src/HOL/Library/Euclidean_Space.thy
author huffman
Tue Jun 02 23:49:46 2009 -0700 (2009-06-02)
changeset 31406 e23dd3aac0fb
parent 31399 d9769f093160
child 31416 f4c079225845
permissions -rw-r--r--
instance ^ :: complete_space
     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   "Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n $ i)"
   557 unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_nth_le])
   558 
   559 lemma LIMSEQ_vector:
   560   fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n::finite"
   561   assumes X: "\<And>i. (\<lambda>n. X n $ i) ----> (a $ i)"
   562   shows "X ----> a"
   563 proof (rule metric_LIMSEQ_I)
   564   fix r :: real assume "0 < r"
   565   then have "0 < r / of_nat CARD('n)" (is "0 < ?s")
   566     by (simp add: divide_pos_pos)
   567   def N \<equiv> "\<lambda>i. LEAST N. \<forall>n\<ge>N. dist (X n $ i) (a $ i) < ?s"
   568   def M \<equiv> "Max (range N)"
   569   have "\<And>i. \<exists>N. \<forall>n\<ge>N. dist (X n $ i) (a $ i) < ?s"
   570     using X `0 < ?s` by (rule metric_LIMSEQ_D)
   571   hence "\<And>i. \<forall>n\<ge>N i. dist (X n $ i) (a $ i) < ?s"
   572     unfolding N_def by (rule LeastI_ex)
   573   hence M: "\<And>i. \<forall>n\<ge>M. dist (X n $ i) (a $ i) < ?s"
   574     unfolding M_def by simp
   575   {
   576     fix n :: nat assume "M \<le> n"
   577     have "dist (X n) a = setL2 (\<lambda>i. dist (X n $ i) (a $ i)) UNIV"
   578       unfolding dist_vector_def ..
   579     also have "\<dots> \<le> setsum (\<lambda>i. dist (X n $ i) (a $ i)) UNIV"
   580       by (rule setL2_le_setsum [OF zero_le_dist])
   581     also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
   582       by (rule setsum_strict_mono, simp_all add: M `M \<le> n`)
   583     also have "\<dots> = r"
   584       by simp
   585     finally have "dist (X n) a < r" .
   586   }
   587   hence "\<forall>n\<ge>M. dist (X n) a < r"
   588     by simp
   589   then show "\<exists>M. \<forall>n\<ge>M. dist (X n) a < r" ..
   590 qed
   591 
   592 lemma Cauchy_vector:
   593   fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n::finite"
   594   assumes X: "\<And>i. Cauchy (\<lambda>n. X n $ i)"
   595   shows "Cauchy (\<lambda>n. X n)"
   596 proof (rule metric_CauchyI)
   597   fix r :: real assume "0 < r"
   598   then have "0 < r / of_nat CARD('n)" (is "0 < ?s")
   599     by (simp add: divide_pos_pos)
   600   def N \<equiv> "\<lambda>i. LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"
   601   def M \<equiv> "Max (range N)"
   602   have "\<And>i. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"
   603     using X `0 < ?s` by (rule metric_CauchyD)
   604   hence "\<And>i. \<forall>m\<ge>N i. \<forall>n\<ge>N i. dist (X m $ i) (X n $ i) < ?s"
   605     unfolding N_def by (rule LeastI_ex)
   606   hence M: "\<And>i. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m $ i) (X n $ i) < ?s"
   607     unfolding M_def by simp
   608   {
   609     fix m n :: nat
   610     assume "M \<le> m" "M \<le> n"
   611     have "dist (X m) (X n) = setL2 (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
   612       unfolding dist_vector_def ..
   613     also have "\<dots> \<le> setsum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
   614       by (rule setL2_le_setsum [OF zero_le_dist])
   615     also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
   616       by (rule setsum_strict_mono, simp_all add: M `M \<le> m` `M \<le> n`)
   617     also have "\<dots> = r"
   618       by simp
   619     finally have "dist (X m) (X n) < r" .
   620   }
   621   hence "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r"
   622     by simp
   623   then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
   624 qed
   625 
   626 instance "^" :: (complete_space, finite) complete_space
   627 proof
   628   fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
   629   have "\<And>i. (\<lambda>n. X n $ i) ----> lim (\<lambda>n. X n $ i)"
   630     using Cauchy_Cart_nth [OF `Cauchy X`]
   631     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
   632   hence "X ----> Cart_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
   633     by (simp add: LIMSEQ_vector)
   634   then show "convergent X"
   635     by (rule convergentI)
   636 qed
   637 
   638 subsection {* Norms *}
   639 
   640 instantiation "^" :: (real_normed_vector, finite) real_normed_vector
   641 begin
   642 
   643 definition vector_norm_def:
   644   "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) UNIV"
   645 
   646 definition vector_sgn_def:
   647   "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
   648 
   649 instance proof
   650   fix a :: real and x y :: "'a ^ 'b"
   651   show "0 \<le> norm x"
   652     unfolding vector_norm_def
   653     by (rule setL2_nonneg)
   654   show "norm x = 0 \<longleftrightarrow> x = 0"
   655     unfolding vector_norm_def
   656     by (simp add: setL2_eq_0_iff Cart_eq)
   657   show "norm (x + y) \<le> norm x + norm y"
   658     unfolding vector_norm_def
   659     apply (rule order_trans [OF _ setL2_triangle_ineq])
   660     apply (simp add: setL2_mono norm_triangle_ineq)
   661     done
   662   show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
   663     unfolding vector_norm_def
   664     by (simp add: norm_scaleR setL2_right_distrib)
   665   show "sgn x = scaleR (inverse (norm x)) x"
   666     by (rule vector_sgn_def)
   667   show "dist x y = norm (x - y)"
   668     unfolding dist_vector_def vector_norm_def
   669     by (simp add: dist_norm)
   670 qed
   671 
   672 end
   673 
   674 lemma norm_nth_le: "norm (x $ i) \<le> norm x"
   675 unfolding vector_norm_def
   676 by (rule member_le_setL2) simp_all
   677 
   678 interpretation Cart_nth: bounded_linear "\<lambda>x. x $ i"
   679 apply default
   680 apply (rule vector_add_component)
   681 apply (rule vector_scaleR_component)
   682 apply (rule_tac x="1" in exI, simp add: norm_nth_le)
   683 done
   684 
   685 instance "^" :: (banach, finite) banach ..
   686 
   687 subsection {* Inner products *}
   688 
   689 instantiation "^" :: (real_inner, finite) real_inner
   690 begin
   691 
   692 definition vector_inner_def:
   693   "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
   694 
   695 instance proof
   696   fix r :: real and x y z :: "'a ^ 'b"
   697   show "inner x y = inner y x"
   698     unfolding vector_inner_def
   699     by (simp add: inner_commute)
   700   show "inner (x + y) z = inner x z + inner y z"
   701     unfolding vector_inner_def
   702     by (simp add: inner_left_distrib setsum_addf)
   703   show "inner (scaleR r x) y = r * inner x y"
   704     unfolding vector_inner_def
   705     by (simp add: inner_scaleR_left setsum_right_distrib)
   706   show "0 \<le> inner x x"
   707     unfolding vector_inner_def
   708     by (simp add: setsum_nonneg)
   709   show "inner x x = 0 \<longleftrightarrow> x = 0"
   710     unfolding vector_inner_def
   711     by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
   712   show "norm x = sqrt (inner x x)"
   713     unfolding vector_inner_def vector_norm_def setL2_def
   714     by (simp add: power2_norm_eq_inner)
   715 qed
   716 
   717 end
   718 
   719 subsection{* Properties of the dot product.  *}
   720 
   721 lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x"
   722   by (vector mult_commute)
   723 lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \<bullet> z = (x \<bullet> z) + (y \<bullet> z)"
   724   by (vector ring_simps)
   725 lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)"
   726   by (vector ring_simps)
   727 lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)"
   728   by (vector ring_simps)
   729 lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)"
   730   by (vector ring_simps)
   731 lemma dot_lmult: "(c *s x) \<bullet> y = (c::'a::ring) * (x \<bullet> y)" by (vector ring_simps)
   732 lemma dot_rmult: "x \<bullet> (c *s y) = (c::'a::comm_ring) * (x \<bullet> y)" by (vector ring_simps)
   733 lemma dot_lneg: "(-x) \<bullet> (y::'a::ring ^ 'n) = -(x \<bullet> y)" by vector
   734 lemma dot_rneg: "(x::'a::ring ^ 'n) \<bullet> (-y) = -(x \<bullet> y)" by vector
   735 lemma dot_lzero[simp]: "0 \<bullet> x = (0::'a::{comm_monoid_add, mult_zero})" by vector
   736 lemma dot_rzero[simp]: "x \<bullet> 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector
   737 lemma dot_pos_le[simp]: "(0::'a\<Colon>ordered_ring_strict) <= x \<bullet> x"
   738   by (simp add: dot_def setsum_nonneg)
   739 
   740 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)"
   741 using fS fp setsum_nonneg[OF fp]
   742 proof (induct set: finite)
   743   case empty thus ?case by simp
   744 next
   745   case (insert x F)
   746   from insert.prems have Fx: "f x \<ge> 0" and Fp: "\<forall> a \<in> F. f a \<ge> 0" by simp_all
   747   from insert.hyps Fp setsum_nonneg[OF Fp]
   748   have h: "setsum f F = 0 \<longleftrightarrow> (\<forall>a \<in>F. f a = 0)" by metis
   749   from add_nonneg_eq_0_iff[OF Fx  setsum_nonneg[OF Fp]] insert.hyps(1,2)
   750   show ?case by (simp add: h)
   751 qed
   752 
   753 lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n::finite) = 0"
   754   by (simp add: dot_def setsum_squares_eq_0_iff Cart_eq)
   755 
   756 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]
   757   by (auto simp add: le_less)
   758 
   759 subsection{* The collapse of the general concepts to dimension one. *}
   760 
   761 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
   762   by (simp add: Cart_eq forall_1)
   763 
   764 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
   765   apply auto
   766   apply (erule_tac x= "x$1" in allE)
   767   apply (simp only: vector_one[symmetric])
   768   done
   769 
   770 lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
   771   by (simp add: vector_norm_def UNIV_1)
   772 
   773 lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
   774   by (simp add: norm_vector_1)
   775 
   776 lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
   777   by (auto simp add: norm_real dist_norm)
   778 
   779 subsection {* A connectedness or intermediate value lemma with several applications. *}
   780 
   781 lemma connected_real_lemma:
   782   fixes f :: "real \<Rightarrow> real ^ 'n::finite"
   783   assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
   784   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"
   785   and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
   786   and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
   787   and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
   788   shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
   789 proof-
   790   let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
   791   have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
   792   have Sub: "\<exists>y. isUb UNIV ?S y"
   793     apply (rule exI[where x= b])
   794     using ab fb e12 by (auto simp add: isUb_def setle_def)
   795   from reals_complete[OF Se Sub] obtain l where
   796     l: "isLub UNIV ?S l"by blast
   797   have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
   798     apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
   799     by (metis linorder_linear)
   800   have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
   801     apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
   802     by (metis linorder_linear not_le)
   803     have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
   804     have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
   805     have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo
   806     {assume le2: "f l \<in> e2"
   807       from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
   808       hence lap: "l - a > 0" using alb by arith
   809       from e2[rule_format, OF le2] obtain e where
   810 	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
   811       from dst[OF alb e(1)] obtain d where
   812 	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
   813       have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1)
   814 	apply ferrack by arith
   815       then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
   816       from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
   817       from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
   818       moreover
   819       have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
   820       ultimately have False using e12 alb d' by auto}
   821     moreover
   822     {assume le1: "f l \<in> e1"
   823     from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
   824       hence blp: "b - l > 0" using alb by arith
   825       from e1[rule_format, OF le1] obtain e where
   826 	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
   827       from dst[OF alb e(1)] obtain d where
   828 	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
   829       have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo
   830       then obtain d' where d': "d' > 0" "d' < d" by metis
   831       from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
   832       hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
   833       with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
   834       with l d' have False
   835 	by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
   836     ultimately show ?thesis using alb by metis
   837 qed
   838 
   839 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"} *}
   840 
   841 lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
   842 proof-
   843   have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
   844   thus ?thesis by (simp add: ring_simps power2_eq_square)
   845 qed
   846 
   847 lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
   848   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)
   849   apply (rule_tac x="s" in exI)
   850   apply auto
   851   apply (erule_tac x=y in allE)
   852   apply auto
   853   done
   854 
   855 lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
   856   using real_sqrt_le_iff[of x "y^2"] by simp
   857 
   858 lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
   859   using real_sqrt_le_mono[of "x^2" y] by simp
   860 
   861 lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
   862   using real_sqrt_less_mono[of "x^2" y] by simp
   863 
   864 lemma sqrt_even_pow2: assumes n: "even n"
   865   shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
   866 proof-
   867   from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2
   868     by (auto simp add: nat_number)
   869   from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
   870     by (simp only: power_mult[symmetric] mult_commute)
   871   then show ?thesis  using m by simp
   872 qed
   873 
   874 lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
   875   apply (cases "x = 0", simp_all)
   876   using sqrt_divide_self_eq[of x]
   877   apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps)
   878   done
   879 
   880 text{* Hence derive more interesting properties of the norm. *}
   881 
   882 text {*
   883   This type-specific version is only here
   884   to make @{text normarith.ML} happy.
   885 *}
   886 lemma norm_0: "norm (0::real ^ _) = 0"
   887   by (rule norm_zero)
   888 
   889 lemma norm_mul[simp]: "norm(a *s x) = abs(a) * norm x"
   890   by (simp add: vector_norm_def vector_component setL2_right_distrib
   891            abs_mult cong: strong_setL2_cong)
   892 lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
   893   by (simp add: vector_norm_def dot_def setL2_def power2_eq_square)
   894 lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"
   895   by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
   896 lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
   897   by (simp add: real_vector_norm_def)
   898 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n::finite)" by (metis norm_eq_zero)
   899 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
   900   by vector
   901 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
   902   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
   903 lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
   904   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
   905 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
   906   by (metis vector_mul_lcancel)
   907 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
   908   by (metis vector_mul_rcancel)
   909 lemma norm_cauchy_schwarz:
   910   fixes x y :: "real ^ 'n::finite"
   911   shows "x \<bullet> y <= norm x * norm y"
   912 proof-
   913   {assume "norm x = 0"
   914     hence ?thesis by (simp add: dot_lzero dot_rzero)}
   915   moreover
   916   {assume "norm y = 0"
   917     hence ?thesis by (simp add: dot_lzero dot_rzero)}
   918   moreover
   919   {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
   920     let ?z = "norm y *s x - norm x *s y"
   921     from h have p: "norm x * norm y > 0" by (metis norm_ge_zero le_less zero_compare_simps)
   922     from dot_pos_le[of ?z]
   923     have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"
   924       apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)
   925       by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym)
   926     hence "x\<bullet>y \<le> (norm x ^2 * norm y ^2) / (norm x * norm y)" using p
   927       by (simp add: field_simps)
   928     hence ?thesis using h by (simp add: power2_eq_square)}
   929   ultimately show ?thesis by metis
   930 qed
   931 
   932 lemma norm_cauchy_schwarz_abs:
   933   fixes x y :: "real ^ 'n::finite"
   934   shows "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
   935   using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
   936   by (simp add: real_abs_def dot_rneg)
   937 
   938 lemma norm_triangle_sub:
   939   fixes x y :: "'a::real_normed_vector"
   940   shows "norm x \<le> norm y  + norm (x - y)"
   941   using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
   942 
   943 lemma norm_triangle_le: "norm(x::real ^'n::finite) + norm y <= e ==> norm(x + y) <= e"
   944   by (metis order_trans norm_triangle_ineq)
   945 lemma norm_triangle_lt: "norm(x::real ^'n::finite) + norm(y) < e ==> norm(x + y) < e"
   946   by (metis basic_trans_rules(21) norm_triangle_ineq)
   947 
   948 lemma setsum_delta:
   949   assumes fS: "finite S"
   950   shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
   951 proof-
   952   let ?f = "(\<lambda>k. if k=a then b k else 0)"
   953   {assume a: "a \<notin> S"
   954     hence "\<forall> k\<in> S. ?f k = 0" by simp
   955     hence ?thesis  using a by simp}
   956   moreover
   957   {assume a: "a \<in> S"
   958     let ?A = "S - {a}"
   959     let ?B = "{a}"
   960     have eq: "S = ?A \<union> ?B" using a by blast
   961     have dj: "?A \<inter> ?B = {}" by simp
   962     from fS have fAB: "finite ?A" "finite ?B" by auto
   963     have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
   964       using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
   965       by simp
   966     then have ?thesis  using a by simp}
   967   ultimately show ?thesis by blast
   968 qed
   969 
   970 lemma component_le_norm: "\<bar>x$i\<bar> <= norm (x::real ^ 'n::finite)"
   971   apply (simp add: vector_norm_def)
   972   apply (rule member_le_setL2, simp_all)
   973   done
   974 
   975 lemma norm_bound_component_le: "norm(x::real ^ 'n::finite) <= e
   976                 ==> \<bar>x$i\<bar> <= e"
   977   by (metis component_le_norm order_trans)
   978 
   979 lemma norm_bound_component_lt: "norm(x::real ^ 'n::finite) < e
   980                 ==> \<bar>x$i\<bar> < e"
   981   by (metis component_le_norm basic_trans_rules(21))
   982 
   983 lemma norm_le_l1: "norm (x:: real ^'n::finite) <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
   984   by (simp add: vector_norm_def setL2_le_setsum)
   985 
   986 lemma real_abs_norm: "\<bar>norm x\<bar> = norm (x :: real ^ _)"
   987   by (rule abs_norm_cancel)
   988 lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n::finite) - norm y\<bar> <= norm(x - y)"
   989   by (rule norm_triangle_ineq3)
   990 lemma norm_le: "norm(x::real ^ _) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
   991   by (simp add: real_vector_norm_def)
   992 lemma norm_lt: "norm(x::real ^ _) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
   993   by (simp add: real_vector_norm_def)
   994 lemma norm_eq: "norm (x::real ^ _) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
   995   by (simp add: order_eq_iff norm_le)
   996 lemma norm_eq_1: "norm(x::real ^ _) = 1 \<longleftrightarrow> x \<bullet> x = 1"
   997   by (simp add: real_vector_norm_def)
   998 
   999 text{* Squaring equations and inequalities involving norms.  *}
  1000 
  1001 lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
  1002   by (simp add: real_vector_norm_def)
  1003 
  1004 lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
  1005   by (auto simp add: real_vector_norm_def)
  1006 
  1007 lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
  1008 proof-
  1009   have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: ring_simps power2_eq_square)
  1010   also have "\<dots> \<longleftrightarrow> \<bar>x\<bar> \<le> \<bar>y\<bar>" apply (simp add: zero_compare_simps real_abs_def not_less) by arith
  1011 finally show ?thesis ..
  1012 qed
  1013 
  1014 lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
  1015   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
  1016   using norm_ge_zero[of x]
  1017   apply arith
  1018   done
  1019 
  1020 lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
  1021   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
  1022   using norm_ge_zero[of x]
  1023   apply arith
  1024   done
  1025 
  1026 lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
  1027   by (metis not_le norm_ge_square)
  1028 lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
  1029   by (metis norm_le_square not_less)
  1030 
  1031 text{* Dot product in terms of the norm rather than conversely. *}
  1032 
  1033 lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
  1034   by (simp add: norm_pow_2 dot_ladd dot_radd dot_sym)
  1035 
  1036 lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
  1037   by (simp add: norm_pow_2 dot_ladd dot_radd dot_lsub dot_rsub dot_sym)
  1038 
  1039 
  1040 text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
  1041 
  1042 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")
  1043 proof
  1044   assume "?lhs" then show ?rhs by simp
  1045 next
  1046   assume ?rhs
  1047   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y\<bullet> y = 0" by simp
  1048   hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
  1049     by (simp add: dot_rsub dot_lsub dot_sym)
  1050   then have "(x - y) \<bullet> (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub)
  1051   then show "x = y" by (simp add: dot_eq_0)
  1052 qed
  1053 
  1054 
  1055 subsection{* General linear decision procedure for normed spaces. *}
  1056 
  1057 lemma norm_cmul_rule_thm: "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(c *s x)"
  1058   apply (clarsimp simp add: norm_mul)
  1059   apply (rule mult_mono1)
  1060   apply simp_all
  1061   done
  1062 
  1063   (* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
  1064 lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n::finite) \<Longrightarrow> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)"
  1065   apply (rule norm_triangle_le) by simp
  1066 
  1067 lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"
  1068   by (simp add: ring_simps)
  1069 
  1070 lemma pth_1: "(x::real^'n) == 1 *s x" by (simp only: vector_smult_lid)
  1071 lemma pth_2: "x - (y::real^'n) == x + -y" by (atomize (full)) simp
  1072 lemma pth_3: "(-x::real^'n) == -1 *s x" by vector
  1073 lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+
  1074 lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector
  1075 lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps)
  1076 lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all
  1077 lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps)
  1078 lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z"
  1079   "c *s x + (d *s x + z) == (c + d) *s x + z"
  1080   "(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+
  1081 lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector
  1082 lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y"
  1083   "(c *s x + z) + d *s y == c *s x + (z + d *s y)"
  1084   "c *s x + (d *s y + z) == c *s x + (d *s y + z)"
  1085   "(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))"
  1086   by ((atomize (full)), vector)+
  1087 lemma pth_c: "(c::real) *s x + d *s y == d *s y + c *s x"
  1088   "(c *s x + z) + d *s y == d *s y + (c *s x + z)"
  1089   "c *s x + (d *s y + z) == d *s y + (c *s x + z)"
  1090   "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+
  1091 lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector
  1092 
  1093 lemma norm_imp_pos_and_ge: "norm (x::real ^ _) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
  1094   by (atomize) (auto simp add: norm_ge_zero)
  1095 
  1096 lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
  1097 
  1098 lemma norm_pths:
  1099   "(x::real ^'n::finite) = y \<longleftrightarrow> norm (x - y) \<le> 0"
  1100   "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
  1101   using norm_ge_zero[of "x - y"] by auto
  1102 
  1103 lemma vector_dist_norm:
  1104   fixes x y :: "real ^ _"
  1105   shows "dist x y = norm (x - y)"
  1106   by (rule dist_norm)
  1107 
  1108 use "normarith.ML"
  1109 
  1110 method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
  1111 *} "Proves simple linear statements about vector norms"
  1112 
  1113 
  1114 
  1115 text{* Hence more metric properties. *}
  1116 
  1117 lemma dist_triangle_alt:
  1118   fixes x y z :: "'a::metric_space"
  1119   shows "dist y z <= dist x y + dist x z"
  1120 using dist_triangle [of y z x] by (simp add: dist_commute)
  1121 
  1122 lemma dist_pos_lt:
  1123   fixes x y :: "'a::metric_space"
  1124   shows "x \<noteq> y ==> 0 < dist x y"
  1125 by (simp add: zero_less_dist_iff)
  1126 
  1127 lemma dist_nz:
  1128   fixes x y :: "'a::metric_space"
  1129   shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
  1130 by (simp add: zero_less_dist_iff)
  1131 
  1132 lemma dist_triangle_le:
  1133   fixes x y z :: "'a::metric_space"
  1134   shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
  1135 by (rule order_trans [OF dist_triangle2])
  1136 
  1137 lemma dist_triangle_lt:
  1138   fixes x y z :: "'a::metric_space"
  1139   shows "dist x z + dist y z < e ==> dist x y < e"
  1140 by (rule le_less_trans [OF dist_triangle2])
  1141 
  1142 lemma dist_triangle_half_l:
  1143   fixes x1 x2 y :: "'a::metric_space"
  1144   shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
  1145 by (rule dist_triangle_lt [where z=y], simp)
  1146 
  1147 lemma dist_triangle_half_r:
  1148   fixes x1 x2 y :: "'a::metric_space"
  1149   shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
  1150 by (rule dist_triangle_half_l, simp_all add: dist_commute)
  1151 
  1152 lemma dist_triangle_add:
  1153   fixes x y x' y' :: "'a::real_normed_vector"
  1154   shows "dist (x + y) (x' + y') <= dist x x' + dist y y'"
  1155 unfolding dist_norm by (rule norm_diff_triangle_ineq)
  1156 
  1157 lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
  1158   unfolding dist_norm vector_ssub_ldistrib[symmetric] norm_mul ..
