assumes fS: "finite S"
  2700   shows "pastecart (setsum f S) (setsum g S) = setsum (\<lambda>i. pastecart (f i) (g i)) S"
  2701   by (simp  add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS] fstcart_pastecart sndcart_pastecart)
  2702 
  2703 lemma setsum_Plus:
  2704   "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
  2705     (\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))"
  2706   unfolding Plus_def
  2707   by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)
  2708 
  2709 lemma setsum_UNIV_sum:
  2710   fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
  2711   shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
  2712   apply (subst UNIV_Plus_UNIV [symmetric])
  2713   apply (rule setsum_Plus [OF finite finite])
  2714   done
  2715 
  2716 lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n::finite + 'm::finite))"
  2717 proof-
  2718   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
  2719     by (simp add: pastecart_fst_snd)
  2720   have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
  2721     by (simp add: dot_def setsum_UNIV_sum pastecart_def setsum_nonneg)
  2722   then show ?thesis
  2723     unfolding th0
  2724     unfolding real_vector_norm_def real_sqrt_le_iff id_def
  2725     by (simp add: dot_def)
  2726 qed
  2727 
  2728 lemma dist_fstcart: "dist(fstcart (x::real^_)) (fstcart y) <= dist x y"
  2729   unfolding dist_norm by (metis fstcart_sub[symmetric] norm_fstcart)
  2730 
  2731 lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n::finite + 'm::finite))"
  2732 proof-
  2733   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
  2734     by (simp add: pastecart_fst_snd)
  2735   have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
  2736     by (simp add: dot_def setsum_UNIV_sum pastecart_def setsum_nonneg)
  2737   then show ?thesis
  2738     unfolding th0
  2739     unfolding real_vector_norm_def real_sqrt_le_iff id_def
  2740     by (simp add: dot_def)
  2741 qed
  2742 
  2743 lemma dist_sndcart: "dist(sndcart (x::real^_)) (sndcart y) <= dist x y"
  2744   unfolding dist_norm by (metis sndcart_sub[symmetric] norm_sndcart)
  2745 
  2746 lemma dot_pastecart: "(pastecart (x1::'a::{times,comm_monoid_add}^'n::finite) (x2::'a::{times,comm_monoid_add}^'m::finite)) \<bullet> (pastecart y1 y2) =  x1 \<bullet> y1 + x2 \<bullet> y2"
  2747   by (simp add: dot_def setsum_UNIV_sum pastecart_def)
  2748 
  2749 lemma norm_pastecart: "norm(pastecart x y) <= norm(x :: real ^ 'm::finite) + norm(y::real^'n::finite)"
  2750   unfolding real_vector_norm_def dot_pastecart real_sqrt_le_iff id_def
  2751   apply (rule power2_le_imp_le)
  2752   apply (simp add: real_sqrt_pow2[OF add_nonneg_nonneg[OF dot_pos_le[of x] dot_pos_le[of y]]])
  2753   apply (auto simp add: power2_eq_square ring_simps)
  2754   apply (simp add: power2_eq_square[symmetric])
  2755   apply (rule mult_nonneg_nonneg)
  2756   apply (simp_all add: real_sqrt_pow2[OF dot_pos_le])
  2757   apply (rule add_nonneg_nonneg)
  2758   apply (simp_all add: real_sqrt_pow2[OF dot_pos_le])
  2759   done
  2760 
  2761 subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
  2762 
  2763 definition hull :: "'a set set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) where
  2764   "S hull s = Inter {t. t \<in> S \<and> s \<subseteq> t}"
  2765 
  2766 lemma hull_same: "s \<in> S \<Longrightarrow> S hull s = s"
  2767   unfolding hull_def by auto
  2768 
  2769 lemma hull_in: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) \<in> S"
  2770 unfolding hull_def subset_iff by auto
  2771 
  2772 lemma hull_eq: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) = s \<longleftrightarrow> s \<in> S"
  2773 using hull_same[of s S] hull_in[of S s] by metis
  2774 
  2775 
  2776 lemma hull_hull: "S hull (S hull s) = S hull s"
  2777   unfolding hull_def by blast
  2778 
  2779 lemma hull_subset: "s \<subseteq> (S hull s)"
  2780   unfolding hull_def by blast
  2781 
  2782 lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
  2783   unfolding hull_def by blast
  2784 
  2785 lemma hull_antimono: "S \<subseteq> T ==> (T hull s) \<subseteq> (S hull s)"
  2786   unfolding hull_def by blast
  2787 
  2788 lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> t \<in> S ==> (S hull s) \<subseteq> t"
  2789   unfolding hull_def by blast
  2790 
  2791 lemma subset_hull: "t \<in> S ==> S hull s \<subseteq> t \<longleftrightarrow>  s \<subseteq> t"
  2792   unfolding hull_def by blast
  2793 
  2794 lemma hull_unique: "s \<subseteq> t \<Longrightarrow> t \<in> S \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> t' \<in> S ==> t \<subseteq> t')
  2795            ==> (S hull s = t)"
  2796 unfolding hull_def by auto
  2797 
  2798 lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
  2799   using hull_minimal[of S "{x. P x}" Q]
  2800   by (auto simp add: subset_eq Collect_def mem_def)
  2801 
  2802 lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" by (metis hull_subset subset_eq)
  2803 
  2804 lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
  2805 unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
  2806 
  2807 lemma hull_union: assumes T: "\<And>T. T \<subseteq> S ==> Inter T \<in> S"
  2808   shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
  2809 apply rule
  2810 apply (rule hull_mono)
  2811 unfolding Un_subset_iff
  2812 apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
  2813 apply (rule hull_minimal)
  2814 apply (metis hull_union_subset)
  2815 apply (metis hull_in T)
  2816 done
  2817 
  2818 lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
  2819   unfolding hull_def by blast
  2820 
  2821 lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
  2822 by (metis hull_redundant_eq)
  2823 
  2824 text{* Archimedian properties and useful consequences. *}
  2825 
  2826 lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
  2827   using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto)
  2828 lemmas real_arch_lt = reals_Archimedean2
  2829 
  2830 lemmas real_arch = reals_Archimedean3
  2831 
  2832 lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
  2833   using reals_Archimedean
  2834   apply (auto simp add: field_simps inverse_positive_iff_positive)
  2835   apply (subgoal_tac "inverse (real n) > 0")
  2836   apply arith
  2837   apply simp
  2838   done
  2839 
  2840 lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
  2841 proof(induct n)
  2842   case 0 thus ?case by simp
  2843 next
  2844   case (Suc n)
  2845   hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
  2846   from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
  2847   from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
  2848   also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
  2849     apply (simp add: ring_simps)
  2850     using mult_left_mono[OF p Suc.prems] by simp
  2851   finally show ?case  by (simp add: real_of_nat_Suc ring_simps)
  2852 qed
  2853 
  2854 lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
  2855 proof-
  2856   from x have x0: "x - 1 > 0" by arith
  2857   from real_arch[OF x0, rule_format, of y]
  2858   obtain n::nat where n:"y < real n * (x - 1)" by metis
  2859   from x0 have x00: "x- 1 \<ge> 0" by arith
  2860   from real_pow_lbound[OF x00, of n] n
  2861   have "y < x^n" by auto
  2862   then show ?thesis by metis
  2863 qed
  2864 
  2865 lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
  2866   using real_arch_pow[of 2 x] by simp
  2867 
  2868 lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
  2869   shows "\<exists>n. x^n < y"
  2870 proof-
  2871   {assume x0: "x > 0"
  2872     from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
  2873     from real_arch_pow[OF ix, of "1/y"]
  2874     obtain n where n: "1/y < (1/x)^n" by blast
  2875     then
  2876     have ?thesis using y x0 by (auto simp add: field_simps power_divide) }
  2877   moreover
  2878   {assume "\<not> x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)}
  2879   ultimately show ?thesis by metis
  2880 qed
  2881 
  2882 lemma forall_pos_mono: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n::nat. n \<noteq> 0 ==> P(inverse(real n))) \<Longrightarrow> (\<And>e. 0 < e ==> P e)"
  2883   by (metis real_arch_inv)
  2884 
  2885 lemma forall_pos_mono_1: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e"
  2886   apply (rule forall_pos_mono)
  2887   apply auto
  2888   apply (atomize)
  2889   apply (erule_tac x="n - 1" in allE)
  2890   apply auto
  2891   done
  2892 
  2893 lemma real_archimedian_rdiv_eq_0: assumes x0: "x \<ge> 0" and c: "c \<ge> 0" and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
  2894   shows "x = 0"
  2895 proof-
  2896   {assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
  2897     from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x"  by blast
  2898     with xc[rule_format, of n] have "n = 0" by arith
  2899     with n c have False by simp}
  2900   then show ?thesis by blast
  2901 qed
  2902 
  2903 (* ------------------------------------------------------------------------- *)
  2904 (* Relate max and min to sup and inf.                                        *)
  2905 (* ------------------------------------------------------------------------- *)
  2906 
  2907 lemma real_max_rsup: "max x y = rsup {x,y}"
  2908 proof-
  2909   have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
  2910   from rsup_finite_le_iff[OF f, of "max x y"] have "rsup {x,y} \<le> max x y" by simp
  2911   moreover
  2912   have "max x y \<le> rsup {x,y}" using rsup_finite_ge_iff[OF f, of "max x y"]
  2913     by (simp add: linorder_linear)
  2914   ultimately show ?thesis by arith
  2915 qed
  2916 
  2917 lemma real_min_rinf: "min x y = rinf {x,y}"
  2918 proof-
  2919   have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
  2920   from rinf_finite_le_iff[OF f, of "min x y"] have "rinf {x,y} \<le> min x y"
  2921     by (simp add: linorder_linear)
  2922   moreover
  2923   have "min x y \<le> rinf {x,y}" using rinf_finite_ge_iff[OF f, of "min x y"]
  2924     by simp
  2925   ultimately show ?thesis by arith
  2926 qed
  2927 
  2928 (* ------------------------------------------------------------------------- *)
  2929 (* Geometric progression.                                                    *)
  2930 (* ------------------------------------------------------------------------- *)
  2931 
  2932 lemma sum_gp_basic: "((1::'a::{field}) - x) * setsum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
  2933   (is "?lhs = ?rhs")
  2934 proof-
  2935   {assume x1: "x = 1" hence ?thesis by simp}
  2936   moreover
  2937   {assume x1: "x\<noteq>1"
  2938     hence x1': "x - 1 \<noteq> 0" "1 - x \<noteq> 0" "x - 1 = - (1 - x)" "- (1 - x) \<noteq> 0" by auto
  2939     from geometric_sum[OF x1, of "Suc n", unfolded x1']
  2940     have "(- (1 - x)) * setsum (\<lambda>i. x^i) {0 .. n} = - (1 - x^(Suc n))"
  2941       unfolding atLeastLessThanSuc_atLeastAtMost
  2942       using x1' apply (auto simp only: field_simps)
  2943       apply (simp add: ring_simps)
  2944       done
  2945     then have ?thesis by (simp add: ring_simps) }
  2946   ultimately show ?thesis by metis
  2947 qed
  2948 
  2949 lemma sum_gp_multiplied: assumes mn: "m <= n"
  2950   shows "((1::'a::{field}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
  2951   (is "?lhs = ?rhs")
  2952 proof-
  2953   let ?S = "{0..(n - m)}"
  2954   from mn have mn': "n - m \<ge> 0" by arith
  2955   let ?f = "op + m"
  2956   have i: "inj_on ?f ?S" unfolding inj_on_def by auto
  2957   have f: "?f ` ?S = {m..n}"
  2958     using mn apply (auto simp add: image_iff Bex_def) by arith
  2959   have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)"
  2960     by (rule ext, simp add: power_add power_mult)
  2961   from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
  2962   have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp
  2963   then show ?thesis unfolding sum_gp_basic using mn
  2964     by (simp add: ring_simps power_add[symmetric])
  2965 qed
  2966 
  2967 lemma sum_gp: "setsum (op ^ (x::'a::{field})) {m .. n} =
  2968    (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m)
  2969                     else (x^ m - x^ (Suc n)) / (1 - x))"
  2970 proof-
  2971   {assume nm: "n < m" hence ?thesis by simp}
  2972   moreover
  2973   {assume "\<not> n < m" hence nm: "m \<le> n" by arith
  2974     {assume x: "x = 1"  hence ?thesis by simp}
  2975     moreover
  2976     {assume x: "x \<noteq> 1" hence nz: "1 - x \<noteq> 0" by simp
  2977       from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)}
  2978     ultimately have ?thesis by metis
  2979   }
  2980   ultimately show ?thesis by metis
  2981 qed
  2982 
  2983 lemma sum_gp_offset: "setsum (op ^ (x::'a::{field})) {m .. m+n} =
  2984   (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
  2985   unfolding sum_gp[of x m "m + n"] power_Suc
  2986   by (simp add: ring_simps power_add)
  2987 
  2988 
  2989 subsection{* A bit of linear algebra. *}
  2990 
  2991 definition "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x\<in> S. \<forall>y \<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in>S. c *s x \<in>S )"
  2992 definition "span S = (subspace hull S)"
  2993 definition "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
  2994 abbreviation "independent s == ~(dependent s)"
  2995 
  2996 (* Closure properties of subspaces.                                          *)
  2997 
  2998 lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
  2999 
  3000 lemma subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
  3001 
  3002 lemma subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
  3003   by (metis subspace_def)
  3004 
  3005 lemma subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *s x \<in> S"
  3006   by (metis subspace_def)
  3007 
  3008 lemma subspace_neg: "subspace S \<Longrightarrow> (x::'a::ring_1^'n) \<in> S \<Longrightarrow> - x \<in> S"
  3009   by (metis vector_sneg_minus1 subspace_mul)
  3010 
  3011 lemma subspace_sub: "subspace S \<Longrightarrow> (x::'a::ring_1^'n) \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
  3012   by (metis diff_def subspace_add subspace_neg)
  3013 
  3014 lemma subspace_setsum:
  3015   assumes sA: "subspace A" and fB: "finite B"
  3016   and f: "\<forall>x\<in> B. f x \<in> A"
  3017   shows "setsum f B \<in> A"
  3018   using  fB f sA
  3019   apply(induct rule: finite_induct[OF fB])
  3020   by (simp add: subspace_def sA, auto simp add: sA subspace_add)
  3021 
  3022 lemma subspace_linear_image:
  3023   assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and sS: "subspace S"
  3024   shows "subspace(f ` S)"
  3025   using lf sS linear_0[OF lf]
  3026   unfolding linear_def subspace_def
  3027   apply (auto simp add: image_iff)
  3028   apply (rule_tac x="x + y" in bexI, auto)
  3029   apply (rule_tac x="c*s x" in bexI, auto)
  3030   done
  3031 
  3032 lemma subspace_linear_preimage: "linear (f::'a::semiring_1^'n \<Rightarrow> _) ==> subspace S ==> subspace {x. f x \<in> S}"
  3033   by (auto simp add: subspace_def linear_def linear_0[of f])
  3034 
  3035 lemma subspace_trivial: "subspace {0::'a::semiring_1 ^_}"
  3036   by (simp add: subspace_def)
  3037 
  3038 lemma subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
  3039   by (simp add: subspace_def)
  3040 
  3041 
  3042 lemma span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
  3043   by (metis span_def hull_mono)
  3044 
  3045 lemma subspace_span: "subspace(span S)"
  3046   unfolding span_def
  3047   apply (rule hull_in[unfolded mem_def])
  3048   apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
  3049   apply auto
  3050   apply (erule_tac x="X" in ballE)
  3051   apply (simp add: mem_def)
  3052   apply blast
  3053   apply (erule_tac x="X" in ballE)
  3054   apply (erule_tac x="X" in ballE)
  3055   apply (erule_tac x="X" in ballE)
  3056   apply (clarsimp simp add: mem_def)
  3057   apply simp
  3058   apply simp
  3059   apply simp
  3060   apply (erule_tac x="X" in ballE)
  3061   apply (erule_tac x="X" in ballE)
  3062   apply (simp add: mem_def)
  3063   apply simp
  3064   apply simp
  3065   done
  3066 
  3067 lemma span_clauses:
  3068   "a \<in> S ==> a \<in> span S"
  3069   "0 \<in> span S"
  3070   "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
  3071   "x \<in> span S \<Longrightarrow> c *s x \<in> span S"
  3072   by (metis span_def hull_subset subset_eq subspace_span subspace_def)+
  3073 
  3074 lemma span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
  3075   and P: "subspace P" and x: "x \<in> span S" shows "P x"
  3076 proof-
  3077   from SP have SP': "S \<subseteq> P" by (simp add: mem_def subset_eq)
  3078   from P have P': "P \<in> subspace" by (simp add: mem_def)
  3079   from x hull_minimal[OF SP' P', unfolded span_def[symmetric]]
  3080   show "P x" by (metis mem_def subset_eq)
  3081 qed
  3082 
  3083 lemma span_empty: "span {} = {(0::'a::semiring_0 ^ 'n)}"
  3084   apply (simp add: span_def)
  3085   apply (rule hull_unique)
  3086   apply (auto simp add: mem_def subspace_def)
  3087   unfolding mem_def[of "0::'a^'n", symmetric]
  3088   apply simp
  3089   done
  3090 
  3091 lemma independent_empty: "independent {}"
  3092   by (simp add: dependent_def)
  3093 
  3094 lemma independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
  3095   apply (clarsimp simp add: dependent_def span_mono)
  3096   apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
  3097   apply force
  3098   apply (rule span_mono)
  3099   apply auto
  3100   done
  3101 
  3102 lemma span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
  3103   by (metis order_antisym span_def hull_minimal mem_def)
  3104 
  3105 lemma span_induct': assumes SP: "\<forall>x \<in> S. P x"
  3106   and P: "subspace P" shows "\<forall>x \<in> span S. P x"
  3107   using span_induct SP P by blast
  3108 
  3109 inductive span_induct_alt_help for S:: "'a::semiring_1^'n \<Rightarrow> bool"
  3110   where
  3111   span_induct_alt_help_0: "span_induct_alt_help S 0"
  3112   | span_induct_alt_help_S: "x \<in> S \<Longrightarrow> span_induct_alt_help S z \<Longrightarrow> span_induct_alt_help S (c *s x + z)"
  3113 
  3114 lemma span_induct_alt':
  3115   assumes h0: "h (0::'a::semiring_1^'n)" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c*s x + y)" shows "\<forall>x \<in> span S. h x"
  3116 proof-
  3117   {fix x:: "'a^'n" assume x: "span_induct_alt_help S x"
  3118     have "h x"
  3119       apply (rule span_induct_alt_help.induct[OF x])
  3120       apply (rule h0)
  3121       apply (rule hS, assumption, assumption)
  3122       done}
  3123   note th0 = this
  3124   {fix x assume x: "x \<in> span S"
  3125 
  3126     have "span_induct_alt_help S x"
  3127       proof(rule span_induct[where x=x and S=S])
  3128 	show "x \<in> span S" using x .
