src/HOL/Library/Euclidean_Space.thy
author huffman
Sun Feb 22 08:52:44 2009 -0800 (2009-02-22)
changeset 30066 9223483cc927
parent 30045 b8ddd7667eed
child 30242 aea5d7fa7ef5
child 30303 9c4f4ac0d038
permissions -rw-r--r--
remove duplicate instance declaration
     1 (* Title:      Library/Euclidean_Space
     2    ID:         $Id: 
     3    Author:     Amine Chaieb, University of Cambridge
     4 *)
     5 
     6 header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
     7 
     8 theory Euclidean_Space
     9   imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Complex_Main 
    10   Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type
    11   Inner_Product
    12   uses ("normarith.ML")
    13 begin
    14 
    15 text{* Some common special cases.*}
    16 
    17 lemma forall_1: "(\<forall>(i::'a::{order,one}). 1 <= i \<and> i <= 1 --> P i) \<longleftrightarrow> P 1"
    18   by (metis order_eq_iff)
    19 lemma forall_dimindex_1: "(\<forall>i \<in> {1..dimindex(UNIV:: 1 set)}. P i) \<longleftrightarrow> P 1"
    20   by (simp add: dimindex_def)
    21 
    22 lemma forall_2: "(\<forall>(i::nat). 1 <= i \<and> i <= 2 --> P i) \<longleftrightarrow> P 1 \<and> P 2"
    23 proof-
    24   have "\<And>i::nat. 1 <= i \<and> i <= 2 \<longleftrightarrow> i = 1 \<or> i = 2" by arith
    25   thus ?thesis by metis
    26 qed
    27 
    28 lemma forall_3: "(\<forall>(i::nat). 1 <= i \<and> i <= 3 --> P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
    29 proof-
    30   have "\<And>i::nat. 1 <= i \<and> i <= 3 \<longleftrightarrow> i = 1 \<or> i = 2 \<or> i = 3" by arith
    31   thus ?thesis by metis
    32 qed
    33 
    34 lemma setsum_singleton[simp]: "setsum f {x} = f x" by simp
    35 lemma setsum_1: "setsum f {(1::'a::{order,one})..1} = f 1" 
    36   by (simp add: atLeastAtMost_singleton)
    37 
    38 lemma setsum_2: "setsum f {1::nat..2} = f 1 + f 2" 
    39   by (simp add: nat_number  atLeastAtMostSuc_conv add_commute)
    40 
    41 lemma setsum_3: "setsum f {1::nat..3} = f 1 + f 2 + f 3" 
    42   by (simp add: nat_number  atLeastAtMostSuc_conv add_commute)
    43 
    44 subsection{* Basic componentwise operations on vectors. *}
    45 
    46 instantiation "^" :: (plus,type) plus
    47 begin
    48 definition  vector_add_def : "op + \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) + (y$i)))" 
    49 instance ..
    50 end
    51 
    52 instantiation "^" :: (times,type) times
    53 begin
    54   definition vector_mult_def : "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))" 
    55   instance ..
    56 end
    57 
    58 instantiation "^" :: (minus,type) minus begin
    59   definition vector_minus_def : "op - \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) - (y$i)))"
    60 instance ..
    61 end
    62 
    63 instantiation "^" :: (uminus,type) uminus begin
    64   definition vector_uminus_def : "uminus \<equiv> (\<lambda> x.  (\<chi> i. - (x$i)))"
    65 instance ..
    66 end
    67 instantiation "^" :: (zero,type) zero begin
    68   definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)" 
    69 instance ..
    70 end
    71 
    72 instantiation "^" :: (one,type) one begin
    73   definition vector_one_def : "1 \<equiv> (\<chi> i. 1)" 
    74 instance ..
    75 end
    76 
    77 instantiation "^" :: (ord,type) ord
    78  begin
    79 definition vector_less_eq_def:
    80   "less_eq (x :: 'a ^'b) y = (ALL i : {1 .. dimindex (UNIV :: 'b set)}.
    81   x$i <= y$i)"
    82 definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i : {1 ..
    83   dimindex (UNIV :: 'b set)}. x$i < y$i)"
    84  
    85 instance by (intro_classes)
    86 end
    87 
    88 instantiation "^" :: (scaleR, type) scaleR
    89 begin
    90 definition vector_scaleR_def: "scaleR = (\<lambda> r x.  (\<chi> i. scaleR r (x$i)))" 
    91 instance ..
    92 end
    93 
    94 text{* Also the scalar-vector multiplication. *}
    95 
    96 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'n" (infixr "*s" 75)
    97   where "c *s x = (\<chi> i. c * (x$i))"
    98 
    99 text{* Constant Vectors *}
   100 
   101 definition "vec x = (\<chi> i. x)"
   102 
   103 text{* Dot products. *}
   104 
   105 definition dot :: "'a::{comm_monoid_add, times} ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" (infix "\<bullet>" 70) where
   106   "x \<bullet> y = setsum (\<lambda>i. x$i * y$i) {1 .. dimindex (UNIV:: 'n set)}"
   107 lemma dot_1[simp]: "(x::'a::{comm_monoid_add, times}^1) \<bullet> y = (x$1) * (y$1)"
   108   by (simp add: dot_def dimindex_def)
   109 
   110 lemma dot_2[simp]: "(x::'a::{comm_monoid_add, times}^2) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2)"
   111   by (simp add: dot_def dimindex_def nat_number)
   112 
   113 lemma dot_3[simp]: "(x::'a::{comm_monoid_add, times}^3) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2) + (x$3) * (y$3)"
   114   by (simp add: dot_def dimindex_def nat_number)
   115 
   116 subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
   117 
   118 lemmas Cart_lambda_beta' = Cart_lambda_beta[rule_format]
   119 method_setup vector = {*
   120 let
   121   val ss1 = HOL_basic_ss addsimps [@{thm dot_def}, @{thm setsum_addf} RS sym, 
   122   @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib}, 
   123   @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
   124   val ss2 = @{simpset} addsimps 
   125              [@{thm vector_add_def}, @{thm vector_mult_def},  
   126               @{thm vector_minus_def}, @{thm vector_uminus_def}, 
   127               @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def}, 
   128               @{thm vector_scaleR_def},
   129               @{thm Cart_lambda_beta'}, @{thm vector_scalar_mult_def}]
   130  fun vector_arith_tac ths = 
   131    simp_tac ss1
   132    THEN' (fn i => rtac @{thm setsum_cong2} i
   133          ORELSE rtac @{thm setsum_0'} i 
   134          ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i)
   135    (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
   136    THEN' asm_full_simp_tac (ss2 addsimps ths)
   137  in
   138   Method.thms_args (Method.SIMPLE_METHOD' o vector_arith_tac)
   139 end
   140 *} "Lifts trivial vector statements to real arith statements"
   141 
   142 lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def)
   143 lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def)
   144 
   145 
   146 
   147 text{* Obvious "component-pushing". *}
   148 
   149 lemma vec_component: " i \<in> {1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (vec x :: 'a ^ 'n)$i = x" 
   150   by (vector vec_def) 
   151 
   152 lemma vector_add_component: 
   153   fixes x y :: "'a::{plus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
   154   shows "(x + y)$i = x$i + y$i"
   155   using i by vector
   156 
   157 lemma vector_minus_component: 
   158   fixes x y :: "'a::{minus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
   159   shows "(x - y)$i = x$i - y$i"
   160   using i  by vector
   161 
   162 lemma vector_mult_component: 
   163   fixes x y :: "'a::{times} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
   164   shows "(x * y)$i = x$i * y$i"
   165   using i by vector
   166 
   167 lemma vector_smult_component: 
   168   fixes y :: "'a::{times} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
   169   shows "(c *s y)$i = c * (y$i)"
   170   using i by vector
   171 
   172 lemma vector_uminus_component: 
   173   fixes x :: "'a::{uminus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
   174   shows "(- x)$i = - (x$i)"
   175   using i by vector
   176 
   177 lemma vector_scaleR_component:
   178   fixes x :: "'a::scaleR ^ 'n"
   179   assumes i: "i \<in> {1 .. dimindex(UNIV :: 'n set)}"
   180   shows "(scaleR r x)$i = scaleR r (x$i)"
   181   using i by vector
   182 
   183 lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
   184 
   185 lemmas vector_component =
   186   vec_component vector_add_component vector_mult_component
   187   vector_smult_component vector_minus_component vector_uminus_component
   188   vector_scaleR_component cond_component
   189 
   190 subsection {* Some frequently useful arithmetic lemmas over vectors. *}
   191 
   192 instance "^" :: (semigroup_add,type) semigroup_add 
   193   apply (intro_classes) by (vector add_assoc)
   194 
   195 
   196 instance "^" :: (monoid_add,type) monoid_add 
   197   apply (intro_classes) by vector+ 
   198 
   199 instance "^" :: (group_add,type) group_add 
   200   apply (intro_classes) by (vector algebra_simps)+ 
   201 
   202 instance "^" :: (ab_semigroup_add,type) ab_semigroup_add 
   203   apply (intro_classes) by (vector add_commute)
   204 
   205 instance "^" :: (comm_monoid_add,type) comm_monoid_add
   206   apply (intro_classes) by vector
   207 
   208 instance "^" :: (ab_group_add,type) ab_group_add 
   209   apply (intro_classes) by vector+
   210 
   211 instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add 
   212   apply (intro_classes)
   213   by (vector Cart_eq)+
   214 
   215 instance "^" :: (cancel_ab_semigroup_add,type) cancel_ab_semigroup_add
   216   apply (intro_classes)
   217   by (vector Cart_eq)
   218 
   219 instance "^" :: (real_vector, type) real_vector
   220   by default (vector scaleR_left_distrib scaleR_right_distrib)+
   221 
   222 instance "^" :: (semigroup_mult,type) semigroup_mult 
   223   apply (intro_classes) by (vector mult_assoc)
   224 
   225 instance "^" :: (monoid_mult,type) monoid_mult 
   226   apply (intro_classes) by vector+
   227 
   228 instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult 
   229   apply (intro_classes) by (vector mult_commute)
   230 
   231 instance "^" :: (ab_semigroup_idem_mult,type) ab_semigroup_idem_mult 
   232   apply (intro_classes) by (vector mult_idem)