  1159 
  1160 lemma dist_triangle_add_half:
  1161   fixes x x' y y' :: "'a::real_normed_vector"
  1162   shows "dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 \<Longrightarrow> dist(x + y) (x' + y') < e"
  1163 by (rule le_less_trans [OF dist_triangle_add], simp)
  1164 
  1165 lemma setsum_component [simp]:
  1166   fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
  1167   shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
  1168   by (cases "finite S", induct S set: finite, simp_all)
  1169 
  1170 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
  1171   by (simp add: Cart_eq)
  1172 
  1173 lemma setsum_clauses:
  1174   shows "setsum f {} = 0"
  1175   and "finite S \<Longrightarrow> setsum f (insert x S) =
  1176                  (if x \<in> S then setsum f S else f x + setsum f S)"
  1177   by (auto simp add: insert_absorb)
  1178 
  1179 lemma setsum_cmul:
  1180   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
  1181   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
  1182   by (simp add: Cart_eq setsum_right_distrib)
  1183 
  1184 lemma setsum_norm:
  1185   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1186   assumes fS: "finite S"
  1187   shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
  1188 proof(induct rule: finite_induct[OF fS])
  1189   case 1 thus ?case by simp
  1190 next
  1191   case (2 x S)
  1192   from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
  1193   also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
  1194     using "2.hyps" by simp
  1195   finally  show ?case  using "2.hyps" by simp
  1196 qed
  1197 
  1198 lemma real_setsum_norm:
  1199   fixes f :: "'a \<Rightarrow> real ^'n::finite"
  1200   assumes fS: "finite S"
  1201   shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
  1202 proof(induct rule: finite_induct[OF fS])
  1203   case 1 thus ?case by simp
  1204 next
  1205   case (2 x S)
  1206   from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
  1207   also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
  1208     using "2.hyps" by simp
  1209   finally  show ?case  using "2.hyps" by simp
  1210 qed
  1211 
  1212 lemma setsum_norm_le:
  1213   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1214   assumes fS: "finite S"
  1215   and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
  1216   shows "norm (setsum f S) \<le> setsum g S"
  1217 proof-
  1218   from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
  1219     by - (rule setsum_mono, simp)
  1220   then show ?thesis using setsum_norm[OF fS, of f] fg
  1221     by arith
  1222 qed
  1223 
  1224 lemma real_setsum_norm_le:
  1225   fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
  1226   assumes fS: "finite S"
  1227   and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
  1228   shows "norm (setsum f S) \<le> setsum g S"
  1229 proof-
  1230   from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
  1231     by - (rule setsum_mono, simp)
  1232   then show ?thesis using real_setsum_norm[OF fS, of f] fg
  1233     by arith
  1234 qed
  1235 
  1236 lemma setsum_norm_bound:
  1237   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1238   assumes fS: "finite S"
  1239   and K: "\<forall>x \<in> S. norm (f x) \<le> K"
  1240   shows "norm (setsum f S) \<le> of_nat (card S) * K"
  1241   using setsum_norm_le[OF fS K] setsum_constant[symmetric]
  1242   by simp
  1243 
  1244 lemma real_setsum_norm_bound:
  1245   fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
  1246   assumes fS: "finite S"
  1247   and K: "\<forall>x \<in> S. norm (f x) \<le> K"
  1248   shows "norm (setsum f S) \<le> of_nat (card S) * K"
  1249   using real_setsum_norm_le[OF fS K] setsum_constant[symmetric]
  1250   by simp
  1251 
  1252 lemma setsum_vmul:
  1253   fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector,semiring, mult_zero}"
  1254   assumes fS: "finite S"
  1255   shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
  1256 proof(induct rule: finite_induct[OF fS])
  1257   case 1 then show ?case by (simp add: vector_smult_lzero)
  1258 next
  1259   case (2 x F)
  1260   from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v"
  1261     by simp
  1262   also have "\<dots> = f x *s v + setsum f F *s v"
  1263     by (simp add: vector_sadd_rdistrib)
  1264   also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp
  1265   finally show ?case .
  1266 qed
  1267 
  1268 (* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \<Rightarrow> real ^'n"]  ---
  1269  Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *)
  1270 
  1271 lemma setsum_add_split: assumes mn: "(m::nat) \<le> n + 1"
  1272   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
  1273 proof-
  1274   let ?A = "{m .. n}"
  1275   let ?B = "{n + 1 .. n + p}"
  1276   have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto
  1277   have d: "?A \<inter> ?B = {}" by auto
  1278   from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto
  1279 qed
  1280 
  1281 lemma setsum_natinterval_left:
  1282   assumes mn: "(m::nat) <= n"
  1283   shows "setsum f {m..n} = f m + setsum f {m + 1..n}"
  1284 proof-
  1285   from mn have "{m .. n} = insert m {m+1 .. n}" by auto
  1286   then show ?thesis by auto
  1287 qed
  1288 
  1289 lemma setsum_natinterval_difff:
  1290   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
  1291   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
  1292           (if m <= n then f m - f(n + 1) else 0)"
  1293 by (induct n, auto simp add: ring_simps not_le le_Suc_eq)
  1294 
  1295 lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
  1296 
  1297 lemma setsum_setsum_restrict:
  1298   "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"
  1299   apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
  1300   by (rule setsum_commute)
  1301 
  1302 lemma setsum_image_gen: assumes fS: "finite S"
  1303   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
  1304 proof-
  1305   {fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto}
  1306   note th0 = this
  1307   have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
  1308     apply (rule setsum_cong2)
  1309     by (simp add: th0)
  1310   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
  1311     apply (rule setsum_setsum_restrict[OF fS])
  1312     by (rule finite_imageI[OF fS])
  1313   finally show ?thesis .
  1314 qed
  1315 
  1316     (* FIXME: Here too need stupid finiteness assumption on T!!! *)
  1317 lemma setsum_group:
  1318   assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
  1319   shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
  1320 
  1321 apply (subst setsum_image_gen[OF fS, of g f])
  1322 apply (rule setsum_mono_zero_right[OF fT fST])
  1323 by (auto intro: setsum_0')
  1324 
  1325 lemma vsum_norm_allsubsets_bound:
  1326   fixes f:: "'a \<Rightarrow> real ^'n::finite"
  1327   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
  1328   shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
  1329 proof-
  1330   let ?d = "real CARD('n)"
  1331   let ?nf = "\<lambda>x. norm (f x)"
  1332   let ?U = "UNIV :: 'n set"
  1333   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"
  1334     by (rule setsum_commute)
  1335   have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
  1336   have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
  1337     apply (rule setsum_mono)
  1338     by (rule norm_le_l1)
  1339   also have "\<dots> \<le> 2 * ?d * e"
  1340     unfolding th0 th1
  1341   proof(rule setsum_bounded)
  1342     fix i assume i: "i \<in> ?U"
  1343     let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
  1344     let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
  1345     have thp: "P = ?Pp \<union> ?Pn" by auto
  1346     have thp0: "?Pp \<inter> ?Pn ={}" by auto
  1347     have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
  1348     have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
  1349       using component_le_norm[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
  1350       by (auto intro: abs_le_D1)
  1351     have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
  1352       using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
  1353       by (auto simp add: setsum_negf intro: abs_le_D1)
  1354     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"
  1355       apply (subst thp)
  1356       apply (rule setsum_Un_zero)
  1357       using fP thp0 by auto
  1358     also have "\<dots> \<le> 2*e" using Pne Ppe by arith
  1359     finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
  1360   qed
  1361   finally show ?thesis .
  1362 qed
  1363 
  1364 lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> (y::'a::{comm_ring}^'n) = setsum (\<lambda>x. f x \<bullet> y) S "
  1365   by (induct rule: finite_induct, auto simp add: dot_lzero dot_ladd dot_radd)
  1366 
  1367 lemma dot_rsum: "finite S \<Longrightarrow> (y::'a::{comm_ring}^'n) \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
  1368   by (induct rule: finite_induct, auto simp add: dot_rzero dot_radd)
  1369 
  1370 subsection{* Basis vectors in coordinate directions. *}
  1371 
  1372 
  1373 definition "basis k = (\<chi> i. if i = k then 1 else 0)"
  1374 
  1375 lemma basis_component [simp]: "basis k $ i = (if k=i then 1 else 0)"
  1376   unfolding basis_def by simp
  1377 
  1378 lemma delta_mult_idempotent:
  1379   "(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)
  1380 
  1381 lemma norm_basis:
  1382   shows "norm (basis k :: real ^'n::finite) = 1"
  1383   apply (simp add: basis_def real_vector_norm_def dot_def)
  1384   apply (vector delta_mult_idempotent)
  1385   using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"]
  1386   apply auto
  1387   done
  1388 
  1389 lemma norm_basis_1: "norm(basis 1 :: real ^'n::{finite,one}) = 1"
  1390   by (rule norm_basis)
  1391 
  1392 lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n::finite). norm x = c"
  1393   apply (rule exI[where x="c *s basis arbitrary"])
  1394   by (simp only: norm_mul norm_basis)
  1395 
  1396 lemma vector_choose_dist: assumes e: "0 <= e"
  1397   shows "\<exists>(y::real^'n::finite). dist x y = e"
  1398 proof-
  1399   from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"
  1400     by blast
  1401   then have "dist x (x - c) = e" by (simp add: dist_norm)
  1402   then show ?thesis by blast
  1403 qed
  1404 
  1405 lemma basis_inj: "inj (basis :: 'n \<Rightarrow> real ^'n::finite)"
  1406   by (simp add: inj_on_def Cart_eq)
  1407 
  1408 lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
  1409   by auto
  1410 
  1411 lemma basis_expansion:
  1412   "setsum (\<lambda>i. (x$i) *s basis i) UNIV = (x::('a::ring_1) ^'n::finite)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
  1413   by (auto simp add: Cart_eq cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
  1414 
  1415 lemma basis_expansion_unique:
  1416   "setsum (\<lambda>i. f i *s basis i) UNIV = (x::('a::comm_ring_1) ^'n::finite) \<longleftrightarrow> (\<forall>i. f i = x$i)"
  1417   by (simp add: Cart_eq setsum_delta cond_value_iff cong del: if_weak_cong)
  1418 
  1419 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
  1420   by auto
  1421 
  1422 lemma dot_basis:
  1423   shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i :: 'a^'n::finite) = (x$i :: 'a::semiring_1)"
  1424   by (auto simp add: dot_def basis_def cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)
  1425 
  1426 lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
  1427   by (auto simp add: Cart_eq)
  1428 
  1429 lemma basis_nonzero:
  1430   shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
  1431   by (simp add: basis_eq_0)
  1432 
  1433 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = (z::'a::semiring_1^'n::finite)"
  1434   apply (auto simp add: Cart_eq dot_basis)
  1435   apply (erule_tac x="basis i" in allE)
  1436   apply (simp add: dot_basis)
  1437   apply (subgoal_tac "y = z")
  1438   apply simp
  1439   apply (simp add: Cart_eq)
  1440   done
  1441 
  1442 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = (y::'a::semiring_1^'n::finite)"
  1443   apply (auto simp add: Cart_eq dot_basis)
  1444   apply (erule_tac x="basis i" in allE)
  1445   apply (simp add: dot_basis)
  1446   apply (subgoal_tac "x = y")
  1447   apply simp
  1448   apply (simp add: Cart_eq)
  1449   done
  1450 
  1451 subsection{* Orthogonality. *}
  1452 
  1453 definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
  1454 
  1455 lemma orthogonal_basis:
  1456   shows "orthogonal (basis i :: 'a^'n::finite) x \<longleftrightarrow> x$i = (0::'a::ring_1)"
  1457   by (auto simp add: orthogonal_def dot_def basis_def cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
  1458 
  1459 lemma orthogonal_basis_basis:
  1460   shows "orthogonal (basis i :: 'a::ring_1^'n::finite) (basis j) \<longleftrightarrow> i \<noteq> j"
  1461   unfolding orthogonal_basis[of i] basis_component[of j] by simp
  1462 
  1463   (* FIXME : Maybe some of these require less than comm_ring, but not all*)
  1464 lemma orthogonal_clauses:
  1465   "orthogonal a (0::'a::comm_ring ^'n)"
  1466   "orthogonal a x ==> orthogonal a (c *s x)"
  1467   "orthogonal a x ==> orthogonal a (-x)"
  1468   "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x + y)"
  1469   "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x - y)"
  1470   "orthogonal 0 a"
  1471   "orthogonal x a ==> orthogonal (c *s x) a"
  1472   "orthogonal x a ==> orthogonal (-x) a"
  1473   "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x + y) a"
  1474   "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x - y) a"
  1475   unfolding orthogonal_def dot_rneg dot_rmult dot_radd dot_rsub
  1476   dot_lzero dot_rzero dot_lneg dot_lmult dot_ladd dot_lsub
  1477   by simp_all
  1478 
  1479 lemma orthogonal_commute: "orthogonal (x::'a::{ab_semigroup_mult,comm_monoid_add} ^'n)y \<longleftrightarrow> orthogonal y x"
  1480   by (simp add: orthogonal_def dot_sym)
  1481 
  1482 subsection{* Explicit vector construction from lists. *}
  1483 
  1484 primrec from_nat :: "nat \<Rightarrow> 'a::{monoid_add,one}"
  1485 where "from_nat 0 = 0" | "from_nat (Suc n) = 1 + from_nat n"
  1486 
  1487 lemma from_nat [simp]: "from_nat = of_nat"
  1488 by (rule ext, induct_tac x, simp_all)
  1489 
  1490 primrec
  1491   list_fun :: "nat \<Rightarrow> _ list \<Rightarrow> _ \<Rightarrow> _"
  1492 where
  1493   "list_fun n [] = (\<lambda>x. 0)"
  1494 | "list_fun n (x # xs) = fun_upd (list_fun (Suc n) xs) (from_nat n) x"
  1495 
  1496 definition "vector l = (\<chi> i. list_fun 1 l i)"
  1497 (*definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"*)
  1498 
  1499 lemma vector_1: "(vector[x]) $1 = x"
  1500   unfolding vector_def by simp
  1501 
  1502 lemma vector_2:
  1503  "(vector[x,y]) $1 = x"
  1504  "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
  1505   unfolding vector_def by simp_all
  1506 
  1507 lemma vector_3:
  1508  "(vector [x,y,z] ::('a::zero)^3)$1 = x"
  1509  "(vector [x,y,z] ::('a::zero)^3)$2 = y"
  1510  "(vector [x,y,z] ::('a::zero)^3)$3 = z"
  1511   unfolding vector_def by simp_all
  1512 
  1513 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
  1514   apply auto
  1515   apply (erule_tac x="v$1" in allE)
  1516   apply (subgoal_tac "vector [v$1] = v")
  1517   apply simp
  1518   apply (vector vector_def)
  1519   apply (simp add: forall_1)
  1520   done
  1521 
  1522 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
  1523   apply auto
  1524   apply (erule_tac x="v$1" in allE)
  1525   apply (erule_tac x="v$2" in allE)
  1526   apply (subgoal_tac "vector [v$1, v$2] = v")
  1527   apply simp
  1528   apply (vector vector_def)
  1529   apply (simp add: forall_2)
  1530   done
  1531 
  1532 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
  1533   apply auto
  1534   apply (erule_tac x="v$1" in allE)
  1535   apply (erule_tac x="v$2" in allE)
  1536   apply (erule_tac x="v$3" in allE)
  1537   apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
  1538   apply simp
  1539   apply (vector vector_def)
  1540   apply (simp add: forall_3)
  1541   done
  1542 
  1543 subsection{* Linear functions. *}
  1544 
  1545 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)"
  1546 
  1547 lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. (c::'a::comm_semiring) *s f x)"
  1548   by (vector linear_def Cart_eq ring_simps)
  1549 
  1550 lemma linear_compose_neg: "linear (f :: 'a ^'n \<Rightarrow> 'a::comm_ring ^'m) ==> linear (\<lambda>x. -(f(x)))" by (vector linear_def Cart_eq)
  1551 
  1552 lemma linear_compose_add: "linear (f :: 'a ^'n \<Rightarrow> 'a::semiring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
  1553   by (vector linear_def Cart_eq ring_simps)
  1554 
  1555 lemma linear_compose_sub: "linear (f :: 'a ^'n \<Rightarrow> 'a::ring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
  1556   by (vector linear_def Cart_eq ring_simps)
  1557 
  1558 lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
  1559   by (simp add: linear_def)
  1560 
  1561 lemma linear_id: "linear id" by (simp add: linear_def id_def)
  1562 
  1563 lemma linear_zero: "linear (\<lambda>x. 0::'a::semiring_1 ^ 'n)" by (simp add: linear_def)
  1564 
  1565 lemma linear_compose_setsum:
  1566   assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a :: 'a::semiring_1 ^ 'n \<Rightarrow> 'a ^ 'm)"
  1567   shows "linear(\<lambda>x. setsum (\<lambda>a. f a x :: 'a::semiring_1 ^'m) S)"
  1568   using lS
  1569   apply (induct rule: finite_induct[OF fS])
  1570   by (auto simp add: linear_zero intro: linear_compose_add)
  1571 
  1572 lemma linear_vmul_component:
  1573   fixes f:: "'a::semiring_1^'m \<Rightarrow> 'a^'n"
  1574   assumes lf: "linear f"
  1575   shows "linear (\<lambda>x. f x $ k *s v)"
  1576   using lf
  1577   apply (auto simp add: linear_def )
  1578   by (vector ring_simps)+
  1579 
  1580 lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)"
  1581   unfolding linear_def
  1582   apply clarsimp
  1583   apply (erule allE[where x="0::'a"])
  1584   apply simp
  1585   done
  1586 
  1587 lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def)
  1588 
  1589 lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \<Rightarrow> _) ==> f (-x) = - f x"
  1590   unfolding vector_sneg_minus1
  1591   using linear_cmul[of f] by auto
  1592 
  1593 lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
  1594 
  1595 lemma linear_sub: "linear (f::'a::ring_1 ^'n \<Rightarrow> _) ==> f(x - y) = f x - f y"
  1596   by (simp add: diff_def linear_add linear_neg)
  1597 
  1598 lemma linear_setsum:
  1599   fixes f:: "'a::semiring_1^'n \<Rightarrow> _"
  1600   assumes lf: "linear f" and fS: "finite S"
  1601   shows "f (setsum g S) = setsum (f o g) S"
  1602 proof (induct rule: finite_induct[OF fS])
  1603   case 1 thus ?case by (simp add: linear_0[OF lf])
  1604 next
  1605   case (2 x F)
  1606   have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
  1607     by simp
  1608   also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
  1609   also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
  1610   finally show ?case .
  1611 qed
  1612 
  1613 lemma linear_setsum_mul:
  1614   fixes f:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m"
  1615   assumes lf: "linear f" and fS: "finite S"
  1616   shows "f (setsum (\<lambda>i. c i *s v i) S) = setsum (\<lambda>i. c i *s f (v i)) S"
  1617   using linear_setsum[OF lf fS, of "\<lambda>i. c i *s v i" , unfolded o_def]
  1618   linear_cmul[OF lf] by simp
  1619 
  1620 lemma linear_injective_0:
  1621   assumes lf: "linear (f:: 'a::ring_1 ^ 'n \<Rightarrow> _)"
  1622   shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
  1623 proof-
  1624   have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
  1625   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
  1626   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
  1627     by (simp add: linear_sub[OF lf])
  1628   also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
  1629   finally show ?thesis .
  1630 qed
  1631 
  1632 lemma linear_bounded:
  1633   fixes f:: "real ^'m::finite \<Rightarrow> real ^'n::finite"
  1634   assumes lf: "linear f"
  1635   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
  1636 proof-
  1637   let ?S = "UNIV:: 'm set"
  1638   let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
  1639   have fS: "finite ?S" by simp
  1640   {fix x:: "real ^ 'm"
  1641     let ?g = "(\<lambda>i. (x$i) *s (basis i) :: real ^ 'm)"
  1642     have "norm (f x) = norm (f (setsum (\<lambda>i. (x$i) *s (basis i)) ?S))"
  1643       by (simp only:  basis_expansion)
  1644     also have "\<dots> = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)"
  1645       using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf]
  1646       by auto
  1647     finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)" .
  1648     {fix i assume i: "i \<in> ?S"
  1649       from component_le_norm[of x i]
  1650       have "norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x"
  1651       unfolding norm_mul
  1652       apply (simp only: mult_commute)
  1653       apply (rule mult_mono)
  1654       by (auto simp add: ring_simps norm_ge_zero) }
  1655     then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
  1656     from real_setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (f (basis i))", OF th]
  1657     have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
  1658   then show ?thesis by blast
  1659 qed
  1660 
  1661 lemma linear_bounded_pos:
  1662   fixes f:: "real ^'n::finite \<Rightarrow> real ^ 'm::finite"
  1663   assumes lf: "linear f"
  1664   shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
  1665 proof-
  1666   from linear_bounded[OF lf] obtain B where
  1667     B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
  1668   let ?K = "\<bar>B\<bar> + 1"
  1669   have Kp: "?K > 0" by arith
  1670     {assume C: "B < 0"
  1671       have "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff)
  1672       with C have "B * norm (1:: real ^ 'n) < 0"
  1673 	by (simp add: zero_compare_simps)
  1674       with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp
  1675     }
  1676     then have Bp: "B \<ge> 0" by ferrack
  1677     {fix x::"real ^ 'n"
  1678       have "norm (f x) \<le> ?K *  norm x"
  1679       using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
  1680       apply (auto simp add: ring_simps split add: abs_split)
  1681       apply (erule order_trans, simp)
  1682       done
  1683   }
  1684   then show ?thesis using Kp by blast
  1685 qed
  1686 
  1687 subsection{* Bilinear functions. *}
  1688 
  1689 definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
  1690 
  1691 lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
  1692   by (simp add: bilinear_def linear_def)
  1693 lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
  1694   by (simp add: bilinear_def linear_def)
  1695 
  1696 lemma bilinear_lmul: "bilinear h ==> h (c *s x) y = c *s (h x y)"
  1697   by (simp add: bilinear_def linear_def)
  1698 
  1699 lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)"
  1700   by (simp add: bilinear_def linear_def)
  1701 
  1702 lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)"
  1703   by (simp only: vector_sneg_minus1 bilinear_lmul)
  1704 
  1705 lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y"
  1706   by (simp only: vector_sneg_minus1 bilinear_rmul)
  1707 
  1708 lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
  1709   using add_imp_eq[of x y 0] by auto
  1710 
  1711 lemma bilinear_lzero:
  1712   fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h 0 x = 0"
  1713   using bilinear_ladd[OF bh, of 0 0 x]
  1714     by (simp add: eq_add_iff ring_simps)
  1715 
  1716 lemma bilinear_rzero:
  1717   fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h x 0 = 0"
  1718   using bilinear_radd[OF bh, of x 0 0 ]
  1719     by (simp add: eq_add_iff ring_simps)
  1720 
  1721 lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ 'n)) z = h x z - h y z"
  1722   by (simp  add: diff_def bilinear_ladd bilinear_lneg)
  1723 
  1724 lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ 'n)) = h z x - h z y"
  1725   by (simp  add: diff_def bilinear_radd bilinear_rneg)
  1726 
  1727 lemma bilinear_setsum:
  1728   fixes h:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m \<Rightarrow> 'a ^ 'k"
  1729   assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
  1730   shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
  1731 proof-
  1732   have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
  1733     apply (rule linear_setsum[unfolded o_def])
  1734     using bh fS by (auto simp add: bilinear_def)
  1735   also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
  1736     apply (rule setsum_cong, simp)
  1737     apply (rule linear_setsum[unfolded o_def])
  1738     using bh fT by (auto simp add: bilinear_def)
  1739   finally show ?thesis unfolding setsum_cartesian_product .
  1740 qed
  1741 
  1742 lemma bilinear_bounded:
  1743   fixes h:: "real ^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real ^ 'k::finite"
  1744   assumes bh: "bilinear h"
  1745   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
  1746 proof-
  1747   let ?M = "UNIV :: 'm set"
  1748   let ?N = "UNIV :: 'n set"
  1749   let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
  1750   have fM: "finite ?M" and fN: "finite ?N" by simp_all
  1751   {fix x:: "real ^ 'm" and  y :: "real^'n"
  1752     have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$i) *s basis i) ?M) (setsum (\<lambda>i. (y$i) *s basis i) ?N))" unfolding basis_expansion ..
  1753     also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$i) *s basis i) ((y$j) *s basis j)) (?M \<times> ?N))"  unfolding bilinear_setsum[OF bh fM fN] ..
  1754     finally have th: "norm (h x y) = \<dots>" .