  3129       next
  3130 	fix x assume xS : "x \<in> S"
  3131 	  from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
  3132 	  show "span_induct_alt_help S x" by simp
  3133 	next
  3134 	have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0)
  3135 	moreover
  3136 	{fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y"
  3137 	  from h
  3138 	  have "span_induct_alt_help S (x + y)"
  3139 	    apply (induct rule: span_induct_alt_help.induct)
  3140 	    apply simp
  3141 	    unfolding add_assoc
  3142 	    apply (rule span_induct_alt_help_S)
  3143 	    apply assumption
  3144 	    apply simp
  3145 	    done}
  3146 	moreover
  3147 	{fix c x assume xt: "span_induct_alt_help S x"
  3148 	  then have "span_induct_alt_help S (c*s x)"
  3149 	    apply (induct rule: span_induct_alt_help.induct)
  3150 	    apply (simp add: span_induct_alt_help_0)
  3151 	    apply (simp add: vector_smult_assoc vector_add_ldistrib)
  3152 	    apply (rule span_induct_alt_help_S)
  3153 	    apply assumption
  3154 	    apply simp
  3155 	    done
  3156 	}
  3157 	ultimately show "subspace (span_induct_alt_help S)"
  3158 	  unfolding subspace_def mem_def Ball_def by blast
  3159       qed}
  3160   with th0 show ?thesis by blast
  3161 qed
  3162 
  3163 lemma span_induct_alt:
  3164   assumes h0: "h (0::'a::semiring_1^'n)" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c*s x + y)" and x: "x \<in> span S"
  3165   shows "h x"
  3166 using span_induct_alt'[of h S] h0 hS x by blast
  3167 
  3168 (* Individual closure properties. *)
  3169 
  3170 lemma span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses)
  3171 
  3172 lemma span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
  3173 
  3174 lemma span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
  3175   by (metis subspace_add subspace_span)
  3176 
  3177 lemma span_mul: "x \<in> span S ==> (c *s x) \<in> span S"
  3178   by (metis subspace_span subspace_mul)
  3179 
  3180 lemma span_neg: "x \<in> span S ==> - (x::'a::ring_1^'n) \<in> span S"
  3181   by (metis subspace_neg subspace_span)
  3182 
  3183 lemma span_sub: "(x::'a::ring_1^'n) \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
  3184   by (metis subspace_span subspace_sub)
  3185 
  3186 lemma span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
  3187   apply (rule subspace_setsum)
  3188   by (metis subspace_span subspace_setsum)+
  3189 
  3190 lemma span_add_eq: "(x::'a::ring_1^'n) \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
  3191   apply (auto simp only: span_add span_sub)
  3192   apply (subgoal_tac "(x + y) - x \<in> span S", simp)
  3193   by (simp only: span_add span_sub)
  3194 
  3195 (* Mapping under linear image. *)
  3196 
  3197 lemma span_linear_image: assumes lf: "linear (f::'a::semiring_1 ^ 'n => _)"
  3198   shows "span (f ` S) = f ` (span S)"
  3199 proof-
  3200   {fix x
  3201     assume x: "x \<in> span (f ` S)"
  3202     have "x \<in> f ` span S"
  3203       apply (rule span_induct[where x=x and S = "f ` S"])
  3204       apply (clarsimp simp add: image_iff)
  3205       apply (frule span_superset)
  3206       apply blast
  3207       apply (simp only: mem_def)
  3208       apply (rule subspace_linear_image[OF lf])
  3209       apply (rule subspace_span)
  3210       apply (rule x)
  3211       done}
  3212   moreover
  3213   {fix x assume x: "x \<in> span S"
  3214     have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_ext)
  3215       unfolding mem_def Collect_def ..
  3216     have "f x \<in> span (f ` S)"
  3217       apply (rule span_induct[where S=S])
  3218       apply (rule span_superset)
  3219       apply simp
  3220       apply (subst th0)
  3221       apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"])
  3222       apply (rule x)
  3223       done}
  3224   ultimately show ?thesis by blast
  3225 qed
  3226 
  3227 (* The key breakdown property. *)
  3228 
  3229 lemma span_breakdown:
  3230   assumes bS: "(b::'a::ring_1 ^ 'n) \<in> S" and aS: "a \<in> span S"
  3231   shows "\<exists>k. a - k*s b \<in> span (S - {b})" (is "?P a")
  3232 proof-
  3233   {fix x assume xS: "x \<in> S"
  3234     {assume ab: "x = b"
  3235       then have "?P x"
  3236 	apply simp
  3237 	apply (rule exI[where x="1"], simp)
  3238 	by (rule span_0)}
  3239     moreover
  3240     {assume ab: "x \<noteq> b"
  3241       then have "?P x"  using xS
  3242 	apply -
  3243 	apply (rule exI[where x=0])
  3244 	apply (rule span_superset)
  3245 	by simp}
  3246     ultimately have "?P x" by blast}
  3247   moreover have "subspace ?P"
  3248     unfolding subspace_def
  3249     apply auto
  3250     apply (simp add: mem_def)
  3251     apply (rule exI[where x=0])
  3252     using span_0[of "S - {b}"]
  3253     apply (simp add: mem_def)
  3254     apply (clarsimp simp add: mem_def)
  3255     apply (rule_tac x="k + ka" in exI)
  3256     apply (subgoal_tac "x + y - (k + ka) *s b = (x - k*s b) + (y - ka *s b)")
  3257     apply (simp only: )
  3258     apply (rule span_add[unfolded mem_def])
  3259     apply assumption+
  3260     apply (vector ring_simps)
  3261     apply (clarsimp simp add: mem_def)
  3262     apply (rule_tac x= "c*k" in exI)
  3263     apply (subgoal_tac "c *s x - (c * k) *s b = c*s (x - k*s b)")
  3264     apply (simp only: )
  3265     apply (rule span_mul[unfolded mem_def])
  3266     apply assumption
  3267     by (vector ring_simps)
  3268   ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
  3269 qed
  3270 
  3271 lemma span_breakdown_eq:
  3272   "(x::'a::ring_1^'n) \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *s a) \<in> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
  3273 proof-
  3274   {assume x: "x \<in> span (insert a S)"
  3275     from x span_breakdown[of "a" "insert a S" "x"]
  3276     have ?rhs apply clarsimp
  3277       apply (rule_tac x= "k" in exI)
  3278       apply (rule set_rev_mp[of _ "span (S - {a})" _])
  3279       apply assumption
  3280       apply (rule span_mono)
  3281       apply blast
  3282       done}
  3283   moreover
  3284   { fix k assume k: "x - k *s a \<in> span S"
  3285     have eq: "x = (x - k *s a) + k *s a" by vector
  3286     have "(x - k *s a) + k *s a \<in> span (insert a S)"
  3287       apply (rule span_add)
  3288       apply (rule set_rev_mp[of _ "span S" _])
  3289       apply (rule k)
  3290       apply (rule span_mono)
  3291       apply blast
  3292       apply (rule span_mul)
  3293       apply (rule span_superset)
  3294       apply blast
  3295       done
  3296     then have ?lhs using eq by metis}
  3297   ultimately show ?thesis by blast
  3298 qed
  3299 
  3300 (* Hence some "reversal" results.*)
  3301 
  3302 lemma in_span_insert:
  3303   assumes a: "(a::'a::field^'n) \<in> span (insert b S)" and na: "a \<notin> span S"
  3304   shows "b \<in> span (insert a S)"
  3305 proof-
  3306   from span_breakdown[of b "insert b S" a, OF insertI1 a]
  3307   obtain k where k: "a - k*s b \<in> span (S - {b})" by auto
  3308   {assume k0: "k = 0"
  3309     with k have "a \<in> span S"
  3310       apply (simp)
  3311       apply (rule set_rev_mp)
  3312       apply assumption
  3313       apply (rule span_mono)
  3314       apply blast
  3315       done
  3316     with na  have ?thesis by blast}
  3317   moreover
  3318   {assume k0: "k \<noteq> 0"
  3319     have eq: "b = (1/k) *s a - ((1/k) *s a - b)" by vector
  3320     from k0 have eq': "(1/k) *s (a - k*s b) = (1/k) *s a - b"
  3321       by (vector field_simps)
  3322     from k have "(1/k) *s (a - k*s b) \<in> span (S - {b})"
  3323       by (rule span_mul)
  3324     hence th: "(1/k) *s a - b \<in> span (S - {b})"
  3325       unfolding eq' .
  3326 
  3327     from k
  3328     have ?thesis
  3329       apply (subst eq)
  3330       apply (rule span_sub)
  3331       apply (rule span_mul)
  3332       apply (rule span_superset)
  3333       apply blast
  3334       apply (rule set_rev_mp)
  3335       apply (rule th)
  3336       apply (rule span_mono)
  3337       using na by blast}
  3338   ultimately show ?thesis by blast
  3339 qed
  3340 
  3341 lemma in_span_delete:
  3342   assumes a: "(a::'a::field^'n) \<in> span S"
  3343   and na: "a \<notin> span (S-{b})"
  3344   shows "b \<in> span (insert a (S - {b}))"
  3345   apply (rule in_span_insert)
  3346   apply (rule set_rev_mp)
  3347   apply (rule a)
  3348   apply (rule span_mono)
  3349   apply blast
  3350   apply (rule na)
  3351   done
  3352 
  3353 (* Transitivity property. *)
  3354 
  3355 lemma span_trans:
  3356   assumes x: "(x::'a::ring_1^'n) \<in> span S" and y: "y \<in> span (insert x S)"
  3357   shows "y \<in> span S"
  3358 proof-
  3359   from span_breakdown[of x "insert x S" y, OF insertI1 y]
  3360   obtain k where k: "y -k*s x \<in> span (S - {x})" by auto
  3361   have eq: "y = (y - k *s x) + k *s x" by vector
  3362   show ?thesis
  3363     apply (subst eq)
  3364     apply (rule span_add)
  3365     apply (rule set_rev_mp)
  3366     apply (rule k)
  3367     apply (rule span_mono)
  3368     apply blast
  3369     apply (rule span_mul)
  3370     by (rule x)
  3371 qed
  3372 
  3373 (* ------------------------------------------------------------------------- *)
  3374 (* An explicit expansion is sometimes needed.                                *)
  3375 (* ------------------------------------------------------------------------- *)
  3376 
  3377 lemma span_explicit:
  3378   "span P = {y::'a::semiring_1^'n. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *s v) S = y}"
  3379   (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
  3380 proof-
  3381   {fix x assume x: "x \<in> ?E"
  3382     then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *s v) S = x"
  3383       by blast
  3384     have "x \<in> span P"
  3385       unfolding u[symmetric]
  3386       apply (rule span_setsum[OF fS])
  3387       using span_mono[OF SP]
  3388       by (auto intro: span_superset span_mul)}
  3389   moreover
  3390   have "\<forall>x \<in> span P. x \<in> ?E"
  3391     unfolding mem_def Collect_def
  3392   proof(rule span_induct_alt')
  3393     show "?h 0"
  3394       apply (rule exI[where x="{}"]) by simp
  3395   next
  3396     fix c x y
  3397     assume x: "x \<in> P" and hy: "?h y"
  3398     from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
  3399       and u: "setsum (\<lambda>v. u v *s v) S = y" by blast
  3400     let ?S = "insert x S"
  3401     let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c)
  3402                   else u y"
  3403     from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
  3404     {assume xS: "x \<in> S"
  3405       have S1: "S = (S - {x}) \<union> {x}"
  3406 	and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
  3407       have "setsum (\<lambda>v. ?u v *s v) ?S =(\<Sum>v\<in>S - {x}. u v *s v) + (u x + c) *s x"
  3408 	using xS
  3409 	by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]
  3410 	  setsum_clauses(2)[OF fS] cong del: if_weak_cong)
  3411       also have "\<dots> = (\<Sum>v\<in>S. u v *s v) + c *s x"
  3412 	apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
  3413 	by (vector ring_simps)
  3414       also have "\<dots> = c*s x + y"
  3415 	by (simp add: add_commute u)
  3416       finally have "setsum (\<lambda>v. ?u v *s v) ?S = c*s x + y" .