   233 
   234 instance "^" :: (comm_monoid_mult,type) comm_monoid_mult 
   235   apply (intro_classes) by vector
   236 
   237 fun vector_power :: "('a::{one,times} ^'n) \<Rightarrow> nat \<Rightarrow> 'a^'n" where
   238   "vector_power x 0 = 1"
   239   | "vector_power x (Suc n) = x * vector_power x n"
   240 
   241 instantiation "^" :: (recpower,type) recpower 
   242 begin
   243   definition vec_power_def: "op ^ \<equiv> vector_power"
   244   instance 
   245   apply (intro_classes) by (simp_all add: vec_power_def) 
   246 end
   247 
   248 instance "^" :: (semiring,type) semiring
   249   apply (intro_classes) by (vector ring_simps)+
   250 
   251 instance "^" :: (semiring_0,type) semiring_0
   252   apply (intro_classes) by (vector ring_simps)+
   253 instance "^" :: (semiring_1,type) semiring_1
   254   apply (intro_classes) apply vector using dimindex_ge_1 by auto 
   255 instance "^" :: (comm_semiring,type) comm_semiring
   256   apply (intro_classes) by (vector ring_simps)+
   257 
   258 instance "^" :: (comm_semiring_0,type) comm_semiring_0 by (intro_classes) 
   259 instance "^" :: (cancel_comm_monoid_add, type) cancel_comm_monoid_add ..
   260 instance "^" :: (semiring_0_cancel,type) semiring_0_cancel by (intro_classes) 
   261 instance "^" :: (comm_semiring_0_cancel,type) comm_semiring_0_cancel by (intro_classes) 
   262 instance "^" :: (ring,type) ring by (intro_classes) 
   263 instance "^" :: (semiring_1_cancel,type) semiring_1_cancel by (intro_classes) 
   264 instance "^" :: (comm_semiring_1,type) comm_semiring_1 by (intro_classes)
   265 
   266 instance "^" :: (ring_1,type) ring_1 ..
   267 
   268 instance "^" :: (real_algebra,type) real_algebra
   269   apply intro_classes
   270   apply (simp_all add: vector_scaleR_def ring_simps)
   271   apply vector
   272   apply vector
   273   done
   274 
   275 instance "^" :: (real_algebra_1,type) real_algebra_1 ..
   276 
   277 lemma of_nat_index: 
   278   "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
   279   apply (induct n)
   280   apply vector
   281   apply vector
   282   done
   283 lemma zero_index[simp]: 
   284   "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (0 :: 'a::zero ^'n)$i = 0" by vector
   285 
   286 lemma one_index[simp]: 
   287   "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (1 :: 'a::one ^'n)$i = 1" by vector
   288 
   289 lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \<noteq> 0"
   290 proof-
   291   have "(1::'a) + of_nat n = 0 \<longleftrightarrow> of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp
   292   also have "\<dots> \<longleftrightarrow> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff) 
   293   finally show ?thesis by simp 
   294 qed
   295 
   296 instance "^" :: (semiring_char_0,type) semiring_char_0 
   297 proof (intro_classes) 
   298   fix m n ::nat
   299   show "(of_nat m :: 'a^'b) = of_nat n \<longleftrightarrow> m = n"
   300   proof(induct m arbitrary: n)
   301     case 0 thus ?case apply vector 
   302       apply (induct n,auto simp add: ring_simps)
   303       using dimindex_ge_1 apply auto
   304       apply vector
   305       by (auto simp add: of_nat_index one_plus_of_nat_neq_0)
   306   next
   307     case (Suc n m)
   308     thus ?case  apply vector
   309       apply (induct m, auto simp add: ring_simps of_nat_index zero_index)
   310       using dimindex_ge_1 apply simp apply blast
   311       apply (simp add: one_plus_of_nat_neq_0)
   312       using dimindex_ge_1 apply simp apply blast
   313       apply (simp add: vector_component one_index of_nat_index)
   314       apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)
   315       using  dimindex_ge_1 apply simp apply blast
   316       apply (simp add: vector_component one_index of_nat_index)
   317       apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)
   318       using dimindex_ge_1 apply simp apply blast
   319       apply (simp add: vector_component one_index of_nat_index)
   320       done
   321   qed
   322 qed
   323 
   324 instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes
   325 instance "^" :: (ring_char_0,type) ring_char_0 by intro_classes
   326 
   327 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"  
   328   by (vector mult_assoc)
   329 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" 
   330   by (vector ring_simps)
   331 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" 
   332   by (vector ring_simps)
   333 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
   334 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
   335 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y" 
   336   by (vector ring_simps)
   337 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
   338 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
   339 lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
   340 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
   341 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x" 
   342   by (vector ring_simps)
   343 
   344 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)" 
   345   apply (auto simp add: vec_def Cart_eq vec_component Cart_lambda_beta )
   346   using dimindex_ge_1 apply auto done
   347 
   348 subsection {* Square root of sum of squares *}
   349 
   350 definition
   351   "setL2 f A = sqrt (\<Sum>i\<in>A. (f i)\<twosuperior>)"
   352 
   353 lemma setL2_cong:
   354   "\<lbrakk>A = B; \<And>x. x \<in> B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
   355   unfolding setL2_def by simp
   356 
   357 lemma strong_setL2_cong:
   358   "\<lbrakk>A = B; \<And>x. x \<in> B =simp=> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
   359   unfolding setL2_def simp_implies_def by simp
   360 
   361 lemma setL2_infinite [simp]: "\<not> finite A \<Longrightarrow> setL2 f A = 0"
   362   unfolding setL2_def by simp
   363 
   364 lemma setL2_empty [simp]: "setL2 f {} = 0"
   365   unfolding setL2_def by simp
   366 
   367 lemma setL2_insert [simp]:
   368   "\<lbrakk>finite F; a \<notin> F\<rbrakk> \<Longrightarrow>
   369     setL2 f (insert a F) = sqrt ((f a)\<twosuperior> + (setL2 f F)\<twosuperior>)"
   370   unfolding setL2_def by (simp add: setsum_nonneg)
   371 
   372 lemma setL2_nonneg [simp]: "0 \<le> setL2 f A"
   373   unfolding setL2_def by (simp add: setsum_nonneg)
   374 
   375 lemma setL2_0': "\<forall>a\<in>A. f a = 0 \<Longrightarrow> setL2 f A = 0"
   376   unfolding setL2_def by simp
   377 
   378 lemma setL2_mono:
   379   assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i"
   380   assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"
   381   shows "setL2 f K \<le> setL2 g K"
   382   unfolding setL2_def
   383   by (simp add: setsum_nonneg setsum_mono power_mono prems)
   384 
   385 lemma setL2_right_distrib:
   386   "0 \<le> r \<Longrightarrow> r * setL2 f A = setL2 (\<lambda>x. r * f x) A"
   387   unfolding setL2_def
   388   apply (simp add: power_mult_distrib)
   389   apply (simp add: setsum_right_distrib [symmetric])
   390   apply (simp add: real_sqrt_mult setsum_nonneg)
   391   done
   392 
   393 lemma setL2_left_distrib:
   394   "0 \<le> r \<Longrightarrow> setL2 f A * r = setL2 (\<lambda>x. f x * r) A"
   395   unfolding setL2_def
   396   apply (simp add: power_mult_distrib)
   397   apply (simp add: setsum_left_distrib [symmetric])
   398   apply (simp add: real_sqrt_mult setsum_nonneg)
   399   done
   400 
   401 lemma setsum_nonneg_eq_0_iff:
   402   fixes f :: "'a \<Rightarrow> 'b::pordered_ab_group_add"
   403   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)"
   404   apply (induct set: finite, simp)
   405   apply (simp add: add_nonneg_eq_0_iff setsum_nonneg)
   406   done
   407 
   408 lemma setL2_eq_0_iff: "finite A \<Longrightarrow> setL2 f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
   409   unfolding setL2_def
   410   by (simp add: setsum_nonneg setsum_nonneg_eq_0_iff)
   411 
   412 lemma setL2_triangle_ineq:
   413   shows "setL2 (\<lambda>i. f i + g i) A \<le> setL2 f A + setL2 g A"
   414 proof (cases "finite A")
   415   case False
   416   thus ?thesis by simp
   417 next
   418   case True
   419   thus ?thesis
   420   proof (induct set: finite)
   421     case empty
   422     show ?case by simp
   423   next
   424     case (insert x F)
   425     hence "sqrt ((f x + g x)\<twosuperior> + (setL2 (\<lambda>i. f i + g i) F)\<twosuperior>) \<le>
   426            sqrt ((f x + g x)\<twosuperior> + (setL2 f F + setL2 g F)\<twosuperior>)"
   427       by (intro real_sqrt_le_mono add_left_mono power_mono insert
   428                 setL2_nonneg add_increasing zero_le_power2)
   429     also have
   430       "\<dots> \<le> sqrt ((f x)\<twosuperior> + (setL2 f F)\<twosuperior>) + sqrt ((g x)\<twosuperior> + (setL2 g F)\<twosuperior>)"
   431       by (rule real_sqrt_sum_squares_triangle_ineq)
   432     finally show ?case
   433       using insert by simp
   434   qed
   435 qed
   436 
   437 lemma sqrt_sum_squares_le_sum:
   438   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) \<le> x + y"
   439   apply (rule power2_le_imp_le)
   440   apply (simp add: power2_sum)
   441   apply (simp add: mult_nonneg_nonneg)
   442   apply (simp add: add_nonneg_nonneg)
   443   done
   444 
   445 lemma setL2_le_setsum [rule_format]:
   446   "(\<forall>i\<in>A. 0 \<le> f i) \<longrightarrow> setL2 f A \<le> setsum f A"
   447   apply (cases "finite A")
   448   apply (induct set: finite)
   449   apply simp
   450   apply clarsimp
   451   apply (erule order_trans [OF sqrt_sum_squares_le_sum])
   452   apply simp
   453   apply simp
   454   apply simp
   455   done
   456 
   457 lemma sqrt_sum_squares_le_sum_abs: "sqrt (x\<twosuperior> + y\<twosuperior>) \<le> \<bar>x\<bar> + \<bar>y\<bar>"