  1755     have "norm (h x y) \<le> ?B * norm x * norm y"
  1756       apply (simp add: setsum_left_distrib th)
  1757       apply (rule real_setsum_norm_le)
  1758       using fN fM
  1759       apply simp
  1760       apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)
  1761       apply (rule mult_mono)
  1762       apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
  1763       apply (rule mult_mono)
  1764       apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
  1765       done}
  1766   then show ?thesis by metis
  1767 qed
  1768 
  1769 lemma bilinear_bounded_pos:
  1770   fixes h:: "real ^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real ^ 'k::finite"
  1771   assumes bh: "bilinear h"
  1772   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
  1773 proof-
  1774   from bilinear_bounded[OF bh] obtain B where
  1775     B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
  1776   let ?K = "\<bar>B\<bar> + 1"
  1777   have Kp: "?K > 0" by arith
  1778   have KB: "B < ?K" by arith
  1779   {fix x::"real ^'m" and y :: "real ^'n"
  1780     from KB Kp
  1781     have "B * norm x * norm y \<le> ?K * norm x * norm y"
  1782       apply -
  1783       apply (rule mult_right_mono, rule mult_right_mono)
  1784       by (auto simp add: norm_ge_zero)
  1785     then have "norm (h x y) \<le> ?K * norm x * norm y"
  1786       using B[rule_format, of x y] by simp}
  1787   with Kp show ?thesis by blast
  1788 qed
  1789 
  1790 subsection{* Adjoints. *}
  1791 
  1792 definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
  1793 
  1794 lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
  1795 
  1796 lemma adjoint_works_lemma:
  1797   fixes f:: "'a::ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
  1798   assumes lf: "linear f"
  1799   shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
  1800 proof-
  1801   let ?N = "UNIV :: 'n set"
  1802   let ?M = "UNIV :: 'm set"
  1803   have fN: "finite ?N" by simp
  1804   have fM: "finite ?M" by simp
  1805   {fix y:: "'a ^ 'm"
  1806     let ?w = "(\<chi> i. (f (basis i) \<bullet> y)) :: 'a ^ 'n"
  1807     {fix x
  1808       have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *s basis i) ?N) \<bullet> y"
  1809 	by (simp only: basis_expansion)
  1810       also have "\<dots> = (setsum (\<lambda>i. (x$i) *s f (basis i)) ?N) \<bullet> y"
  1811 	unfolding linear_setsum[OF lf fN]
  1812 	by (simp add: linear_cmul[OF lf])
  1813       finally have "f x \<bullet> y = x \<bullet> ?w"
  1814 	apply (simp only: )
  1815 	apply (simp add: dot_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] ring_simps)
  1816 	done}
  1817   }
  1818   then show ?thesis unfolding adjoint_def
  1819     some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
  1820     using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
  1821     by metis
  1822 qed
  1823 
  1824 lemma adjoint_works:
  1825   fixes f:: "'a::ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
  1826   assumes lf: "linear f"
  1827   shows "x \<bullet> adjoint f y = f x \<bullet> y"
  1828   using adjoint_works_lemma[OF lf] by metis
  1829 
  1830 
  1831 lemma adjoint_linear:
  1832   fixes f :: "'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
  1833   assumes lf: "linear f"
  1834   shows "linear (adjoint f)"
  1835   by (simp add: linear_def vector_eq_ldot[symmetric] dot_radd dot_rmult adjoint_works[OF lf])
  1836 
  1837 lemma adjoint_clauses:
  1838   fixes f:: "'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
  1839   assumes lf: "linear f"
  1840   shows "x \<bullet> adjoint f y = f x \<bullet> y"
  1841   and "adjoint f y \<bullet> x = y \<bullet> f x"
  1842   by (simp_all add: adjoint_works[OF lf] dot_sym )
  1843 
  1844 lemma adjoint_adjoint:
  1845   fixes f:: "'a::comm_ring_1 ^ 'n::finite \<Rightarrow> 'a ^ 'm::finite"
  1846   assumes lf: "linear f"
  1847   shows "adjoint (adjoint f) = f"
  1848   apply (rule ext)
  1849   by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf])
  1850 
  1851 lemma adjoint_unique:
  1852   fixes f:: "'a::comm_ring_1 ^ 'n::finite \<Rightarrow> 'a ^ 'm::finite"
  1853   assumes lf: "linear f" and u: "\<forall>x y. f' x \<bullet> y = x \<bullet> f y"
  1854   shows "f' = adjoint f"
  1855   apply (rule ext)
  1856   using u
  1857   by (simp add: vector_eq_rdot[symmetric] adjoint_clauses[OF lf])
  1858 
  1859 text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
  1860 
  1861 consts generic_mult :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" (infixr "\<star>" 75)
  1862 
  1863 defs (overloaded)
  1864 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)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
  1865 
  1866 abbreviation
  1867   matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)
  1868   where "m ** m' == m\<star> m'"
  1869 
  1870 defs (overloaded)
  1871   matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
  1872 
  1873 abbreviation
  1874   matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)
  1875   where
  1876   "m *v v == m \<star> v"
  1877 
  1878 defs (overloaded)
  1879   vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (x$i)) (UNIV :: 'm set)) :: 'a^'n"
  1880 
  1881 abbreviation
  1882   vactor_matrix_mult' :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)
  1883   where
  1884   "v v* m == v \<star> m"
  1885 
  1886 definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
  1887 definition "(transp::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
  1888 definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
  1889 definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
  1890 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
  1891 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
  1892 
  1893 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
  1894 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \<star> B) + (A \<star> C)"
  1895   by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps)
  1896 
  1897 lemma setsum_delta':
  1898   assumes fS: "finite S" shows
  1899   "setsum (\<lambda>k. if a = k then b k else 0) S =
  1900      (if a\<in> S then b a else 0)"
  1901   using setsum_delta[OF fS, of a b, symmetric]
  1902   by (auto intro: setsum_cong)
  1903 
  1904 lemma matrix_mul_lid:
  1905   fixes A :: "'a::semiring_1 ^ 'm ^ 'n::finite"
  1906   shows "mat 1 ** A = A"
  1907   apply (simp add: matrix_matrix_mult_def mat_def)
  1908   apply vector
  1909   by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite]  mult_1_left mult_zero_left if_True UNIV_I)
  1910 
  1911 
  1912 lemma matrix_mul_rid:
  1913   fixes A :: "'a::semiring_1 ^ 'm::finite ^ 'n"
  1914   shows "A ** mat 1 = A"
  1915   apply (simp add: matrix_matrix_mult_def mat_def)
  1916   apply vector
  1917   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)
  1918 
  1919 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
  1920   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
  1921   apply (subst setsum_commute)
  1922   apply simp
  1923   done
  1924 
  1925 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
  1926   apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
  1927   apply (subst setsum_commute)
  1928   apply simp
  1929   done
  1930 
  1931 lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n::finite)"
  1932   apply (vector matrix_vector_mult_def mat_def)
  1933   by (simp add: cond_value_iff cond_application_beta
  1934     setsum_delta' cong del: if_weak_cong)
  1935 
  1936 lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)"
  1937   by (simp add: matrix_matrix_mult_def transp_def Cart_eq mult_commute)
  1938 
  1939 lemma matrix_eq:
  1940   fixes A B :: "'a::semiring_1 ^ 'n::finite ^ 'm"
  1941   shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
  1942   apply auto
  1943   apply (subst Cart_eq)
  1944   apply clarify
  1945   apply (clarsimp simp add: matrix_vector_mult_def basis_def cond_value_iff cond_application_beta Cart_eq cong del: if_weak_cong)
  1946   apply (erule_tac x="basis ia" in allE)
  1947   apply (erule_tac x="i" in allE)
  1948   by (auto simp add: basis_def cond_value_iff cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong)
  1949 
  1950 lemma matrix_vector_mul_component:
  1951   shows "((A::'a::semiring_1^'n'^'m) *v x)$k = (A$k) \<bullet> x"
  1952   by (simp add: matrix_vector_mult_def dot_def)
  1953 
  1954 lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) v* A) \<bullet> y = x \<bullet> (A *v y)"
  1955   apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
  1956   apply (subst setsum_commute)
  1957   by simp
  1958 
  1959 lemma transp_mat: "transp (mat n) = mat n"
  1960   by (vector transp_def mat_def)
  1961 
  1962 lemma transp_transp: "transp(transp A) = A"
  1963   by (vector transp_def)
  1964 
  1965 lemma row_transp:
  1966   fixes A:: "'a::semiring_1^'n^'m"
  1967   shows "row i (transp A) = column i A"
  1968   by (simp add: row_def column_def transp_def Cart_eq)
  1969 
  1970 lemma column_transp:
  1971   fixes A:: "'a::semiring_1^'n^'m"
  1972   shows "column i (transp A) = row i A"
  1973   by (simp add: row_def column_def transp_def Cart_eq)
  1974 
  1975 lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A"
  1976 by (auto simp add: rows_def columns_def row_transp intro: set_ext)
  1977 
  1978 lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp)
  1979 
  1980 text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
  1981 
  1982 lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
  1983   by (simp add: matrix_vector_mult_def dot_def)
  1984 
  1985 lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
  1986   by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute)
  1987 
  1988 lemma vector_componentwise:
  1989   "(x::'a::ring_1^'n::finite) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) (UNIV :: 'n set))"
  1990   apply (subst basis_expansion[symmetric])
  1991   by (vector Cart_eq setsum_component)
  1992 
  1993 lemma linear_componentwise:
  1994   fixes f:: "'a::ring_1 ^ 'm::finite \<Rightarrow> 'a ^ 'n"
  1995   assumes lf: "linear f"
  1996   shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
  1997 proof-
  1998   let ?M = "(UNIV :: 'm set)"
  1999   let ?N = "(UNIV :: 'n set)"
  2000   have fM: "finite ?M" by simp
  2001   have "?rhs = (setsum (\<lambda>i.(x$i) *s f (basis i) ) ?M)$j"
  2002     unfolding vector_smult_component[symmetric]
  2003     unfolding setsum_component[of "(\<lambda>i.(x$i) *s f (basis i :: 'a^'m))" ?M]
  2004     ..
  2005   then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion ..
  2006 qed
  2007 
  2008 text{* Inverse matrices  (not necessarily square) *}
  2009 
  2010 definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
  2011 
  2012 definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
  2013         (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
  2014 
  2015 text{* Correspondence between matrices and linear operators. *}
  2016 
  2017 definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
  2018 where "matrix f = (\<chi> i j. (f(basis j))$i)"
  2019 
  2020 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::'a::comm_semiring_1 ^ 'n))"
  2021   by (simp add: linear_def matrix_vector_mult_def Cart_eq ring_simps setsum_right_distrib setsum_addf)
  2022 
  2023 lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n::finite)"
  2024 apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute)
  2025 apply clarify
  2026 apply (rule linear_componentwise[OF lf, symmetric])
  2027 done
  2028 
  2029 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)
  2030 
  2031 lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n::finite)) = A"
  2032   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
  2033 
  2034 lemma matrix_compose:
  2035   assumes lf: "linear (f::'a::comm_ring_1^'n::finite \<Rightarrow> 'a^'m::finite)"
  2036   and lg: "linear (g::'a::comm_ring_1^'m::finite \<Rightarrow> 'a^'k)"
  2037   shows "matrix (g o f) = matrix g ** matrix f"
  2038   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
  2039   by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
  2040 
  2041 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)"
  2042   by (simp add: matrix_vector_mult_def transp_def Cart_eq mult_commute)
  2043 
  2044 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::'a::comm_ring_1^'n::finite^'m::finite) *v x) = (\<lambda>x. transp A *v x)"
  2045   apply (rule adjoint_unique[symmetric])
  2046   apply (rule matrix_vector_mul_linear)
  2047   apply (simp add: transp_def dot_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
  2048   apply (subst setsum_commute)
  2049   apply (auto simp add: mult_ac)
  2050   done
  2051 
  2052 lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n::finite \<Rightarrow> 'a ^ 'm::finite)"
  2053   shows "matrix(adjoint f) = transp(matrix f)"
  2054   apply (subst matrix_vector_mul[OF lf])
  2055   unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
  2056 
  2057 subsection{* Interlude: Some properties of real sets *}
  2058 
  2059 lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
  2060   shows "\<forall>n \<ge> m. d n < e m"
  2061   using prems apply auto
  2062   apply (erule_tac x="n" in allE)
  2063   apply (erule_tac x="n" in allE)
  2064   apply auto
  2065   done
  2066 
  2067 
  2068 lemma real_convex_bound_lt:
  2069   assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"
  2070   and uv: "u + v = 1"
  2071   shows "u * x + v * y < a"
  2072 proof-
  2073   have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
  2074   have "a = a * (u + v)" unfolding uv  by simp
  2075   hence th: "u * a + v * a = a" by (simp add: ring_simps)
  2076   from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_compare_simps)
  2077   from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_compare_simps)
  2078   from xa ya u v have "u * x + v * y < u * a + v * a"
  2079     apply (cases "u = 0", simp_all add: uv')
  2080     apply(rule mult_strict_left_mono)
  2081     using uv' apply simp_all
  2082 
  2083     apply (rule add_less_le_mono)
  2084     apply(rule mult_strict_left_mono)
  2085     apply simp_all
  2086     apply (rule mult_left_mono)
  2087     apply simp_all
  2088     done
  2089   thus ?thesis unfolding th .
  2090 qed
  2091 
  2092 lemma real_convex_bound_le:
  2093   assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"
  2094   and uv: "u + v = 1"
  2095   shows "u * x + v * y \<le> a"
  2096 proof-
  2097   from xa ya u v have "u * x + v * y \<le> u * a + v * a" by (simp add: add_mono mult_left_mono)
  2098   also have "\<dots> \<le> (u + v) * a" by (simp add: ring_simps)
  2099   finally show ?thesis unfolding uv by simp
  2100 qed
  2101 
  2102 lemma infinite_enumerate: assumes fS: "infinite S"
  2103   shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
  2104 unfolding subseq_def
  2105 using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
  2106 
  2107 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)"
  2108 apply auto
  2109 apply (rule_tac x="d/2" in exI)
  2110 apply auto
  2111 done
  2112 
  2113 
  2114 lemma triangle_lemma:
  2115   assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
  2116   shows "x <= y + z"
  2117 proof-
  2118   have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y  by (simp add: zero_compare_simps)
  2119   with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square ring_simps)
  2120   from y z have yz: "y + z \<ge> 0" by arith
  2121   from power2_le_imp_le[OF th yz] show ?thesis .
  2122 qed
  2123 
  2124 
  2125 lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
  2126    (\<exists>x::'a ^ 'n. \<forall>i. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")
  2127 proof-
  2128   let ?S = "(UNIV :: 'n set)"
  2129   {assume H: "?rhs"
  2130     then have ?lhs by auto}
  2131   moreover
  2132   {assume H: "?lhs"
  2133     then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
  2134     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
  2135     {fix i
  2136       from f have "P i (f i)" by metis
  2137       then have "P i (?x$i)" by auto
  2138     }
  2139     hence "\<forall>i. P i (?x$i)" by metis
  2140     hence ?rhs by metis }
  2141   ultimately show ?thesis by metis
  2142 qed
  2143 
  2144 (* Supremum and infimum of real sets *)
  2145 
  2146 
  2147 definition rsup:: "real set \<Rightarrow> real" where
  2148   "rsup S = (SOME a. isLub UNIV S a)"
  2149 
  2150 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)
  2151 
  2152 lemma rsup: assumes Se: "S \<noteq> {}" and b: "\<exists>b. S *<= b"
  2153   shows "isLub UNIV S (rsup S)"
  2154 using Se b
  2155 unfolding rsup_def
  2156 apply clarify
  2157 apply (rule someI_ex)
  2158 apply (rule reals_complete)
  2159 by (auto simp add: isUb_def setle_def)
  2160 
  2161 lemma rsup_le: assumes Se: "S \<noteq> {}" and Sb: "S *<= b" shows "rsup S \<le> b"
  2162 proof-
  2163   from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def)
  2164   from rsup[OF Se] Sb have "isLub UNIV S (rsup S)"  by blast
  2165   then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def)
  2166 qed
  2167 
  2168 lemma rsup_finite_Max: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2169   shows "rsup S = Max S"
  2170 using fS Se
  2171 proof-
  2172   let ?m = "Max S"
  2173   from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis
  2174   with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def)
  2175   from Max_in[OF fS Se] lub have mrS: "?m \<le> rsup S"
  2176     by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)
  2177   moreover
  2178   have "rsup S \<le> ?m" using Sm lub
  2179     by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
  2180   ultimately  show ?thesis by arith
  2181 qed
  2182 
  2183 lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2184   shows "rsup S \<in> S"
  2185   using rsup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis
  2186 
  2187 lemma rsup_finite_Ub: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2188   shows "isUb S S (rsup S)"
  2189   using rsup_finite_Max[OF fS Se] rsup_finite_in[OF fS Se] Max_ge[OF fS]
  2190   unfolding isUb_def setle_def by metis
  2191 
  2192 lemma rsup_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2193   shows "a \<le> rsup S \<longleftrightarrow> (\<exists> x \<in> S. a \<le> x)"
  2194 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2195 
  2196 lemma rsup_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2197   shows "a \<ge> rsup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)"
  2198 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2199 
  2200 lemma rsup_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2201   shows "a < rsup S \<longleftrightarrow> (\<exists> x \<in> S. a < x)"
  2202 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2203 
  2204 lemma rsup_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2205   shows "a > rsup S \<longleftrightarrow> (\<forall> 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_unique: assumes b: "S *<= b" and S: "\<forall>b' < b. \<exists>x \<in> S. b' < x"
  2209   shows "rsup S = b"
  2210 using b S
  2211 unfolding setle_def rsup_alt
  2212 apply -
  2213 apply (rule some_equality)
  2214 apply (metis  linorder_not_le order_eq_iff[symmetric])+
  2215 done
  2216 
  2217 lemma rsup_le_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. T *<= b) \<Longrightarrow> rsup S \<le> rsup T"
  2218   apply (rule rsup_le)
  2219   apply simp
  2220   using rsup[of T] by (auto simp add: isLub_def leastP_def setge_def setle_def isUb_def)
  2221 
  2222 lemma isUb_def': "isUb R S = (\<lambda>x. S *<= x \<and> x \<in> R)"
  2223   apply (rule ext)
  2224   by (metis isUb_def)
  2225 
  2226 lemma UNIV_trivial: "UNIV x" using UNIV_I[of x] by (metis mem_def)
  2227 lemma rsup_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
  2228   shows "a \<le> rsup S \<and> rsup S \<le> b"
  2229 proof-
  2230   from rsup[OF Se] u have lub: "isLub UNIV S (rsup S)" by blast
  2231   hence b: "rsup S \<le> b" using u by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
  2232   from Se obtain y where y: "y \<in> S" by blast
  2233   from lub l have "a \<le> rsup S" apply (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
  2234     apply (erule ballE[where x=y])
  2235     apply (erule ballE[where x=y])
  2236     apply arith
  2237     using y apply auto
  2238     done
  2239   with b show ?thesis by blast
  2240 qed
  2241 
  2242 lemma rsup_abs_le: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rsup S\<bar> \<le> a"
  2243   unfolding abs_le_interval_iff  using rsup_bounds[of S "-a" a]
  2244   by (auto simp add: setge_def setle_def)
  2245 
  2246 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"
  2247 proof-
  2248   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
  2249   show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th
  2250     by  (auto simp add: setge_def setle_def)
  2251 qed
  2252 
  2253 definition rinf:: "real set \<Rightarrow> real" where
  2254   "rinf S = (SOME a. isGlb UNIV S a)"
  2255 
  2256 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)
  2257 
  2258 lemma reals_complete_Glb: assumes Se: "\<exists>x. x \<in> S" and lb: "\<exists> y. isLb UNIV S y"
  2259   shows "\<exists>(t::real). isGlb UNIV S t"
  2260 proof-
  2261   let ?M = "uminus ` S"
  2262   from lb have th: "\<exists>y. isUb UNIV ?M y" apply (auto simp add: isUb_def isLb_def setle_def setge_def)
  2263     by (rule_tac x="-y" in exI, auto)
  2264   from Se have Me: "\<exists>x. x \<in> ?M" by blast
  2265   from reals_complete[OF Me th] obtain t where t: "isLub UNIV ?M t" by blast
  2266   have "isGlb UNIV S (- t)" using t
  2267     apply (auto simp add: isLub_def isGlb_def leastP_def greatestP_def setle_def setge_def isUb_def isLb_def)
  2268     apply (erule_tac x="-y" in allE)
  2269     apply auto
  2270     done
  2271   then show ?thesis by metis
  2272 qed
  2273 
  2274 lemma rinf: assumes Se: "S \<noteq> {}" and b: "\<exists>b. b <=* S"
  2275   shows "isGlb UNIV S (rinf S)"
  2276 using Se b
  2277 unfolding rinf_def
  2278 apply clarify
  2279 apply (rule someI_ex)
  2280 apply (rule reals_complete_Glb)
  2281 apply (auto simp add: isLb_def setle_def setge_def)
  2282 done
  2283 
  2284 lemma rinf_ge: assumes Se: "S \<noteq> {}" and Sb: "b <=* S" shows "rinf S \<ge> b"
  2285 proof-
  2286   from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def)
  2287   from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)"  by blast
  2288   then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def)
  2289 qed
  2290 
  2291 lemma rinf_finite_Min: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2292   shows "rinf S = Min S"
  2293 using fS Se
  2294 proof-
  2295   let ?m = "Min S"
  2296   from Min_le[OF fS] have Sm: "\<forall> x\<in> S. x \<ge> ?m" by metis
  2297   with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def)
  2298   from Min_in[OF fS Se] glb have mrS: "?m \<ge> rinf S"
  2299     by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def)
  2300   moreover
  2301   have "rinf S \<ge> ?m" using Sm glb
  2302     by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def)
  2303   ultimately  show ?thesis by arith
  2304 qed
  2305 
  2306 lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2307   shows "rinf S \<in> S"
  2308   using rinf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis
  2309 
  2310 lemma rinf_finite_Lb: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2311   shows "isLb S S (rinf S)"
  2312   using rinf_finite_Min[OF fS Se] rinf_finite_in[OF fS Se] Min_le[OF fS]
  2313   unfolding isLb_def setge_def by metis
  2314 
  2315 lemma rinf_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2316   shows "a \<le> rinf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)"
  2317 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2318 
  2319 lemma rinf_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2320   shows "a \<ge> rinf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)"
  2321 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2322 
  2323 lemma rinf_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2324   shows "a < rinf S \<longleftrightarrow> (\<forall> x \<in> S. a < x)"
  2325 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2326 
  2327 lemma rinf_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2328   shows "a > rinf S \<longleftrightarrow> (\<exists> 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_unique: assumes b: "b <=* S" and S: "\<forall>b' > b. \<exists>x \<in> S. b' > x"
  2332   shows "rinf S = b"
  2333 using b S
  2334 unfolding setge_def rinf_alt
  2335 apply -
  2336 apply (rule some_equality)
  2337 apply (metis  linorder_not_le order_eq_iff[symmetric])+
  2338 done
  2339 
  2340 lemma rinf_ge_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. b <=* T) \<Longrightarrow> rinf S >= rinf T"
  2341   apply (rule rinf_ge)
  2342   apply simp
  2343   using rinf[of T] by (auto simp add: isGlb_def greatestP_def setge_def setle_def isLb_def)
  2344 
  2345 lemma isLb_def': "isLb R S = (\<lambda>x. x <=* S \<and> x \<in> R)"
  2346   apply (rule ext)
  2347   by (metis isLb_def)
  2348 
  2349 lemma rinf_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
  2350   shows "a \<le> rinf S \<and> rinf S \<le> b"
  2351 proof-
  2352   from rinf[OF Se] l have lub: "isGlb UNIV S (rinf S)" by blast
  2353   hence b: "a \<le> rinf S" using l by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
  2354   from Se obtain y where y: "y \<in> S" by blast
  2355   from lub u have "b \<ge> rinf S" apply (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
  2356     apply (erule ballE[where x=y])
  2357     apply (erule ballE[where x=y])
  2358     apply arith
  2359     using y apply auto
  2360     done
  2361   with b show ?thesis by blast
  2362 qed
  2363 
  2364 lemma rinf_abs_ge: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rinf S\<bar> \<le> a"
  2365   unfolding abs_le_interval_iff  using rinf_bounds[of S "-a" a]
  2366   by (auto simp add: setge_def setle_def)
  2367 
  2368 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"
  2369 proof-
  2370   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
  2371   show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th
  2372     by  (auto simp add: setge_def setle_def)
  2373 qed
  2374 
  2375 
  2376 
  2377 subsection{* Operator norm. *}
  2378 
  2379 definition "onorm f = rsup {norm (f x)| x. norm x = 1}"
  2380 
  2381 lemma norm_bound_generalize:
  2382   fixes f:: "real ^'n::finite \<Rightarrow> real^'m::finite"
  2383   assumes lf: "linear f"
  2384   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")
  2385 proof-
  2386   {assume H: ?rhs
  2387     {fix x :: "real^'n" assume x: "norm x = 1"
  2388       from H[rule_format, of x] x have "norm (f x) \<le> b" by simp}
  2389     then have ?lhs by blast }
  2390 
  2391   moreover
  2392   {assume H: ?lhs
  2393     from H[rule_format, of "basis arbitrary"]
  2394     have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis arbitrary)"]
  2395       by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])
  2396     {fix x :: "real ^'n"
  2397       {assume "x = 0"
  2398 	then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}
  2399       moreover
  2400       {assume x0: "x \<noteq> 0"
  2401 	hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
  2402 	let ?c = "1/ norm x"
  2403 	have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul)
  2404 	with H have "norm (f(?c*s x)) \<le> b" by blast
  2405 	hence "?c * norm (f x) \<le> b"
  2406 	  by (simp add: linear_cmul[OF lf] norm_mul)
  2407 	hence "norm (f x) \<le> b * norm x"
  2408 	  using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
  2409       ultimately have "norm (f x) \<le> b * norm x" by blast}
  2410     then have ?rhs by blast}
  2411   ultimately show ?thesis by blast
  2412 qed
  2413 
  2414 lemma onorm:
  2415   fixes f:: "real ^'n::finite \<Rightarrow> real ^'m::finite"
  2416   assumes lf: "linear f"
  2417   shows "norm (f x) <= onorm f * norm x"
  2418   and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
  2419 proof-
  2420   {
  2421     let ?S = "{norm (f x) |x. norm x = 1}"
  2422     have Se: "?S \<noteq> {}" using  norm_basis by auto
  2423     from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
  2424       unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
  2425     {from rsup[OF Se b, unfolded onorm_def[symmetric]]
  2426       show "norm (f x) <= onorm f * norm x"
  2427 	apply -
  2428 	apply (rule spec[where x = x])
  2429 	unfolding norm_bound_generalize[OF lf, symmetric]
  2430 	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
  2431     {
  2432       show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
  2433 	using rsup[OF Se b, unfolded onorm_def[symmetric]]
  2434 	unfolding norm_bound_generalize[OF lf, symmetric]
  2435 	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
  2436   }
  2437 qed
  2438 