  3417     then have "?Q ?S ?u (c*s x + y)" using th0 by blast}
  3418   moreover
  3419   {assume xS: "x \<notin> S"
  3420     have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *s v) = y"
  3421       unfolding u[symmetric]
  3422       apply (rule setsum_cong2)
  3423       using xS by auto
  3424     have "?Q ?S ?u (c*s x + y)" using fS xS th0
  3425       by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)}
  3426   ultimately have "?Q ?S ?u (c*s x + y)"
  3427     by (cases "x \<in> S", simp, simp)
  3428     then show "?h (c*s x + y)"
  3429       apply -
  3430       apply (rule exI[where x="?S"])
  3431       apply (rule exI[where x="?u"]) by metis
  3432   qed
  3433   ultimately show ?thesis by blast
  3434 qed
  3435 
  3436 lemma dependent_explicit:
  3437   "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>(v::'a::{idom,field}^'n) \<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *s v) S = 0))" (is "?lhs = ?rhs")
  3438 proof-
  3439   {assume dP: "dependent P"
  3440     then obtain a S u where aP: "a \<in> P" and fS: "finite S"
  3441       and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *s v) S = a"
  3442       unfolding dependent_def span_explicit by blast
  3443     let ?S = "insert a S"
  3444     let ?u = "\<lambda>y. if y = a then - 1 else u y"
  3445     let ?v = a
  3446     from aP SP have aS: "a \<notin> S" by blast
  3447     from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
  3448     have s0: "setsum (\<lambda>v. ?u v *s v) ?S = 0"
  3449       using fS aS
  3450       apply (simp add: vector_smult_lneg vector_smult_lid setsum_clauses ring_simps )
  3451       apply (subst (2) ua[symmetric])
  3452       apply (rule setsum_cong2)
  3453       by auto
  3454     with th0 have ?rhs
  3455       apply -
  3456       apply (rule exI[where x= "?S"])
  3457       apply (rule exI[where x= "?u"])
  3458       by clarsimp}
  3459   moreover
  3460   {fix S u v assume fS: "finite S"
  3461       and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
  3462     and u: "setsum (\<lambda>v. u v *s v) S = 0"
  3463     let ?a = v
  3464     let ?S = "S - {v}"
  3465     let ?u = "\<lambda>i. (- u i) / u v"
  3466     have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"       using fS SP vS by auto
  3467     have "setsum (\<lambda>v. ?u v *s v) ?S = setsum (\<lambda>v. (- (inverse (u ?a))) *s (u v *s v)) S - ?u v *s v"
  3468       using fS vS uv
  3469       by (simp add: setsum_diff1 vector_smult_lneg divide_inverse
  3470 	vector_smult_assoc field_simps)
  3471     also have "\<dots> = ?a"
  3472       unfolding setsum_cmul u
  3473       using uv by (simp add: vector_smult_lneg)
  3474     finally  have "setsum (\<lambda>v. ?u v *s v) ?S = ?a" .
  3475     with th0 have ?lhs
  3476       unfolding dependent_def span_explicit
  3477       apply -
  3478       apply (rule bexI[where x= "?a"])
  3479       apply simp_all
  3480       apply (rule exI[where x= "?S"])
  3481       by auto}
  3482   ultimately show ?thesis by blast
  3483 qed
  3484 
  3485 
  3486 lemma span_finite:
  3487   assumes fS: "finite S"
  3488   shows "span S = {(y::'a::semiring_1^'n). \<exists>u. setsum (\<lambda>v. u v *s v) S = y}"
  3489   (is "_ = ?rhs")
  3490 proof-
  3491   {fix y assume y: "y \<in> span S"
  3492     from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and
  3493       u: "setsum (\<lambda>v. u v *s v) S' = y" unfolding span_explicit by blast
  3494     let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
  3495     from setsum_restrict_set[OF fS, of "\<lambda>v. u v *s v" S', symmetric] SS'
  3496     have "setsum (\<lambda>v. ?u v *s v) S = setsum (\<lambda>v. u v *s v) S'"
  3497       unfolding cond_value_iff cond_application_beta
  3498       apply (simp add: cond_value_iff cong del: if_weak_cong)
  3499       apply (rule setsum_cong)
  3500       apply auto
  3501       done
  3502     hence "setsum (\<lambda>v. ?u v *s v) S = y" by (metis u)
  3503     hence "y \<in> ?rhs" by auto}
  3504   moreover
  3505   {fix y u assume u: "setsum (\<lambda>v. u v *s v) S = y"
  3506     then have "y \<in> span S" using fS unfolding span_explicit by auto}
  3507   ultimately show ?thesis by blast
  3508 qed
  3509 
  3510 
  3511 (* Standard bases are a spanning set, and obviously finite.                  *)
  3512 
  3513 lemma span_stdbasis:"span {basis i :: 'a::ring_1^'n::finite | i. i \<in> (UNIV :: 'n set)} = UNIV"
  3514 apply (rule set_ext)
  3515 apply auto
  3516 apply (subst basis_expansion[symmetric])
  3517 apply (rule span_setsum)
  3518 apply simp
  3519 apply auto
  3520 apply (rule span_mul)
  3521 apply (rule span_superset)
  3522 apply (auto simp add: Collect_def mem_def)
  3523 done
  3524 
  3525 lemma has_size_stdbasis: "{basis i ::real ^'n::finite | i. i \<in> (UNIV :: 'n set)} hassize CARD('n)" (is "?S hassize ?n")
  3526 proof-
  3527   have eq: "?S = basis ` UNIV" by blast
  3528   show ?thesis unfolding eq
  3529     apply (rule hassize_image_inj[OF basis_inj])
  3530     by (simp add: hassize_def)
  3531 qed
  3532 
  3533 lemma finite_stdbasis: "finite {basis i ::real^'n::finite |i. i\<in> (UNIV:: 'n set)}"
  3534   using has_size_stdbasis[unfolded hassize_def]
  3535   ..
  3536 
  3537 lemma card_stdbasis: "card {basis i ::real^'n::finite |i. i\<in> (UNIV :: 'n set)} = CARD('n)"
  3538   using has_size_stdbasis[unfolded hassize_def]
  3539   ..
  3540 
  3541 lemma independent_stdbasis_lemma:
  3542   assumes x: "(x::'a::semiring_1 ^ 'n) \<in> span (basis ` S)"
  3543   and iS: "i \<notin> S"
  3544   shows "(x$i) = 0"
  3545 proof-
  3546   let ?U = "UNIV :: 'n set"
  3547   let ?B = "basis ` S"
  3548   let ?P = "\<lambda>(x::'a^'n). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0"
  3549  {fix x::"'a^'n" assume xS: "x\<in> ?B"
  3550    from xS have "?P x" by auto}
  3551  moreover
  3552  have "subspace ?P"
  3553    by (auto simp add: subspace_def Collect_def mem_def)
  3554  ultimately show ?thesis
  3555    using x span_induct[of ?B ?P x] iS by blast
  3556 qed
  3557 
  3558 lemma independent_stdbasis: "independent {basis i ::real^'n::finite |i. i\<in> (UNIV :: 'n set)}"
  3559 proof-
  3560   let ?I = "UNIV :: 'n set"
  3561   let ?b = "basis :: _ \<Rightarrow> real ^'n"
  3562   let ?B = "?b ` ?I"
  3563   have eq: "{?b i|i. i \<in> ?I} = ?B"
  3564     by auto
  3565   {assume d: "dependent ?B"
  3566     then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"
  3567       unfolding dependent_def by auto
  3568     have eq1: "?B - {?b k} = ?B - ?b ` {k}"  by simp
  3569     have eq2: "?B - {?b k} = ?b ` (?I - {k})"
  3570       unfolding eq1
  3571       apply (rule inj_on_image_set_diff[symmetric])
  3572       apply (rule basis_inj) using k(1) by auto
  3573     from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
  3574     from independent_stdbasis_lemma[OF th0, of k, simplified]
  3575     have False by simp}
  3576   then show ?thesis unfolding eq dependent_def ..
  3577 qed
  3578 
  3579 (* This is useful for building a basis step-by-step.                         *)
  3580 
  3581 lemma independent_insert:
  3582   "independent(insert (a::'a::field ^'n) S) \<longleftrightarrow>
  3583       (if a \<in> S then independent S
  3584                 else independent S \<and> a \<notin> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
  3585 proof-
  3586   {assume aS: "a \<in> S"
  3587     hence ?thesis using insert_absorb[OF aS] by simp}
  3588   moreover
  3589   {assume aS: "a \<notin> S"
  3590     {assume i: ?lhs
  3591       then have ?rhs using aS
  3592 	apply simp
  3593 	apply (rule conjI)
  3594 	apply (rule independent_mono)
  3595 	apply assumption
  3596 	apply blast
  3597 	by (simp add: dependent_def)}
  3598     moreover
  3599     {assume i: ?rhs
  3600       have ?lhs using i aS
  3601 	apply simp
  3602 	apply (auto simp add: dependent_def)
  3603 	apply (case_tac "aa = a", auto)
  3604 	apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
  3605 	apply simp
  3606 	apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
  3607 	apply (subgoal_tac "insert aa (S - {aa}) = S")
  3608 	apply simp
  3609 	apply blast
  3610 	apply (rule in_span_insert)
  3611 	apply assumption
  3612 	apply blast
  3613 	apply blast
  3614 	done}
  3615     ultimately have ?thesis by blast}
  3616   ultimately show ?thesis by blast
  3617 qed
  3618 
  3619 (* The degenerate case of the Exchange Lemma.  *)
  3620 
  3621 lemma mem_delete: "x \<in> (A - {a}) \<longleftrightarrow> x \<noteq> a \<and> x \<in> A"
  3622   by blast
  3623 
  3624 lemma span_span: "span (span A) = span A"
  3625   unfolding span_def hull_hull ..
  3626 
  3627 lemma span_inc: "S \<subseteq> span S"
  3628   by (metis subset_eq span_superset)
  3629 
  3630 lemma spanning_subset_independent:
  3631   assumes BA: "B \<subseteq> A" and iA: "independent (A::('a::field ^'n) set)"
  3632   and AsB: "A \<subseteq> span B"
  3633   shows "A = B"
  3634 proof
  3635   from BA show "B \<subseteq> A" .
  3636 next
  3637   from span_mono[OF BA] span_mono[OF AsB]
  3638   have sAB: "span A = span B" unfolding span_span by blast
  3639 
  3640   {fix x assume x: "x \<in> A"
  3641     from iA have th0: "x \<notin> span (A - {x})"
  3642       unfolding dependent_def using x by blast
  3643     from x have xsA: "x \<in> span A" by (blast intro: span_superset)
  3644     have "A - {x} \<subseteq> A" by blast
  3645     hence th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
  3646     {assume xB: "x \<notin> B"
  3647       from xB BA have "B \<subseteq> A -{x}" by blast
  3648       hence "span B \<subseteq> span (A - {x})" by (metis span_mono)
  3649       with th1 th0 sAB have "x \<notin> span A" by blast
  3650       with x have False by (metis span_superset)}
  3651     then have "x \<in> B" by blast}
  3652   then show "A \<subseteq> B" by blast
  3653 qed
  3654 
  3655 (* The general case of the Exchange Lemma, the key to what follows.  *)
  3656 
  3657 lemma exchange_lemma:
  3658   assumes f:"finite (t:: ('a::field^'n) set)" and i: "independent s"
  3659   and sp:"s \<subseteq> span t"
  3660   shows "\<exists>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
  3661 using f i sp
  3662 proof(induct c\<equiv>"card(t - s)" arbitrary: s t rule: nat_less_induct)
  3663   fix n:: nat and s t :: "('a ^'n) set"
  3664   assume H: " \<forall>m<n. \<forall>(x:: ('a ^'n) set) xa.
  3665                 finite xa \<longrightarrow>
  3666                 independent x \<longrightarrow>
  3667                 x \<subseteq> span xa \<longrightarrow>
  3668                 m = card (xa - x) \<longrightarrow>
  3669                 (\<exists>t'. (t' hassize card xa) \<and>
  3670                       x \<subseteq> t' \<and> t' \<subseteq> x \<union> xa \<and> x \<subseteq> span t')"
  3671     and ft: "finite t" and s: "independent s" and sp: "s \<subseteq> span t"
  3672     and n: "n = card (t - s)"
  3673   let ?P = "\<lambda>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
  3674   let ?ths = "\<exists>t'. ?P t'"
  3675   {assume st: "s \<subseteq> t"
  3676     from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
  3677       by (auto simp add: hassize_def intro: span_superset)}
  3678   moreover
  3679   {assume st: "t \<subseteq> s"
  3680 
  3681     from spanning_subset_independent[OF st s sp]
  3682       st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
  3683       by (auto simp add: hassize_def intro: span_superset)}
  3684   moreover
  3685   {assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
  3686     from st(2) obtain b where b: "b \<in> t" "b \<notin> s" by blast
  3687       from b have "t - {b} - s \<subset> t - s" by blast
  3688       then have cardlt: "card (t - {b} - s) < n" using n ft
  3689  	by (auto intro: psubset_card_mono)
  3690       from b ft have ct0: "card t \<noteq> 0" by auto
  3691     {assume stb: "s \<subseteq> span(t -{b})"
  3692       from ft have ftb: "finite (t -{b})" by auto
  3693       from H[rule_format, OF cardlt ftb s stb]
  3694       obtain u where u: "u hassize card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" by blast
  3695       let ?w = "insert b u"
  3696       have th0: "s \<subseteq> insert b u" using u by blast
  3697       from u(3) b have "u \<subseteq> s \<union> t" by blast
  3698       then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
  3699       have bu: "b \<notin> u" using b u by blast
  3700       from u(1) have fu: "finite u" by (simp add: hassize_def)
  3701       from u(1) ft b have "u hassize (card t - 1)" by auto
  3702       then
  3703       have th2: "insert b u hassize card t"
  3704 	using  card_insert_disjoint[OF fu bu] ct0 by (auto simp add: hassize_def)
  3705       from u(4) have "s \<subseteq> span u" .