   458   apply (rule power2_le_imp_le)
   459   apply (simp add: power2_sum)
   460   apply (simp add: mult_nonneg_nonneg)
   461   apply (simp add: add_nonneg_nonneg)
   462   done
   463 
   464 lemma setL2_le_setsum_abs: "setL2 f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
   465   apply (cases "finite A")
   466   apply (induct set: finite)
   467   apply simp
   468   apply simp
   469   apply (rule order_trans [OF sqrt_sum_squares_le_sum_abs])
   470   apply simp
   471   apply simp
   472   done
   473 
   474 lemma setL2_mult_ineq_lemma:
   475   fixes a b c d :: real
   476   shows "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
   477 proof -
   478   have "0 \<le> (a * d - b * c)\<twosuperior>" by simp
   479   also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * d) * (b * c)"
   480     by (simp only: power2_diff power_mult_distrib)
   481   also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * c) * (b * d)"
   482     by simp
   483   finally show "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
   484     by simp
   485 qed
   486 
   487 lemma setL2_mult_ineq: "(\<Sum>i\<in>A. \<bar>f i\<bar> * \<bar>g i\<bar>) \<le> setL2 f A * setL2 g A"
   488   apply (cases "finite A")
   489   apply (induct set: finite)
   490   apply simp
   491   apply (rule power2_le_imp_le, simp)
   492   apply (rule order_trans)
   493   apply (rule power_mono)
   494   apply (erule add_left_mono)
   495   apply (simp add: add_nonneg_nonneg mult_nonneg_nonneg setsum_nonneg)
   496   apply (simp add: power2_sum)
   497   apply (simp add: power_mult_distrib)
   498   apply (simp add: right_distrib left_distrib)
   499   apply (rule ord_le_eq_trans)
   500   apply (rule setL2_mult_ineq_lemma)
   501   apply simp
   502   apply (intro mult_nonneg_nonneg setL2_nonneg)
   503   apply simp
   504   done
   505 
   506 lemma member_le_setL2: "\<lbrakk>finite A; i \<in> A\<rbrakk> \<Longrightarrow> f i \<le> setL2 f A"
   507   apply (rule_tac s="insert i (A - {i})" and t="A" in subst)
   508   apply fast
   509   apply (subst setL2_insert)
   510   apply simp
   511   apply simp
   512   apply simp
   513   done
   514 
   515 subsection {* Norms *}
   516 
   517 instantiation "^" :: (real_normed_vector, type) real_normed_vector
   518 begin
   519 
   520 definition vector_norm_def:
   521   "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) {1 .. dimindex (UNIV:: 'b set)}"
   522 
   523 definition vector_sgn_def:
   524   "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
   525 
   526 instance proof
   527   fix a :: real and x y :: "'a ^ 'b"
   528   show "0 \<le> norm x"
   529     unfolding vector_norm_def
   530     by (rule setL2_nonneg)
   531   show "norm x = 0 \<longleftrightarrow> x = 0"
   532     unfolding vector_norm_def
   533     by (simp add: setL2_eq_0_iff Cart_eq)
   534   show "norm (x + y) \<le> norm x + norm y"
   535     unfolding vector_norm_def
   536     apply (rule order_trans [OF _ setL2_triangle_ineq])
   537     apply (rule setL2_mono)
   538     apply (simp add: vector_component norm_triangle_ineq)
   539     apply simp
   540     done
   541   show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
   542     unfolding vector_norm_def
   543     by (simp add: vector_component norm_scaleR setL2_right_distrib
   544              cong: strong_setL2_cong)
   545   show "sgn x = scaleR (inverse (norm x)) x"
   546     by (rule vector_sgn_def)
   547 qed
   548 
   549 end
   550 
   551 subsection {* Inner products *}
   552 
   553 instantiation "^" :: (real_inner, type) real_inner
   554 begin
   555 
   556 definition vector_inner_def:
   557   "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) {1 .. dimindex(UNIV::'b set)}"
   558 
   559 instance proof
   560   fix r :: real and x y z :: "'a ^ 'b"
   561   show "inner x y = inner y x"
   562     unfolding vector_inner_def
   563     by (simp add: inner_commute)
   564   show "inner (x + y) z = inner x z + inner y z"
   565     unfolding vector_inner_def
   566     by (vector inner_left_distrib)
   567   show "inner (scaleR r x) y = r * inner x y"
   568     unfolding vector_inner_def
   569     by (vector inner_scaleR_left)
   570   show "0 \<le> inner x x"
   571     unfolding vector_inner_def
   572     by (simp add: setsum_nonneg)
   573   show "inner x x = 0 \<longleftrightarrow> x = 0"
   574     unfolding vector_inner_def
   575     by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
   576   show "norm x = sqrt (inner x x)"
   577     unfolding vector_inner_def vector_norm_def setL2_def
   578     by (simp add: power2_norm_eq_inner)
   579 qed
   580 
   581 end
   582 
   583 subsection{* Properties of the dot product.  *}
   584 
   585 lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x" 
   586   by (vector mult_commute)
   587 lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \<bullet> z = (x \<bullet> z) + (y \<bullet> z)"
   588   by (vector ring_simps)
   589 lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)" 
   590   by (vector ring_simps)
   591 lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)" 
   592   by (vector ring_simps)
   593 lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)" 
   594   by (vector ring_simps)
   595 lemma dot_lmult: "(c *s x) \<bullet> y = (c::'a::ring) * (x \<bullet> y)" by (vector ring_simps)
   596 lemma dot_rmult: "x \<bullet> (c *s y) = (c::'a::comm_ring) * (x \<bullet> y)" by (vector ring_simps)
   597 lemma dot_lneg: "(-x) \<bullet> (y::'a::ring ^ 'n) = -(x \<bullet> y)" by vector
   598 lemma dot_rneg: "(x::'a::ring ^ 'n) \<bullet> (-y) = -(x \<bullet> y)" by vector
   599 lemma dot_lzero[simp]: "0 \<bullet> x = (0::'a::{comm_monoid_add, mult_zero})" by vector
   600 lemma dot_rzero[simp]: "x \<bullet> 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector
   601 lemma dot_pos_le[simp]: "(0::'a\<Colon>ordered_ring_strict) <= x \<bullet> x"
   602   by (simp add: dot_def setsum_nonneg)
   603 
   604 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)"
   605 using fS fp setsum_nonneg[OF fp]
   606 proof (induct set: finite)
   607   case empty thus ?case by simp
   608 next
   609   case (insert x F)
   610   from insert.prems have Fx: "f x \<ge> 0" and Fp: "\<forall> a \<in> F. f a \<ge> 0" by simp_all
   611   from insert.hyps Fp setsum_nonneg[OF Fp]
   612   have h: "setsum f F = 0 \<longleftrightarrow> (\<forall>a \<in>F. f a = 0)" by metis
   613   from sum_nonneg_eq_zero_iff[OF Fx  setsum_nonneg[OF Fp]] insert.hyps(1,2)
   614   show ?case by (simp add: h)
   615 qed
   616 
   617 lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) = 0"
   618 proof-
   619   {assume f: "finite (UNIV :: 'n set)"
   620     let ?S = "{Suc 0 .. card (UNIV :: 'n set)}"
   621     have fS: "finite ?S" using f by simp
   622     have fp: "\<forall> i\<in> ?S. x$i * x$i>= 0" by simp
   623     have ?thesis by (vector dimindex_def f setsum_squares_eq_0_iff[OF fS fp])}
   624   moreover
   625   {assume "\<not> finite (UNIV :: 'n set)" then have ?thesis by (vector dimindex_def)}
   626   ultimately show ?thesis by metis
   627 qed
   628 
   629 lemma dot_pos_lt: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x] 
   630   by (auto simp add: le_less) 
   631 
   632 subsection{* The collapse of the general concepts to dimension one. *}
   633 
   634 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
   635   by (vector dimindex_def)
   636 
   637 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
   638   apply auto
   639   apply (erule_tac x= "x$1" in allE)
   640   apply (simp only: vector_one[symmetric])
   641   done
   642 
   643 lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
   644   by (simp add: vector_norm_def dimindex_def)
   645 
   646 lemma norm_real: "norm(x::real ^ 1) = abs(x$1)" 
   647   by (simp add: norm_vector_1)
   648 
   649 text{* Metric *}
   650 
   651 text {* FIXME: generalize to arbitrary @{text real_normed_vector} types *}
   652 definition dist:: "real ^ 'n \<Rightarrow> real ^ 'n \<Rightarrow> real" where 
   653   "dist x y = norm (x - y)"
   654 
   655 lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
   656   using dimindex_ge_1[of "UNIV :: 1 set"]
   657   by (auto simp add: norm_real dist_def vector_component Cart_lambda_beta[where ?'a = "1"] )
   658 
   659 subsection {* A connectedness or intermediate value lemma with several applications. *}
   660 
   661 lemma connected_real_lemma:
   662   fixes f :: "real \<Rightarrow> real ^ 'n"
   663   assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
   664   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"
   665   and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
   666   and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
   667   and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
   668   shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
   669 proof-
   670   let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
   671   have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa) 
   672   have Sub: "\<exists>y. isUb UNIV ?S y" 
   673     apply (rule exI[where x= b])
   674     using ab fb e12 by (auto simp add: isUb_def setle_def)  
   675   from reals_complete[OF Se Sub] obtain l where 
   676     l: "isLub UNIV ?S l"by blast
   677   have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
   678     apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)    
   679     by (metis linorder_linear)
   680   have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
   681     apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
   682     by (metis linorder_linear not_le)
   683     have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