  2439 lemma onorm_pos_le: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" shows "0 <= onorm f"
  2440   using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis arbitrary"], unfolded norm_basis] by simp
  2441 
  2442 lemma onorm_eq_0: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)"
  2443   shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
  2444   using onorm[OF lf]
  2445   apply (auto simp add: onorm_pos_le)
  2446   apply atomize
  2447   apply (erule allE[where x="0::real"])
  2448   using onorm_pos_le[OF lf]
  2449   apply arith
  2450   done
  2451 
  2452 lemma onorm_const: "onorm(\<lambda>x::real^'n::finite. (y::real ^ 'm::finite)) = norm y"
  2453 proof-
  2454   let ?f = "\<lambda>x::real^'n. (y::real ^ 'm)"
  2455   have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
  2456     by(auto intro: vector_choose_size set_ext)
  2457   show ?thesis
  2458     unfolding onorm_def th
  2459     apply (rule rsup_unique) by (simp_all  add: setle_def)
  2460 qed
  2461 
  2462 lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n::finite \<Rightarrow> real ^'m::finite)"
  2463   shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"
  2464   unfolding onorm_eq_0[OF lf, symmetric]
  2465   using onorm_pos_le[OF lf] by arith
  2466 
  2467 lemma onorm_compose:
  2468   assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)"
  2469   and lg: "linear (g::real^'k::finite \<Rightarrow> real^'n::finite)"
  2470   shows "onorm (f o g) <= onorm f * onorm g"
  2471   apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
  2472   unfolding o_def
  2473   apply (subst mult_assoc)
  2474   apply (rule order_trans)
  2475   apply (rule onorm(1)[OF lf])
  2476   apply (rule mult_mono1)
  2477   apply (rule onorm(1)[OF lg])
  2478   apply (rule onorm_pos_le[OF lf])
  2479   done
  2480 
  2481 lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real^'m::finite)"
  2482   shows "onorm (\<lambda>x. - f x) \<le> onorm f"
  2483   using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
  2484   unfolding norm_minus_cancel by metis
  2485 
  2486 lemma onorm_neg: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real^'m::finite)"
  2487   shows "onorm (\<lambda>x. - f x) = onorm f"
  2488   using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]]
  2489   by simp
  2490 
  2491 lemma onorm_triangle:
  2492   assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" and lg: "linear g"
  2493   shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
  2494   apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
  2495   apply (rule order_trans)
  2496   apply (rule norm_triangle_ineq)
  2497   apply (simp add: distrib)
  2498   apply (rule add_mono)
  2499   apply (rule onorm(1)[OF lf])
  2500   apply (rule onorm(1)[OF lg])
  2501   done
  2502 
  2503 lemma onorm_triangle_le: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
  2504   \<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"
  2505   apply (rule order_trans)
  2506   apply (rule onorm_triangle)
  2507   apply assumption+
  2508   done
  2509 
  2510 lemma onorm_triangle_lt: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
  2511   ==> onorm(\<lambda>x. f x + g x) < e"
  2512   apply (rule order_le_less_trans)
  2513   apply (rule onorm_triangle)
  2514   by assumption+
  2515 
  2516 (* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)
  2517 
  2518 definition vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x = (\<chi> i. x)"
  2519 
  2520 definition dest_vec1:: "'a ^1 \<Rightarrow> 'a"
  2521   where "dest_vec1 x = (x$1)"
  2522 
  2523 lemma vec1_component[simp]: "(vec1 x)$1 = x"
  2524   by (simp add: vec1_def)
  2525 
  2526 lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
  2527   by (simp_all add: vec1_def dest_vec1_def Cart_eq forall_1)
  2528 
  2529 lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))" by (metis vec1_dest_vec1)
  2530 
  2531 lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))" by (metis vec1_dest_vec1)
  2532 
  2533 lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))"  by (metis vec1_dest_vec1)
  2534 
  2535 lemma exists_dest_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(dest_vec1 x))"by (metis vec1_dest_vec1)
  2536 
  2537 lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
  2538 
  2539 lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
  2540 
  2541 lemma vec1_in_image_vec1: "vec1 x \<in> (vec1 ` S) \<longleftrightarrow> x \<in> S" by auto
  2542 
  2543 lemma vec1_vec: "vec1 x = vec x" by (vector vec1_def)
  2544 
  2545 lemma vec1_add: "vec1(x + y) = vec1 x + vec1 y" by (vector vec1_def)
  2546 lemma vec1_sub: "vec1(x - y) = vec1 x - vec1 y" by (vector vec1_def)
  2547 lemma vec1_cmul: "vec1(c* x) = c *s vec1 x " by (vector vec1_def)
  2548 lemma vec1_neg: "vec1(- x) = - vec1 x " by (vector vec1_def)
  2549 
  2550 lemma vec1_setsum: assumes fS: "finite S"
  2551   shows "vec1(setsum f S) = setsum (vec1 o f) S"
  2552   apply (induct rule: finite_induct[OF fS])
  2553   apply (simp add: vec1_vec)
  2554   apply (auto simp add: vec1_add)
  2555   done
  2556 
  2557 lemma dest_vec1_lambda: "dest_vec1(\<chi> i. x i) = x 1"
  2558   by (simp add: dest_vec1_def)
  2559 
  2560 lemma dest_vec1_vec: "dest_vec1(vec x) = x"
  2561   by (simp add: vec1_vec[symmetric])
  2562 
  2563 lemma dest_vec1_add: "dest_vec1(x + y) = dest_vec1 x + dest_vec1 y"
  2564  by (metis vec1_dest_vec1 vec1_add)
  2565 
  2566 lemma dest_vec1_sub: "dest_vec1(x - y) = dest_vec1 x - dest_vec1 y"
  2567  by (metis vec1_dest_vec1 vec1_sub)
  2568 
  2569 lemma dest_vec1_cmul: "dest_vec1(c*sx) = c * dest_vec1 x"
  2570  by (metis vec1_dest_vec1 vec1_cmul)
  2571 
  2572 lemma dest_vec1_neg: "dest_vec1(- x) = - dest_vec1 x"
  2573  by (metis vec1_dest_vec1 vec1_neg)
  2574 
  2575 lemma dest_vec1_0[simp]: "dest_vec1 0 = 0" by (metis vec_0 dest_vec1_vec)
  2576 
  2577 lemma dest_vec1_sum: assumes fS: "finite S"
  2578   shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"
  2579   apply (induct rule: finite_induct[OF fS])
  2580   apply (simp add: dest_vec1_vec)
  2581   apply (auto simp add: dest_vec1_add)
  2582   done
  2583 
  2584 lemma norm_vec1: "norm(vec1 x) = abs(x)"
  2585   by (simp add: vec1_def norm_real)
  2586 
  2587 lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)"
  2588   by (simp only: dist_real vec1_component)
  2589 lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
  2590   by (metis vec1_dest_vec1 norm_vec1)
  2591 
  2592 lemma linear_vmul_dest_vec1:
  2593   fixes f:: "'a::semiring_1^'n \<Rightarrow> 'a^1"
  2594   shows "linear f \<Longrightarrow> linear (\<lambda>x. dest_vec1(f x) *s v)"
  2595   unfolding dest_vec1_def
  2596   apply (rule linear_vmul_component)
  2597   by auto
  2598 
  2599 lemma linear_from_scalars:
  2600   assumes lf: "linear (f::'a::comm_ring_1 ^1 \<Rightarrow> 'a^'n)"
  2601   shows "f = (\<lambda>x. dest_vec1 x *s column 1 (matrix f))"
  2602   apply (rule ext)
  2603   apply (subst matrix_works[OF lf, symmetric])
  2604   apply (auto simp add: Cart_eq matrix_vector_mult_def dest_vec1_def column_def  mult_commute UNIV_1)
  2605   done
  2606 
  2607 lemma linear_to_scalars: assumes lf: "linear (f::'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a^1)"
  2608   shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
  2609   apply (rule ext)
  2610   apply (subst matrix_works[OF lf, symmetric])
  2611   apply (simp add: Cart_eq matrix_vector_mult_def vec1_def row_def dot_def mult_commute forall_1)
  2612   done
  2613 
  2614 lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
  2615   by (simp add: dest_vec1_eq[symmetric])
  2616 
  2617 lemma setsum_scalars: assumes fS: "finite S"
  2618   shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)"
  2619   unfolding vec1_setsum[OF fS] by simp
  2620 
  2621 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"
  2622   apply (cases "dest_vec1 x \<le> dest_vec1 y")
  2623   apply simp
  2624   apply (subgoal_tac "dest_vec1 y \<le> dest_vec1 x")
  2625   apply (auto)
  2626   done
  2627 
  2628 text{* Pasting vectors. *}
  2629 
  2630 lemma linear_fstcart: "linear fstcart"
  2631   by (auto simp add: linear_def Cart_eq)
  2632 
  2633 lemma linear_sndcart: "linear sndcart"
  2634   by (auto simp add: linear_def Cart_eq)
  2635 
  2636 lemma fstcart_vec[simp]: "fstcart(vec x) = vec x"
  2637   by (simp add: Cart_eq)
  2638 
  2639 lemma fstcart_add[simp]:"fstcart(x + y) = fstcart (x::'a::{plus,times}^('b + 'c)) + fstcart y"
  2640   by (simp add: Cart_eq)
  2641 
  2642 lemma fstcart_cmul[simp]:"fstcart(c*s x) = c*s fstcart (x::'a::{plus,times}^('b + 'c))"
  2643   by (simp add: Cart_eq)
  2644 
  2645 lemma fstcart_neg[simp]:"fstcart(- x) = - fstcart (x::'a::ring_1^('b + 'c))"
  2646   by (simp add: Cart_eq)
  2647 
  2648 lemma fstcart_sub[simp]:"fstcart(x - y) = fstcart (x::'a::ring_1^('b + 'c)) - fstcart y"
  2649   by (simp add: Cart_eq)
  2650 
  2651 lemma fstcart_setsum:
  2652   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
  2653   assumes fS: "finite S"
  2654   shows "fstcart (setsum f S) = setsum (\<lambda>i. fstcart (f i)) S"
  2655   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
  2656 
  2657 lemma sndcart_vec[simp]: "sndcart(vec x) = vec x"
  2658   by (simp add: Cart_eq)
  2659 
  2660 lemma sndcart_add[simp]:"sndcart(x + y) = sndcart (x::'a::{plus,times}^('b + 'c)) + sndcart y"
  2661   by (simp add: Cart_eq)
  2662 
  2663 lemma sndcart_cmul[simp]:"sndcart(c*s x) = c*s sndcart (x::'a::{plus,times}^('b + 'c))"
  2664   by (simp add: Cart_eq)
  2665 
  2666 lemma sndcart_neg[simp]:"sndcart(- x) = - sndcart (x::'a::ring_1^('b + 'c))"
  2667   by (simp add: Cart_eq)
  2668 
  2669 lemma sndcart_sub[simp]:"sndcart(x - y) = sndcart (x::'a::ring_1^('b + 'c)) - sndcart y"
  2670   by (simp add: Cart_eq)
  2671 
  2672 lemma sndcart_setsum:
  2673   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
  2674   assumes fS: "finite S"
  2675   shows "sndcart (setsum f S) = setsum (\<lambda>i. sndcart (f i)) S"
  2676   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
  2677 
  2678 lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x"
  2679   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
  2680 
  2681 lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)"
  2682   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
  2683 
  2684 lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1"
  2685   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
  2686 
  2687 lemma pastecart_neg[simp]: "pastecart (- (x::'a::ring_1^_)) (- y) = - pastecart x y"
  2688   unfolding vector_sneg_minus1 pastecart_cmul ..
  2689 
  2690 lemma pastecart_sub: "pastecart (x1::'a::ring_1^_) y1 - pastecart x2 y2 = pastecart (x1 - x2) (y1 - y2)"
  2691   by (simp add: diff_def pastecart_neg[symmetric] del: pastecart_neg)
  2692 
  2693 lemma pastecart_setsum:
  2694   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
  2695   assumes fS: "finite S"
  2696   shows "pastecart (setsum f S) (setsum g S) = setsum (\<lambda>i. pastecart (f i) (g i)) S"
  2697   by (simp  add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS] fstcart_pastecart sndcart_pastecart)
  2698 
  2699 lemma setsum_Plus:
  2700   "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
  2701     (\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))"
  2702   unfolding Plus_def
  2703   by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)
  2704 
  2705 lemma setsum_UNIV_sum:
  2706   fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
  2707   shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
  2708   apply (subst UNIV_Plus_UNIV [symmetric])
  2709   apply (rule setsum_Plus [OF finite finite])
  2710   done
  2711 
  2712 lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n::finite + 'm::finite))"
  2713 proof-
  2714   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
  2715     by (simp add: pastecart_fst_snd)
  2716   have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
  2717     by (simp add: dot_def setsum_UNIV_sum pastecart_def setsum_nonneg)
  2718   then show ?thesis
  2719     unfolding th0
  2720     unfolding real_vector_norm_def real_sqrt_le_iff id_def
  2721     by (simp add: dot_def)
  2722 qed
  2723 
  2724 lemma dist_fstcart: "dist(fstcart (x::real^_)) (fstcart y) <= dist x y"
  2725   unfolding dist_norm by (metis fstcart_sub[symmetric] norm_fstcart)
  2726 
  2727 lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n::finite + 'm::finite))"
  2728 proof-
  2729   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
  2730     by (simp add: pastecart_fst_snd)
  2731   have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
  2732     by (simp add: dot_def setsum_UNIV_sum pastecart_def setsum_nonneg)
  2733   then show ?thesis
  2734     unfolding th0
  2735     unfolding real_vector_norm_def real_sqrt_le_iff id_def
  2736     by (simp add: dot_def)
  2737 qed
  2738 
  2739 lemma dist_sndcart: "dist(sndcart (x::real^_)) (sndcart y) <= dist x y"
  2740   unfolding dist_norm by (metis sndcart_sub[symmetric] norm_sndcart)
  2741 
  2742 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"
  2743   by (simp add: dot_def setsum_UNIV_sum pastecart_def)
  2744 
  2745 text {* TODO: move to NthRoot *}
  2746 lemma sqrt_add_le_add_sqrt:
  2747   assumes x: "0 \<le> x" and y: "0 \<le> y"
  2748   shows "sqrt (x + y) \<le> sqrt x + sqrt y"
  2749 apply (rule power2_le_imp_le)
  2750 apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
  2751 apply (simp add: mult_nonneg_nonneg x y)
  2752 apply (simp add: add_nonneg_nonneg x y)
  2753 done
  2754 
  2755 lemma norm_pastecart: "norm (pastecart x y) <= norm x + norm y"
  2756   unfolding vector_norm_def setL2_def setsum_UNIV_sum
  2757   by (simp add: sqrt_add_le_add_sqrt setsum_nonneg)
  2758 
  2759 subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
  2760 
  2761 definition hull :: "'a set set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) where
  2762   "S hull s = Inter {t. t \<in> S \<and> s \<subseteq> t}"
  2763 
  2764 lemma hull_same: "s \<in> S \<Longrightarrow> S hull s = s"
  2765   unfolding hull_def by auto
  2766 
  2767 lemma hull_in: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) \<in> S"
  2768 unfolding hull_def subset_iff by auto
  2769 
  2770 lemma hull_eq: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) = s \<longleftrightarrow> s \<in> S"
  2771 using hull_same[of s S] hull_in[of S s] by metis
  2772 
  2773 
  2774 lemma hull_hull: "S hull (S hull s) = S hull s"
  2775   unfolding hull_def by blast
  2776 
  2777 lemma hull_subset: "s \<subseteq> (S hull s)"
  2778   unfolding hull_def by blast
  2779 
  2780 lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
  2781   unfolding hull_def by blast
  2782 
  2783 lemma hull_antimono: "S \<subseteq> T ==> (T hull s) \<subseteq> (S hull s)"
  2784   unfolding hull_def by blast
  2785 
  2786 lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> t \<in> S ==> (S hull s) \<subseteq> t"
  2787   unfolding hull_def by blast
  2788 
  2789 lemma subset_hull: "t \<in> S ==> S hull s \<subseteq> t \<longleftrightarrow>  s \<subseteq> t"
  2790   unfolding hull_def by blast
  2791 
  2792 lemma hull_unique: "s \<subseteq> t \<Longrightarrow> t \<in> S \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> t' \<in> S ==> t \<subseteq> t')
  2793            ==> (S hull s = t)"
  2794 unfolding hull_def by auto
  2795 
  2796 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"
  2797   using hull_minimal[of S "{x. P x}" Q]
  2798   by (auto simp add: subset_eq Collect_def mem_def)
  2799 
  2800 lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" by (metis hull_subset subset_eq)
  2801 
  2802 lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
  2803 unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
  2804 
  2805 lemma hull_union: assumes T: "\<And>T. T \<subseteq> S ==> Inter T \<in> S"
  2806   shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
  2807 apply rule
  2808 apply (rule hull_mono)
  2809 unfolding Un_subset_iff
  2810 apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
  2811 apply (rule hull_minimal)
  2812 apply (metis hull_union_subset)
  2813 apply (metis hull_in T)
  2814 done
  2815 
  2816 lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
  2817   unfolding hull_def by blast
  2818 
  2819 lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
  2820 by (metis hull_redundant_eq)
  2821 
  2822 text{* Archimedian properties and useful consequences. *}
  2823 
  2824 lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
  2825   using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto)
  2826 lemmas real_arch_lt = reals_Archimedean2
  2827 
  2828 lemmas real_arch = reals_Archimedean3
  2829 
  2830 lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
  2831   using reals_Archimedean
  2832   apply (auto simp add: field_simps inverse_positive_iff_positive)
  2833   apply (subgoal_tac "inverse (real n) > 0")
  2834   apply arith
  2835   apply simp
  2836   done
  2837 
  2838 lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
  2839 proof(induct n)
  2840   case 0 thus ?case by simp
  2841 next
  2842   case (Suc n)
  2843   hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
  2844   from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
  2845   from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
  2846   also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
  2847     apply (simp add: ring_simps)
  2848     using mult_left_mono[OF p Suc.prems] by simp
  2849   finally show ?case  by (simp add: real_of_nat_Suc ring_simps)
  2850 qed
  2851 
  2852 lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
  2853 proof-
  2854   from x have x0: "x - 1 > 0" by arith
  2855   from real_arch[OF x0, rule_format, of y]
  2856   obtain n::nat where n:"y < real n * (x - 1)" by metis
  2857   from x0 have x00: "x- 1 \<ge> 0" by arith
  2858   from real_pow_lbound[OF x00, of n] n
  2859   have "y < x^n" by auto
  2860   then show ?thesis by metis
  2861 qed
  2862 
  2863 lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
  2864   using real_arch_pow[of 2 x] by simp
  2865 
  2866 lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
  2867   shows "\<exists>n. x^n < y"
  2868 proof-
  2869   {assume x0: "x > 0"
  2870     from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
  2871     from real_arch_pow[OF ix, of "1/y"]
  2872     obtain n where n: "1/y < (1/x)^n" by blast
  2873     then
  2874     have ?thesis using y x0 by (auto simp add: field_simps power_divide) }
  2875   moreover
  2876   {assume "\<not> x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)}
  2877   ultimately show ?thesis by metis
  2878 qed
  2879 
  2880 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)"
  2881   by (metis real_arch_inv)
  2882 
  2883 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"
  2884   apply (rule forall_pos_mono)
  2885   apply auto
  2886   apply (atomize)
  2887   apply (erule_tac x="n - 1" in allE)
  2888   apply auto
  2889   done
  2890 
  2891 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"
  2892   shows "x = 0"
  2893 proof-
  2894   {assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
  2895     from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x"  by blast
  2896     with xc[rule_format, of n] have "n = 0" by arith
  2897     with n c have False by simp}
  2898   then show ?thesis by blast
  2899 qed
  2900 
  2901 (* ------------------------------------------------------------------------- *)
  2902 (* Relate max and min to sup and inf.                                        *)
  2903 (* ------------------------------------------------------------------------- *)
  2904 
  2905 lemma real_max_rsup: "max x y = rsup {x,y}"
  2906 proof-
  2907   have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
  2908   from rsup_finite_le_iff[OF f, of "max x y"] have "rsup {x,y} \<le> max x y" by simp
  2909   moreover
  2910   have "max x y \<le> rsup {x,y}" using rsup_finite_ge_iff[OF f, of "max x y"]
  2911     by (simp add: linorder_linear)
  2912   ultimately show ?thesis by arith
  2913 qed
  2914 
  2915 lemma real_min_rinf: "min x y = rinf {x,y}"
  2916 proof-
  2917   have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
  2918   from rinf_finite_le_iff[OF f, of "min x y"] have "rinf {x,y} \<le> min x y"
  2919     by (simp add: linorder_linear)
  2920   moreover
  2921   have "min x y \<le> rinf {x,y}" using rinf_finite_ge_iff[OF f, of "min x y"]
  2922     by simp
  2923   ultimately show ?thesis by arith
  2924 qed
  2925 
  2926 (* ------------------------------------------------------------------------- *)
  2927 (* Geometric progression.                                                    *)
  2928 (* ------------------------------------------------------------------------- *)
  2929 
  2930 lemma sum_gp_basic: "((1::'a::{field}) - x) * setsum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
  2931   (is "?lhs = ?rhs")
  2932 proof-
  2933   {assume x1: "x = 1" hence ?thesis by simp}
  2934   moreover
  2935   {assume x1: "x\<noteq>1"
  2936     hence x1': "x - 1 \<noteq> 0" "1 - x \<noteq> 0" "x - 1 = - (1 - x)" "- (1 - x) \<noteq> 0" by auto
  2937     from geometric_sum[OF x1, of "Suc n", unfolded x1']
  2938     have "(- (1 - x)) * setsum (\<lambda>i. x^i) {0 .. n} = - (1 - x^(Suc n))"
  2939       unfolding atLeastLessThanSuc_atLeastAtMost
  2940       using x1' apply (auto simp only: field_simps)
  2941       apply (simp add: ring_simps)
  2942       done
  2943     then have ?thesis by (simp add: ring_simps) }
  2944   ultimately show ?thesis by metis
  2945 qed
  2946 
  2947 lemma sum_gp_multiplied: assumes mn: "m <= n"
  2948   shows "((1::'a::{field}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
  2949   (is "?lhs = ?rhs")
  2950 proof-
  2951   let ?S = "{0..(n - m)}"
  2952   from mn have mn': "n - m \<ge> 0" by arith
  2953   let ?f = "op + m"
  2954   have i: "inj_on ?f ?S" unfolding inj_on_def by auto
  2955   have f: "?f ` ?S = {m..n}"
  2956     using mn apply (auto simp add: image_iff Bex_def) by arith
  2957   have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)"
  2958     by (rule ext, simp add: power_add power_mult)
  2959   from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
  2960   have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp
  2961   then show ?thesis unfolding sum_gp_basic using mn
  2962     by (simp add: ring_simps power_add[symmetric])
  2963 qed
  2964 
  2965 lemma sum_gp: "setsum (op ^ (x::'a::{field})) {m .. n} =
  2966    (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m)
  2967                     else (x^ m - x^ (Suc n)) / (1 - x))"
  2968 proof-
  2969   {assume nm: "n < m" hence ?thesis by simp}
  2970   moreover
  2971   {assume "\<not> n < m" hence nm: "m \<le> n" by arith
  2972     {assume x: "x = 1"  hence ?thesis by simp}
  2973     moreover
  2974     {assume x: "x \<noteq> 1" hence nz: "1 - x \<noteq> 0" by simp
  2975       from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)}
  2976     ultimately have ?thesis by metis
  2977   }
  2978   ultimately show ?thesis by metis
  2979 qed
  2980 
  2981 lemma sum_gp_offset: "setsum (op ^ (x::'a::{field})) {m .. m+n} =
  2982   (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
  2983   unfolding sum_gp[of x m "m + n"] power_Suc
  2984   by (simp add: ring_simps power_add)
  2985 
  2986 
  2987 subsection{* A bit of linear algebra. *}
  2988 
  2989 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 )"
  2990 definition "span S = (subspace hull S)"
  2991 definition "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
  2992 abbreviation "independent s == ~(dependent s)"
  2993 
  2994 (* Closure properties of subspaces.                                          *)
  2995 
  2996 lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
  2997 
  2998 lemma subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
  2999 
  3000 lemma subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
  3001   by (metis subspace_def)
  3002 
  3003 lemma subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *s x \<in> S"
  3004   by (metis subspace_def)
  3005 
  3006 lemma subspace_neg: "subspace S \<Longrightarrow> (x::'a::ring_1^'n) \<in> S \<Longrightarrow> - x \<in> S"
  3007   by (metis vector_sneg_minus1 subspace_mul)
  3008 
  3009 lemma subspace_sub: "subspace S \<Longrightarrow> (x::'a::ring_1^'n) \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
  3010   by (metis diff_def subspace_add subspace_neg)
  3011 
  3012 lemma subspace_setsum:
  3013   assumes sA: "subspace A" and fB: "finite B"
  3014   and f: "\<forall>x\<in> B. f x \<in> A"
  3015   shows "setsum f B \<in> A"
  3016   using  fB f sA
  3017   apply(induct rule: finite_induct[OF fB])
  3018   by (simp add: subspace_def sA, auto simp add: sA subspace_add)
  3019 
  3020 lemma subspace_linear_image:
  3021   assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and sS: "subspace S"
  3022   shows "subspace(f ` S)"
  3023   using lf sS linear_0[OF lf]
  3024   unfolding linear_def subspace_def
  3025   apply (auto simp add: image_iff)
  3026   apply (rule_tac x="x + y" in bexI, auto)
  3027   apply (rule_tac x="c*s x" in bexI, auto)
  3028   done
  3029 
  3030 lemma subspace_linear_preimage: "linear (f::'a::semiring_1^'n \<Rightarrow> _) ==> subspace S ==> subspace {x. f x \<in> S}"
  3031   by (auto simp add: subspace_def linear_def linear_0[of f])
  3032 
  3033 lemma subspace_trivial: "subspace {0::'a::semiring_1 ^_}"
  3034   by (simp add: subspace_def)
  3035 
  3036 lemma subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
  3037   by (simp add: subspace_def)
  3038 
  3039 
  3040 lemma span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
  3041   by (metis span_def hull_mono)
  3042 
  3043 lemma subspace_span: "subspace(span S)"
  3044   unfolding span_def
  3045   apply (rule hull_in[unfolded mem_def])
  3046   apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
  3047   apply auto
  3048   apply (erule_tac x="X" in ballE)
  3049   apply (simp add: mem_def)
  3050   apply blast
  3051   apply (erule_tac x="X" in ballE)
  3052   apply (erule_tac x="X" in ballE)
  3053   apply (erule_tac x="X" in ballE)
  3054   apply (clarsimp simp add: mem_def)
  3055   apply simp
  3056   apply simp
  3057   apply simp
  3058   apply (erule_tac x="X" in ballE)
  3059   apply (erule_tac x="X" in ballE)
  3060   apply (simp add: mem_def)
  3061   apply simp
  3062   apply simp
  3063   done
  3064 
  3065 lemma span_clauses:
  3066   "a \<in> S ==> a \<in> span S"
  3067   "0 \<in> span S"
  3068   "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
  3069   "x \<in> span S \<Longrightarrow> c *s x \<in> span S"
  3070   by (metis span_def hull_subset subset_eq subspace_span subspace_def)+
  3071 
  3072 lemma span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
  3073   and P: "subspace P" and x: "x \<in> span S" shows "P x"
  3074 proof-
  3075   from SP have SP': "S \<subseteq> P" by (simp add: mem_def subset_eq)
  3076   from P have P': "P \<in> subspace" by (simp add: mem_def)
  3077   from x hull_minimal[OF SP' P', unfolded span_def[symmetric]]
  3078   show "P x" by (metis mem_def subset_eq)
  3079 qed
  3080 
  3081 lemma span_empty: "span {} = {(0::'a::semiring_0 ^ 'n)}"
  3082   apply (simp add: span_def)
  3083   apply (rule hull_unique)
  3084   apply (auto simp add: mem_def subspace_def)
  3085   unfolding mem_def[of "0::'a^'n", symmetric]
  3086   apply simp
  3087   done
  3088 
  3089 lemma independent_empty: "independent {}"
  3090   by (simp add: dependent_def)
  3091 
  3092 lemma independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
  3093   apply (clarsimp simp add: dependent_def span_mono)
  3094   apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
  3095   apply force
  3096   apply (rule span_mono)
  3097   apply auto
  3098   done
  3099 
  3100 lemma span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
  3101   by (metis order_antisym span_def hull_minimal mem_def)
  3102 
  3103 lemma span_induct': assumes SP: "\<forall>x \<in> S. P x"
  3104   and P: "subspace P" shows "\<forall>x \<in> span S. P x"
  3105   using span_induct SP P by blast
  3106 
  3107 inductive span_induct_alt_help for S:: "'a::semiring_1^'n \<Rightarrow> bool"
  3108   where
  3109   span_induct_alt_help_0: "span_induct_alt_help S 0"
  3110   | 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)"
  3111 
  3112 lemma span_induct_alt':
  3113   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"
  3114 proof-
  3115   {fix x:: "'a^'n" assume x: "span_induct_alt_help S x"
  3116     have "h x"
  3117       apply (rule span_induct_alt_help.induct[OF x])
  3118       apply (rule h0)
  3119       apply (rule hS, assumption, assumption)
  3120       done}
  3121   note th0 = this
  3122   {fix x assume x: "x \<in> span S"
  3123 
  3124     have "span_induct_alt_help S x"
  3125       proof(rule span_induct[where x=x and S=S])
  3126 	show "x \<in> span S" using x .