  3706       also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
  3707       finally have th3: "s \<subseteq> span (insert b u)" .      from th0 th1 th2 th3 have th: "?P ?w"  by blast
  3708       from th have ?ths by blast}
  3709     moreover
  3710     {assume stb: "\<not> s \<subseteq> span(t -{b})"
  3711       from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
  3712       have ab: "a \<noteq> b" using a b by blast
  3713       have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
  3714       have mlt: "card ((insert a (t - {b})) - s) < n"
  3715 	using cardlt ft n  a b by auto
  3716       have ft': "finite (insert a (t - {b}))" using ft by auto
  3717       {fix x assume xs: "x \<in> s"
  3718 	have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
  3719 	from b(1) have "b \<in> span t" by (simp add: span_superset)
  3720 	have bs: "b \<in> span (insert a (t - {b}))"
  3721 	  by (metis in_span_delete a sp mem_def subset_eq)
  3722 	from xs sp have "x \<in> span t" by blast
  3723 	with span_mono[OF t]
  3724 	have x: "x \<in> span (insert b (insert a (t - {b})))" ..
  3725 	from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))"  .}
  3726       then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
  3727 
  3728       from H[rule_format, OF mlt ft' s sp' refl] obtain u where
  3729 	u: "u hassize card (insert a (t -{b}))" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
  3730 	"s \<subseteq> span u" by blast
  3731       from u a b ft at ct0 have "?P u" by (auto simp add: hassize_def)
  3732       then have ?ths by blast }
  3733     ultimately have ?ths by blast
  3734   }
  3735   ultimately
  3736   show ?ths  by blast
  3737 qed
  3738 
  3739 (* This implies corresponding size bounds.                                   *)
  3740 
  3741 lemma independent_span_bound:
  3742   assumes f: "finite t" and i: "independent (s::('a::field^'n) set)" and sp:"s \<subseteq> span t"
  3743   shows "finite s \<and> card s \<le> card t"
  3744   by (metis exchange_lemma[OF f i sp] hassize_def finite_subset card_mono)
  3745 
  3746 lemma finite_Atleast_Atmost[simp]: "finite {f x |x. x\<in> {(i::'a::finite_intvl_succ) .. j}}"
  3747 proof-
  3748   have eq: "{f x |x. x\<in> {i .. j}} = f ` {i .. j}" by auto
  3749   show ?thesis unfolding eq
  3750     apply (rule finite_imageI)
  3751     apply (rule finite_intvl)
  3752     done
  3753 qed
  3754 
  3755 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
  3756 proof-
  3757   have eq: "{f x |x. x\<in> UNIV} = f ` UNIV" by auto
  3758   show ?thesis unfolding eq
  3759     apply (rule finite_imageI)
  3760     apply (rule finite)
  3761     done
  3762 qed
  3763 
  3764 
  3765 lemma independent_bound:
  3766   fixes S:: "(real^'n::finite) set"
  3767   shows "independent S \<Longrightarrow> finite S \<and> card S <= CARD('n)"
  3768   apply (subst card_stdbasis[symmetric])
  3769   apply (rule independent_span_bound)
  3770   apply (rule finite_Atleast_Atmost_nat)
  3771   apply assumption
  3772   unfolding span_stdbasis
  3773   apply (rule subset_UNIV)
  3774   done
  3775 
  3776 lemma dependent_biggerset: "(finite (S::(real ^'n::finite) set) ==> card S > CARD('n)) ==> dependent S"
  3777   by (metis independent_bound not_less)
  3778 
  3779 (* Hence we can create a maximal independent subset.                         *)
  3780 
  3781 lemma maximal_independent_subset_extend:
  3782   assumes sv: "(S::(real^'n::finite) set) \<subseteq> V" and iS: "independent S"
  3783   shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
  3784   using sv iS
  3785 proof(induct d\<equiv> "CARD('n) - card S" arbitrary: S rule: nat_less_induct)
  3786   fix n and S:: "(real^'n) set"
  3787   assume H: "\<forall>m<n. \<forall>S \<subseteq> V. independent S \<longrightarrow> m = CARD('n) - card S \<longrightarrow>
  3788               (\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B)"
  3789     and sv: "S \<subseteq> V" and i: "independent S" and n: "n = CARD('n) - card S"
  3790   let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
  3791   let ?ths = "\<exists>x. ?P x"
  3792   let ?d = "CARD('n)"
  3793   {assume "V \<subseteq> span S"
  3794     then have ?ths  using sv i by blast }
  3795   moreover
  3796   {assume VS: "\<not> V \<subseteq> span S"
  3797     from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
  3798     from a have aS: "a \<notin> S" by (auto simp add: span_superset)
  3799     have th0: "insert a S \<subseteq> V" using a sv by blast
  3800     from independent_insert[of a S]  i a
  3801     have th1: "independent (insert a S)" by auto
  3802     have mlt: "?d - card (insert a S) < n"
  3803       using aS a n independent_bound[OF th1]
  3804       by auto
  3805 
  3806     from H[rule_format, OF mlt th0 th1 refl]
  3807     obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
  3808       by blast
  3809     from B have "?P B" by auto
  3810     then have ?ths by blast}
  3811   ultimately show ?ths by blast
  3812 qed
  3813 
  3814 lemma maximal_independent_subset:
  3815   "\<exists>(B:: (real ^'n::finite) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
  3816   by (metis maximal_independent_subset_extend[of "{}:: (real ^'n) set"] empty_subsetI independent_empty)
  3817 
  3818 (* Notion of dimension.                                                      *)
  3819 
  3820 definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n))"
  3821 
  3822 lemma basis_exists:  "\<exists>B. (B :: (real ^'n::finite) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize dim V)"
  3823 unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n)"]
  3824 unfolding hassize_def
  3825 using maximal_independent_subset[of V] independent_bound
  3826 by auto
  3827 
  3828 (* Consequences of independence or spanning for cardinality.                 *)
  3829 
  3830 lemma independent_card_le_dim: "(B::(real ^'n::finite) set) \<subseteq> V \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B \<le> dim V"
  3831 by (metis basis_exists[of V] independent_span_bound[where ?'a=real] hassize_def subset_trans)
  3832 
  3833 lemma span_card_ge_dim:  "(B::(real ^'n::finite) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
  3834   by (metis basis_exists[of V] independent_span_bound hassize_def subset_trans)
  3835 
  3836 lemma basis_card_eq_dim:
  3837   "B \<subseteq> (V:: (real ^'n::finite) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
  3838   by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_mono)
  3839 
  3840 lemma dim_unique: "(B::(real ^'n::finite) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> B hassize n \<Longrightarrow> dim V = n"
  3841   by (metis basis_card_eq_dim hassize_def)
  3842 
  3843 (* More lemmas about dimension.                                              *)
  3844 
  3845 lemma dim_univ: "dim (UNIV :: (real^'n::finite) set) = CARD('n)"
  3846   apply (rule dim_unique[of "{basis i |i. i\<in> (UNIV :: 'n set)}"])
  3847   by (auto simp only: span_stdbasis has_size_stdbasis independent_stdbasis)
  3848 
  3849 lemma dim_subset:
  3850   "(S:: (real ^'n::finite) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
  3851   using basis_exists[of T] basis_exists[of S]
  3852   by (metis independent_span_bound[where ?'a = real and ?'n = 'n] subset_eq hassize_def)
  3853 
  3854 lemma dim_subset_univ: "dim (S:: (real^'n::finite) set) \<le> CARD('n)"
  3855   by (metis dim_subset subset_UNIV dim_univ)
  3856 
  3857 (* Converses to those.                                                       *)
  3858 
  3859 lemma card_ge_dim_independent:
  3860   assumes BV:"(B::(real ^'n::finite) set) \<subseteq> V" and iB:"independent B" and dVB:"dim V \<le> card B"
  3861   shows "V \<subseteq> span B"
  3862 proof-
  3863   {fix a assume aV: "a \<in> V"
  3864     {assume aB: "a \<notin> span B"
  3865       then have iaB: "independent (insert a B)" using iB aV  BV by (simp add: independent_insert)
  3866       from aV BV have th0: "insert a B \<subseteq> V" by blast
  3867       from aB have "a \<notin>B" by (auto simp add: span_superset)
  3868       with independent_card_le_dim[OF th0 iaB] dVB  have False by auto}
  3869     then have "a \<in> span B"  by blast}
  3870   then show ?thesis by blast
  3871 qed
  3872 
  3873 lemma card_le_dim_spanning:
  3874   assumes BV: "(B:: (real ^'n::finite) set) \<subseteq> V" and VB: "V \<subseteq> span B"
  3875   and fB: "finite B" and dVB: "dim V \<ge> card B"
  3876   shows "independent B"
  3877 proof-
  3878   {fix a assume a: "a \<in> B" "a \<in> span (B -{a})"
  3879     from a fB have c0: "card B \<noteq> 0" by auto
  3880     from a fB have cb: "card (B -{a}) = card B - 1" by auto
  3881     from BV a have th0: "B -{a} \<subseteq> V" by blast
  3882     {fix x assume x: "x \<in> V"
  3883       from a have eq: "insert a (B -{a}) = B" by blast
  3884       from x VB have x': "x \<in> span B" by blast
  3885       from span_trans[OF a(2), unfolded eq, OF x']
  3886       have "x \<in> span (B -{a})" . }
  3887     then have th1: "V \<subseteq> span (B -{a})" by blast
  3888     have th2: "finite (B -{a})" using fB by auto
  3889     from span_card_ge_dim[OF th0 th1 th2]
  3890     have c: "dim V \<le> card (B -{a})" .
  3891     from c c0 dVB cb have False by simp}
  3892   then show ?thesis unfolding dependent_def by blast
  3893 qed
  3894 
  3895 lemma card_eq_dim: "(B:: (real ^'n::finite) set) \<subseteq> V \<Longrightarrow> B hassize dim V \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
  3896   by (metis hassize_def order_eq_iff card_le_dim_spanning
  3897     card_ge_dim_independent)
  3898 
  3899 (* ------------------------------------------------------------------------- *)
  3900 (* More general size bound lemmas.                                           *)
  3901 (* ------------------------------------------------------------------------- *)
  3902 
  3903 lemma independent_bound_general:
  3904   "independent (S:: (real^'n::finite) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
  3905   by (metis independent_card_le_dim independent_bound subset_refl)
  3906 
  3907 lemma dependent_biggerset_general: "(finite (S:: (real^'n::finite) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
  3908   using independent_bound_general[of S] by (metis linorder_not_le)
  3909 
  3910 lemma dim_span: "dim (span (S:: (real ^'n::finite) set)) = dim S"
  3911 proof-
  3912   have th0: "dim S \<le> dim (span S)"
  3913     by (auto simp add: subset_eq intro: dim_subset span_superset)
  3914   from basis_exists[of S]
  3915   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
  3916   from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
  3917   have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc)
  3918   have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span)
  3919   from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
  3920     using fB(2)  by arith
  3921 qed
  3922 
  3923 lemma subset_le_dim: "(S:: (real ^'n::finite) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
  3924   by (metis dim_span dim_subset)
  3925 
  3926 lemma span_eq_dim: "span (S:: (real ^'n::finite) set) = span T ==> dim S = dim T"
  3927   by (metis dim_span)
  3928 
  3929 lemma spans_image:
  3930   assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and VB: "V \<subseteq> span B"
  3931   shows "f ` V \<subseteq> span (f ` B)"
  3932   unfolding span_linear_image[OF lf]
  3933   by (metis VB image_mono)
  3934 
  3935 lemma dim_image_le:
  3936   fixes f :: "real^'n::finite \<Rightarrow> real^'m::finite"
  3937   assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S:: (real ^'n::finite) set)"
  3938 proof-
  3939   from basis_exists[of S] obtain B where
  3940     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
  3941   from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
  3942   have "dim (f ` S) \<le> card (f ` B)"
  3943     apply (rule span_card_ge_dim)
  3944     using lf B fB by (auto simp add: span_linear_image spans_image subset_image_iff)
  3945   also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
  3946   finally show ?thesis .
  3947 qed
  3948 
  3949 (* Relation between bases and injectivity/surjectivity of map.               *)
  3950 
  3951 lemma spanning_surjective_image:
  3952   assumes us: "UNIV \<subseteq> span (S:: ('a::semiring_1 ^'n) set)"
  3953   and lf: "linear f" and sf: "surj f"
  3954   shows "UNIV \<subseteq> span (f ` S)"
  3955 proof-
  3956   have "UNIV \<subseteq> f ` UNIV" using sf by (auto simp add: surj_def)
  3957   also have " \<dots> \<subseteq> span (f ` S)" using spans_image[OF lf us] .
  3958 finally show ?thesis .