   684     have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
   685     have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo
   686     {assume le2: "f l \<in> e2"
   687       from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
   688       hence lap: "l - a > 0" using alb by arith
   689       from e2[rule_format, OF le2] obtain e where 
   690 	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
   691       from dst[OF alb e(1)] obtain d where 
   692 	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
   693       have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1) 
   694 	apply ferrack by arith
   695       then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
   696       from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
   697       from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
   698       moreover
   699       have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
   700       ultimately have False using e12 alb d' by auto}
   701     moreover
   702     {assume le1: "f l \<in> e1"
   703     from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
   704       hence blp: "b - l > 0" using alb by arith
   705       from e1[rule_format, OF le1] obtain e where 
   706 	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
   707       from dst[OF alb e(1)] obtain d where 
   708 	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
   709       have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo 
   710       then obtain d' where d': "d' > 0" "d' < d" by metis
   711       from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
   712       hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
   713       with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
   714       with l d' have False 
   715 	by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
   716     ultimately show ?thesis using alb by metis
   717 qed
   718 
   719 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"} *}
   720 
   721 lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
   722 proof-
   723   have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith 
   724   thus ?thesis by (simp add: ring_simps power2_eq_square)
   725 qed
   726 
   727 lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
   728   using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_def, rule_format, of e x] apply (auto simp add: power2_eq_square)
   729   apply (rule_tac x="s" in exI)
   730   apply auto
   731   apply (erule_tac x=y in allE)
   732   apply auto
   733   done
   734 
   735 lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
   736   using real_sqrt_le_iff[of x "y^2"] by simp
   737 
   738 lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
   739   using real_sqrt_le_mono[of "x^2" y] by simp
   740 
   741 lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
   742   using real_sqrt_less_mono[of "x^2" y] by simp
   743 
   744 lemma sqrt_even_pow2: assumes n: "even n" 
   745   shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
   746 proof-
   747   from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2 
   748     by (auto simp add: nat_number) 
   749   from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
   750     by (simp only: power_mult[symmetric] mult_commute)
   751   then show ?thesis  using m by simp 
   752 qed
   753 
   754 lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
   755   apply (cases "x = 0", simp_all)
   756   using sqrt_divide_self_eq[of x]
   757   apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps)
   758   done
   759 
   760 text{* Hence derive more interesting properties of the norm. *}
   761 
   762 lemma norm_0: "norm (0::real ^ 'n) = 0"
   763   by (rule norm_zero)
   764 
   765 lemma norm_mul: "norm(a *s x) = abs(a) * norm x"
   766   by (simp add: vector_norm_def vector_component setL2_right_distrib
   767            abs_mult cong: strong_setL2_cong)
   768 lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
   769   by (simp add: vector_norm_def dot_def setL2_def power2_eq_square)
   770 lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"
   771   by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
   772 lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
   773   by (simp add: real_vector_norm_def)
   774 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
   775 lemma vector_mul_eq_0: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
   776   by vector
   777 lemma vector_mul_lcancel: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
   778   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
   779 lemma vector_mul_rcancel: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
   780   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
   781 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
   782   by (metis vector_mul_lcancel)
   783 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
   784   by (metis vector_mul_rcancel)
   785 lemma norm_cauchy_schwarz: "x \<bullet> y <= norm x * norm y"
   786 proof-
   787   {assume "norm x = 0"
   788     hence ?thesis by (simp add: dot_lzero dot_rzero)}
   789   moreover
   790   {assume "norm y = 0" 
   791     hence ?thesis by (simp add: dot_lzero dot_rzero)}
   792   moreover
   793   {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
   794     let ?z = "norm y *s x - norm x *s y"
   795     from h have p: "norm x * norm y > 0" by (metis norm_ge_zero le_less zero_compare_simps)
   796     from dot_pos_le[of ?z]
   797     have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"
   798       apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)
   799       by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym)
   800     hence "x\<bullet>y \<le> (norm x ^2 * norm y ^2) / (norm x * norm y)" using p
   801       by (simp add: field_simps)
   802     hence ?thesis using h by (simp add: power2_eq_square)}
   803   ultimately show ?thesis by metis
   804 qed
   805 
   806 lemma norm_cauchy_schwarz_abs: "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
   807   using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
   808   by (simp add: real_abs_def dot_rneg)
   809 
   810 lemma norm_triangle_sub: "norm (x::real ^'n) <= norm(y) + norm(x - y)"
   811   using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
   812 lemma norm_triangle_le: "norm(x::real ^'n) + norm y <= e ==> norm(x + y) <= e"
   813   by (metis order_trans norm_triangle_ineq)
   814 lemma norm_triangle_lt: "norm(x::real ^'n) + norm(y) < e ==> norm(x + y) < e"
   815   by (metis basic_trans_rules(21) norm_triangle_ineq)
   816 
   817 lemma setsum_delta: 
   818   assumes fS: "finite S"
   819   shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
   820 proof-
   821   let ?f = "(\<lambda>k. if k=a then b k else 0)"
   822   {assume a: "a \<notin> S"
   823     hence "\<forall> k\<in> S. ?f k = 0" by simp
   824     hence ?thesis  using a by simp}
   825   moreover 
   826   {assume a: "a \<in> S"
   827     let ?A = "S - {a}"
   828     let ?B = "{a}"
   829     have eq: "S = ?A \<union> ?B" using a by blast 
   830     have dj: "?A \<inter> ?B = {}" by simp
   831     from fS have fAB: "finite ?A" "finite ?B" by auto  
   832     have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
   833       using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
   834       by simp
   835     then have ?thesis  using a by simp}
   836   ultimately show ?thesis by blast
   837 qed
   838   
   839 lemma component_le_norm: "i \<in> {1 .. dimindex(UNIV :: 'n set)} ==> \<bar>x$i\<bar> <= norm (x::real ^ 'n)"
   840   apply (simp add: vector_norm_def)
   841   apply (rule member_le_setL2, simp_all)
   842   done
   843 
   844 lemma norm_bound_component_le: "norm(x::real ^ 'n) <= e
   845                 ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> <= e"
   846   by (metis component_le_norm order_trans)
   847 
   848 lemma norm_bound_component_lt: "norm(x::real ^ 'n) < e
   849                 ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> < e"
   850   by (metis component_le_norm basic_trans_rules(21))
   851 
   852 lemma norm_le_l1: "norm (x:: real ^'n) <= setsum(\<lambda>i. \<bar>x$i\<bar>) {1..dimindex(UNIV::'n set)}"
   853   by (simp add: vector_norm_def setL2_le_setsum)
   854 
   855 lemma real_abs_norm: "\<bar> norm x\<bar> = norm (x :: real ^'n)" 
   856   by (rule abs_norm_cancel)
   857 lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n) - norm y\<bar> <= norm(x - y)"
   858   by (rule norm_triangle_ineq3)
   859 lemma norm_le: "norm(x::real ^ 'n) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
   860   by (simp add: real_vector_norm_def)
   861 lemma norm_lt: "norm(x::real ^'n) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
   862   by (simp add: real_vector_norm_def)
   863 lemma norm_eq: "norm (x::real ^'n) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
   864   by (simp add: order_eq_iff norm_le)
   865 lemma norm_eq_1: "norm(x::real ^ 'n) = 1 \<longleftrightarrow> x \<bullet> x = 1"
   866   by (simp add: real_vector_norm_def)
   867 
   868 text{* Squaring equations and inequalities involving norms.  *}
   869 
   870 lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
   871   by (simp add: real_vector_norm_def  dot_pos_le )
   872 
   873 lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
   874   by (auto simp add: real_vector_norm_def)
   875 
   876 lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
   877 proof-
   878   have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: ring_simps power2_eq_square)
   879   also have "\<dots> \<longleftrightarrow> \<bar>x\<bar> \<le> \<bar>y\<bar>" apply (simp add: zero_compare_simps real_abs_def not_less) by arith
   880 finally show ?thesis ..