  3127       next
  3128 	fix x assume xS : "x \<in> S"
  3129 	  from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
  3130 	  show "span_induct_alt_help S x" by simp
  3131 	next
  3132 	have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0)
  3133 	moreover
  3134 	{fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y"
  3135 	  from h
  3136 	  have "span_induct_alt_help S (x + y)"
  3137 	    apply (induct rule: span_induct_alt_help.induct)
  3138 	    apply simp
  3139 	    unfolding add_assoc
  3140 	    apply (rule span_induct_alt_help_S)
  3141 	    apply assumption
  3142 	    apply simp
  3143 	    done}
  3144 	moreover
  3145 	{fix c x assume xt: "span_induct_alt_help S x"
  3146 	  then have "span_induct_alt_help S (c*s x)"
  3147 	    apply (induct rule: span_induct_alt_help.induct)
  3148 	    apply (simp add: span_induct_alt_help_0)
  3149 	    apply (simp add: vector_smult_assoc vector_add_ldistrib)
  3150 	    apply (rule span_induct_alt_help_S)
  3151 	    apply assumption
  3152 	    apply simp
  3153 	    done
  3154 	}
  3155 	ultimately show "subspace (span_induct_alt_help S)"
  3156 	  unfolding subspace_def mem_def Ball_def by blast
  3157       qed}
  3158   with th0 show ?thesis by blast
  3159 qed
  3160 
  3161 lemma span_induct_alt:
  3162   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"
  3163   shows "h x"
  3164 using span_induct_alt'[of h S] h0 hS x by blast
  3165 
  3166 (* Individual closure properties. *)
  3167 
  3168 lemma span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses)
  3169 
  3170 lemma span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
  3171 
  3172 lemma span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
  3173   by (metis subspace_add subspace_span)
  3174 
  3175 lemma span_mul: "x \<in> span S ==> (c *s x) \<in> span S"
  3176   by (metis subspace_span subspace_mul)
  3177 
  3178 lemma span_neg: "x \<in> span S ==> - (x::'a::ring_1^'n) \<in> span S"
  3179   by (metis subspace_neg subspace_span)
  3180 
  3181 lemma span_sub: "(x::'a::ring_1^'n) \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
  3182   by (metis subspace_span subspace_sub)
  3183 
  3184 lemma span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
  3185   apply (rule subspace_setsum)
  3186   by (metis subspace_span subspace_setsum)+
  3187 
  3188 lemma span_add_eq: "(x::'a::ring_1^'n) \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
  3189   apply (auto simp only: span_add span_sub)
  3190   apply (subgoal_tac "(x + y) - x \<in> span S", simp)
  3191   by (simp only: span_add span_sub)
  3192 
  3193 (* Mapping under linear image. *)
  3194 
  3195 lemma span_linear_image: assumes lf: "linear (f::'a::semiring_1 ^ 'n => _)"
  3196   shows "span (f ` S) = f ` (span S)"
  3197 proof-
  3198   {fix x
  3199     assume x: "x \<in> span (f ` S)"
  3200     have "x \<in> f ` span S"
  3201       apply (rule span_induct[where x=x and S = "f ` S"])
  3202       apply (clarsimp simp add: image_iff)
  3203       apply (frule span_superset)
  3204       apply blast
  3205       apply (simp only: mem_def)
  3206       apply (rule subspace_linear_image[OF lf])
  3207       apply (rule subspace_span)
  3208       apply (rule x)
  3209       done}
  3210   moreover
  3211   {fix x assume x: "x \<in> span S"
  3212     have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_ext)
  3213       unfolding mem_def Collect_def ..
  3214     have "f x \<in> span (f ` S)"
  3215       apply (rule span_induct[where S=S])
  3216       apply (rule span_superset)
  3217       apply simp
  3218       apply (subst th0)
  3219       apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"])
  3220       apply (rule x)
  3221       done}
  3222   ultimately show ?thesis by blast
  3223 qed
  3224 
  3225 (* The key breakdown property. *)
  3226 
  3227 lemma span_breakdown:
  3228   assumes bS: "(b::'a::ring_1 ^ 'n) \<in> S" and aS: "a \<in> span S"
  3229   shows "\<exists>k. a - k*s b \<in> span (S - {b})" (is "?P a")
  3230 proof-
  3231   {fix x assume xS: "x \<in> S"
  3232     {assume ab: "x = b"
  3233       then have "?P x"
  3234 	apply simp
  3235 	apply (rule exI[where x="1"], simp)
  3236 	by (rule span_0)}
  3237     moreover
  3238     {assume ab: "x \<noteq> b"
  3239       then have "?P x"  using xS
  3240 	apply -
  3241 	apply (rule exI[where x=0])
  3242 	apply (rule span_superset)
  3243 	by simp}
  3244     ultimately have "?P x" by blast}
  3245   moreover have "subspace ?P"
  3246     unfolding subspace_def
  3247     apply auto
  3248     apply (simp add: mem_def)
  3249     apply (rule exI[where x=0])
  3250     using span_0[of "S - {b}"]
  3251     apply (simp add: mem_def)
  3252     apply (clarsimp simp add: mem_def)
  3253     apply (rule_tac x="k + ka" in exI)
  3254     apply (subgoal_tac "x + y - (k + ka) *s b = (x - k*s b) + (y - ka *s b)")
  3255     apply (simp only: )
  3256     apply (rule span_add[unfolded mem_def])
  3257     apply assumption+
  3258     apply (vector ring_simps)
  3259     apply (clarsimp simp add: mem_def)
  3260     apply (rule_tac x= "c*k" in exI)
  3261     apply (subgoal_tac "c *s x - (c * k) *s b = c*s (x - k*s b)")
  3262     apply (simp only: )
  3263     apply (rule span_mul[unfolded mem_def])
  3264     apply assumption
  3265     by (vector ring_simps)
  3266   ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
  3267 qed
  3268 
  3269 lemma span_breakdown_eq:
  3270   "(x::'a::ring_1^'n) \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *s a) \<in> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
  3271 proof-
  3272   {assume x: "x \<in> span (insert a S)"
  3273     from x span_breakdown[of "a" "insert a S" "x"]
  3274     have ?rhs apply clarsimp
  3275       apply (rule_tac x= "k" in exI)
  3276       apply (rule set_rev_mp[of _ "span (S - {a})" _])
  3277       apply assumption
  3278       apply (rule span_mono)
  3279       apply blast
  3280       done}
  3281   moreover
  3282   { fix k assume k: "x - k *s a \<in> span S"
  3283     have eq: "x = (x - k *s a) + k *s a" by vector
  3284     have "(x - k *s a) + k *s a \<in> span (insert a S)"
  3285       apply (rule span_add)
  3286       apply (rule set_rev_mp[of _ "span S" _])
  3287       apply (rule k)
  3288       apply (rule span_mono)
  3289       apply blast
  3290       apply (rule span_mul)
  3291       apply (rule span_superset)
  3292       apply blast
  3293       done
  3294     then have ?lhs using eq by metis}
  3295   ultimately show ?thesis by blast
  3296 qed
  3297 
  3298 (* Hence some "reversal" results.*)
  3299 
  3300 lemma in_span_insert:
  3301   assumes a: "(a::'a::field^'n) \<in> span (insert b S)" and na: "a \<notin> span S"
  3302   shows "b \<in> span (insert a S)"
  3303 proof-
  3304   from span_breakdown[of b "insert b S" a, OF insertI1 a]
  3305   obtain k where k: "a - k*s b \<in> span (S - {b})" by auto
  3306   {assume k0: "k = 0"
  3307     with k have "a \<in> span S"
  3308       apply (simp)
  3309       apply (rule set_rev_mp)
  3310       apply assumption
  3311       apply (rule span_mono)
  3312       apply blast
  3313       done
  3314     with na  have ?thesis by blast}
  3315   moreover
  3316   {assume k0: "k \<noteq> 0"
  3317     have eq: "b = (1/k) *s a - ((1/k) *s a - b)" by vector
  3318     from k0 have eq': "(1/k) *s (a - k*s b) = (1/k) *s a - b"
  3319       by (vector field_simps)
  3320     from k have "(1/k) *s (a - k*s b) \<in> span (S - {b})"
  3321       by (rule span_mul)
  3322     hence th: "(1/k) *s a - b \<in> span (S - {b})"
  3323       unfolding eq' .
  3324 
  3325     from k
  3326     have ?thesis
  3327       apply (subst eq)
  3328       apply (rule span_sub)
  3329       apply (rule span_mul)
  3330       apply (rule span_superset)
  3331       apply blast
  3332       apply (rule set_rev_mp)
  3333       apply (rule th)
  3334       apply (rule span_mono)
  3335       using na by blast}
  3336   ultimately show ?thesis by blast
  3337 qed
  3338 
  3339 lemma in_span_delete:
  3340   assumes a: "(a::'a::field^'n) \<in> span S"
  3341   and na: "a \<notin> span (S-{b})"
  3342   shows "b \<in> span (insert a (S - {b}))"
  3343   apply (rule in_span_insert)
  3344   apply (rule set_rev_mp)
  3345   apply (rule a)
  3346   apply (rule span_mono)
  3347   apply blast
  3348   apply (rule na)
  3349   done
  3350 
  3351 (* Transitivity property. *)
  3352 
  3353 lemma span_trans:
  3354   assumes x: "(x::'a::ring_1^'n) \<in> span S" and y: "y \<in> span (insert x S)"
  3355   shows "y \<in> span S"
  3356 proof-
  3357   from span_breakdown[of x "insert x S" y, OF insertI1 y]
  3358   obtain k where k: "y -k*s x \<in> span (S - {x})" by auto
  3359   have eq: "y = (y - k *s x) + k *s x" by vector
  3360   show ?thesis
  3361     apply (subst eq)
  3362     apply (rule span_add)
  3363     apply (rule set_rev_mp)
  3364     apply (rule k)
  3365     apply (rule span_mono)
  3366     apply blast
  3367     apply (rule span_mul)
  3368     by (rule x)
  3369 qed
  3370 
  3371 (* ------------------------------------------------------------------------- *)
  3372 (* An explicit expansion is sometimes needed.                                *)
  3373 (* ------------------------------------------------------------------------- *)
  3374 
  3375 lemma span_explicit:
  3376   "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}"
  3377   (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
  3378 proof-
  3379   {fix x assume x: "x \<in> ?E"
  3380     then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *s v) S = x"
  3381       by blast
  3382     have "x \<in> span P"
  3383       unfolding u[symmetric]
  3384       apply (rule span_setsum[OF fS])
  3385       using span_mono[OF SP]
  3386       by (auto intro: span_superset span_mul)}
  3387   moreover
  3388   have "\<forall>x \<in> span P. x \<in> ?E"
  3389     unfolding mem_def Collect_def
  3390   proof(rule span_induct_alt')
  3391     show "?h 0"
  3392       apply (rule exI[where x="{}"]) by simp
  3393   next
  3394     fix c x y
  3395     assume x: "x \<in> P" and hy: "?h y"
  3396     from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
  3397       and u: "setsum (\<lambda>v. u v *s v) S = y" by blast
  3398     let ?S = "insert x S"
  3399     let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c)
  3400                   else u y"
  3401     from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
  3402     {assume xS: "x \<in> S"
  3403       have S1: "S = (S - {x}) \<union> {x}"
  3404 	and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
  3405       have "setsum (\<lambda>v. ?u v *s v) ?S =(\<Sum>v\<in>S - {x}. u v *s v) + (u x + c) *s x"
  3406 	using xS
  3407 	by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]
  3408 	  setsum_clauses(2)[OF fS] cong del: if_weak_cong)
  3409       also have "\<dots> = (\<Sum>v\<in>S. u v *s v) + c *s x"
  3410 	apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
  3411 	by (vector ring_simps)
  3412       also have "\<dots> = c*s x + y"
  3413 	by (simp add: add_commute u)
  3414       finally have "setsum (\<lambda>v. ?u v *s v) ?S = c*s x + y" .
  3415     then have "?Q ?S ?u (c*s x + y)" using th0 by blast}
  3416   moreover
  3417   {assume xS: "x \<notin> S"
  3418     have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *s v) = y"
  3419       unfolding u[symmetric]
  3420       apply (rule setsum_cong2)
  3421       using xS by auto
  3422     have "?Q ?S ?u (c*s x + y)" using fS xS th0
  3423       by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)}
  3424   ultimately have "?Q ?S ?u (c*s x + y)"
  3425     by (cases "x \<in> S", simp, simp)
  3426     then show "?h (c*s x + y)"
  3427       apply -
  3428       apply (rule exI[where x="?S"])
  3429       apply (rule exI[where x="?u"]) by metis
  3430   qed
  3431   ultimately show ?thesis by blast
  3432 qed
  3433 
  3434 lemma dependent_explicit:
  3435   "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")
  3436 proof-
  3437   {assume dP: "dependent P"
  3438     then obtain a S u where aP: "a \<in> P" and fS: "finite S"
  3439       and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *s v) S = a"
  3440       unfolding dependent_def span_explicit by blast
  3441     let ?S = "insert a S"
  3442     let ?u = "\<lambda>y. if y = a then - 1 else u y"
  3443     let ?v = a
  3444     from aP SP have aS: "a \<notin> S" by blast
  3445     from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
  3446     have s0: "setsum (\<lambda>v. ?u v *s v) ?S = 0"
  3447       using fS aS
  3448       apply (simp add: vector_smult_lneg vector_smult_lid setsum_clauses ring_simps )
  3449       apply (subst (2) ua[symmetric])
  3450       apply (rule setsum_cong2)
  3451       by auto
  3452     with th0 have ?rhs
  3453       apply -
  3454       apply (rule exI[where x= "?S"])
  3455       apply (rule exI[where x= "?u"])
  3456       by clarsimp}
  3457   moreover
  3458   {fix S u v assume fS: "finite S"
  3459       and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
  3460     and u: "setsum (\<lambda>v. u v *s v) S = 0"
  3461     let ?a = v
  3462     let ?S = "S - {v}"
  3463     let ?u = "\<lambda>i. (- u i) / u v"
  3464     have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"       using fS SP vS by auto
  3465     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"
  3466       using fS vS uv
  3467       by (simp add: setsum_diff1 vector_smult_lneg divide_inverse
  3468 	vector_smult_assoc field_simps)
  3469     also have "\<dots> = ?a"
  3470       unfolding setsum_cmul u
  3471       using uv by (simp add: vector_smult_lneg)
  3472     finally  have "setsum (\<lambda>v. ?u v *s v) ?S = ?a" .
  3473     with th0 have ?lhs
  3474       unfolding dependent_def span_explicit
  3475       apply -
  3476       apply (rule bexI[where x= "?a"])
  3477       apply simp_all
  3478       apply (rule exI[where x= "?S"])
  3479       by auto}
  3480   ultimately show ?thesis by blast
  3481 qed
  3482 
  3483 
  3484 lemma span_finite:
  3485   assumes fS: "finite S"
  3486   shows "span S = {(y::'a::semiring_1^'n). \<exists>u. setsum (\<lambda>v. u v *s v) S = y}"
  3487   (is "_ = ?rhs")
  3488 proof-
  3489   {fix y assume y: "y \<in> span S"
  3490     from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and
  3491       u: "setsum (\<lambda>v. u v *s v) S' = y" unfolding span_explicit by blast
  3492     let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
  3493     from setsum_restrict_set[OF fS, of "\<lambda>v. u v *s v" S', symmetric] SS'
  3494     have "setsum (\<lambda>v. ?u v *s v) S = setsum (\<lambda>v. u v *s v) S'"
  3495       unfolding cond_value_iff cond_application_beta
  3496       apply (simp add: cond_value_iff cong del: if_weak_cong)
  3497       apply (rule setsum_cong)
  3498       apply auto
  3499       done
  3500     hence "setsum (\<lambda>v. ?u v *s v) S = y" by (metis u)
  3501     hence "y \<in> ?rhs" by auto}
  3502   moreover
  3503   {fix y u assume u: "setsum (\<lambda>v. u v *s v) S = y"
  3504     then have "y \<in> span S" using fS unfolding span_explicit by auto}
  3505   ultimately show ?thesis by blast
  3506 qed
  3507 
  3508 
  3509 (* Standard bases are a spanning set, and obviously finite.                  *)
  3510 
  3511 lemma span_stdbasis:"span {basis i :: 'a::ring_1^'n::finite | i. i \<in> (UNIV :: 'n set)} = UNIV"
  3512 apply (rule set_ext)
  3513 apply auto
  3514 apply (subst basis_expansion[symmetric])
  3515 apply (rule span_setsum)
  3516 apply simp
  3517 apply auto
  3518 apply (rule span_mul)
  3519 apply (rule span_superset)
  3520 apply (auto simp add: Collect_def mem_def)
  3521 done
  3522 
  3523 lemma has_size_stdbasis: "{basis i ::real ^'n::finite | i. i \<in> (UNIV :: 'n set)} hassize CARD('n)" (is "?S hassize ?n")
  3524 proof-
  3525   have eq: "?S = basis ` UNIV" by blast
  3526   show ?thesis unfolding eq
  3527     apply (rule hassize_image_inj[OF basis_inj])
  3528     by (simp add: hassize_def)
  3529 qed
  3530 
  3531 lemma finite_stdbasis: "finite {basis i ::real^'n::finite |i. i\<in> (UNIV:: 'n set)}"
  3532   using has_size_stdbasis[unfolded hassize_def]
  3533   ..
  3534 
  3535 lemma card_stdbasis: "card {basis i ::real^'n::finite |i. i\<in> (UNIV :: 'n set)} = CARD('n)"
  3536   using has_size_stdbasis[unfolded hassize_def]
  3537   ..
  3538 
  3539 lemma independent_stdbasis_lemma:
  3540   assumes x: "(x::'a::semiring_1 ^ 'n) \<in> span (basis ` S)"
  3541   and iS: "i \<notin> S"
  3542   shows "(x$i) = 0"
  3543 proof-
  3544   let ?U = "UNIV :: 'n set"
  3545   let ?B = "basis ` S"
  3546   let ?P = "\<lambda>(x::'a^'n). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0"
  3547  {fix x::"'a^'n" assume xS: "x\<in> ?B"
  3548    from xS have "?P x" by auto}
  3549  moreover
  3550  have "subspace ?P"
  3551    by (auto simp add: subspace_def Collect_def mem_def)
  3552  ultimately show ?thesis
  3553    using x span_induct[of ?B ?P x] iS by blast
  3554 qed
  3555 
  3556 lemma independent_stdbasis: "independent {basis i ::real^'n::finite |i. i\<in> (UNIV :: 'n set)}"
  3557 proof-
  3558   let ?I = "UNIV :: 'n set"
  3559   let ?b = "basis :: _ \<Rightarrow> real ^'n"
  3560   let ?B = "?b ` ?I"
  3561   have eq: "{?b i|i. i \<in> ?I} = ?B"
  3562     by auto
  3563   {assume d: "dependent ?B"
  3564     then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"
  3565       unfolding dependent_def by auto
  3566     have eq1: "?B - {?b k} = ?B - ?b ` {k}"  by simp
  3567     have eq2: "?B - {?b k} = ?b ` (?I - {k})"
  3568       unfolding eq1
  3569       apply (rule inj_on_image_set_diff[symmetric])
  3570       apply (rule basis_inj) using k(1) by auto
  3571     from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
  3572     from independent_stdbasis_lemma[OF th0, of k, simplified]
  3573     have False by simp}
  3574   then show ?thesis unfolding eq dependent_def ..