  3959 qed
  3960 
  3961 lemma independent_injective_image:
  3962   assumes iS: "independent (S::('a::semiring_1^'n) set)" and lf: "linear f" and fi: "inj f"
  3963   shows "independent (f ` S)"
  3964 proof-
  3965   {fix a assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
  3966     have eq: "f ` S - {f a} = f ` (S - {a})" using fi
  3967       by (auto simp add: inj_on_def)
  3968     from a have "f a \<in> f ` span (S -{a})"
  3969       unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
  3970     hence "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
  3971     with a(1) iS  have False by (simp add: dependent_def) }
  3972   then show ?thesis unfolding dependent_def by blast
  3973 qed
  3974 
  3975 (* ------------------------------------------------------------------------- *)
  3976 (* Picking an orthogonal replacement for a spanning set.                     *)
  3977 (* ------------------------------------------------------------------------- *)
  3978     (* FIXME : Move to some general theory ?*)
  3979 definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
  3980 
  3981 lemma vector_sub_project_orthogonal: "(b::'a::ordered_field^'n::finite) \<bullet> (x - ((b \<bullet> x) / (b\<bullet>b)) *s b) = 0"
  3982   apply (cases "b = 0", simp)
  3983   apply (simp add: dot_rsub dot_rmult)
  3984   unfolding times_divide_eq_right[symmetric]
  3985   by (simp add: field_simps dot_eq_0)
  3986 
  3987 lemma basis_orthogonal:
  3988   fixes B :: "(real ^'n::finite) set"
  3989   assumes fB: "finite B"
  3990   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
  3991   (is " \<exists>C. ?P B C")
  3992 proof(induct rule: finite_induct[OF fB])
  3993   case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
  3994 next
  3995   case (2 a B)
  3996   note fB = `finite B` and aB = `a \<notin> B`
  3997   from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
  3998   obtain C where C: "finite C" "card C \<le> card B"
  3999     "span C = span B" "pairwise orthogonal C" by blast
  4000   let ?a = "a - setsum (\<lambda>x. (x\<bullet>a / (x\<bullet>x)) *s x) C"
  4001   let ?C = "insert ?a C"
  4002   from C(1) have fC: "finite ?C" by simp
  4003   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" by (simp add: card_insert_if)
  4004   {fix x k
  4005     have th0: "\<And>(a::'b::comm_ring) b c. a - (b - c) = c + (a - b)" by (simp add: ring_simps)
  4006     have "x - k *s (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *s x)) \<in> span C \<longleftrightarrow> x - k *s a \<in> span C"
  4007       apply (simp only: vector_ssub_ldistrib th0)
  4008       apply (rule span_add_eq)
  4009       apply (rule span_mul)
  4010       apply (rule span_setsum[OF C(1)])
  4011       apply clarify
  4012       apply (rule span_mul)
  4013       by (rule span_superset)}
  4014   then have SC: "span ?C = span (insert a B)"
  4015     unfolding expand_set_eq span_breakdown_eq C(3)[symmetric] by auto
  4016   thm pairwise_def
  4017   {fix x y assume xC: "x \<in> ?C" and yC: "y \<in> ?C" and xy: "x \<noteq> y"
  4018     {assume xa: "x = ?a" and ya: "y = ?a"
  4019       have "orthogonal x y" using xa ya xy by blast}
  4020     moreover
  4021     {assume xa: "x = ?a" and ya: "y \<noteq> ?a" "y \<in> C"
  4022       from ya have Cy: "C = insert y (C - {y})" by blast
  4023       have fth: "finite (C - {y})" using C by simp
  4024       have "orthogonal x y"
  4025 	using xa ya
  4026 	unfolding orthogonal_def xa dot_lsub dot_rsub diff_eq_0_iff_eq
  4027 	apply simp
  4028 	apply (subst Cy)
  4029 	using C(1) fth
  4030 	apply (simp only: setsum_clauses)
  4031 	thm dot_ladd
  4032 	apply (auto simp add: dot_ladd dot_radd dot_lmult dot_rmult dot_eq_0 dot_sym[of y a] dot_lsum[OF fth])
  4033 	apply (rule setsum_0')
  4034 	apply clarsimp
  4035 	apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
  4036 	by auto}
  4037     moreover
  4038     {assume xa: "x \<noteq> ?a" "x \<in> C" and ya: "y = ?a"
  4039       from xa have Cx: "C = insert x (C - {x})" by blast
  4040       have fth: "finite (C - {x})" using C by simp
  4041       have "orthogonal x y"
  4042 	using xa ya
  4043 	unfolding orthogonal_def ya dot_rsub dot_lsub diff_eq_0_iff_eq
  4044 	apply simp
  4045 	apply (subst Cx)
  4046 	using C(1) fth
  4047 	apply (simp only: setsum_clauses)
  4048 	apply (subst dot_sym[of x])
  4049 	apply (auto simp add: dot_radd dot_rmult dot_eq_0 dot_sym[of x a] dot_rsum[OF fth])
  4050 	apply (rule setsum_0')
  4051 	apply clarsimp
  4052 	apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
  4053 	by auto}
  4054     moreover
  4055     {assume xa: "x \<in> C" and ya: "y \<in> C"
  4056       have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast}
  4057     ultimately have "orthogonal x y" using xC yC by blast}
  4058   then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast
  4059   from fC cC SC CPO have "?P (insert a B) ?C" by blast
  4060   then show ?case by blast
  4061 qed
  4062 
  4063 lemma orthogonal_basis_exists:
  4064   fixes V :: "(real ^'n::finite) set"
  4065   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (B hassize dim V) \<and> pairwise orthogonal B"
  4066 proof-
  4067   from basis_exists[of V] obtain B where B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "B hassize dim V" by blast
  4068   from B have fB: "finite B" "card B = dim V" by (simp_all add: hassize_def)
  4069   from basis_orthogonal[OF fB(1)] obtain C where
  4070     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
  4071   from C B
  4072   have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans)
  4073   from span_mono[OF B(3)]  C have SVC: "span V \<subseteq> span C" by (simp add: span_span)
  4074   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
  4075   have iC: "independent C" by (simp add: dim_span)
  4076   from C fB have "card C \<le> dim V" by simp
  4077   moreover have "dim V \<le> card C" using span_card_ge_dim[OF CSV SVC C(1)]
  4078     by (simp add: dim_span)
  4079   ultimately have CdV: "C hassize dim V" unfolding hassize_def using C(1) by simp
  4080   from C B CSV CdV iC show ?thesis by auto
  4081 qed
  4082 
  4083 lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
  4084   by (metis set_eq_subset span_mono span_span span_inc) (* FIXME: slow *)
  4085 
  4086 (* ------------------------------------------------------------------------- *)
  4087 (* Low-dimensional subset is in a hyperplane (weak orthogonal complement).   *)
  4088 (* ------------------------------------------------------------------------- *)
  4089 
  4090 lemma span_not_univ_orthogonal:
  4091   assumes sU: "span S \<noteq> UNIV"
  4092   shows "\<exists>(a:: real ^'n::finite). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
  4093 proof-
  4094   from sU obtain a where a: "a \<notin> span S" by blast
  4095   from orthogonal_basis_exists obtain B where
  4096     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "B hassize dim S" "pairwise orthogonal B"
  4097     by blast
  4098   from B have fB: "finite B" "card B = dim S" by (simp_all add: hassize_def)
  4099   from span_mono[OF B(2)] span_mono[OF B(3)]
  4100   have sSB: "span S = span B" by (simp add: span_span)
  4101   let ?a = "a - setsum (\<lambda>b. (a\<bullet>b / (b\<bullet>b)) *s b) B"
  4102   have "setsum (\<lambda>b. (a\<bullet>b / (b\<bullet>b)) *s b) B \<in> span S"
  4103     unfolding sSB
  4104     apply (rule span_setsum[OF fB(1)])
  4105     apply clarsimp
  4106     apply (rule span_mul)
  4107     by (rule span_superset)
  4108   with a have a0:"?a  \<noteq> 0" by auto
  4109   have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
  4110   proof(rule span_induct')
  4111     show "subspace (\<lambda>x. ?a \<bullet> x = 0)"
  4112       by (auto simp add: subspace_def mem_def dot_radd dot_rmult)
  4113   next
  4114     {fix x assume x: "x \<in> B"
  4115       from x have B': "B = insert x (B - {x})" by blast
  4116       have fth: "finite (B - {x})" using fB by simp
  4117       have "?a \<bullet> x = 0"
  4118 	apply (subst B') using fB fth
  4119 	unfolding setsum_clauses(2)[OF fth]
  4120 	apply simp
  4121 	apply (clarsimp simp add: dot_lsub dot_ladd dot_lmult dot_lsum dot_eq_0)
  4122 	apply (rule setsum_0', rule ballI)
  4123 	unfolding dot_sym
  4124 	by (auto simp add: x field_simps dot_eq_0 intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
  4125     then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
  4126   qed
  4127   with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
  4128 qed
  4129 
  4130 lemma span_not_univ_subset_hyperplane:
  4131   assumes SU: "span S \<noteq> (UNIV ::(real^'n::finite) set)"
  4132   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
  4133   using span_not_univ_orthogonal[OF SU] by auto
  4134 
  4135 lemma lowdim_subset_hyperplane:
  4136   assumes d: "dim S < CARD('n::finite)"
  4137   shows "\<exists>(a::real ^'n::finite). a  \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
  4138 proof-
  4139   {assume "span S = UNIV"
  4140     hence "dim (span S) = dim (UNIV :: (real ^'n) set)" by simp
  4141     hence "dim S = CARD('n)" by (simp add: dim_span dim_univ)
  4142     with d have False by arith}
  4143   hence th: "span S \<noteq> UNIV" by blast
  4144   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
  4145 qed
  4146 
  4147 (* We can extend a linear basis-basis injection to the whole set.            *)
  4148 
  4149 lemma linear_indep_image_lemma:
  4150   assumes lf: "linear f" and fB: "finite B"
  4151   and ifB: "independent (f ` B)"
  4152   and fi: "inj_on f B" and xsB: "x \<in> span B"
  4153   and fx: "f (x::'a::field^'n) = 0"
  4154   shows "x = 0"
  4155   using fB ifB fi xsB fx
  4156 proof(induct arbitrary: x rule: finite_induct[OF fB])
  4157   case 1 thus ?case by (auto simp add:  span_empty)
  4158 next
  4159   case (2 a b x)
  4160   have fb: "finite b" using "2.prems" by simp
  4161   have th0: "f ` b \<subseteq> f ` (insert a b)"
  4162     apply (rule image_mono) by blast
  4163   from independent_mono[ OF "2.prems"(2) th0]
  4164   have ifb: "independent (f ` b)"  .
  4165   have fib: "inj_on f b"
  4166     apply (rule subset_inj_on [OF "2.prems"(3)])
  4167     by blast
  4168   from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
  4169   obtain k where k: "x - k*s a \<in> span (b -{a})" by blast
  4170   have "f (x - k*s a) \<in> span (f ` b)"
  4171     unfolding span_linear_image[OF lf]
  4172     apply (rule imageI)
  4173     using k span_mono[of "b-{a}" b] by blast
  4174   hence "f x - k*s f a \<in> span (f ` b)"
  4175     by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
  4176   hence th: "-k *s f a \<in> span (f ` b)"
  4177     using "2.prems"(5) by (simp add: vector_smult_lneg)
  4178   {assume k0: "k = 0"
  4179     from k0 k have "x \<in> span (b -{a})" by simp
  4180     then have "x \<in> span b" using span_mono[of "b-{a}" b]
  4181       by blast}
  4182   moreover
  4183   {assume k0: "k \<noteq> 0"
  4184     from span_mul[OF th, of "- 1/ k"] k0
  4185     have th1: "f a \<in> span (f ` b)"
  4186       by (auto simp add: vector_smult_assoc)
  4187     from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
  4188     have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
  4189     from "2.prems"(2)[unfolded dependent_def bex_simps(10), rule_format, of "f a"]
  4190     have "f a \<notin> span (f ` b)" using tha
  4191       using "2.hyps"(2)
  4192       "2.prems"(3) by auto
  4193     with th1 have False by blast
  4194     then have "x \<in> span b" by blast}
  4195   ultimately have xsb: "x \<in> span b" by blast
  4196   from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)]
  4197   show "x = 0" .
  4198 qed
  4199 
  4200 (* We can extend a linear mapping from basis.                                *)
  4201 
  4202 lemma linear_independent_extend_lemma:
  4203   assumes fi: "finite B" and ib: "independent B"
  4204   shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g ((x::'a::field^'n) + y) = g x + g y)
  4205            \<and> (\<forall>x\<in> span B. \<forall>c. g (c*s x) = c *s g x)
  4206            \<and> (\<forall>x\<in> B. g x = f x)"
  4207 using ib fi
  4208 proof(induct rule: finite_induct[OF fi])
  4209   case 1 thus ?case by (auto simp add: span_empty)
  4210 next
  4211   case (2 a b)
  4212   from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
  4213     by (simp_all add: independent_insert)
  4214   from "2.hyps"(3)[OF ibf] obtain g where
  4215     g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
  4216     "\<forall>x\<in>span b. \<forall>c. g (c *s x) = c *s g x" "\<forall>x\<in>b. g x = f x" by blast
  4217   let ?h = "\<lambda>z. SOME k. (z - k *s a) \<in> span b"
  4218   {fix z assume z: "z \<in> span (insert a b)"
  4219     have th0: "z - ?h z *s a \<in> span b"
  4220       apply (rule someI_ex)
  4221       unfolding span_breakdown_eq[symmetric]
  4222       using z .