   881 qed
   882 
   883 lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
   884   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
   885   using norm_ge_zero[of x]
   886   apply arith
   887   done
   888 
   889 lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2" 
   890   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
   891   using norm_ge_zero[of x]
   892   apply arith
   893   done
   894 
   895 lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
   896   by (metis not_le norm_ge_square)
   897 lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
   898   by (metis norm_le_square not_less)
   899 
   900 text{* Dot product in terms of the norm rather than conversely. *}
   901 
   902 lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
   903   by (simp add: norm_pow_2 dot_ladd dot_radd dot_sym)
   904 
   905 lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
   906   by (simp add: norm_pow_2 dot_ladd dot_radd dot_lsub dot_rsub dot_sym)
   907 
   908 
   909 text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
   910 
   911 lemma vector_eq: "(x:: real ^ 'n) = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y\<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
   912 proof
   913   assume "?lhs" then show ?rhs by simp
   914 next
   915   assume ?rhs
   916   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y\<bullet> y = 0" by simp
   917   hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" 
   918     by (simp add: dot_rsub dot_lsub dot_sym)
   919   then have "(x - y) \<bullet> (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub)
   920   then show "x = y" by (simp add: dot_eq_0)
   921 qed
   922 
   923 
   924 subsection{* General linear decision procedure for normed spaces. *}
   925 
   926 lemma norm_cmul_rule_thm: "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(c *s x)"
   927   apply (clarsimp simp add: norm_mul)
   928   apply (rule mult_mono1)
   929   apply simp_all
   930   done
   931 
   932 lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n) \<Longrightarrow> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)"
   933   apply (rule norm_triangle_le) by simp
   934 
   935 lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"
   936   by (simp add: ring_simps)
   937 
   938 lemma pth_1: "(x::real^'n) == 1 *s x" by (simp only: vector_smult_lid)
   939 lemma pth_2: "x - (y::real^'n) == x + -y" by (atomize (full)) simp
   940 lemma pth_3: "(-x::real^'n) == -1 *s x" by vector
   941 lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+
   942 lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector
   943 lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps)
   944 lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all 
   945 lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps) 
   946 lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z"
   947   "c *s x + (d *s x + z) == (c + d) *s x + z"
   948   "(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+
   949 lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector
   950 lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y" 
   951   "(c *s x + z) + d *s y == c *s x + (z + d *s y)"
   952   "c *s x + (d *s y + z) == c *s x + (d *s y + z)"
   953   "(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))"
   954   by ((atomize (full)), vector)+
   955 lemma pth_c: "(c::real) *s x + d *s y == d *s y + c *s x"
   956   "(c *s x + z) + d *s y == d *s y + (c *s x + z)"
   957   "c *s x + (d *s y + z) == d *s y + (c *s x + z)"
   958   "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+
   959 lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector
   960 
   961 lemma norm_imp_pos_and_ge: "norm (x::real ^ 'n) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
   962   by (atomize) (auto simp add: norm_ge_zero)
   963 
   964 lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
   965 
   966 lemma norm_pths: 
   967   "(x::real ^'n) = y \<longleftrightarrow> norm (x - y) \<le> 0"
   968   "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
   969   using norm_ge_zero[of "x - y"] by auto
   970 
   971 use "normarith.ML"
   972 
   973 method_setup norm = {* Method.ctxt_args (Method.SIMPLE_METHOD' o NormArith.norm_arith_tac)
   974 *} "Proves simple linear statements about vector norms"
   975 
   976 
   977 
   978 text{* Hence more metric properties. *}
   979 
   980 lemma dist_refl: "dist x x = 0" by norm
   981 
   982 lemma dist_sym: "dist x y = dist y x"by norm
   983 
   984 lemma dist_pos_le: "0 <= dist x y" by norm
   985 
   986 lemma dist_triangle: "dist x z <= dist x y + dist y z" by norm
   987 
   988 lemma dist_triangle_alt: "dist y z <= dist x y + dist x z" by norm
   989 
   990 lemma dist_eq_0: "dist x y = 0 \<longleftrightarrow> x = y" by norm
   991 
   992 lemma dist_pos_lt: "x \<noteq> y ==> 0 < dist x y" by norm 
   993 lemma dist_nz:  "x \<noteq> y \<longleftrightarrow> 0 < dist x y" by norm 
   994 
   995 lemma dist_triangle_le: "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e" by norm 
   996 
   997 lemma dist_triangle_lt: "dist x z + dist y z < e ==> dist x y < e" by norm 
   998 
   999 lemma dist_triangle_half_l: "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 ==> dist x1 x2 < e" by norm 
  1000 
  1001 lemma dist_triangle_half_r: "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 ==> dist x1 x2 < e" by norm 
  1002 
  1003 lemma dist_triangle_add: "dist (x + y) (x' + y') <= dist x x' + dist y y'"
  1004   by norm 
  1005 
  1006 lemma dist_mul: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y" 
  1007   unfolding dist_def vector_ssub_ldistrib[symmetric] norm_mul .. 