  3575 qed
  3576 
  3577 (* This is useful for building a basis step-by-step.                         *)
  3578 
  3579 lemma independent_insert:
  3580   "independent(insert (a::'a::field ^'n) S) \<longleftrightarrow>
  3581       (if a \<in> S then independent S
  3582                 else independent S \<and> a \<notin> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
  3583 proof-
  3584   {assume aS: "a \<in> S"
  3585     hence ?thesis using insert_absorb[OF aS] by simp}
  3586   moreover
  3587   {assume aS: "a \<notin> S"
  3588     {assume i: ?lhs
  3589       then have ?rhs using aS
  3590 	apply simp
  3591 	apply (rule conjI)
  3592 	apply (rule independent_mono)
  3593 	apply assumption
  3594 	apply blast
  3595 	by (simp add: dependent_def)}
  3596     moreover
  3597     {assume i: ?rhs
  3598       have ?lhs using i aS
  3599 	apply simp
  3600 	apply (auto simp add: dependent_def)
  3601 	apply (case_tac "aa = a", auto)
  3602 	apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
  3603 	apply simp
  3604 	apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
  3605 	apply (subgoal_tac "insert aa (S - {aa}) = S")
  3606 	apply simp
  3607 	apply blast
  3608 	apply (rule in_span_insert)
  3609 	apply assumption
  3610 	apply blast
  3611 	apply blast
  3612 	done}
  3613     ultimately have ?thesis by blast}
  3614   ultimately show ?thesis by blast
  3615 qed
  3616 
  3617 (* The degenerate case of the Exchange Lemma.  *)
  3618 
  3619 lemma mem_delete: "x \<in> (A - {a}) \<longleftrightarrow> x \<noteq> a \<and> x \<in> A"
  3620   by blast
  3621 
  3622 lemma span_span: "span (span A) = span A"
  3623   unfolding span_def hull_hull ..
  3624 
  3625 lemma span_inc: "S \<subseteq> span S"
  3626   by (metis subset_eq span_superset)
  3627 
  3628 lemma spanning_subset_independent:
  3629   assumes BA: "B \<subseteq> A" and iA: "independent (A::('a::field ^'n) set)"
  3630   and AsB: "A \<subseteq> span B"
  3631   shows "A = B"
  3632 proof
  3633   from BA show "B \<subseteq> A" .
  3634 next
  3635   from span_mono[OF BA] span_mono[OF AsB]
  3636   have sAB: "span A = span B" unfolding span_span by blast
  3637 
  3638   {fix x assume x: "x \<in> A"
  3639     from iA have th0: "x \<notin> span (A - {x})"
  3640       unfolding dependent_def using x by blast
  3641     from x have xsA: "x \<in> span A" by (blast intro: span_superset)
  3642     have "A - {x} \<subseteq> A" by blast
  3643     hence th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
  3644     {assume xB: "x \<notin> B"
  3645       from xB BA have "B \<subseteq> A -{x}" by blast
  3646       hence "span B \<subseteq> span (A - {x})" by (metis span_mono)
  3647       with th1 th0 sAB have "x \<notin> span A" by blast
  3648       with x have False by (metis span_superset)}
  3649     then have "x \<in> B" by blast}
  3650   then show "A \<subseteq> B" by blast
  3651 qed
  3652 
  3653 (* The general case of the Exchange Lemma, the key to what follows.  *)
  3654 
  3655 lemma exchange_lemma:
  3656   assumes f:"finite (t:: ('a::field^'n) set)" and i: "independent s"
  3657   and sp:"s \<subseteq> span t"
  3658   shows "\<exists>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
  3659 using f i sp
  3660 proof(induct c\<equiv>"card(t - s)" arbitrary: s t rule: nat_less_induct)
  3661   fix n:: nat and s t :: "('a ^'n) set"
  3662   assume H: " \<forall>m<n. \<forall>(x:: ('a ^'n) set) xa.
  3663                 finite xa \<longrightarrow>
  3664                 independent x \<longrightarrow>
  3665                 x \<subseteq> span xa \<longrightarrow>
  3666                 m = card (xa - x) \<longrightarrow>
  3667                 (\<exists>t'. (t' hassize card xa) \<and>
  3668                       x \<subseteq> t' \<and> t' \<subseteq> x \<union> xa \<and> x \<subseteq> span t')"
  3669     and ft: "finite t" and s: "independent s" and sp: "s \<subseteq> span t"
  3670     and n: "n = card (t - s)"
  3671   let ?P = "\<lambda>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
  3672   let ?ths = "\<exists>t'. ?P t'"
  3673   {assume st: "s \<subseteq> t"
  3674     from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
  3675       by (auto simp add: hassize_def intro: span_superset)}
  3676   moreover
  3677   {assume st: "t \<subseteq> s"
  3678 
  3679     from spanning_subset_independent[OF st s sp]
  3680       st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
  3681       by (auto simp add: hassize_def intro: span_superset)}
  3682   moreover
  3683   {assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
  3684     from st(2) obtain b where b: "b \<in> t" "b \<notin> s" by blast
  3685       from b have "t - {b} - s \<subset> t - s" by blast
  3686       then have cardlt: "card (t - {b} - s) < n" using n ft
  3687  	by (auto intro: psubset_card_mono)
  3688       from b ft have ct0: "card t \<noteq> 0" by auto
  3689     {assume stb: "s \<subseteq> span(t -{b})"
  3690       from ft have ftb: "finite (t -{b})" by auto
  3691       from H[rule_format, OF cardlt ftb s stb]
  3692       obtain u where u: "u hassize card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" by blast
  3693       let ?w = "insert b u"
  3694       have th0: "s \<subseteq> insert b u" using u by blast
  3695       from u(3) b have "u \<subseteq> s \<union> t" by blast
  3696       then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
  3697       have bu: "b \<notin> u" using b u by blast
  3698       from u(1) have fu: "finite u" by (simp add: hassize_def)
  3699       from u(1) ft b have "u hassize (card t - 1)" by auto
  3700       then
  3701       have th2: "insert b u hassize card t"
  3702 	using  card_insert_disjoint[OF fu bu] ct0 by (auto simp add: hassize_def)
  3703       from u(4) have "s \<subseteq> span u" .
  3704       also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
  3705       finally have th3: "s \<subseteq> span (insert b u)" .      from th0 th1 th2 th3 have th: "?P ?w"  by blast
  3706       from th have ?ths by blast}
  3707     moreover
  3708     {assume stb: "\<not> s \<subseteq> span(t -{b})"
  3709       from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
  3710       have ab: "a \<noteq> b" using a b by blast
  3711       have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
  3712       have mlt: "card ((insert a (t - {b})) - s) < n"
  3713 	using cardlt ft n  a b by auto
  3714       have ft': "finite (insert a (t - {b}))" using ft by auto
  3715       {fix x assume xs: "x \<in> s"
  3716 	have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
  3717 	from b(1) have "b \<in> span t" by (simp add: span_superset)
  3718 	have bs: "b \<in> span (insert a (t - {b}))"
  3719 	  by (metis in_span_delete a sp mem_def subset_eq)
  3720 	from xs sp have "x \<in> span t" by blast
  3721 	with span_mono[OF t]
  3722 	have x: "x \<in> span (insert b (insert a (t - {b})))" ..
  3723 	from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))"  .}
  3724       then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
  3725 
  3726       from H[rule_format, OF mlt ft' s sp' refl] obtain u where
  3727 	u: "u hassize card (insert a (t -{b}))" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
  3728 	"s \<subseteq> span u" by blast
  3729       from u a b ft at ct0 have "?P u" by (auto simp add: hassize_def)
  3730       then have ?ths by blast }
  3731     ultimately have ?ths by blast
  3732   }
  3733   ultimately
  3734   show ?ths  by blast
  3735 qed
  3736 
  3737 (* This implies corresponding size bounds.                                   *)
  3738 
  3739 lemma independent_span_bound:
  3740   assumes f: "finite t" and i: "independent (s::('a::field^'n) set)" and sp:"s \<subseteq> span t"
  3741   shows "finite s \<and> card s \<le> card t"
  3742   by (metis exchange_lemma[OF f i sp] hassize_def finite_subset card_mono)
  3743 
  3744 lemma finite_Atleast_Atmost[simp]: "finite {f x |x. x\<in> {(i::'a::finite_intvl_succ) .. j}}"
  3745 proof-
  3746   have eq: "{f x |x. x\<in> {i .. j}} = f ` {i .. j}" by auto
  3747   show ?thesis unfolding eq
  3748     apply (rule finite_imageI)
  3749     apply (rule finite_intvl)
  3750     done
  3751 qed
  3752 
  3753 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
  3754 proof-
  3755   have eq: "{f x |x. x\<in> UNIV} = f ` UNIV" by auto
  3756   show ?thesis unfolding eq
  3757     apply (rule finite_imageI)
  3758     apply (rule finite)
  3759     done
  3760 qed
  3761 
  3762 
  3763 lemma independent_bound:
  3764   fixes S:: "(real^'n::finite) set"
  3765   shows "independent S \<Longrightarrow> finite S \<and> card S <= CARD('n)"
  3766   apply (subst card_stdbasis[symmetric])
  3767   apply (rule independent_span_bound)
  3768   apply (rule finite_Atleast_Atmost_nat)
  3769   apply assumption
  3770   unfolding span_stdbasis
  3771   apply (rule subset_UNIV)
  3772   done
  3773 
  3774 lemma dependent_biggerset: "(finite (S::(real ^'n::finite) set) ==> card S > CARD('n)) ==> dependent S"
  3775   by (metis independent_bound not_less)
  3776 
  3777 (* Hence we can create a maximal independent subset.                         *)
  3778 
  3779 lemma maximal_independent_subset_extend:
  3780   assumes sv: "(S::(real^'n::finite) set) \<subseteq> V" and iS: "independent S"
  3781   shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
  3782   using sv iS
  3783 proof(induct d\<equiv> "CARD('n) - card S" arbitrary: S rule: nat_less_induct)
  3784   fix n and S:: "(real^'n) set"
  3785   assume H: "\<forall>m<n. \<forall>S \<subseteq> V. independent S \<longrightarrow> m = CARD('n) - card S \<longrightarrow>
  3786               (\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B)"
  3787     and sv: "S \<subseteq> V" and i: "independent S" and n: "n = CARD('n) - card S"
  3788   let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
  3789   let ?ths = "\<exists>x. ?P x"
  3790   let ?d = "CARD('n)"
  3791   {assume "V \<subseteq> span S"
  3792     then have ?ths  using sv i by blast }
  3793   moreover
  3794   {assume VS: "\<not> V \<subseteq> span S"
  3795     from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
  3796     from a have aS: "a \<notin> S" by (auto simp add: span_superset)
  3797     have th0: "insert a S \<subseteq> V" using a sv by blast
  3798     from independent_insert[of a S]  i a
  3799     have th1: "independent (insert a S)" by auto
  3800     have mlt: "?d - card (insert a S) < n"
  3801       using aS a n independent_bound[OF th1]
  3802       by auto
  3803 
  3804     from H[rule_format, OF mlt th0 th1 refl]
  3805     obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
  3806       by blast
  3807     from B have "?P B" by auto
  3808     then have ?ths by blast}
  3809   ultimately show ?ths by blast
  3810 qed
  3811 
  3812 lemma maximal_independent_subset:
  3813   "\<exists>(B:: (real ^'n::finite) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
  3814   by (metis maximal_independent_subset_extend[of "{}:: (real ^'n) set"] empty_subsetI independent_empty)
  3815 
  3816 (* Notion of dimension.                                                      *)
  3817 
  3818 definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n))"
  3819 
  3820 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)"
  3821 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)"]
  3822 unfolding hassize_def
  3823 using maximal_independent_subset[of V] independent_bound
  3824 by auto
  3825 
  3826 (* Consequences of independence or spanning for cardinality.                 *)
  3827 
  3828 lemma independent_card_le_dim: "(B::(real ^'n::finite) set) \<subseteq> V \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B \<le> dim V"
  3829 by (metis basis_exists[of V] independent_span_bound[where ?'a=real] hassize_def subset_trans)
  3830 
  3831 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"
  3832   by (metis basis_exists[of V] independent_span_bound hassize_def subset_trans)
  3833 
  3834 lemma basis_card_eq_dim:
  3835   "B \<subseteq> (V:: (real ^'n::finite) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
  3836   by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_mono)
  3837 
  3838 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"
  3839   by (metis basis_card_eq_dim hassize_def)
  3840 
  3841 (* More lemmas about dimension.                                              *)
  3842 
  3843 lemma dim_univ: "dim (UNIV :: (real^'n::finite) set) = CARD('n)"
  3844   apply (rule dim_unique[of "{basis i |i. i\<in> (UNIV :: 'n set)}"])
  3845   by (auto simp only: span_stdbasis has_size_stdbasis independent_stdbasis)
  3846 
  3847 lemma dim_subset:
  3848   "(S:: (real ^'n::finite) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
  3849   using basis_exists[of T] basis_exists[of S]
  3850   by (metis independent_span_bound[where ?'a = real and ?'n = 'n] subset_eq hassize_def)
  3851 
  3852 lemma dim_subset_univ: "dim (S:: (real^'n::finite) set) \<le> CARD('n)"
  3853   by (metis dim_subset subset_UNIV dim_univ)
  3854 
  3855 (* Converses to those.                                                       *)
  3856 
  3857 lemma card_ge_dim_independent:
  3858   assumes BV:"(B::(real ^'n::finite) set) \<subseteq> V" and iB:"independent B" and dVB:"dim V \<le> card B"
  3859   shows "V \<subseteq> span B"
  3860 proof-
  3861   {fix a assume aV: "a \<in> V"
  3862     {assume aB: "a \<notin> span B"
  3863       then have iaB: "independent (insert a B)" using iB aV  BV by (simp add: independent_insert)
  3864       from aV BV have th0: "insert a B \<subseteq> V" by blast
  3865       from aB have "a \<notin>B" by (auto simp add: span_superset)
  3866       with independent_card_le_dim[OF th0 iaB] dVB  have False by auto}
  3867     then have "a \<in> span B"  by blast}
  3868   then show ?thesis by blast
  3869 qed
  3870 
  3871 lemma card_le_dim_spanning:
  3872   assumes BV: "(B:: (real ^'n::finite) set) \<subseteq> V" and VB: "V \<subseteq> span B"
  3873   and fB: "finite B" and dVB: "dim V \<ge> card B"
  3874   shows "independent B"
  3875 proof-
  3876   {fix a assume a: "a \<in> B" "a \<in> span (B -{a})"
  3877     from a fB have c0: "card B \<noteq> 0" by auto
  3878     from a fB have cb: "card (B -{a}) = card B - 1" by auto
  3879     from BV a have th0: "B -{a} \<subseteq> V" by blast
  3880     {fix x assume x: "x \<in> V"
  3881       from a have eq: "insert a (B -{a}) = B" by blast
  3882       from x VB have x': "x \<in> span B" by blast
  3883       from span_trans[OF a(2), unfolded eq, OF x']
  3884       have "x \<in> span (B -{a})" . }
  3885     then have th1: "V \<subseteq> span (B -{a})" by blast
  3886     have th2: "finite (B -{a})" using fB by auto
  3887     from span_card_ge_dim[OF th0 th1 th2]
  3888     have c: "dim V \<le> card (B -{a})" .
  3889     from c c0 dVB cb have False by simp}
  3890   then show ?thesis unfolding dependent_def by blast
  3891 qed
  3892 
  3893 lemma card_eq_dim: "(B:: (real ^'n::finite) set) \<subseteq> V \<Longrightarrow> B hassize dim V \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
  3894   by (metis hassize_def order_eq_iff card_le_dim_spanning
  3895     card_ge_dim_independent)
  3896 
  3897 (* ------------------------------------------------------------------------- *)
  3898 (* More general size bound lemmas.                                           *)
  3899 (* ------------------------------------------------------------------------- *)
  3900 
  3901 lemma independent_bound_general:
  3902   "independent (S:: (real^'n::finite) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
  3903   by (metis independent_card_le_dim independent_bound subset_refl)
  3904 
  3905 lemma dependent_biggerset_general: "(finite (S:: (real^'n::finite) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
  3906   using independent_bound_general[of S] by (metis linorder_not_le)
  3907 
  3908 lemma dim_span: "dim (span (S:: (real ^'n::finite) set)) = dim S"
  3909 proof-
  3910   have th0: "dim S \<le> dim (span S)"
  3911     by (auto simp add: subset_eq intro: dim_subset span_superset)
  3912   from basis_exists[of S]
  3913   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
  3914   from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
  3915   have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc)
  3916   have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span)
  3917   from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
  3918     using fB(2)  by arith
  3919 qed
  3920 
  3921 lemma subset_le_dim: "(S:: (real ^'n::finite) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
  3922   by (metis dim_span dim_subset)
  3923 
  3924 lemma span_eq_dim: "span (S:: (real ^'n::finite) set) = span T ==> dim S = dim T"
  3925   by (metis dim_span)
  3926 
  3927 lemma spans_image:
  3928   assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and VB: "V \<subseteq> span B"
  3929   shows "f ` V \<subseteq> span (f ` B)"
  3930   unfolding span_linear_image[OF lf]
  3931   by (metis VB image_mono)
  3932 
  3933 lemma dim_image_le:
  3934   fixes f :: "real^'n::finite \<Rightarrow> real^'m::finite"
  3935   assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S:: (real ^'n::finite) set)"
  3936 proof-
  3937   from basis_exists[of S] obtain B where
  3938     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
  3939   from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
  3940   have "dim (f ` S) \<le> card (f ` B)"
  3941     apply (rule span_card_ge_dim)
  3942     using lf B fB by (auto simp add: span_linear_image spans_image subset_image_iff)
  3943   also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
  3944   finally show ?thesis .
  3945 qed
  3946 
  3947 (* Relation between bases and injectivity/surjectivity of map.               *)
  3948 
  3949 lemma spanning_surjective_image:
  3950   assumes us: "UNIV \<subseteq> span (S:: ('a::semiring_1 ^'n) set)"
  3951   and lf: "linear f" and sf: "surj f"
  3952   shows "UNIV \<subseteq> span (f ` S)"
  3953 proof-
  3954   have "UNIV \<subseteq> f ` UNIV" using sf by (auto simp add: surj_def)
  3955   also have " \<dots> \<subseteq> span (f ` S)" using spans_image[OF lf us] .
  3956 finally show ?thesis .
  3957 qed
  3958 
  3959 lemma independent_injective_image:
  3960   assumes iS: "independent (S::('a::semiring_1^'n) set)" and lf: "linear f" and fi: "inj f"
  3961   shows "independent (f ` S)"
  3962 proof-
  3963   {fix a assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
  3964     have eq: "f ` S - {f a} = f ` (S - {a})" using fi
  3965       by (auto simp add: inj_on_def)
  3966     from a have "f a \<in> f ` span (S -{a})"
  3967       unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
  3968     hence "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
  3969     with a(1) iS  have False by (simp add: dependent_def) }
  3970   then show ?thesis unfolding dependent_def by blast
  3971 qed
  3972 
  3973 (* ------------------------------------------------------------------------- *)
  3974 (* Picking an orthogonal replacement for a spanning set.                     *)
  3975 (* ------------------------------------------------------------------------- *)
  3976     (* FIXME : Move to some general theory ?*)
  3977 definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
  3978 
  3979 lemma vector_sub_project_orthogonal: "(b::'a::ordered_field^'n::finite) \<bullet> (x - ((b \<bullet> x) / (b\<bullet>b)) *s b) = 0"
  3980   apply (cases "b = 0", simp)
  3981   apply (simp add: dot_rsub dot_rmult)
  3982   unfolding times_divide_eq_right[symmetric]
  3983   by (simp add: field_simps dot_eq_0)
  3984 
  3985 lemma basis_orthogonal:
  3986   fixes B :: "(real ^'n::finite) set"
  3987   assumes fB: "finite B"
  3988   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
  3989   (is " \<exists>C. ?P B C")
  3990 proof(induct rule: finite_induct[OF fB])
  3991   case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
  3992 next
  3993   case (2 a B)
  3994   note fB = `finite B` and aB = `a \<notin> B`
  3995   from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
  3996   obtain C where C: "finite C" "card C \<le> card B"
  3997     "span C = span B" "pairwise orthogonal C" by blast
  3998   let ?a = "a - setsum (\<lambda>x. (x\<bullet>a / (x\<bullet>x)) *s x) C"
  3999   let ?C = "insert ?a C"
  4000   from C(1) have fC: "finite ?C" by simp
  4001   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" by (simp add: card_insert_if)
  4002   {fix x k
  4003     have th0: "\<And>(a::'b::comm_ring) b c. a - (b - c) = c + (a - b)" by (simp add: ring_simps)
  4004     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"
  4005       apply (simp only: vector_ssub_ldistrib th0)
  4006       apply (rule span_add_eq)
  4007       apply (rule span_mul)
  4008       apply (rule span_setsum[OF C(1)])
  4009       apply clarify
  4010       apply (rule span_mul)
  4011       by (rule span_superset)}
  4012   then have SC: "span ?C = span (insert a B)"
  4013     unfolding expand_set_eq span_breakdown_eq C(3)[symmetric] by auto
  4014   thm pairwise_def
  4015   {fix x y assume xC: "x \<in> ?C" and yC: "y \<in> ?C" and xy: "x \<noteq> y"
  4016     {assume xa: "x = ?a" and ya: "y = ?a"
  4017       have "orthogonal x y" using xa ya xy by blast}
  4018     moreover
  4019     {assume xa: "x = ?a" and ya: "y \<noteq> ?a" "y \<in> C"
  4020       from ya have Cy: "C = insert y (C - {y})" by blast
  4021       have fth: "finite (C - {y})" using C by simp
  4022       have "orthogonal x y"
  4023 	using xa ya
  4024 	unfolding orthogonal_def xa dot_lsub dot_rsub diff_eq_0_iff_eq
  4025 	apply simp
  4026 	apply (subst Cy)
  4027 	using C(1) fth
  4028 	apply (simp only: setsum_clauses)
  4029 	thm dot_ladd
  4030 	apply (auto simp add: dot_ladd dot_radd dot_lmult dot_rmult dot_eq_0 dot_sym[of y a] dot_lsum[OF fth])
  4031 	apply (rule setsum_0')
  4032 	apply clarsimp
  4033 	apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
  4034 	by auto}
  4035     moreover
  4036     {assume xa: "x \<noteq> ?a" "x \<in> C" and ya: "y = ?a"
  4037       from xa have Cx: "C = insert x (C - {x})" by blast
  4038       have fth: "finite (C - {x})" using C by simp
  4039       have "orthogonal x y"
  4040 	using xa ya
  4041 	unfolding orthogonal_def ya dot_rsub dot_lsub diff_eq_0_iff_eq
  4042 	apply simp
  4043 	apply (subst Cx)
  4044 	using C(1) fth
  4045 	apply (simp only: setsum_clauses)
  4046 	apply (subst dot_sym[of x])
  4047 	apply (auto simp add: dot_radd dot_rmult dot_eq_0 dot_sym[of x a] dot_rsum[OF fth])
  4048 	apply (rule setsum_0')
  4049 	apply clarsimp
  4050 	apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
  4051 	by auto}
  4052     moreover
  4053     {assume xa: "x \<in> C" and ya: "y \<in> C"
  4054       have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast}
  4055     ultimately have "orthogonal x y" using xC yC by blast}
  4056   then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast
  4057   from fC cC SC CPO have "?P (insert a B) ?C" by blast
  4058   then show ?case by blast
  4059 qed
  4060 
  4061 lemma orthogonal_basis_exists:
  4062   fixes V :: "(real ^'n::finite) set"
  4063   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (B hassize dim V) \<and> pairwise orthogonal B"
  4064 proof-
  4065   from basis_exists[of V] obtain B where B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "B hassize dim V" by blast
  4066   from B have fB: "finite B" "card B = dim V" by (simp_all add: hassize_def)
  4067   from basis_orthogonal[OF fB(1)] obtain C where
  4068     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
  4069   from C B
  4070   have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans)
  4071   from span_mono[OF B(3)]  C have SVC: "span V \<subseteq> span C" by (simp add: span_span)
  4072   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
  4073   have iC: "independent C" by (simp add: dim_span)
  4074   from C fB have "card C \<le> dim V" by simp
  4075   moreover have "dim V \<le> card C" using span_card_ge_dim[OF CSV SVC C(1)]
  4076     by (simp add: dim_span)
  4077   ultimately have CdV: "C hassize dim V" unfolding hassize_def using C(1) by simp
  4078   from C B CSV CdV iC show ?thesis by auto
  4079 qed
  4080 
  4081 lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
  4082   by (metis set_eq_subset span_mono span_span span_inc) (* FIXME: slow *)
  4083 
  4084 (* ------------------------------------------------------------------------- *)
  4085 (* Low-dimensional subset is in a hyperplane (weak orthogonal complement).   *)
  4086 (* ------------------------------------------------------------------------- *)
  4087 
  4088 lemma span_not_univ_orthogonal:
  4089   assumes sU: "span S \<noteq> UNIV"
  4090   shows "\<exists>(a:: real ^'n::finite). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
  4091 proof-
  4092   from sU obtain a where a: "a \<notin> span S" by blast
  4093   from orthogonal_basis_exists obtain B where
  4094     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "B hassize dim S" "pairwise orthogonal B"
  4095     by blast
  4096   from B have fB: "finite B" "card B = dim S" by (simp_all add: hassize_def)
  4097   from span_mono[OF B(2)] span_mono[OF B(3)]
  4098   have sSB: "span S = span B" by (simp add: span_span)
  4099   let ?a = "a - setsum (\<lambda>b. (a\<bullet>b / (b\<bullet>b)) *s b) B"
  4100   have "setsum (\<lambda>b. (a\<bullet>b / (b\<bullet>b)) *s b) B \<in> span S"
  4101     unfolding sSB
  4102     apply (rule span_setsum[OF fB(1)])
  4103     apply clarsimp
  4104     apply (rule span_mul)
  4105     by (rule span_superset)
  4106   with a have a0:"?a  \<noteq> 0" by auto
  4107   have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
  4108   proof(rule span_induct')
  4109     show "subspace (\<lambda>x. ?a \<bullet> x = 0)"
  4110       by (auto simp add: subspace_def mem_def dot_radd dot_rmult)
  4111   next
  4112     {fix x assume x: "x \<in> B"
  4113       from x have B': "B = insert x (B - {x})" by blast
  4114       have fth: "finite (B - {x})" using fB by simp
  4115       have "?a \<bullet> x = 0"
  4116 	apply (subst B') using fB fth
  4117 	unfolding setsum_clauses(2)[OF fth]
  4118 	apply simp
  4119 	apply (clarsimp simp add: dot_lsub dot_ladd dot_lmult dot_lsum dot_eq_0)
  4120 	apply (rule setsum_0', rule ballI)
  4121 	unfolding dot_sym
  4122 	by (auto simp add: x field_simps dot_eq_0 intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
  4123     then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
  4124   qed
  4125   with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
  4126 qed
  4127 
  4128 lemma span_not_univ_subset_hyperplane:
  4129   assumes SU: "span S \<noteq> (UNIV ::(real^'n::finite) set)"
  4130   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
  4131   using span_not_univ_orthogonal[OF SU] by auto
  4132 
  4133 lemma lowdim_subset_hyperplane:
  4134   assumes d: "dim S < CARD('n::finite)"
  4135   shows "\<exists>(a::real ^'n::finite). a  \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
  4136 proof-
  4137   {assume "span S = UNIV"
  4138     hence "dim (span S) = dim (UNIV :: (real ^'n) set)" by simp
  4139     hence "dim S = CARD('n)" by (simp add: dim_span dim_univ)
  4140     with d have False by arith}
  4141   hence th: "span S \<noteq> UNIV" by blast
  4142   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
  4143 qed
  4144 
  4145 (* We can extend a linear basis-basis injection to the whole set.            *)
  4146 
  4147 lemma linear_indep_image_lemma:
  4148   assumes lf: "linear f" and fB: "finite B"
  4149   and ifB: "independent (f ` B)"
  4150   and fi: "inj_on f B" and xsB: "x \<in> span B"
  4151   and fx: "f (x::'a::field^'n) = 0"
  4152   shows "x = 0"
  4153   using fB ifB fi xsB fx
  4154 proof(induct arbitrary: x rule: finite_induct[OF fB])
  4155   case 1 thus ?case by (auto simp add:  span_empty)
  4156 next
  4157   case (2 a b x)
  4158   have fb: "finite b" using "2.prems" by simp
  4159   have th0: "f ` b \<subseteq> f ` (insert a b)"
  4160     apply (rule image_mono) by blast
  4161   from independent_mono[ OF "2.prems"(2) th0]
  4162   have ifb: "independent (f ` b)"  .