  4223     {fix k assume k: "z - k *s a \<in> span b"
  4224       have eq: "z - ?h z *s a - (z - k*s a) = (k - ?h z) *s a"
  4225 	by (simp add: ring_simps vector_sadd_rdistrib[symmetric])
  4226       from span_sub[OF th0 k]
  4227       have khz: "(k - ?h z) *s a \<in> span b" by (simp add: eq)
  4228       {assume "k \<noteq> ?h z" hence k0: "k - ?h z \<noteq> 0" by simp
  4229 	from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
  4230 	have "a \<in> span b" by (simp add: vector_smult_assoc)
  4231 	with "2.prems"(1) "2.hyps"(2) have False
  4232 	  by (auto simp add: dependent_def)}
  4233       then have "k = ?h z" by blast}
  4234     with th0 have "z - ?h z *s a \<in> span b \<and> (\<forall>k. z - k *s a \<in> span b \<longrightarrow> k = ?h z)" by blast}
  4235   note h = this
  4236   let ?g = "\<lambda>z. ?h z *s f a + g (z - ?h z *s a)"
  4237   {fix x y assume x: "x \<in> span (insert a b)" and y: "y \<in> span (insert a b)"
  4238     have tha: "\<And>(x::'a^'n) y a k l. (x + y) - (k + l) *s a = (x - k *s a) + (y - l *s a)"
  4239       by (vector ring_simps)
  4240     have addh: "?h (x + y) = ?h x + ?h y"
  4241       apply (rule conjunct2[OF h, rule_format, symmetric])
  4242       apply (rule span_add[OF x y])
  4243       unfolding tha
  4244       by (metis span_add x y conjunct1[OF h, rule_format])
  4245     have "?g (x + y) = ?g x + ?g y"
  4246       unfolding addh tha
  4247       g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
  4248       by (simp add: vector_sadd_rdistrib)}
  4249   moreover
  4250   {fix x:: "'a^'n" and c:: 'a  assume x: "x \<in> span (insert a b)"
  4251     have tha: "\<And>(x::'a^'n) c k a. c *s x - (c * k) *s a = c *s (x - k *s a)"
  4252       by (vector ring_simps)
  4253     have hc: "?h (c *s x) = c * ?h x"
  4254       apply (rule conjunct2[OF h, rule_format, symmetric])
  4255       apply (metis span_mul x)
  4256       by (metis tha span_mul x conjunct1[OF h])
  4257     have "?g (c *s x) = c*s ?g x"
  4258       unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
  4259       by (vector ring_simps)}
  4260   moreover
  4261   {fix x assume x: "x \<in> (insert a b)"
  4262     {assume xa: "x = a"
  4263       have ha1: "1 = ?h a"
  4264 	apply (rule conjunct2[OF h, rule_format])
  4265 	apply (metis span_superset insertI1)
  4266 	using conjunct1[OF h, OF span_superset, OF insertI1]
  4267 	by (auto simp add: span_0)
  4268 
  4269       from xa ha1[symmetric] have "?g x = f x"
  4270 	apply simp
  4271 	using g(2)[rule_format, OF span_0, of 0]
  4272 	by simp}
  4273     moreover
  4274     {assume xb: "x \<in> b"
  4275       have h0: "0 = ?h x"
  4276 	apply (rule conjunct2[OF h, rule_format])
  4277 	apply (metis  span_superset insertI1 xb x)
  4278 	apply simp
  4279 	apply (metis span_superset xb)
  4280 	done
  4281       have "?g x = f x"
  4282 	by (simp add: h0[symmetric] g(3)[rule_format, OF xb])}
  4283     ultimately have "?g x = f x" using x by blast }
  4284   ultimately show ?case apply - apply (rule exI[where x="?g"]) by blast
  4285 qed
  4286 
  4287 lemma linear_independent_extend:
  4288   assumes iB: "independent (B:: (real ^'n::finite) set)"
  4289   shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
  4290 proof-
  4291   from maximal_independent_subset_extend[of B UNIV] iB
  4292   obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
  4293 
  4294   from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
  4295   obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y)
  4296            \<and> (\<forall>x\<in> span C. \<forall>c. g (c*s x) = c *s g x)
  4297            \<and> (\<forall>x\<in> C. g x = f x)" by blast
  4298   from g show ?thesis unfolding linear_def using C
  4299     apply clarsimp by blast
  4300 qed
  4301 
  4302 (* Can construct an isomorphism between spaces of same dimension.            *)
  4303 
  4304 lemma card_le_inj: assumes fA: "finite A" and fB: "finite B"
  4305   and c: "card A \<le> card B" shows "(\<exists>f. f ` A \<subseteq> B \<and> inj_on f A)"
  4306 using fB c
  4307 proof(induct arbitrary: B rule: finite_induct[OF fA])
  4308   case 1 thus ?case by simp
  4309 next
  4310   case (2 x s t)
  4311   thus ?case
  4312   proof(induct rule: finite_induct[OF "2.prems"(1)])
  4313     case 1    then show ?case by simp
  4314   next
  4315     case (2 y t)
  4316     from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
  4317     from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where
  4318       f: "f ` s \<subseteq> t \<and> inj_on f s" by blast
  4319     from f "2.prems"(2) "2.hyps"(2) show ?case
  4320       apply -
  4321       apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
  4322       by (auto simp add: inj_on_def)
  4323   qed
  4324 qed
  4325 
  4326 lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and
  4327   c: "card A = card B"
  4328   shows "A = B"
  4329 proof-
  4330   from fB AB have fA: "finite A" by (auto intro: finite_subset)
  4331   from fA fB have fBA: "finite (B - A)" by auto
  4332   have e: "A \<inter> (B - A) = {}" by blast
  4333   have eq: "A \<union> (B - A) = B" using AB by blast
  4334   from card_Un_disjoint[OF fA fBA e, unfolded eq c]
  4335   have "card (B - A) = 0" by arith
  4336   hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
  4337   with AB show "A = B" by blast
  4338 qed
  4339 
  4340 lemma subspace_isomorphism:
  4341   assumes s: "subspace (S:: (real ^'n::finite) set)"
  4342   and t: "subspace (T :: (real ^ 'm::finite) set)"
  4343   and d: "dim S = dim T"
  4344   shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
  4345 proof-
  4346   from basis_exists[of S] obtain B where
  4347     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
  4348   from basis_exists[of T] obtain C where
  4349     C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "C hassize dim T" by blast
  4350   from B(4) C(4) card_le_inj[of B C] d obtain f where
  4351     f: "f ` B \<subseteq> C" "inj_on f B" unfolding hassize_def by auto
  4352   from linear_independent_extend[OF B(2)] obtain g where
  4353     g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
  4354   from B(4) have fB: "finite B" by (simp add: hassize_def)
  4355   from C(4) have fC: "finite C" by (simp add: hassize_def)
  4356   from inj_on_iff_eq_card[OF fB, of f] f(2)
  4357   have "card (f ` B) = card B" by simp
  4358   with B(4) C(4) have ceq: "card (f ` B) = card C" using d
  4359     by (simp add: hassize_def)
  4360   have "g ` B = f ` B" using g(2)
  4361     by (auto simp add: image_iff)
  4362   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
  4363   finally have gBC: "g ` B = C" .
  4364   have gi: "inj_on g B" using f(2) g(2)
  4365     by (auto simp add: inj_on_def)
  4366   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
  4367   {fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
  4368     from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
  4369     from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
  4370     have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
  4371     have "x=y" using g0[OF th1 th0] by simp }
  4372   then have giS: "inj_on g S"
  4373     unfolding inj_on_def by blast
  4374   from span_subspace[OF B(1,3) s]
  4375   have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
  4376   also have "\<dots> = span C" unfolding gBC ..
  4377   also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
  4378   finally have gS: "g ` S = T" .
  4379   from g(1) gS giS show ?thesis by blast
  4380 qed
  4381 
  4382 (* linear functions are equal on a subspace if they are on a spanning set.   *)
  4383 
  4384 lemma subspace_kernel:
  4385   assumes lf: "linear (f::'a::semiring_1 ^'n \<Rightarrow> _)"
  4386   shows "subspace {x. f x = 0}"
  4387 apply (simp add: subspace_def)
  4388 by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
  4389 
  4390 lemma linear_eq_0_span:
  4391   assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
  4392   shows "\<forall>x \<in> span B. f x = (0::'a::semiring_1 ^'n)"
  4393 proof
  4394   fix x assume x: "x \<in> span B"
  4395   let ?P = "\<lambda>x. f x = 0"
  4396   from subspace_kernel[OF lf] have "subspace ?P" unfolding Collect_def .
  4397   with x f0 span_induct[of B "?P" x] show "f x = 0" by blast
  4398 qed
  4399 
  4400 lemma linear_eq_0:
  4401   assumes lf: "linear f" and SB: "S \<subseteq> span B" and f0: "\<forall>x\<in>B. f x = 0"
  4402   shows "\<forall>x \<in> S. f x = (0::'a::semiring_1^'n)"
  4403   by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
  4404 
  4405 lemma linear_eq:
  4406   assumes lf: "linear (f::'a::ring_1^'n \<Rightarrow> _)" and lg: "linear g" and S: "S \<subseteq> span B"
  4407   and fg: "\<forall> x\<in> B. f x = g x"
  4408   shows "\<forall>x\<in> S. f x = g x"
  4409 proof-
  4410   let ?h = "\<lambda>x. f x - g x"
  4411   from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
  4412   from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
  4413   show ?thesis by simp
  4414 qed
  4415 
  4416 lemma linear_eq_stdbasis:
  4417   assumes lf: "linear (f::'a::ring_1^'m::finite \<Rightarrow> 'a^'n::finite)" and lg: "linear g"
  4418   and fg: "\<forall>i. f (basis i) = g(basis i)"
  4419   shows "f = g"
  4420 proof-
  4421   let ?U = "UNIV :: 'm set"
  4422   let ?I = "{basis i:: 'a^'m|i. i \<in> ?U}"
  4423   {fix x assume x: "x \<in> (UNIV :: ('a^'m) set)"
  4424     from equalityD2[OF span_stdbasis]
  4425     have IU: " (UNIV :: ('a^'m) set) \<subseteq> span ?I" by blast
  4426     from linear_eq[OF lf lg IU] fg x
  4427     have "f x = g x" unfolding Collect_def  Ball_def mem_def by metis}
  4428   then show ?thesis by (auto intro: ext)
  4429 qed
  4430 
  4431 (* Similar results for bilinear functions.                                   *)
  4432 
  4433 lemma bilinear_eq:
  4434   assumes bf: "bilinear (f:: 'a::ring^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)"
  4435   and bg: "bilinear g"
  4436   and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
  4437   and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
  4438   shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
  4439 proof-
  4440   let ?P = "\<lambda>x. \<forall>y\<in> span C. f x y = g x y"
  4441   from bf bg have sp: "subspace ?P"
  4442     unfolding bilinear_def linear_def subspace_def bf bg
  4443     by(auto simp add: span_0 mem_def bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def intro:  bilinear_ladd[OF bf])
  4444 
  4445   have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
  4446     apply -
  4447     apply (rule ballI)
  4448     apply (rule span_induct[of B ?P])
  4449     defer
  4450     apply (rule sp)
  4451     apply assumption
  4452     apply (clarsimp simp add: Ball_def)
  4453     apply (rule_tac P="\<lambda>y. f xa y = g xa y" and S=C in span_induct)
  4454     using fg
  4455     apply (auto simp add: subspace_def)
  4456     using bf bg unfolding bilinear_def linear_def
  4457     by(auto simp add: span_0 mem_def bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def intro:  bilinear_ladd[OF bf])
  4458   then show ?thesis using SB TC by (auto intro: ext)
  4459 qed
  4460 
  4461 lemma bilinear_eq_stdbasis:
  4462   assumes bf: "bilinear (f:: 'a::ring_1^'m::finite \<Rightarrow> 'a^'n::finite \<Rightarrow> 'a^'p)"
  4463   and bg: "bilinear g"
  4464   and fg: "\<forall>i j. f (basis i) (basis j) = g (basis i) (basis j)"
  4465   shows "f = g"
  4466 proof-
  4467   from fg have th: "\<forall>x \<in> {basis i| i. i\<in> (UNIV :: 'm set)}. \<forall>y\<in>  {basis j |j. j \<in> (UNIV :: 'n set)}. f x y = g x y" by blast
  4468   from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext)
  4469 qed
  4470 
  4471 (* Detailed theorems about left and right invertibility in general case.     *)
  4472 
  4473 lemma left_invertible_transp:
  4474   "(\<exists>(B::'a^'n^'m). B ** transp (A::'a^'n^'m) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B::'a^'m^'n). A ** B = mat 1)"
  4475   by (metis matrix_transp_mul transp_mat transp_transp)
  4476 
  4477 lemma right_invertible_transp:
  4478   "(\<exists>(B::'a^'n^'m). transp (A::'a^'n^'m) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B::'a^'m^'n). B ** A = mat 1)"
  4479   by (metis matrix_transp_mul transp_mat transp_transp)
  4480 
  4481 lemma linear_injective_left_inverse:
  4482   assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" and fi: "inj f"
  4483   shows "\<exists>g. linear g \<and> g o f = id"
  4484 proof-
  4485   from linear_independent_extend[OF independent_injective_image, OF independent_stdbasis, OF lf fi]
  4486   obtain h:: "real ^'m \<Rightarrow> real ^'n" where h: "linear h" " \<forall>x \<in> f ` {basis i|i. i \<in> (UNIV::'n set)}. h x = inv f x" by blast
  4487   from h(2)
  4488   have th: "\<forall>i. (h \<circ> f) (basis i) = id (basis i)"
  4489     using inv_o_cancel[OF fi, unfolded stupid_ext[symmetric] id_def o_def]
  4490     by auto
  4491 
  4492   from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
  4493   have "h o f = id" .
  4494   then show ?thesis using h(1) by blast
  4495 qed
  4496 
  4497 lemma linear_surjective_right_inverse:
  4498   assumes lf: "linear (f:: real ^'m::finite \<Rightarrow> real ^'n::finite)" and sf: "surj f"
  4499   shows "\<exists>g. linear g \<and> f o g = id"
  4500 proof-
  4501   from linear_independent_extend[OF independent_stdbasis]
  4502   obtain h:: "real ^'n \<Rightarrow> real ^'m" where
  4503     h: "linear h" "\<forall> x\<in> {basis i| i. i\<in> (UNIV :: 'n set)}. h x = inv f x" by blast
  4504   from h(2)
  4505   have th: "\<forall>i. (f o h) (basis i) = id (basis i)"
  4506     using sf
  4507     apply (auto simp add: surj_iff o_def stupid_ext[symmetric])
  4508     apply (erule_tac x="basis i" in allE)
  4509     by auto
  4510 
  4511   from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
  4512   have "f o h = id" .
  4513   then show ?thesis using h(1) by blast
  4514 qed
  4515 
  4516 lemma matrix_left_invertible_injective:
  4517 "(\<exists>B. (B::real^'m^'n) ** (A::real^'n::finite^'m::finite) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
  4518 proof-
  4519   {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
  4520     from xy have "B*v (A *v x) = B *v (A*v y)" by simp
  4521     hence "x = y"
  4522       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .}
  4523   moreover
  4524   {assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
  4525     hence i: "inj (op *v A)" unfolding inj_on_def by auto
  4526     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
  4527     obtain g where g: "linear g" "g o op *v A = id" by blast
  4528     have "matrix g ** A = mat 1"
  4529       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
  4530       using g(2) by (simp add: o_def id_def stupid_ext)
  4531     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast}
  4532   ultimately show ?thesis by blast
  4533 qed
  4534 
  4535 lemma matrix_left_invertible_ker:
  4536   "(\<exists>B. (B::real ^'m::finite^'n::finite) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
  4537   unfolding matrix_left_invertible_injective
  4538   using linear_injective_0[OF matrix_vector_mul_linear, of A]
  4539   by (simp add: inj_on_def)
  4540 
  4541 lemma matrix_right_invertible_surjective:
  4542 "(\<exists>B. (A::real^'n::finite^'m::finite) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
  4543 proof-
  4544   {fix B :: "real ^'m^'n"  assume AB: "A ** B = mat 1"
  4545     {fix x :: "real ^ 'm"
  4546       have "A *v (B *v x) = x"
  4547 	by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)}
  4548     hence "surj (op *v A)" unfolding surj_def by metis }
  4549   moreover
  4550   {assume sf: "surj (op *v A)"
  4551     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
  4552     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
  4553       by blast
  4554 
  4555     have "A ** (matrix g) = mat 1"
  4556       unfolding matrix_eq  matrix_vector_mul_lid
  4557 	matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
  4558       using g(2) unfolding o_def stupid_ext[symmetric] id_def
  4559       .