  1008 
  1009 lemma dist_triangle_add_half: " dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 ==> dist(x + y) (x' + y') < e" by norm 
  1010 
  1011 lemma dist_le_0: "dist x y <= 0 \<longleftrightarrow> x = y" by norm 
  1012 
  1013 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
  1014   apply vector
  1015   apply auto
  1016   apply (cases "finite S")
  1017   apply (rule finite_induct[of S])
  1018   apply (auto simp add: vector_component zero_index)
  1019   done
  1020 
  1021 lemma setsum_clauses: 
  1022   shows "setsum f {} = 0"
  1023   and "finite S \<Longrightarrow> setsum f (insert x S) =
  1024                  (if x \<in> S then setsum f S else f x + setsum f S)"
  1025   by (auto simp add: insert_absorb)
  1026 
  1027 lemma setsum_cmul: 
  1028   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
  1029   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
  1030   by (simp add: setsum_eq Cart_eq Cart_lambda_beta vector_component setsum_right_distrib)
  1031 
  1032 lemma setsum_component: 
  1033   fixes f:: " 'a \<Rightarrow> ('b::semiring_1) ^'n"
  1034   assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
  1035   shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
  1036   using i by (simp add: setsum_eq Cart_lambda_beta)
  1037 
  1038   (* This needs finiteness assumption due to the definition of fold!!! *)
  1039 
  1040 lemma setsum_superset:
  1041   assumes fb: "finite B" and ab: "A \<subseteq> B" 
  1042   and f0: "\<forall>x \<in> B - A. f x = 0"
  1043   shows "setsum f B = setsum f A"
  1044 proof-
  1045   from ab fb have fa: "finite A" by (metis finite_subset)
  1046   from fb have fba: "finite (B - A)" by (metis finite_Diff)
  1047   have d: "A \<inter> (B - A) = {}" by blast
  1048   from ab have b: "B = A \<union> (B - A)" by blast
  1049   from setsum_Un_disjoint[OF fa fba d, of f] b
  1050     setsum_0'[OF f0]
  1051   show "setsum f B = setsum f A" by simp
  1052 qed
  1053 
  1054 lemma setsum_restrict_set:
  1055   assumes fA: "finite A"
  1056   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
  1057 proof-
  1058   from fA have fab: "finite (A \<inter> B)" by auto
  1059   have aba: "A \<inter> B \<subseteq> A" by blast
  1060   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
  1061   from setsum_superset[OF fA aba, of ?g]
  1062   show ?thesis by simp
  1063 qed
  1064 
  1065 lemma setsum_cases:
  1066   assumes fA: "finite A"
  1067   shows "setsum (\<lambda>x. if x \<in> B then f x else g x) A =
  1068          setsum f (A \<inter> B) + setsum g (A \<inter> - B)"
  1069 proof-
  1070   have a: "A = A \<inter> B \<union> A \<inter> -B" "(A \<inter> B) \<inter> (A \<inter> -B) = {}" 
  1071     by blast+
  1072   from fA 
  1073   have f: "finite (A \<inter> B)" "finite (A \<inter> -B)" by auto
  1074   let ?g = "\<lambda>x. if x \<in> B then f x else g x"
  1075   from setsum_Un_disjoint[OF f a(2), of ?g] a(1)
  1076   show ?thesis by simp
  1077 qed
  1078 
  1079 lemma setsum_norm: 
  1080   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1081   assumes fS: "finite S"
  1082   shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
  1083 proof(induct rule: finite_induct[OF fS])
  1084   case 1 thus ?case by simp
  1085 next
  1086   case (2 x S)
  1087   from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
  1088   also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
  1089     using "2.hyps" by simp
  1090   finally  show ?case  using "2.hyps" by simp
  1091 qed
  1092 
  1093 lemma real_setsum_norm: 
  1094   fixes f :: "'a \<Rightarrow> real ^'n"
  1095   assumes fS: "finite S"
  1096   shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
  1097 proof(induct rule: finite_induct[OF fS])
  1098   case 1 thus ?case by simp
  1099 next
  1100   case (2 x S)
  1101   from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
  1102   also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
  1103     using "2.hyps" by simp
  1104   finally  show ?case  using "2.hyps" by simp
  1105 qed
  1106 
  1107 lemma setsum_norm_le: 
  1108   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1109   assumes fS: "finite S"
  1110   and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
  1111   shows "norm (setsum f S) \<le> setsum g S"
  1112 proof-
  1113   from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S" 
  1114     by - (rule setsum_mono, simp)
  1115   then show ?thesis using setsum_norm[OF fS, of f] fg
  1116     by arith
  1117 qed
  1118 
  1119 lemma real_setsum_norm_le: 
  1120   fixes f :: "'a \<Rightarrow> real ^ 'n"
  1121   assumes fS: "finite S"
  1122   and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
  1123   shows "norm (setsum f S) \<le> setsum g S"
  1124 proof-
  1125   from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S" 
  1126     by - (rule setsum_mono, simp)
  1127   then show ?thesis using real_setsum_norm[OF fS, of f] fg
  1128     by arith
  1129 qed
  1130 
  1131 lemma setsum_norm_bound:
  1132   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1133   assumes fS: "finite S"
  1134   and K: "\<forall>x \<in> S. norm (f x) \<le> K"
  1135   shows "norm (setsum f S) \<le> of_nat (card S) * K"
  1136   using setsum_norm_le[OF fS K] setsum_constant[symmetric]
  1137   by simp
  1138 
  1139 lemma real_setsum_norm_bound:
  1140   fixes f :: "'a \<Rightarrow> real ^ 'n"
  1141   assumes fS: "finite S"
  1142   and K: "\<forall>x \<in> S. norm (f x) \<le> K"
  1143   shows "norm (setsum f S) \<le> of_nat (card S) * K"
  1144   using real_setsum_norm_le[OF fS K] setsum_constant[symmetric]
  1145   by simp
  1146 
  1147 lemma setsum_vmul:
  1148   fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector,semiring, mult_zero}"
  1149   assumes fS: "finite S"
  1150   shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
  1151 proof(induct rule: finite_induct[OF fS])
  1152   case 1 then show ?case by (simp add: vector_smult_lzero)
  1153 next
  1154   case (2 x F)
  1155   from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v" 
  1156     by simp
  1157   also have "\<dots> = f x *s v + setsum f F *s v" 
  1158     by (simp add: vector_sadd_rdistrib)
  1159   also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp
  1160   finally show ?case .
  1161 qed
  1162 
  1163 (* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \<Rightarrow> real ^'n"]  ---
  1164  Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *)
  1165 
  1166 lemma setsum_add_split: assumes mn: "(m::nat) \<le> n + 1"
  1167   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
  1168 proof-
  1169   let ?A = "{m .. n}"
  1170   let ?B = "{n + 1 .. n + p}"
  1171   have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto 
  1172   have d: "?A \<inter> ?B = {}" by auto
  1173   from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto
  1174 qed
  1175 
  1176 lemma setsum_reindex_nonzero: 
  1177   assumes fS: "finite S"
  1178   and nz: "\<And> x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
  1179   shows "setsum h (f ` S) = setsum (h o f) S"
  1180 using nz
  1181 proof(induct rule: finite_induct[OF fS])
  1182   case 1 thus ?case by simp
  1183 next
  1184   case (2 x F) 
  1185   {assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
  1186     then obtain y where y: "y \<in> F" "f x = f y" by auto 
  1187     from "2.hyps" y have xy: "x \<noteq> y" by auto
  1188     
  1189     from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
  1190     have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
  1191     also have "\<dots> = setsum (h o f) (insert x F)" 
  1192       using "2.hyps" "2.prems" h0  by auto 
  1193     finally have ?case .}
  1194   moreover
  1195   {assume fxF: "f x \<notin> f ` F"
  1196     have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" 
  1197       using fxF "2.hyps" by simp 
  1198     also have "\<dots> = setsum (h o f) (insert x F)"  
  1199       using "2.hyps" "2.prems" fxF
  1200       apply auto apply metis done
  1201     finally have ?case .}
  1202   ultimately show ?case by blast
  1203 qed
  1204 
  1205 lemma setsum_Un_nonzero:
  1206   assumes fS: "finite S" and fF: "finite F"
  1207   and f: "\<forall> x\<in> S \<inter> F . f x = (0::'a::ab_group_add)"
  1208   shows "setsum f (S \<union> F) = setsum f S + setsum f F"
  1209   using setsum_Un[OF fS fF, of f] setsum_0'[OF f] by simp
  1210 
  1211 lemma setsum_natinterval_left:
  1212   assumes mn: "(m::nat) <= n" 
  1213   shows "setsum f {m..n} = f m + setsum f {m + 1..n}"
  1214 proof-
  1215   from mn have "{m .. n} = insert m {m+1 .. n}" by auto
  1216   then show ?thesis by auto
  1217 qed
  1218 
  1219 lemma setsum_natinterval_difff: 
  1220   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
  1221   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
  1222           (if m <= n then f m - f(n + 1) else 0)"
  1223 by (induct n, auto simp add: ring_simps not_le le_Suc_eq)
  1224 
  1225 lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
  1226 
  1227 lemma setsum_setsum_restrict:
  1228   "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"
  1229   apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
  1230   by (rule setsum_commute)
  1231 
  1232 lemma setsum_image_gen: assumes fS: "finite S"
  1233   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
  1234 proof-
  1235   {fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto}
  1236   note th0 = this
  1237   have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S" 
  1238     apply (rule setsum_cong2) 
  1239     by (simp add: th0)
  1240   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
  1241     apply (rule setsum_setsum_restrict[OF fS])
  1242     by (rule finite_imageI[OF fS])
  1243   finally show ?thesis .