  4163   have fib: "inj_on f b"
  4164     apply (rule subset_inj_on [OF "2.prems"(3)])
  4165     by blast
  4166   from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
  4167   obtain k where k: "x - k*s a \<in> span (b -{a})" by blast
  4168   have "f (x - k*s a) \<in> span (f ` b)"
  4169     unfolding span_linear_image[OF lf]
  4170     apply (rule imageI)
  4171     using k span_mono[of "b-{a}" b] by blast
  4172   hence "f x - k*s f a \<in> span (f ` b)"
  4173     by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
  4174   hence th: "-k *s f a \<in> span (f ` b)"
  4175     using "2.prems"(5) by (simp add: vector_smult_lneg)
  4176   {assume k0: "k = 0"
  4177     from k0 k have "x \<in> span (b -{a})" by simp
  4178     then have "x \<in> span b" using span_mono[of "b-{a}" b]
  4179       by blast}
  4180   moreover
  4181   {assume k0: "k \<noteq> 0"
  4182     from span_mul[OF th, of "- 1/ k"] k0
  4183     have th1: "f a \<in> span (f ` b)"
  4184       by (auto simp add: vector_smult_assoc)
  4185     from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
  4186     have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
  4187     from "2.prems"(2)[unfolded dependent_def bex_simps(10), rule_format, of "f a"]
  4188     have "f a \<notin> span (f ` b)" using tha
  4189       using "2.hyps"(2)
  4190       "2.prems"(3) by auto
  4191     with th1 have False by blast
  4192     then have "x \<in> span b" by blast}
  4193   ultimately have xsb: "x \<in> span b" by blast
  4194   from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)]
  4195   show "x = 0" .
  4196 qed
  4197 
  4198 (* We can extend a linear mapping from basis.                                *)
  4199 
  4200 lemma linear_independent_extend_lemma:
  4201   assumes fi: "finite B" and ib: "independent B"
  4202   shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g ((x::'a::field^'n) + y) = g x + g y)
  4203            \<and> (\<forall>x\<in> span B. \<forall>c. g (c*s x) = c *s g x)
  4204            \<and> (\<forall>x\<in> B. g x = f x)"
  4205 using ib fi
  4206 proof(induct rule: finite_induct[OF fi])
  4207   case 1 thus ?case by (auto simp add: span_empty)
  4208 next
  4209   case (2 a b)
  4210   from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
  4211     by (simp_all add: independent_insert)
  4212   from "2.hyps"(3)[OF ibf] obtain g where
  4213     g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
  4214     "\<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
  4215   let ?h = "\<lambda>z. SOME k. (z - k *s a) \<in> span b"
  4216   {fix z assume z: "z \<in> span (insert a b)"
  4217     have th0: "z - ?h z *s a \<in> span b"
  4218       apply (rule someI_ex)
  4219       unfolding span_breakdown_eq[symmetric]
  4220       using z .
  4221     {fix k assume k: "z - k *s a \<in> span b"
  4222       have eq: "z - ?h z *s a - (z - k*s a) = (k - ?h z) *s a"
  4223 	by (simp add: ring_simps vector_sadd_rdistrib[symmetric])
  4224       from span_sub[OF th0 k]
  4225       have khz: "(k - ?h z) *s a \<in> span b" by (simp add: eq)
  4226       {assume "k \<noteq> ?h z" hence k0: "k - ?h z \<noteq> 0" by simp
  4227 	from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
  4228 	have "a \<in> span b" by (simp add: vector_smult_assoc)
  4229 	with "2.prems"(1) "2.hyps"(2) have False
  4230 	  by (auto simp add: dependent_def)}
  4231       then have "k = ?h z" by blast}
  4232     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}
  4233   note h = this
  4234   let ?g = "\<lambda>z. ?h z *s f a + g (z - ?h z *s a)"
  4235   {fix x y assume x: "x \<in> span (insert a b)" and y: "y \<in> span (insert a b)"
  4236     have tha: "\<And>(x::'a^'n) y a k l. (x + y) - (k + l) *s a = (x - k *s a) + (y - l *s a)"
  4237       by (vector ring_simps)
  4238     have addh: "?h (x + y) = ?h x + ?h y"
  4239       apply (rule conjunct2[OF h, rule_format, symmetric])
  4240       apply (rule span_add[OF x y])
  4241       unfolding tha
  4242       by (metis span_add x y conjunct1[OF h, rule_format])
  4243     have "?g (x + y) = ?g x + ?g y"
  4244       unfolding addh tha
  4245       g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
  4246       by (simp add: vector_sadd_rdistrib)}
  4247   moreover
  4248   {fix x:: "'a^'n" and c:: 'a  assume x: "x \<in> span (insert a b)"
  4249     have tha: "\<And>(x::'a^'n) c k a. c *s x - (c * k) *s a = c *s (x - k *s a)"
  4250       by (vector ring_simps)
  4251     have hc: "?h (c *s x) = c * ?h x"
  4252       apply (rule conjunct2[OF h, rule_format, symmetric])
  4253       apply (metis span_mul x)
  4254       by (metis tha span_mul x conjunct1[OF h])
  4255     have "?g (c *s x) = c*s ?g x"
  4256       unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
  4257       by (vector ring_simps)}
  4258   moreover
  4259   {fix x assume x: "x \<in> (insert a b)"
  4260     {assume xa: "x = a"
  4261       have ha1: "1 = ?h a"
  4262 	apply (rule conjunct2[OF h, rule_format])
  4263 	apply (metis span_superset insertI1)
  4264 	using conjunct1[OF h, OF span_superset, OF insertI1]
  4265 	by (auto simp add: span_0)
  4266 
  4267       from xa ha1[symmetric] have "?g x = f x"
  4268 	apply simp
  4269 	using g(2)[rule_format, OF span_0, of 0]
  4270 	by simp}
  4271     moreover
  4272     {assume xb: "x \<in> b"
  4273       have h0: "0 = ?h x"
  4274 	apply (rule conjunct2[OF h, rule_format])
  4275 	apply (metis  span_superset insertI1 xb x)
  4276 	apply simp
  4277 	apply (metis span_superset xb)
  4278 	done
  4279       have "?g x = f x"
  4280 	by (simp add: h0[symmetric] g(3)[rule_format, OF xb])}
  4281     ultimately have "?g x = f x" using x by blast }
  4282   ultimately show ?case apply - apply (rule exI[where x="?g"]) by blast
  4283 qed
  4284 
  4285 lemma linear_independent_extend:
  4286   assumes iB: "independent (B:: (real ^'n::finite) set)"
  4287   shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
  4288 proof-
  4289   from maximal_independent_subset_extend[of B UNIV] iB
  4290   obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
  4291 
  4292   from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
  4293   obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y)
  4294            \<and> (\<forall>x\<in> span C. \<forall>c. g (c*s x) = c *s g x)
  4295            \<and> (\<forall>x\<in> C. g x = f x)" by blast
  4296   from g show ?thesis unfolding linear_def using C
  4297     apply clarsimp by blast
  4298 qed
  4299 
  4300 (* Can construct an isomorphism between spaces of same dimension.            *)
  4301 
  4302 lemma card_le_inj: assumes fA: "finite A" and fB: "finite B"
  4303   and c: "card A \<le> card B" shows "(\<exists>f. f ` A \<subseteq> B \<and> inj_on f A)"
  4304 using fB c
  4305 proof(induct arbitrary: B rule: finite_induct[OF fA])
  4306   case 1 thus ?case by simp
  4307 next
  4308   case (2 x s t)
  4309   thus ?case
  4310   proof(induct rule: finite_induct[OF "2.prems"(1)])
  4311     case 1    then show ?case by simp
  4312   next
  4313     case (2 y t)
  4314     from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
  4315     from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where
  4316       f: "f ` s \<subseteq> t \<and> inj_on f s" by blast
  4317     from f "2.prems"(2) "2.hyps"(2) show ?case
  4318       apply -
  4319       apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
  4320       by (auto simp add: inj_on_def)
  4321   qed
  4322 qed
  4323 
  4324 lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and
  4325   c: "card A = card B"
  4326   shows "A = B"
  4327 proof-
  4328   from fB AB have fA: "finite A" by (auto intro: finite_subset)
  4329   from fA fB have fBA: "finite (B - A)" by auto
  4330   have e: "A \<inter> (B - A) = {}" by blast
  4331   have eq: "A \<union> (B - A) = B" using AB by blast
  4332   from card_Un_disjoint[OF fA fBA e, unfolded eq c]
  4333   have "card (B - A) = 0" by arith
  4334   hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
  4335   with AB show "A = B" by blast
  4336 qed
  4337 
  4338 lemma subspace_isomorphism:
  4339   assumes s: "subspace (S:: (real ^'n::finite) set)"
  4340   and t: "subspace (T :: (real ^ 'm::finite) set)"
  4341   and d: "dim S = dim T"
  4342   shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
  4343 proof-
  4344   from basis_exists[of S] obtain B where
  4345     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
  4346   from basis_exists[of T] obtain C where
  4347     C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "C hassize dim T" by blast
  4348   from B(4) C(4) card_le_inj[of B C] d obtain f where
  4349     f: "f ` B \<subseteq> C" "inj_on f B" unfolding hassize_def by auto
  4350   from linear_independent_extend[OF B(2)] obtain g where
  4351     g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
  4352   from B(4) have fB: "finite B" by (simp add: hassize_def)
  4353   from C(4) have fC: "finite C" by (simp add: hassize_def)
  4354   from inj_on_iff_eq_card[OF fB, of f] f(2)
  4355   have "card (f ` B) = card B" by simp
  4356   with B(4) C(4) have ceq: "card (f ` B) = card C" using d
  4357     by (simp add: hassize_def)
  4358   have "g ` B = f ` B" using g(2)
  4359     by (auto simp add: image_iff)
  4360   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
  4361   finally have gBC: "g ` B = C" .
  4362   have gi: "inj_on g B" using f(2) g(2)
  4363     by (auto simp add: inj_on_def)
  4364   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
  4365   {fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
  4366     from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
  4367     from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
  4368     have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
  4369     have "x=y" using g0[OF th1 th0] by simp }
  4370   then have giS: "inj_on g S"
  4371     unfolding inj_on_def by blast
  4372   from span_subspace[OF B(1,3) s]
  4373   have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
  4374   also have "\<dots> = span C" unfolding gBC ..
  4375   also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
  4376   finally have gS: "g ` S = T" .
  4377   from g(1) gS giS show ?thesis by blast
  4378 qed
  4379 
  4380 (* linear functions are equal on a subspace if they are on a spanning set.   *)
  4381 
  4382 lemma subspace_kernel:
  4383   assumes lf: "linear (f::'a::semiring_1 ^'n \<Rightarrow> _)"
  4384   shows "subspace {x. f x = 0}"
  4385 apply (simp add: subspace_def)
  4386 by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
  4387 
  4388 lemma linear_eq_0_span:
  4389   assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
  4390   shows "\<forall>x \<in> span B. f x = (0::'a::semiring_1 ^'n)"
  4391 proof
  4392   fix x assume x: "x \<in> span B"
  4393   let ?P = "\<lambda>x. f x = 0"
  4394   from subspace_kernel[OF lf] have "subspace ?P" unfolding Collect_def .
  4395   with x f0 span_induct[of B "?P" x] show "f x = 0" by blast
  4396 qed
  4397 
  4398 lemma linear_eq_0:
  4399   assumes lf: "linear f" and SB: "S \<subseteq> span B" and f0: "\<forall>x\<in>B. f x = 0"
  4400   shows "\<forall>x \<in> S. f x = (0::'a::semiring_1^'n)"
  4401   by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
  4402 
  4403 lemma linear_eq:
  4404   assumes lf: "linear (f::'a::ring_1^'n \<Rightarrow> _)" and lg: "linear g" and S: "S \<subseteq> span B"
  4405   and fg: "\<forall> x\<in> B. f x = g x"
  4406   shows "\<forall>x\<in> S. f x = g x"
  4407 proof-
  4408   let ?h = "\<lambda>x. f x - g x"
  4409   from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
  4410   from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
  4411   show ?thesis by simp
  4412 qed
  4413 
  4414 lemma linear_eq_stdbasis:
  4415   assumes lf: "linear (f::'a::ring_1^'m::finite \<Rightarrow> 'a^'n::finite)" and lg: "linear g"
  4416   and fg: "\<forall>i. f (basis i) = g(basis i)"
  4417   shows "f = g"
  4418 proof-
  4419   let ?U = "UNIV :: 'm set"
  4420   let ?I = "{basis i:: 'a^'m|i. i \<in> ?U}"
  4421   {fix x assume x: "x \<in> (UNIV :: ('a^'m) set)"
  4422     from equalityD2[OF span_stdbasis]
  4423     have IU: " (UNIV :: ('a^'m) set) \<subseteq> span ?I" by blast
  4424     from linear_eq[OF lf lg IU] fg x
  4425     have "f x = g x" unfolding Collect_def  Ball_def mem_def by metis}
  4426   then show ?thesis by (auto intro: ext)
  4427 qed
  4428 
  4429 (* Similar results for bilinear functions.                                   *)
  4430 
  4431 lemma bilinear_eq:
  4432   assumes bf: "bilinear (f:: 'a::ring^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)"
  4433   and bg: "bilinear g"
  4434   and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
  4435   and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
  4436   shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
  4437 proof-
  4438   let ?P = "\<lambda>x. \<forall>y\<in> span C. f x y = g x y"
  4439   from bf bg have sp: "subspace ?P"
  4440     unfolding bilinear_def linear_def subspace_def bf bg
  4441     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])
  4442 
  4443   have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
  4444     apply -
  4445     apply (rule ballI)
  4446     apply (rule span_induct[of B ?P])
  4447     defer
  4448     apply (rule sp)
  4449     apply assumption
  4450     apply (clarsimp simp add: Ball_def)
  4451     apply (rule_tac P="\<lambda>y. f xa y = g xa y" and S=C in span_induct)
  4452     using fg
  4453     apply (auto simp add: subspace_def)
  4454     using bf bg unfolding bilinear_def linear_def
  4455     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])
  4456   then show ?thesis using SB TC by (auto intro: ext)
  4457 qed
  4458 
  4459 lemma bilinear_eq_stdbasis:
  4460   assumes bf: "bilinear (f:: 'a::ring_1^'m::finite \<Rightarrow> 'a^'n::finite \<Rightarrow> 'a^'p)"
  4461   and bg: "bilinear g"
  4462   and fg: "\<forall>i j. f (basis i) (basis j) = g (basis i) (basis j)"
  4463   shows "f = g"
  4464 proof-
  4465   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
  4466   from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext)
  4467 qed
  4468 
  4469 (* Detailed theorems about left and right invertibility in general case.     *)
  4470 
  4471 lemma left_invertible_transp:
  4472   "(\<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)"
  4473   by (metis matrix_transp_mul transp_mat transp_transp)
  4474 
  4475 lemma right_invertible_transp:
  4476   "(\<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)"
  4477   by (metis matrix_transp_mul transp_mat transp_transp)
  4478 
  4479 lemma linear_injective_left_inverse:
  4480   assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" and fi: "inj f"
  4481   shows "\<exists>g. linear g \<and> g o f = id"
  4482 proof-
  4483   from linear_independent_extend[OF independent_injective_image, OF independent_stdbasis, OF lf fi]
  4484   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
  4485   from h(2)
  4486   have th: "\<forall>i. (h \<circ> f) (basis i) = id (basis i)"
  4487     using inv_o_cancel[OF fi, unfolded stupid_ext[symmetric] id_def o_def]
  4488     by auto
  4489 
  4490   from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
  4491   have "h o f = id" .
  4492   then show ?thesis using h(1) by blast
  4493 qed
  4494 
  4495 lemma linear_surjective_right_inverse:
  4496   assumes lf: "linear (f:: real ^'m::finite \<Rightarrow> real ^'n::finite)" and sf: "surj f"
  4497   shows "\<exists>g. linear g \<and> f o g = id"
  4498 proof-
  4499   from linear_independent_extend[OF independent_stdbasis]
  4500   obtain h:: "real ^'n \<Rightarrow> real ^'m" where
  4501     h: "linear h" "\<forall> x\<in> {basis i| i. i\<in> (UNIV :: 'n set)}. h x = inv f x" by blast
  4502   from h(2)
  4503   have th: "\<forall>i. (f o h) (basis i) = id (basis i)"
  4504     using sf
  4505     apply (auto simp add: surj_iff o_def stupid_ext[symmetric])
  4506     apply (erule_tac x="basis i" in allE)
  4507     by auto
  4508 
  4509   from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
  4510   have "f o h = id" .
  4511   then show ?thesis using h(1) by blast
  4512 qed
  4513 
  4514 lemma matrix_left_invertible_injective:
  4515 "(\<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)"
  4516 proof-
  4517   {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
  4518     from xy have "B*v (A *v x) = B *v (A*v y)" by simp
  4519     hence "x = y"
  4520       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .}
  4521   moreover
  4522   {assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
  4523     hence i: "inj (op *v A)" unfolding inj_on_def by auto
  4524     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
  4525     obtain g where g: "linear g" "g o op *v A = id" by blast
  4526     have "matrix g ** A = mat 1"
  4527       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
  4528       using g(2) by (simp add: o_def id_def stupid_ext)
  4529     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast}
  4530   ultimately show ?thesis by blast
  4531 qed
  4532 
  4533 lemma matrix_left_invertible_ker:
  4534   "(\<exists>B. (B::real ^'m::finite^'n::finite) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
  4535   unfolding matrix_left_invertible_injective
  4536   using linear_injective_0[OF matrix_vector_mul_linear, of A]
  4537   by (simp add: inj_on_def)
  4538 
  4539 lemma matrix_right_invertible_surjective:
  4540 "(\<exists>B. (A::real^'n::finite^'m::finite) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
  4541 proof-
  4542   {fix B :: "real ^'m^'n"  assume AB: "A ** B = mat 1"
  4543     {fix x :: "real ^ 'm"
  4544       have "A *v (B *v x) = x"
  4545 	by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)}
  4546     hence "surj (op *v A)" unfolding surj_def by metis }
  4547   moreover
  4548   {assume sf: "surj (op *v A)"
  4549     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
  4550     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
  4551       by blast
  4552 
  4553     have "A ** (matrix g) = mat 1"
  4554       unfolding matrix_eq  matrix_vector_mul_lid
  4555 	matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
  4556       using g(2) unfolding o_def stupid_ext[symmetric] id_def
  4557       .
  4558     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
  4559   }
  4560   ultimately show ?thesis unfolding surj_def by blast
  4561 qed
  4562 
  4563 lemma matrix_left_invertible_independent_columns:
  4564   fixes A :: "real^'n::finite^'m::finite"
  4565   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))"
  4566    (is "?lhs \<longleftrightarrow> ?rhs")
  4567 proof-
  4568   let ?U = "UNIV :: 'n set"
  4569   {assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
  4570     {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0"
  4571       and i: "i \<in> ?U"
  4572       let ?x = "\<chi> i. c i"
  4573       have th0:"A *v ?x = 0"
  4574 	using c
  4575 	unfolding matrix_mult_vsum Cart_eq
  4576 	by auto
  4577       from k[rule_format, OF th0] i
  4578       have "c i = 0" by (vector Cart_eq)}
  4579     hence ?rhs by blast}
  4580   moreover
  4581   {assume H: ?rhs
  4582     {fix x assume x: "A *v x = 0"
  4583       let ?c = "\<lambda>i. ((x$i ):: real)"
  4584       from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
  4585       have "x = 0" by vector}}
  4586   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
  4587 qed
  4588 
  4589 lemma matrix_right_invertible_independent_rows:
  4590   fixes A :: "real^'n::finite^'m::finite"
  4591   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))"
  4592   unfolding left_invertible_transp[symmetric]
  4593     matrix_left_invertible_independent_columns
  4594   by (simp add: column_transp)
  4595 
  4596 lemma matrix_right_invertible_span_columns:
  4597   "(\<exists>(B::real ^'n::finite^'m::finite). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")
  4598 proof-
  4599   let ?U = "UNIV :: 'm set"
  4600   have fU: "finite ?U" by simp
  4601   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
  4602     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
  4603     apply (subst eq_commute) ..
  4604   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
  4605   {assume h: ?lhs
  4606     {fix x:: "real ^'n"
  4607 	from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m"
  4608 	  where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
  4609 	have "x \<in> span (columns A)"
  4610 	  unfolding y[symmetric]
  4611 	  apply (rule span_setsum[OF fU])
  4612 	  apply clarify
  4613 	  apply (rule span_mul)
  4614 	  apply (rule span_superset)
  4615 	  unfolding columns_def
  4616 	  by blast}
  4617     then have ?rhs unfolding rhseq by blast}
  4618   moreover
  4619   {assume h:?rhs
  4620     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
  4621     {fix y have "?P y"
  4622       proof(rule span_induct_alt[of ?P "columns A"])
  4623 	show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
  4624 	  apply (rule exI[where x=0])
  4625 	  by (simp add: zero_index vector_smult_lzero)
  4626       next
  4627 	fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
  4628 	from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
  4629 	  unfolding columns_def by blast
  4630 	from y2 obtain x:: "real ^'m" where
  4631 	  x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
  4632 	let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
  4633 	show "?P (c*s y1 + y2)"
  4634 	  proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] cond_value_iff right_distrib cond_application_beta cong del: if_weak_cong)
  4635 	    fix j
  4636 	    have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
  4637            else (x$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)
  4638 	      by (simp add: ring_simps)
  4639 	    have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
  4640            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)$j))) ?U"
  4641 	      apply (rule setsum_cong[OF refl])
  4642 	      using th by blast
  4643 	    also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
  4644 	      by (simp add: setsum_addf)
  4645 	    also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
  4646 	      unfolding setsum_delta[OF fU]
  4647 	      using i(1) by simp
  4648 	    finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
  4649            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
  4650 	  qed
  4651 	next
  4652 	  show "y \<in> span (columns A)" unfolding h by blast
  4653 	qed}
  4654     then have ?lhs unfolding lhseq ..}
  4655   ultimately show ?thesis by blast
  4656 qed
  4657 
  4658 lemma matrix_left_invertible_span_rows:
  4659   "(\<exists>(B::real^'m::finite^'n::finite). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
  4660   unfolding right_invertible_transp[symmetric]
  4661   unfolding columns_transp[symmetric]
  4662   unfolding matrix_right_invertible_span_columns
  4663  ..