  4560     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
  4561   }
  4562   ultimately show ?thesis unfolding surj_def by blast
  4563 qed
  4564 
  4565 lemma matrix_left_invertible_independent_columns:
  4566   fixes A :: "real^'n::finite^'m::finite"
  4567   shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
  4568    (is "?lhs \<longleftrightarrow> ?rhs")
  4569 proof-
  4570   let ?U = "UNIV :: 'n set"
  4571   {assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
  4572     {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0"
  4573       and i: "i \<in> ?U"
  4574       let ?x = "\<chi> i. c i"
  4575       have th0:"A *v ?x = 0"
  4576 	using c
  4577 	unfolding matrix_mult_vsum Cart_eq
  4578 	by auto
  4579       from k[rule_format, OF th0] i
  4580       have "c i = 0" by (vector Cart_eq)}
  4581     hence ?rhs by blast}
  4582   moreover
  4583   {assume H: ?rhs
  4584     {fix x assume x: "A *v x = 0"
  4585       let ?c = "\<lambda>i. ((x$i ):: real)"
  4586       from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
  4587       have "x = 0" by vector}}
  4588   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
  4589 qed
  4590 
  4591 lemma matrix_right_invertible_independent_rows:
  4592   fixes A :: "real^'n::finite^'m::finite"
  4593   shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
  4594   unfolding left_invertible_transp[symmetric]
  4595     matrix_left_invertible_independent_columns
  4596   by (simp add: column_transp)
  4597 
  4598 lemma matrix_right_invertible_span_columns:
  4599   "(\<exists>(B::real ^'n::finite^'m::finite). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")
  4600 proof-
  4601   let ?U = "UNIV :: 'm set"
  4602   have fU: "finite ?U" by simp
  4603   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
  4604     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
  4605     apply (subst eq_commute) ..
  4606   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
  4607   {assume h: ?lhs
  4608     {fix x:: "real ^'n"
  4609 	from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m"
  4610 	  where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
  4611 	have "x \<in> span (columns A)"
  4612 	  unfolding y[symmetric]
  4613 	  apply (rule span_setsum[OF fU])
  4614 	  apply clarify
  4615 	  apply (rule span_mul)
  4616 	  apply (rule span_superset)
  4617 	  unfolding columns_def
  4618 	  by blast}
  4619     then have ?rhs unfolding rhseq by blast}
  4620   moreover
  4621   {assume h:?rhs
  4622     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
  4623     {fix y have "?P y"
  4624       proof(rule span_induct_alt[of ?P "columns A"])
  4625 	show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
  4626 	  apply (rule exI[where x=0])
  4627 	  by (simp add: zero_index vector_smult_lzero)
  4628       next
  4629 	fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
  4630 	from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
  4631 	  unfolding columns_def by blast
  4632 	from y2 obtain x:: "real ^'m" where
  4633 	  x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
  4634 	let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
  4635 	show "?P (c*s y1 + y2)"
  4636 	  proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] cond_value_iff right_distrib cond_application_beta cong del: if_weak_cong)
  4637 	    fix j
  4638 	    have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
  4639            else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))" using i(1)
  4640 	      by (simp add: ring_simps)
  4641 	    have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
  4642            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)$j))) ?U"
  4643 	      apply (rule setsum_cong[OF refl])
  4644 	      using th by blast
  4645 	    also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
  4646 	      by (simp add: setsum_addf)
  4647 	    also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
  4648 	      unfolding setsum_delta[OF fU]
  4649 	      using i(1) by simp
  4650 	    finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
  4651            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
  4652 	  qed
  4653 	next
  4654 	  show "y \<in> span (columns A)" unfolding h by blast
  4655 	qed}
  4656     then have ?lhs unfolding lhseq ..}
  4657   ultimately show ?thesis by blast
  4658 qed
  4659 
  4660 lemma matrix_left_invertible_span_rows:
  4661   "(\<exists>(B::real^'m::finite^'n::finite). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
  4662   unfolding right_invertible_transp[symmetric]
  4663   unfolding columns_transp[symmetric]
  4664   unfolding matrix_right_invertible_span_columns
  4665  ..
  4666 
  4667 (* An injective map real^'n->real^'n is also surjective.                       *)
  4668 
  4669 lemma linear_injective_imp_surjective:
  4670   assumes lf: "linear (f:: real ^'n::finite \<Rightarrow> real ^'n)" and fi: "inj f"
  4671   shows "surj f"
  4672 proof-
  4673   let ?U = "UNIV :: (real ^'n) set"
  4674   from basis_exists[of ?U] obtain B
  4675     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U"
  4676     by blast
  4677   from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
  4678   have th: "?U \<subseteq> span (f ` B)"
  4679     apply (rule card_ge_dim_independent)
  4680     apply blast
  4681     apply (rule independent_injective_image[OF B(2) lf fi])
  4682     apply (rule order_eq_refl)
  4683     apply (rule sym)
  4684     unfolding d
  4685     apply (rule card_image)
  4686     apply (rule subset_inj_on[OF fi])
  4687     by blast
  4688   from th show ?thesis
  4689     unfolding span_linear_image[OF lf] surj_def
  4690     using B(3) by blast
  4691 qed
  4692 
  4693 (* And vice versa.                                                           *)
  4694 
  4695 lemma surjective_iff_injective_gen:
  4696   assumes fS: "finite S" and fT: "finite T" and c: "card S = card T"
  4697   and ST: "f ` S \<subseteq> T"
  4698   shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs")
  4699 proof-
  4700   {assume h: "?lhs"
  4701     {fix x y assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
  4702       from x fS have S0: "card S \<noteq> 0" by auto
  4703       {assume xy: "x \<noteq> y"
  4704 	have th: "card S \<le> card (f ` (S - {y}))"
  4705 	  unfolding c
  4706 	  apply (rule card_mono)
  4707 	  apply (rule finite_imageI)
  4708 	  using fS apply simp
  4709 	  using h xy x y f unfolding subset_eq image_iff
  4710 	  apply auto
  4711 	  apply (case_tac "xa = f x")
  4712 	  apply (rule bexI[where x=x])
  4713 	  apply auto
  4714 	  done
  4715 	also have " \<dots> \<le> card (S -{y})"
  4716 	  apply (rule card_image_le)
  4717 	  using fS by simp
  4718 	also have "\<dots> \<le> card S - 1" using y fS by simp
  4719 	finally have False  using S0 by arith }
  4720       then have "x = y" by blast}
  4721     then have ?rhs unfolding inj_on_def by blast}
  4722   moreover
  4723   {assume h: ?rhs
  4724     have "f ` S = T"
  4725       apply (rule card_subset_eq[OF fT ST])
  4726       unfolding card_image[OF h] using c .
  4727     then have ?lhs by blast}
  4728   ultimately show ?thesis by blast
  4729 qed
  4730 
  4731 lemma linear_surjective_imp_injective:
  4732   assumes lf: "linear (f::real ^'n::finite => real ^'n)" and sf: "surj f"
  4733   shows "inj f"
  4734 proof-
  4735   let ?U = "UNIV :: (real ^'n) set"
  4736   from basis_exists[of ?U] obtain B
  4737     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U"
  4738     by blast
  4739   {fix x assume x: "x \<in> span B" and fx: "f x = 0"
  4740     from B(4) have fB: "finite B" by (simp add: hassize_def)
  4741     from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
  4742     have fBi: "independent (f ` B)"
  4743       apply (rule card_le_dim_spanning[of "f ` B" ?U])
  4744       apply blast
  4745       using sf B(3)
  4746       unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
  4747       apply blast
  4748       using fB apply (blast intro: finite_imageI)
  4749       unfolding d
  4750       apply (rule card_image_le)
  4751       apply (rule fB)
  4752       done
  4753     have th0: "dim ?U \<le> card (f ` B)"
  4754       apply (rule span_card_ge_dim)
  4755       apply blast
  4756       unfolding span_linear_image[OF lf]
  4757       apply (rule subset_trans[where B = "f ` UNIV"])
  4758       using sf unfolding surj_def apply blast
  4759       apply (rule image_mono)
  4760       apply (rule B(3))
  4761       apply (metis finite_imageI fB)
  4762       done
  4763 
  4764     moreover have "card (f ` B) \<le> card B"
  4765       by (rule card_image_le, rule fB)
  4766     ultimately have th1: "card B = card (f ` B)" unfolding d by arith
  4767     have fiB: "inj_on f B"
  4768       unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast
  4769     from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
  4770     have "x = 0" by blast}
  4771   note th = this
  4772   from th show ?thesis unfolding linear_injective_0[OF lf]
  4773     using B(3) by blast
  4774 qed
  4775 
  4776 (* Hence either is enough for isomorphism.                                   *)
  4777 
  4778 lemma left_right_inverse_eq:
  4779   assumes fg: "f o g = id" and gh: "g o h = id"
  4780   shows "f = h"
  4781 proof-
  4782   have "f = f o (g o h)" unfolding gh by simp
  4783   also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
  4784   finally show "f = h" unfolding fg by simp
  4785 qed
  4786 
  4787 lemma isomorphism_expand:
  4788   "f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
  4789   by (simp add: expand_fun_eq o_def id_def)
  4790 
  4791 lemma linear_injective_isomorphism:
  4792   assumes lf: "linear (f :: real^'n::finite \<Rightarrow> real ^'n)" and fi: "inj f"
  4793   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
  4794 unfolding isomorphism_expand[symmetric]
  4795 using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi]
  4796 by (metis left_right_inverse_eq)
  4797 
  4798 lemma linear_surjective_isomorphism:
  4799   assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'n)" and sf: "surj f"
  4800   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
  4801 unfolding isomorphism_expand[symmetric]
  4802 using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
  4803 by (metis left_right_inverse_eq)
  4804 
  4805 (* Left and right inverses are the same for R^N->R^N.                        *)
  4806 
  4807 lemma linear_inverse_left:
  4808   assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'n)" and lf': "linear f'"
  4809   shows "f o f' = id \<longleftrightarrow> f' o f = id"
  4810 proof-
  4811   {fix f f':: "real ^'n \<Rightarrow> real ^'n"
  4812     assume lf: "linear f" "linear f'" and f: "f o f' = id"
  4813     from f have sf: "surj f"
  4814 
  4815       apply (auto simp add: o_def stupid_ext[symmetric] id_def surj_def)
  4816       by metis
  4817     from linear_surjective_isomorphism[OF lf(1) sf] lf f
  4818     have "f' o f = id" unfolding stupid_ext[symmetric] o_def id_def
  4819       by metis}
  4820   then show ?thesis using lf lf' by metis
  4821 qed
  4822 
  4823 (* Moreover, a one-sided inverse is automatically linear.                    *)
  4824 
  4825 lemma left_inverse_linear:
  4826   assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'n)" and gf: "g o f = id"
  4827   shows "linear g"
  4828 proof-
  4829   from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def stupid_ext[symmetric])
  4830     by metis
  4831   from linear_injective_isomorphism[OF lf fi]
  4832   obtain h:: "real ^'n \<Rightarrow> real ^'n" where
  4833     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
  4834   have "h = g" apply (rule ext) using gf h(2,3)
  4835     apply (simp add: o_def id_def stupid_ext[symmetric])
  4836     by metis
  4837   with h(1) show ?thesis by blast
  4838 qed
  4839 
  4840 lemma right_inverse_linear:
  4841   assumes lf: "linear (f:: real ^'n::finite \<Rightarrow> real ^'n)" and gf: "f o g = id"
  4842   shows "linear g"
  4843 proof-
  4844   from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def stupid_ext[symmetric])
  4845     by metis
  4846   from linear_surjective_isomorphism[OF lf fi]
  4847   obtain h:: "real ^'n \<Rightarrow> real ^'n" where
  4848     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
  4849   have "h = g" apply (rule ext) using gf h(2,3)
  4850     apply (simp add: o_def id_def stupid_ext[symmetric])
  4851     by metis
  4852   with h(1) show ?thesis by blast
  4853 qed
  4854 
  4855 (* The same result in terms of square matrices.                              *)
  4856 
  4857 lemma matrix_left_right_inverse:
  4858   fixes A A' :: "real ^'n::finite^'n"
  4859   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
  4860 proof-
  4861   {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"
  4862     have sA: "surj (op *v A)"
  4863       unfolding surj_def
  4864       apply clarify
  4865       apply (rule_tac x="(A' *v y)" in exI)
  4866       by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
  4867     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
  4868     obtain f' :: "real ^'n \<Rightarrow> real ^'n"
  4869       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
  4870     have th: "matrix f' ** A = mat 1"
  4871       by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
  4872     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
  4873     hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
  4874     hence "matrix f' ** A = A' ** A" by simp
  4875     hence "A' ** A = mat 1" by (simp add: th)}
  4876   then show ?thesis by blast
  4877 qed
  4878 
  4879 (* Considering an n-element vector as an n-by-1 or 1-by-n matrix.            *)
  4880 
  4881 definition "rowvector v = (\<chi> i j. (v$j))"
  4882 
  4883 definition "columnvector v = (\<chi> i j. (v$i))"
  4884 
  4885 lemma transp_columnvector:
  4886  "transp(columnvector v) = rowvector v"
  4887   by (simp add: transp_def rowvector_def columnvector_def Cart_eq)
  4888 
  4889 lemma transp_rowvector: "transp(rowvector v) = columnvector v"
  4890   by (simp add: transp_def columnvector_def rowvector_def Cart_eq)
  4891 
  4892 lemma dot_rowvector_columnvector:
  4893   "columnvector (A *v v) = A ** columnvector v"
  4894   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
  4895 
  4896 lemma dot_matrix_product: "(x::'a::semiring_1^'n::finite) \<bullet> y = (((rowvector x ::'a^'n^1) ** (columnvector y :: 'a^1^'n))$1)$1"
  4897   by (vector matrix_matrix_mult_def rowvector_def columnvector_def dot_def)
  4898 
  4899 lemma dot_matrix_vector_mul:
  4900   fixes A B :: "real ^'n::finite ^'n" and x y :: "real ^'n"
  4901   shows "(A *v x) \<bullet> (B *v y) =
  4902       (((rowvector x :: real^'n^1) ** ((transp A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
  4903 unfolding dot_matrix_product transp_columnvector[symmetric]
  4904   dot_rowvector_columnvector matrix_transp_mul matrix_mul_assoc ..