  1244 qed
  1245 
  1246     (* FIXME: Here too need stupid finiteness assumption on T!!! *)
  1247 lemma setsum_group:
  1248   assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
  1249   shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
  1250   
  1251 apply (subst setsum_image_gen[OF fS, of g f])
  1252 apply (rule setsum_superset[OF fT fST])
  1253 by (auto intro: setsum_0')
  1254 
  1255 (* FIXME: Change the name to fold_image\<dots> *)
  1256 lemma (in comm_monoid_mult) fold_1': "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
  1257   apply (induct set: finite)
  1258   apply simp by (auto simp add: fold_image_insert)
  1259 
  1260 lemma (in comm_monoid_mult) fold_union_nonzero:
  1261   assumes fS: "finite S" and fT: "finite T"
  1262   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
  1263   shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
  1264 proof-
  1265   have "fold_image op * f 1 (S \<inter> T) = 1" 
  1266     apply (rule fold_1')
  1267     using fS fT I0 by auto 
  1268   with fold_image_Un_Int[OF fS fT] show ?thesis by simp
  1269 qed
  1270 
  1271 lemma setsum_union_nonzero:  
  1272   assumes fS: "finite S" and fT: "finite T"
  1273   and I0: "\<forall>x \<in> S\<inter>T. f x = 0"
  1274   shows "setsum f (S \<union> T) = setsum f S  + setsum f T"
  1275   using fS fT
  1276   apply (simp add: setsum_def)
  1277   apply (rule comm_monoid_add.fold_union_nonzero)
  1278   using I0 by auto
  1279 
  1280 lemma setprod_union_nonzero:  
  1281   assumes fS: "finite S" and fT: "finite T"
  1282   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
  1283   shows "setprod f (S \<union> T) = setprod f S  * setprod f T"
  1284   using fS fT
  1285   apply (simp add: setprod_def)
  1286   apply (rule fold_union_nonzero)
  1287   using I0 by auto
  1288 
  1289 lemma setsum_unions_nonzero: 
  1290   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
  1291   and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"
  1292   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
  1293   using fSS f0
  1294 proof(induct rule: finite_induct[OF fS])
  1295   case 1 thus ?case by simp
  1296 next
  1297   case (2 T F)
  1298   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F" 
  1299     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by (auto simp add: finite_insert)
  1300   from fTF have fUF: "finite (\<Union>F)" by (auto intro: finite_Union)
  1301   from "2.prems" TF fTF
  1302   show ?case 
  1303     by (auto simp add: H[symmetric] intro: setsum_union_nonzero[OF fTF(1) fUF, of f])
  1304 qed
  1305 
  1306   (* FIXME : Copied from Pocklington --- should be moved to Finite_Set!!!!!!!! *)
  1307 
  1308 
  1309 lemma (in comm_monoid_mult) fold_related: 
  1310   assumes Re: "R e e" 
  1311   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
  1312   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
  1313   shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
  1314   using fS by (rule finite_subset_induct) (insert assms, auto)
  1315 
  1316   (* FIXME: I think we can get rid of the finite assumption!! *)	
  1317 lemma (in comm_monoid_mult) 
  1318   fold_eq_general:
  1319   assumes fS: "finite S"
  1320   and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
  1321   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
  1322   shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
  1323 proof-
  1324   from h f12 have hS: "h ` S = S'" by auto
  1325   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
  1326     from f12 h H  have "x = y" by auto }
  1327   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
  1328   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
  1329   from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
  1330   also have "\<dots> = fold_image (op *) (f2 o h) e S" 
  1331     using fold_image_reindex[OF fS hinj, of f2 e] .
  1332   also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
  1333     by blast
  1334   finally show ?thesis ..
  1335 qed
  1336 
  1337 lemma (in comm_monoid_mult) fold_eq_general_inverses:
  1338   assumes fS: "finite S" 
  1339   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
  1340   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
  1341   shows "fold_image (op *) f e S = fold_image (op *) g e T"
  1342   using fold_eq_general[OF fS, of T h g f e] kh hk by metis
  1343 
  1344 lemma setsum_eq_general_reverses:
  1345   assumes fS: "finite S" and fT: "finite T"
  1346   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
  1347   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
  1348   shows "setsum f S = setsum g T"
  1349   apply (simp add: setsum_def fS fT)
  1350   apply (rule comm_monoid_add.fold_eq_general_inverses[OF fS])
  1351   apply (erule kh)
  1352   apply (erule hk)
  1353   done
  1354 
  1355 lemma vsum_norm_allsubsets_bound:
  1356   fixes f:: "'a \<Rightarrow> real ^'n"
  1357   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e" 
  1358   shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real (dimindex(UNIV :: 'n set)) *  e"
  1359 proof-
  1360   let ?d = "real (dimindex (UNIV ::'n set))"
  1361   let ?nf = "\<lambda>x. norm (f x)"
  1362   let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
  1363   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"
  1364     by (rule setsum_commute)
  1365   have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
  1366   have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
  1367     apply (rule setsum_mono)
  1368     by (rule norm_le_l1)
  1369   also have "\<dots> \<le> 2 * ?d * e"
  1370     unfolding th0 th1
  1371   proof(rule setsum_bounded)
  1372     fix i assume i: "i \<in> ?U"
  1373     let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
  1374     let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
  1375     have thp: "P = ?Pp \<union> ?Pn" by auto
  1376     have thp0: "?Pp \<inter> ?Pn ={}" by auto
  1377     have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
  1378     have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
  1379       using i component_le_norm[OF i, of "setsum (\<lambda>x. f x) ?Pp"]  fPs[OF PpP]
  1380       by (auto simp add: setsum_component intro: abs_le_D1)
  1381     have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
  1382       using i component_le_norm[OF i, of "setsum (\<lambda>x. - f x) ?Pn"]  fPs[OF PnP]
  1383       by (auto simp add: setsum_negf setsum_component vector_component intro: abs_le_D1)
  1384     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" 
  1385       apply (subst thp)
  1386       apply (rule setsum_Un_nonzero) 
  1387       using fP thp0 by auto
  1388     also have "\<dots> \<le> 2*e" using Pne Ppe by arith
  1389     finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
  1390   qed
  1391   finally show ?thesis .
  1392 qed
  1393 
  1394 lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> (y::'a::{comm_ring}^'n) = setsum (\<lambda>x. f x \<bullet> y) S "
  1395   by (induct rule: finite_induct, auto simp add: dot_lzero dot_ladd)
  1396 
  1397 lemma dot_rsum: "finite S \<Longrightarrow> (y::'a::{comm_ring}^'n) \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
  1398   by (induct rule: finite_induct, auto simp add: dot_rzero dot_radd)
  1399 
  1400 subsection{* Basis vectors in coordinate directions. *}
  1401 
  1402 
  1403 definition "basis k = (\<chi> i. if i = k then 1 else 0)"
  1404 
  1405 lemma delta_mult_idempotent: 
  1406   "(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)
  1407 
  1408 lemma norm_basis:
  1409   assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
  1410   shows "norm (basis k :: real ^'n) = 1"
  1411   using k 
  1412   apply (simp add: basis_def real_vector_norm_def dot_def)
  1413   apply (vector delta_mult_idempotent)
  1414   using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "k" "\<lambda>k. 1::real"]
  1415   apply auto
  1416   done
  1417 
  1418 lemma norm_basis_1: "norm(basis 1 :: real ^'n) = 1"
  1419   apply (simp add: basis_def real_vector_norm_def dot_def)
  1420   apply (vector delta_mult_idempotent)
  1421   using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "1" "\<lambda>k. 1::real"] dimindex_nonzero[of "UNIV :: 'n set"]
  1422   apply auto
  1423   done
  1424 
  1425 lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c"
  1426   apply (rule exI[where x="c *s basis 1"])
  1427   by (simp only: norm_mul norm_basis_1)
  1428 
  1429 lemma vector_choose_dist: assumes e: "0 <= e" 
  1430   shows "\<exists>(y::real^'n). dist x y = e"
  1431 proof-
  1432   from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"
  1433     by blast
  1434   then have "dist x (x - c) = e" by (simp add: dist_def)
  1435   then show ?thesis by blast
  1436 qed
  1437 
  1438 lemma basis_inj: "inj_on (basis :: nat \<Rightarrow> real ^'n) {1 .. dimindex (UNIV :: 'n set)}"