  4664 
  4665 (* An injective map real^'n->real^'n is also surjective.                       *)
  4666 
  4667 lemma linear_injective_imp_surjective:
  4668   assumes lf: "linear (f:: real ^'n::finite \<Rightarrow> real ^'n)" and fi: "inj f"
  4669   shows "surj f"
  4670 proof-
  4671   let ?U = "UNIV :: (real ^'n) set"
  4672   from basis_exists[of ?U] obtain B
  4673     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U"
  4674     by blast
  4675   from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
  4676   have th: "?U \<subseteq> span (f ` B)"
  4677     apply (rule card_ge_dim_independent)
  4678     apply blast
  4679     apply (rule independent_injective_image[OF B(2) lf fi])
  4680     apply (rule order_eq_refl)
  4681     apply (rule sym)
  4682     unfolding d
  4683     apply (rule card_image)
  4684     apply (rule subset_inj_on[OF fi])
  4685     by blast
  4686   from th show ?thesis
  4687     unfolding span_linear_image[OF lf] surj_def
  4688     using B(3) by blast
  4689 qed
  4690 
  4691 (* And vice versa.                                                           *)
  4692 
  4693 lemma surjective_iff_injective_gen:
  4694   assumes fS: "finite S" and fT: "finite T" and c: "card S = card T"
  4695   and ST: "f ` S \<subseteq> T"
  4696   shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs")
  4697 proof-
  4698   {assume h: "?lhs"
  4699     {fix x y assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
  4700       from x fS have S0: "card S \<noteq> 0" by auto
  4701       {assume xy: "x \<noteq> y"
  4702 	have th: "card S \<le> card (f ` (S - {y}))"
  4703 	  unfolding c
  4704 	  apply (rule card_mono)
  4705 	  apply (rule finite_imageI)
  4706 	  using fS apply simp
  4707 	  using h xy x y f unfolding subset_eq image_iff
  4708 	  apply auto
  4709 	  apply (case_tac "xa = f x")
  4710 	  apply (rule bexI[where x=x])
  4711 	  apply auto
  4712 	  done
  4713 	also have " \<dots> \<le> card (S -{y})"
  4714 	  apply (rule card_image_le)
  4715 	  using fS by simp
  4716 	also have "\<dots> \<le> card S - 1" using y fS by simp
  4717 	finally have False  using S0 by arith }
  4718       then have "x = y" by blast}
  4719     then have ?rhs unfolding inj_on_def by blast}
  4720   moreover
  4721   {assume h: ?rhs
  4722     have "f ` S = T"
  4723       apply (rule card_subset_eq[OF fT ST])
  4724       unfolding card_image[OF h] using c .
  4725     then have ?lhs by blast}
  4726   ultimately show ?thesis by blast
  4727 qed
  4728 
  4729 lemma linear_surjective_imp_injective:
  4730   assumes lf: "linear (f::real ^'n::finite => real ^'n)" and sf: "surj f"
  4731   shows "inj f"
  4732 proof-
  4733   let ?U = "UNIV :: (real ^'n) set"
  4734   from basis_exists[of ?U] obtain B
  4735     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U"
  4736     by blast
  4737   {fix x assume x: "x \<in> span B" and fx: "f x = 0"
  4738     from B(4) have fB: "finite B" by (simp add: hassize_def)
  4739     from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
  4740     have fBi: "independent (f ` B)"
  4741       apply (rule card_le_dim_spanning[of "f ` B" ?U])
  4742       apply blast
  4743       using sf B(3)
  4744       unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
  4745       apply blast
  4746       using fB apply (blast intro: finite_imageI)
  4747       unfolding d
  4748       apply (rule card_image_le)
  4749       apply (rule fB)
  4750       done
  4751     have th0: "dim ?U \<le> card (f ` B)"
  4752       apply (rule span_card_ge_dim)
  4753       apply blast
  4754       unfolding span_linear_image[OF lf]
  4755       apply (rule subset_trans[where B = "f ` UNIV"])
  4756       using sf unfolding surj_def apply blast
  4757       apply (rule image_mono)
  4758       apply (rule B(3))
  4759       apply (metis finite_imageI fB)
  4760       done
  4761 
  4762     moreover have "card (f ` B) \<le> card B"
  4763       by (rule card_image_le, rule fB)
  4764     ultimately have th1: "card B = card (f ` B)" unfolding d by arith
  4765     have fiB: "inj_on f B"
  4766       unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast
  4767     from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
  4768     have "x = 0" by blast}
  4769   note th = this
  4770   from th show ?thesis unfolding linear_injective_0[OF lf]
  4771     using B(3) by blast
  4772 qed
  4773 
  4774 (* Hence either is enough for isomorphism.                                   *)
  4775 
  4776 lemma left_right_inverse_eq:
  4777   assumes fg: "f o g = id" and gh: "g o h = id"
  4778   shows "f = h"
  4779 proof-
  4780   have "f = f o (g o h)" unfolding gh by simp
  4781   also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
  4782   finally show "f = h" unfolding fg by simp
  4783 qed
  4784 
  4785 lemma isomorphism_expand:
  4786   "f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
  4787   by (simp add: expand_fun_eq o_def id_def)
  4788 
  4789 lemma linear_injective_isomorphism:
  4790   assumes lf: "linear (f :: real^'n::finite \<Rightarrow> real ^'n)" and fi: "inj f"
  4791   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
  4792 unfolding isomorphism_expand[symmetric]
  4793 using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi]
  4794 by (metis left_right_inverse_eq)
  4795 
  4796 lemma linear_surjective_isomorphism:
  4797   assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'n)" and sf: "surj f"
  4798   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
  4799 unfolding isomorphism_expand[symmetric]
  4800 using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
  4801 by (metis left_right_inverse_eq)
  4802 
  4803 (* Left and right inverses are the same for R^N->R^N.                        *)
  4804 
  4805 lemma linear_inverse_left:
  4806   assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'n)" and lf': "linear f'"
  4807   shows "f o f' = id \<longleftrightarrow> f' o f = id"
  4808 proof-
  4809   {fix f f':: "real ^'n \<Rightarrow> real ^'n"
  4810     assume lf: "linear f" "linear f'" and f: "f o f' = id"
  4811     from f have sf: "surj f"
  4812 
  4813       apply (auto simp add: o_def stupid_ext[symmetric] id_def surj_def)
  4814       by metis
  4815     from linear_surjective_isomorphism[OF lf(1) sf] lf f
  4816     have "f' o f = id" unfolding stupid_ext[symmetric] o_def id_def
  4817       by metis}
  4818   then show ?thesis using lf lf' by metis
  4819 qed
  4820 
  4821 (* Moreover, a one-sided inverse is automatically linear.                    *)
  4822 
  4823 lemma left_inverse_linear:
  4824   assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'n)" and gf: "g o f = id"
  4825   shows "linear g"
  4826 proof-
  4827   from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def stupid_ext[symmetric])
  4828     by metis
  4829   from linear_injective_isomorphism[OF lf fi]
  4830   obtain h:: "real ^'n \<Rightarrow> real ^'n" where
  4831     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
  4832   have "h = g" apply (rule ext) using gf h(2,3)
  4833     apply (simp add: o_def id_def stupid_ext[symmetric])
  4834     by metis
  4835   with h(1) show ?thesis by blast
  4836 qed
  4837 
  4838 lemma right_inverse_linear:
  4839   assumes lf: "linear (f:: real ^'n::finite \<Rightarrow> real ^'n)" and gf: "f o g = id"
  4840   shows "linear g"
  4841 proof-
  4842   from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def stupid_ext[symmetric])
  4843     by metis
  4844   from linear_surjective_isomorphism[OF lf fi]
  4845   obtain h:: "real ^'n \<Rightarrow> real ^'n" where
  4846     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
  4847   have "h = g" apply (rule ext) using gf h(2,3)
  4848     apply (simp add: o_def id_def stupid_ext[symmetric])
  4849     by metis
  4850   with h(1) show ?thesis by blast
  4851 qed
  4852 
  4853 (* The same result in terms of square matrices.                              *)
  4854 
  4855 lemma matrix_left_right_inverse:
  4856   fixes A A' :: "real ^'n::finite^'n"
  4857   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
  4858 proof-
  4859   {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"
  4860     have sA: "surj (op *v A)"
  4861       unfolding surj_def
  4862       apply clarify
  4863       apply (rule_tac x="(A' *v y)" in exI)
  4864       by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
  4865     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
  4866     obtain f' :: "real ^'n \<Rightarrow> real ^'n"
  4867       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
  4868     have th: "matrix f' ** A = mat 1"
  4869       by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
  4870     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
  4871     hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
  4872     hence "matrix f' ** A = A' ** A" by simp
  4873     hence "A' ** A = mat 1" by (simp add: th)}
  4874   then show ?thesis by blast
  4875 qed
  4876 
  4877 (* Considering an n-element vector as an n-by-1 or 1-by-n matrix.            *)
  4878 
  4879 definition "rowvector v = (\<chi> i j. (v$j))"
  4880 
  4881 definition "columnvector v = (\<chi> i j. (v$i))"
  4882 
  4883 lemma transp_columnvector:
  4884  "transp(columnvector v) = rowvector v"
  4885   by (simp add: transp_def rowvector_def columnvector_def Cart_eq)
  4886 
  4887 lemma transp_rowvector: "transp(rowvector v) = columnvector v"
  4888   by (simp add: transp_def columnvector_def rowvector_def Cart_eq)
  4889 
  4890 lemma dot_rowvector_columnvector:
  4891   "columnvector (A *v v) = A ** columnvector v"
  4892   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
  4893 
  4894 lemma dot_matrix_product: "(x::'a::semiring_1^'n::finite) \<bullet> y = (((rowvector x ::'a^'n^1) ** (columnvector y :: 'a^1^'n))$1)$1"
  4895   by (vector matrix_matrix_mult_def rowvector_def columnvector_def dot_def)
  4896 
  4897 lemma dot_matrix_vector_mul:
  4898   fixes A B :: "real ^'n::finite ^'n" and x y :: "real ^'n"
  4899   shows "(A *v x) \<bullet> (B *v y) =
  4900       (((rowvector x :: real^'n^1) ** ((transp A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
  4901 unfolding dot_matrix_product transp_columnvector[symmetric]
  4902   dot_rowvector_columnvector matrix_transp_mul matrix_mul_assoc ..
  4903 
  4904 (* Infinity norm.                                                            *)
  4905 
  4906 definition "infnorm (x::real^'n::finite) = rsup {abs(x$i) |i. i\<in> (UNIV :: 'n set)}"
  4907 
  4908 lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> (UNIV :: 'n set)"
  4909   by auto
  4910 
  4911 lemma infnorm_set_image:
  4912   "{abs(x$i) |i. i\<in> (UNIV :: 'n set)} =
  4913   (\<lambda>i. abs(x$i)) ` (UNIV :: 'n set)" by blast
  4914 
  4915 lemma infnorm_set_lemma:
  4916   shows "finite {abs((x::'a::abs ^'n::finite)$i) |i. i\<in> (UNIV :: 'n set)}"
  4917   and "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"
  4918   unfolding infnorm_set_image
  4919   by (auto intro: finite_imageI)
  4920 
  4921 lemma infnorm_pos_le: "0 \<le> infnorm (x::real^'n::finite)"
  4922   unfolding infnorm_def
  4923   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
  4924   unfolding infnorm_set_image
  4925   by auto
  4926 
  4927 lemma infnorm_triangle: "infnorm ((x::real^'n::finite) + y) \<le> infnorm x + infnorm y"
  4928 proof-
  4929   have th: "\<And>x y (z::real). x - y <= z \<longleftrightarrow> x - z <= y" by arith
  4930   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
  4931   have th2: "\<And>x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith
  4932   show ?thesis
  4933   unfolding infnorm_def
  4934   unfolding rsup_finite_le_iff[ OF infnorm_set_lemma]
  4935   apply (subst diff_le_eq[symmetric])
  4936   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
  4937   unfolding infnorm_set_image bex_simps
  4938   apply (subst th)
  4939   unfolding th1
  4940   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
  4941 
  4942   unfolding infnorm_set_image ball_simps bex_simps
  4943   apply simp
  4944   apply (metis th2)
  4945   done
  4946 qed
  4947 
  4948 lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::real ^'n::finite) = 0"
  4949 proof-
  4950   have "infnorm x <= 0 \<longleftrightarrow> x = 0"
  4951     unfolding infnorm_def
  4952     unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
  4953     unfolding infnorm_set_image ball_simps
  4954     by vector
  4955   then show ?thesis using infnorm_pos_le[of x] by simp
  4956 qed
  4957 
  4958 lemma infnorm_0: "infnorm 0 = 0"
  4959   by (simp add: infnorm_eq_0)
  4960 
  4961 lemma infnorm_neg: "infnorm (- x) = infnorm x"
  4962   unfolding infnorm_def
  4963   apply (rule cong[of "rsup" "rsup"])
  4964   apply blast
  4965   apply (rule set_ext)
  4966   apply auto
  4967   done
  4968 
  4969 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
  4970 proof-
  4971   have "y - x = - (x - y)" by simp
  4972   then show ?thesis  by (metis infnorm_neg)
  4973 qed
  4974 
  4975 lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
  4976 proof-
  4977   have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
  4978     by arith
  4979   from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
  4980   have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
  4981     "infnorm y \<le> infnorm (x - y) + infnorm x"
  4982     by (simp_all add: ring_simps infnorm_neg diff_def[symmetric])
  4983   from th[OF ths]  show ?thesis .
  4984 qed
  4985 
  4986 lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
  4987   using infnorm_pos_le[of x] by arith
  4988 
  4989 lemma component_le_infnorm:
  4990   shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n::finite)"
  4991 proof-
  4992   let ?U = "UNIV :: 'n set"
  4993   let ?S = "{\<bar>x$i\<bar> |i. i\<in> ?U}"
  4994   have fS: "finite ?S" unfolding image_Collect[symmetric]
  4995     apply (rule finite_imageI) unfolding Collect_def mem_def by simp
  4996   have S0: "?S \<noteq> {}" by blast
  4997   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
  4998   from rsup_finite_in[OF fS S0] rsup_finite_Ub[OF fS S0]
  4999   show ?thesis unfolding infnorm_def isUb_def setle_def
  5000     unfolding infnorm_set_image ball_simps by auto
  5001 qed
  5002 
  5003 lemma infnorm_mul_lemma: "infnorm(a *s x) <= \<bar>a\<bar> * infnorm x"
  5004   apply (subst infnorm_def)
  5005   unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
  5006   unfolding infnorm_set_image ball_simps
  5007   apply (simp add: abs_mult)
  5008   apply (rule allI)
  5009   apply (cut_tac component_le_infnorm[of x])
  5010   apply (rule mult_mono)
  5011   apply auto
  5012   done
  5013 
  5014 lemma infnorm_mul: "infnorm(a *s x) = abs a * infnorm x"
  5015 proof-
  5016   {assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) }
  5017   moreover
  5018   {assume a0: "a \<noteq> 0"
  5019     from a0 have th: "(1/a) *s (a *s x) = x"
  5020       by (simp add: vector_smult_assoc)
  5021     from a0 have ap: "\<bar>a\<bar> > 0" by arith
  5022     from infnorm_mul_lemma[of "1/a" "a *s x"]
  5023     have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*s x)"
  5024       unfolding th by simp
  5025     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)
  5026     then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*s x)"
  5027       using ap by (simp add: field_simps)
  5028     with infnorm_mul_lemma[of a x] have ?thesis by arith }
  5029   ultimately show ?thesis by blast
  5030 qed
  5031 
  5032 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
  5033   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
  5034 
  5035 (* Prove that it differs only up to a bound from Euclidean norm.             *)
  5036 
  5037 lemma infnorm_le_norm: "infnorm x \<le> norm x"
  5038   unfolding infnorm_def rsup_finite_le_iff[OF infnorm_set_lemma]
  5039   unfolding infnorm_set_image  ball_simps
  5040   by (metis component_le_norm)
  5041 lemma card_enum: "card {1 .. n} = n" by auto
  5042 lemma norm_le_infnorm: "norm(x) <= sqrt(real CARD('n)) * infnorm(x::real ^'n::finite)"
  5043 proof-
  5044   let ?d = "CARD('n)"
  5045   have "real ?d \<ge> 0" by simp
  5046   hence d2: "(sqrt (real ?d))^2 = real ?d"
  5047     by (auto intro: real_sqrt_pow2)
  5048   have th: "sqrt (real ?d) * infnorm x \<ge> 0"
  5049     by (simp add: zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le)
  5050   have th1: "x\<bullet>x \<le> (sqrt (real ?d) * infnorm x)^2"
  5051     unfolding power_mult_distrib d2
  5052     apply (subst power2_abs[symmetric])
  5053     unfolding real_of_nat_def dot_def power2_eq_square[symmetric]
  5054     apply (subst power2_abs[symmetric])
  5055     apply (rule setsum_bounded)
  5056     apply (rule power_mono)
  5057     unfolding abs_of_nonneg[OF infnorm_pos_le]
  5058     unfolding infnorm_def  rsup_finite_ge_iff[OF infnorm_set_lemma]
  5059     unfolding infnorm_set_image bex_simps
  5060     apply blast
  5061     by (rule abs_ge_zero)
  5062   from real_le_lsqrt[OF dot_pos_le th th1]
  5063   show ?thesis unfolding real_vector_norm_def id_def .
  5064 qed
  5065 
  5066 (* Equality in Cauchy-Schwarz and triangle inequalities.                     *)
  5067 
  5068 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")
  5069 proof-
  5070   {assume h: "x = 0"
  5071     hence ?thesis by simp}
  5072   moreover
  5073   {assume h: "y = 0"
  5074     hence ?thesis by simp}
  5075   moreover
  5076   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
  5077     from dot_eq_0[of "norm y *s x - norm x *s y"]
  5078     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)"
  5079       using x y
  5080       unfolding dot_rsub dot_lsub dot_lmult dot_rmult
  5081       unfolding norm_pow_2[symmetric] power2_eq_square diff_eq_0_iff_eq apply (simp add: dot_sym)
  5082       apply (simp add: ring_simps)
  5083       apply metis
  5084       done
  5085     also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
  5086       by (simp add: ring_simps dot_sym)
  5087     also have "\<dots> \<longleftrightarrow> ?lhs" using x y
  5088       apply simp
  5089       by metis
  5090     finally have ?thesis by blast}
  5091   ultimately show ?thesis by blast
  5092 qed
  5093 
  5094 lemma norm_cauchy_schwarz_abs_eq:
  5095   fixes x y :: "real ^ 'n::finite"
  5096   shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
  5097                 norm x *s y = norm y *s x \<or> norm(x) *s y = - norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
  5098 proof-
  5099   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
  5100   have "?rhs \<longleftrightarrow> norm x *s y = norm y *s x \<or> norm (- x) *s y = norm y *s (- x)"
  5101     apply simp by vector
  5102   also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or>
  5103      (-x) \<bullet> y = norm x * norm y)"
  5104     unfolding norm_cauchy_schwarz_eq[symmetric]
  5105     unfolding norm_minus_cancel
  5106       norm_mul by blast
  5107   also have "\<dots> \<longleftrightarrow> ?lhs"
  5108     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] dot_lneg
  5109     by arith
  5110   finally show ?thesis ..
  5111 qed
  5112 
  5113 lemma norm_triangle_eq:
  5114   fixes x y :: "real ^ 'n::finite"
  5115   shows "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *s y = norm y *s x"
  5116 proof-
  5117   {assume x: "x =0 \<or> y =0"
  5118     hence ?thesis by (cases "x=0", simp_all)}
  5119   moreover
  5120   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
  5121     hence "norm x \<noteq> 0" "norm y \<noteq> 0"
  5122       by simp_all
  5123     hence n: "norm x > 0" "norm y > 0"
  5124       using norm_ge_zero[of x] norm_ge_zero[of y]
  5125       by arith+
  5126     have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
  5127     have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
  5128       apply (rule th) using n norm_ge_zero[of "x + y"]
  5129       by arith
  5130     also have "\<dots> \<longleftrightarrow> norm x *s y = norm y *s x"
  5131       unfolding norm_cauchy_schwarz_eq[symmetric]
  5132       unfolding norm_pow_2 dot_ladd dot_radd
  5133       by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym ring_simps)
  5134     finally have ?thesis .}
  5135   ultimately show ?thesis by blast
  5136 qed
  5137 
  5138 (* Collinearity.*)
  5139 
  5140 definition "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *s u)"
  5141 
  5142 lemma collinear_empty:  "collinear {}" by (simp add: collinear_def)
  5143 
  5144 lemma collinear_sing: "collinear {(x::'a::ring_1^'n)}"
  5145   apply (simp add: collinear_def)
  5146   apply (rule exI[where x=0])
  5147   by simp
  5148 
  5149 lemma collinear_2: "collinear {(x::'a::ring_1^'n),y}"
  5150   apply (simp add: collinear_def)
  5151   apply (rule exI[where x="x - y"])
  5152   apply auto
  5153   apply (rule exI[where x=0], simp)
  5154   apply (rule exI[where x=1], simp)
  5155   apply (rule exI[where x="- 1"], simp add: vector_sneg_minus1[symmetric])
  5156   apply (rule exI[where x=0], simp)
  5157   done
  5158 
  5159 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")
  5160 proof-
  5161   {assume "x=0 \<or> y = 0" hence ?thesis
  5162       by (cases "x = 0", simp_all add: collinear_2 insert_commute)}
  5163   moreover
  5164   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
  5165     {assume h: "?lhs"
  5166       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
  5167       from u[rule_format, of x 0] u[rule_format, of y 0]
  5168       obtain cx and cy where
  5169 	cx: "x = cx*s u" and cy: "y = cy*s u"
  5170 	by auto
  5171       from cx x have cx0: "cx \<noteq> 0" by auto
  5172       from cy y have cy0: "cy \<noteq> 0" by auto
  5173       let ?d = "cy / cx"
  5174       from cx cy cx0 have "y = ?d *s x"
  5175 	by (simp add: vector_smult_assoc)
  5176       hence ?rhs using x y by blast}
  5177     moreover
  5178     {assume h: "?rhs"
  5179       then obtain c where c: "y = c*s x" using x y by blast
  5180       have ?lhs unfolding collinear_def c
  5181 	apply (rule exI[where x=x])
  5182 	apply auto
  5183 	apply (rule exI[where x="- 1"], simp only: vector_smult_lneg vector_smult_lid)
  5184 	apply (rule exI[where x= "-c"], simp only: vector_smult_lneg)
  5185 	apply (rule exI[where x=1], simp)
  5186 	apply (rule exI[where x="1 - c"], simp add: vector_smult_lneg vector_sub_rdistrib)
  5187 	apply (rule exI[where x="c - 1"], simp add: vector_smult_lneg vector_sub_rdistrib)
  5188 	done}
  5189     ultimately have ?thesis by blast}
  5190   ultimately show ?thesis by blast
  5191 qed
  5192 
  5193 lemma norm_cauchy_schwarz_equal:
  5194   fixes x y :: "real ^ 'n::finite"
  5195   shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {(0::real^'n),x,y}"
  5196 unfolding norm_cauchy_schwarz_abs_eq
  5197 apply (cases "x=0", simp_all add: collinear_2)
  5198 apply (cases "y=0", simp_all add: collinear_2 insert_commute)
  5199 unfolding collinear_lemma
  5200 apply simp
  5201 apply (subgoal_tac "norm x \<noteq> 0")
  5202 apply (subgoal_tac "norm y \<noteq> 0")
  5203 apply (rule iffI)
  5204 apply (cases "norm x *s y = norm y *s x")
  5205 apply (rule exI[where x="(1/norm x) * norm y"])
  5206 apply (drule sym)
  5207 unfolding vector_smult_assoc[symmetric]
  5208 apply (simp add: vector_smult_assoc field_simps)
  5209 apply (rule exI[where x="(1/norm x) * - norm y"])
  5210 apply clarify
  5211 apply (drule sym)
  5212 unfolding vector_smult_assoc[symmetric]
  5213 apply (simp add: vector_smult_assoc field_simps)
  5214 apply (erule exE)
  5215 apply (erule ssubst)
  5216 unfolding vector_smult_assoc
  5217 unfolding norm_mul
  5218 apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
  5219 apply (case_tac "c <= 0", simp add: ring_simps)
  5220 apply (simp add: ring_simps)
  5221 apply (case_tac "c <= 0", simp add: ring_simps)
  5222 apply (simp add: ring_simps)
  5223 apply simp
  5224 apply simp
  5225 done
  5226 
  5227 end