  4905 
  4906 (* Infinity norm.                                                            *)
  4907 
  4908 definition "infnorm (x::real^'n::finite) = rsup {abs(x$i) |i. i\<in> (UNIV :: 'n set)}"
  4909 
  4910 lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> (UNIV :: 'n set)"
  4911   by auto
  4912 
  4913 lemma infnorm_set_image:
  4914   "{abs(x$i) |i. i\<in> (UNIV :: 'n set)} =
  4915   (\<lambda>i. abs(x$i)) ` (UNIV :: 'n set)" by blast
  4916 
  4917 lemma infnorm_set_lemma:
  4918   shows "finite {abs((x::'a::abs ^'n::finite)$i) |i. i\<in> (UNIV :: 'n set)}"
  4919   and "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"
  4920   unfolding infnorm_set_image
  4921   by (auto intro: finite_imageI)
  4922 
  4923 lemma infnorm_pos_le: "0 \<le> infnorm (x::real^'n::finite)"
  4924   unfolding infnorm_def
  4925   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
  4926   unfolding infnorm_set_image
  4927   by auto
  4928 
  4929 lemma infnorm_triangle: "infnorm ((x::real^'n::finite) + y) \<le> infnorm x + infnorm y"
  4930 proof-
  4931   have th: "\<And>x y (z::real). x - y <= z \<longleftrightarrow> x - z <= y" by arith
  4932   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
  4933   have th2: "\<And>x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith
  4934   show ?thesis
  4935   unfolding infnorm_def
  4936   unfolding rsup_finite_le_iff[ OF infnorm_set_lemma]
  4937   apply (subst diff_le_eq[symmetric])
  4938   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
  4939   unfolding infnorm_set_image bex_simps
  4940   apply (subst th)
  4941   unfolding th1
  4942   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
  4943 
  4944   unfolding infnorm_set_image ball_simps bex_simps
  4945   apply simp
  4946   apply (metis th2)
  4947   done
  4948 qed
  4949 
  4950 lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::real ^'n::finite) = 0"
  4951 proof-
  4952   have "infnorm x <= 0 \<longleftrightarrow> x = 0"
  4953     unfolding infnorm_def
  4954     unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
  4955     unfolding infnorm_set_image ball_simps
  4956     by vector
  4957   then show ?thesis using infnorm_pos_le[of x] by simp
  4958 qed
  4959 
  4960 lemma infnorm_0: "infnorm 0 = 0"
  4961   by (simp add: infnorm_eq_0)
  4962 
  4963 lemma infnorm_neg: "infnorm (- x) = infnorm x"
  4964   unfolding infnorm_def
  4965   apply (rule cong[of "rsup" "rsup"])
  4966   apply blast
  4967   apply (rule set_ext)
  4968   apply auto
  4969   done
  4970 
  4971 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
  4972 proof-
  4973   have "y - x = - (x - y)" by simp
  4974   then show ?thesis  by (metis infnorm_neg)
  4975 qed
  4976 
  4977 lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
  4978 proof-
  4979   have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
  4980     by arith
  4981   from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
  4982   have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
  4983     "infnorm y \<le> infnorm (x - y) + infnorm x"
  4984     by (simp_all add: ring_simps infnorm_neg diff_def[symmetric])
  4985   from th[OF ths]  show ?thesis .
  4986 qed
  4987 
  4988 lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
  4989   using infnorm_pos_le[of x] by arith
  4990 
  4991 lemma component_le_infnorm:
  4992   shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n::finite)"
  4993 proof-
  4994   let ?U = "UNIV :: 'n set"
  4995   let ?S = "{\<bar>x$i\<bar> |i. i\<in> ?U}"
  4996   have fS: "finite ?S" unfolding image_Collect[symmetric]
  4997     apply (rule finite_imageI) unfolding Collect_def mem_def by simp
  4998   have S0: "?S \<noteq> {}" by blast
  4999   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
  5000   from rsup_finite_in[OF fS S0] rsup_finite_Ub[OF fS S0]
  5001   show ?thesis unfolding infnorm_def isUb_def setle_def
  5002     unfolding infnorm_set_image ball_simps by auto
  5003 qed
  5004 
  5005 lemma infnorm_mul_lemma: "infnorm(a *s x) <= \<bar>a\<bar> * infnorm x"
  5006   apply (subst infnorm_def)
  5007   unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
  5008   unfolding infnorm_set_image ball_simps
  5009   apply (simp add: abs_mult)
  5010   apply (rule allI)
  5011   apply (cut_tac component_le_infnorm[of x])
  5012   apply (rule mult_mono)
  5013   apply auto
  5014   done
  5015 
  5016 lemma infnorm_mul: "infnorm(a *s x) = abs a * infnorm x"
  5017 proof-
  5018   {assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) }
  5019   moreover
  5020   {assume a0: "a \<noteq> 0"
  5021     from a0 have th: "(1/a) *s (a *s x) = x"
  5022       by (simp add: vector_smult_assoc)
  5023     from a0 have ap: "\<bar>a\<bar> > 0" by arith
  5024     from infnorm_mul_lemma[of "1/a" "a *s x"]
  5025     have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*s x)"
  5026       unfolding th by simp
  5027     with ap have "\<bar>a\<bar> * infnorm x \<le> \<bar>a\<bar> * (1/\<bar>a\<bar> * infnorm (a *s x))" by (simp add: field_simps)
  5028     then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*s x)"
  5029       using ap by (simp add: field_simps)
  5030     with infnorm_mul_lemma[of a x] have ?thesis by arith }
  5031   ultimately show ?thesis by blast
  5032 qed
  5033 
  5034 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
  5035   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
  5036 
  5037 (* Prove that it differs only up to a bound from Euclidean norm.             *)
  5038 
  5039 lemma infnorm_le_norm: "infnorm x \<le> norm x"
  5040   unfolding infnorm_def rsup_finite_le_iff[OF infnorm_set_lemma]
  5041   unfolding infnorm_set_image  ball_simps
  5042   by (metis component_le_norm)
  5043 lemma card_enum: "card {1 .. n} = n" by auto
  5044 lemma norm_le_infnorm: "norm(x) <= sqrt(real CARD('n)) * infnorm(x::real ^'n::finite)"
  5045 proof-
  5046   let ?d = "CARD('n)"
  5047   have "real ?d \<ge> 0" by simp
  5048   hence d2: "(sqrt (real ?d))^2 = real ?d"
  5049     by (auto intro: real_sqrt_pow2)
  5050   have th: "sqrt (real ?d) * infnorm x \<ge> 0"
  5051     by (simp add: zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le)
  5052   have th1: "x\<bullet>x \<le> (sqrt (real ?d) * infnorm x)^2"
  5053     unfolding power_mult_distrib d2
  5054     apply (subst power2_abs[symmetric])
  5055     unfolding real_of_nat_def dot_def power2_eq_square[symmetric]
  5056     apply (subst power2_abs[symmetric])
  5057     apply (rule setsum_bounded)
  5058     apply (rule power_mono)
  5059     unfolding abs_of_nonneg[OF infnorm_pos_le]
  5060     unfolding infnorm_def  rsup_finite_ge_iff[OF infnorm_set_lemma]
  5061     unfolding infnorm_set_image bex_simps
  5062     apply blast
  5063     by (rule abs_ge_zero)
  5064   from real_le_lsqrt[OF dot_pos_le th th1]
  5065   show ?thesis unfolding real_vector_norm_def id_def .
  5066 qed
  5067 
  5068 (* Equality in Cauchy-Schwarz and triangle inequalities.                     *)
  5069 
  5070 lemma norm_cauchy_schwarz_eq: "(x::real ^'n::finite) \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *s y = norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
  5071 proof-
  5072   {assume h: "x = 0"
  5073     hence ?thesis by simp}
  5074   moreover
  5075   {assume h: "y = 0"
  5076     hence ?thesis by simp}
  5077   moreover
  5078   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
  5079     from dot_eq_0[of "norm y *s x - norm x *s y"]
  5080     have "?rhs \<longleftrightarrow> (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) - norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
  5081       using x y
  5082       unfolding dot_rsub dot_lsub dot_lmult dot_rmult
  5083       unfolding norm_pow_2[symmetric] power2_eq_square diff_eq_0_iff_eq apply (simp add: dot_sym)
  5084       apply (simp add: ring_simps)
  5085       apply metis
  5086       done
  5087     also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
  5088       by (simp add: ring_simps dot_sym)
  5089     also have "\<dots> \<longleftrightarrow> ?lhs" using x y
  5090       apply simp
  5091       by metis
  5092     finally have ?thesis by blast}
  5093   ultimately show ?thesis by blast
  5094 qed
  5095 
  5096 lemma norm_cauchy_schwarz_abs_eq:
  5097   fixes x y :: "real ^ 'n::finite"
  5098   shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
  5099                 norm x *s y = norm y *s x \<or> norm(x) *s y = - norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
  5100 proof-
  5101   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
  5102   have "?rhs \<longleftrightarrow> norm x *s y = norm y *s x \<or> norm (- x) *s y = norm y *s (- x)"
  5103     apply simp by vector
  5104   also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or>
  5105      (-x) \<bullet> y = norm x * norm y)"
  5106     unfolding norm_cauchy_schwarz_eq[symmetric]
  5107     unfolding norm_minus_cancel
  5108       norm_mul by blast
  5109   also have "\<dots> \<longleftrightarrow> ?lhs"
  5110     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] dot_lneg
  5111     by arith
  5112   finally show ?thesis ..
  5113 qed
  5114 
  5115 lemma norm_triangle_eq:
  5116   fixes x y :: "real ^ 'n::finite"
  5117   shows "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *s y = norm y *s x"
  5118 proof-
  5119   {assume x: "x =0 \<or> y =0"
  5120     hence ?thesis by (cases "x=0", simp_all)}
  5121   moreover
  5122   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
  5123     hence "norm x \<noteq> 0" "norm y \<noteq> 0"
  5124       by simp_all
  5125     hence n: "norm x > 0" "norm y > 0"
  5126       using norm_ge_zero[of x] norm_ge_zero[of y]
  5127       by arith+
  5128     have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
  5129     have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
  5130       apply (rule th) using n norm_ge_zero[of "x + y"]
  5131       by arith
  5132     also have "\<dots> \<longleftrightarrow> norm x *s y = norm y *s x"
  5133       unfolding norm_cauchy_schwarz_eq[symmetric]
  5134       unfolding norm_pow_2 dot_ladd dot_radd
  5135       by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym ring_simps)
  5136     finally have ?thesis .}
  5137   ultimately show ?thesis by blast
  5138 qed
  5139 
  5140 (* Collinearity.*)
  5141 
  5142 definition "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *s u)"
  5143 
  5144 lemma collinear_empty:  "collinear {}" by (simp add: collinear_def)
  5145 
  5146 lemma collinear_sing: "collinear {(x::'a::ring_1^'n)}"
  5147   apply (simp add: collinear_def)
  5148   apply (rule exI[where x=0])
  5149   by simp
  5150 
  5151 lemma collinear_2: "collinear {(x::'a::ring_1^'n),y}"
  5152   apply (simp add: collinear_def)
  5153   apply (rule exI[where x="x - y"])
  5154   apply auto
  5155   apply (rule exI[where x=0], simp)
  5156   apply (rule exI[where x=1], simp)
  5157   apply (rule exI[where x="- 1"], simp add: vector_sneg_minus1[symmetric])
  5158   apply (rule exI[where x=0], simp)
  5159   done
  5160 
  5161 lemma collinear_lemma: "collinear {(0::real^'n),x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *s x)" (is "?lhs \<longleftrightarrow> ?rhs")
  5162 proof-
  5163   {assume "x=0 \<or> y = 0" hence ?thesis
  5164       by (cases "x = 0", simp_all add: collinear_2 insert_commute)}
  5165   moreover
  5166   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
  5167     {assume h: "?lhs"
  5168       then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *s u" unfolding collinear_def by blast
  5169       from u[rule_format, of x 0] u[rule_format, of y 0]
  5170       obtain cx and cy where
  5171 	cx: "x = cx*s u" and cy: "y = cy*s u"
  5172 	by auto
  5173       from cx x have cx0: "cx \<noteq> 0" by auto
  5174       from cy y have cy0: "cy \<noteq> 0" by auto
  5175       let ?d = "cy / cx"
  5176       from cx cy cx0 have "y = ?d *s x"
  5177 	by (simp add: vector_smult_assoc)
  5178       hence ?rhs using x y by blast}
  5179     moreover
  5180     {assume h: "?rhs"
  5181       then obtain c where c: "y = c*s x" using x y by blast
  5182       have ?lhs unfolding collinear_def c
  5183 	apply (rule exI[where x=x])
  5184 	apply auto
  5185 	apply (rule exI[where x="- 1"], simp only: vector_smult_lneg vector_smult_lid)
  5186 	apply (rule exI[where x= "-c"], simp only: vector_smult_lneg)
  5187 	apply (rule exI[where x=1], simp)
  5188 	apply (rule exI[where x="1 - c"], simp add: vector_smult_lneg vector_sub_rdistrib)
  5189 	apply (rule exI[where x="c - 1"], simp add: vector_smult_lneg vector_sub_rdistrib)
  5190 	done}
  5191     ultimately have ?thesis by blast}
  5192   ultimately show ?thesis by blast
  5193 qed
  5194 
  5195 lemma norm_cauchy_schwarz_equal:
  5196   fixes x y :: "real ^ 'n::finite"
  5197   shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {(0::real^'n),x,y}"
  5198 unfolding norm_cauchy_schwarz_abs_eq
  5199 apply (cases "x=0", simp_all add: collinear_2)
  5200 apply (cases "y=0", simp_all add: collinear_2 insert_commute)
  5201 unfolding collinear_lemma
  5202 apply simp
  5203 apply (subgoal_tac "norm x \<noteq> 0")
  5204 apply (subgoal_tac "norm y \<noteq> 0")
  5205 apply (rule iffI)
  5206 apply (cases "norm x *s y = norm y *s x")
  5207 apply (rule exI[where x="(1/norm x) * norm y"])
  5208 apply (drule sym)
  5209 unfolding vector_smult_assoc[symmetric]
  5210 apply (simp add: vector_smult_assoc field_simps)
  5211 apply (rule exI[where x="(1/norm x) * - norm y"])
  5212 apply clarify
  5213 apply (drule sym)
  5214 unfolding vector_smult_assoc[symmetric]
  5215 apply (simp add: vector_smult_assoc field_simps)
  5216 apply (erule exE)
  5217 apply (erule ssubst)
  5218 unfolding vector_smult_assoc
  5219 unfolding norm_mul
  5220 apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
  5221 apply (case_tac "c <= 0", simp add: ring_simps)
  5222 apply (simp add: ring_simps)
  5223 apply (case_tac "c <= 0", simp add: ring_simps)
  5224 apply (simp add: ring_simps)
  5225 apply simp
  5226 apply simp
  5227 done
  5228 
  5229 end