  1439   by (auto simp add: inj_on_def basis_def Cart_eq Cart_lambda_beta)
  1440 
  1441 lemma basis_component: "i \<in> {1 .. dimindex(UNIV:: 'n set)} ==> (basis k ::('a::semiring_1)^'n)$i = (if k=i then 1 else 0)"
  1442   by (simp add: basis_def Cart_lambda_beta)
  1443 
  1444 lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
  1445   by auto
  1446 
  1447 lemma basis_expansion:
  1448   "setsum (\<lambda>i. (x$i) *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
  1449   by (auto simp add: Cart_eq basis_component[where ?'n = "'n"] setsum_component vector_component cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
  1450 
  1451 lemma basis_expansion_unique: 
  1452   "setsum (\<lambda>i. f i *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i\<in>{1 .. dimindex(UNIV:: 'n set)}. f i = x$i)"
  1453   by (simp add: Cart_eq setsum_component vector_component basis_component setsum_delta cond_value_iff cong del: if_weak_cong)
  1454 
  1455 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
  1456   by auto
  1457 
  1458 lemma dot_basis:
  1459   assumes i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
  1460   shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i :: 'a^'n) = (x$i :: 'a::semiring_1)"
  1461   using i
  1462   by (auto simp add: dot_def basis_def Cart_lambda_beta cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)
  1463 
  1464 lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> i \<notin> {1..dimindex(UNIV ::'n set)}"
  1465   by (auto simp add: Cart_eq basis_component zero_index)
  1466 
  1467 lemma basis_nonzero: 
  1468   assumes k: "k \<in> {1 .. dimindex(UNIV ::'n set)}"
  1469   shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
  1470   using k by (simp add: basis_eq_0)
  1471 
  1472 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = (z::'a::semiring_1^'n)"
  1473   apply (auto simp add: Cart_eq dot_basis)
  1474   apply (erule_tac x="basis i" in allE)
  1475   apply (simp add: dot_basis)
  1476   apply (subgoal_tac "y = z")
  1477   apply simp
  1478   apply vector
  1479   done
  1480 
  1481 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = (y::'a::semiring_1^'n)"
  1482   apply (auto simp add: Cart_eq dot_basis)
  1483   apply (erule_tac x="basis i" in allE)
  1484   apply (simp add: dot_basis)
  1485   apply (subgoal_tac "x = y")
  1486   apply simp
  1487   apply vector
  1488   done
  1489 
  1490 subsection{* Orthogonality. *}
  1491 
  1492 definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
  1493 
  1494 lemma orthogonal_basis:
  1495   assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}" 
  1496   shows "orthogonal (basis i :: 'a^'n) x \<longleftrightarrow> x$i = (0::'a::ring_1)"
  1497   using i
  1498   by (auto simp add: orthogonal_def dot_def basis_def Cart_lambda_beta cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
  1499 
  1500 lemma orthogonal_basis_basis:
  1501   assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}" 
  1502   and j: "j \<in> {1 .. dimindex(UNIV ::'n set)}" 
  1503   shows "orthogonal (basis i :: 'a::ring_1^'n) (basis j) \<longleftrightarrow> i \<noteq> j" 
  1504   unfolding orthogonal_basis[OF i] basis_component[OF i] by simp
  1505 
  1506   (* FIXME : Maybe some of these require less than comm_ring, but not all*)
  1507 lemma orthogonal_clauses:
  1508   "orthogonal a (0::'a::comm_ring ^'n)"
  1509   "orthogonal a x ==> orthogonal a (c *s x)"
  1510   "orthogonal a x ==> orthogonal a (-x)"
  1511   "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x + y)"
  1512   "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x - y)"
  1513   "orthogonal 0 a"
  1514   "orthogonal x a ==> orthogonal (c *s x) a"
  1515   "orthogonal x a ==> orthogonal (-x) a"
  1516   "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x + y) a"
  1517   "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x - y) a"
  1518   unfolding orthogonal_def dot_rneg dot_rmult dot_radd dot_rsub
  1519   dot_lzero dot_rzero dot_lneg dot_lmult dot_ladd dot_lsub
  1520   by simp_all
  1521 
  1522 lemma orthogonal_commute: "orthogonal (x::'a::{ab_semigroup_mult,comm_monoid_add} ^'n)y \<longleftrightarrow> orthogonal y x"
  1523   by (simp add: orthogonal_def dot_sym)
  1524 
  1525 subsection{* Explicit vector construction from lists. *}
  1526 
  1527 lemma Cart_lambda_beta_1[simp]: "(Cart_lambda g)$1 = g 1"
  1528   apply (rule Cart_lambda_beta[rule_format])
  1529   using dimindex_ge_1 apply auto done
  1530 
  1531 lemma Cart_lambda_beta_1'[simp]: "(Cart_lambda g)$(Suc 0) = g 1"
  1532   by (simp only: One_nat_def[symmetric] Cart_lambda_beta_1)
  1533 
  1534 definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"
  1535 
  1536 lemma vector_1: "(vector[x]) $1 = x"
  1537   using dimindex_ge_1
  1538   by (auto simp add: vector_def Cart_lambda_beta[rule_format])
  1539 lemma dimindex_2[simp]: "2 \<in> {1 .. dimindex (UNIV :: 2 set)}"
  1540   by (auto simp add: dimindex_def)
  1541 lemma dimindex_2'[simp]: "2 \<in> {Suc 0 .. dimindex (UNIV :: 2 set)}"
  1542   by (auto simp add: dimindex_def)
  1543 lemma dimindex_3[simp]: "2 \<in> {1 .. dimindex (UNIV :: 3 set)}" "3 \<in> {1 .. dimindex (UNIV :: 3 set)}"
  1544   by (auto simp add: dimindex_def)
  1545 
  1546 lemma dimindex_3'[simp]: "2 \<in> {Suc 0 .. dimindex (UNIV :: 3 set)}" "3 \<in> {Suc 0 .. dimindex (UNIV :: 3 set)}"
  1547   by (auto simp add: dimindex_def)
  1548 
  1549 lemma vector_2:
  1550  "(vector[x,y]) $1 = x"
  1551  "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
  1552   apply (simp add: vector_def)
  1553   using Cart_lambda_beta[rule_format, OF dimindex_2, of "\<lambda>i. if i \<le> length [x,y] then [x,y] ! (i - 1) else (0::'a)"]
  1554   apply (simp only: vector_def )
  1555   apply auto
  1556   done
  1557 
  1558 lemma vector_3:
  1559  "(vector [x,y,z] ::('a::zero)^3)$1 = x"
  1560  "(vector [x,y,z] ::('a::zero)^3)$2 = y"
  1561  "(vector [x,y,z] ::('a::zero)^3)$3 = z"
  1562 apply (simp_all add: vector_def Cart_lambda_beta dimindex_3)
  1563   using Cart_lambda_beta[rule_format, OF dimindex_3(1), of "\<lambda>i. if i \<le> length [x,y,z] then [x,y,z] ! (i - 1) else (0::'a)"]   using Cart_lambda_beta[rule_format, OF dimindex_3(2), of "\<lambda>i. if i \<le> length [x,y,z] then [x,y,z] ! (i - 1) else (0::'a)"]
  1564   by simp_all
  1565 
  1566 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
  1567   apply auto
  1568   apply (erule_tac x="v$1" in allE)
  1569   apply (subgoal_tac "vector [v$1] = v")
  1570   apply simp
  1571   by (vector vector_def dimindex_def)
  1572 
  1573 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
  1574   apply auto
  1575   apply (erule_tac x="v$1" in allE)
  1576   apply (erule_tac x="v$2" in allE)
  1577   apply (subgoal_tac "vector [v$1, v$2] = v")
  1578   apply simp
  1579   apply (vector vector_def dimindex_def)
  1580   apply auto
  1581   apply (subgoal_tac "i = 1 \<or> i =2", auto)
  1582   done
  1583 
  1584 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
  1585   apply auto
  1586   apply (erule_tac x="v$1" in allE)
  1587   apply (erule_tac x="v$2" in allE)
  1588   apply (erule_tac x="v$3" in allE)
  1589   apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
  1590   apply simp
  1591   apply (vector vector_def dimindex_def)
  1592   apply auto
  1593   apply (subgoal_tac "i = 1 \<or> i =2 \<or> i = 3", auto)
  1594   done
  1595 
  1596 subsection{* Linear functions. *}
  1597 
  1598 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)"
  1599 
  1600 lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. (c::'a::comm_semiring) *s f x)"
  1601   by (vector linear_def Cart_eq Cart_lambda_beta[rule_format] ring_simps)
  1602 
  1603 lemma linear_compose_neg: "linear (f :: 'a ^'n \<Rightarrow> 'a::comm_ring ^'m) ==> linear (\<lambda>x. -(f(x)))" by (vector linear_def Cart_eq)
  1604 
  1605 lemma linear_compose_add: "linear (f :: 'a ^'n \<Rightarrow> 'a::semiring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
  1606   by (vector linear_def Cart_eq ring_simps)
  1607 
  1608 lemma linear_compose_sub: "linear (f :: 'a ^'n \<Rightarrow> 'a::ring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
  1609   by (vector linear_def Cart_eq ring_simps)
  1610 
  1611 lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
  1612   by (simp add: linear_def)
  1613 
  1614 lemma linear_id: "linear id" by (simp add: linear_def id_def)
  1615 
  1616 lemma linear_zero: "linear (\<lambda>x. 0::'a::semiring_1 ^ 'n)" by (simp add: linear_def)
  1617 
  1618 lemma linear_compose_setsum:
  1619   assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a :: 'a::semiring_1 ^ 'n \<Rightarrow> 'a ^ 'm)"
  1620   shows "linear(\<lambda>x. setsum (\<lambda>a. f a x :: 'a::semiring_1 ^'m) S)"
  1621   using lS
  1622   apply (induct rule: finite_induct[OF fS])
  1623   by (auto simp add: linear_zero intro: linear_compose_add)
  1624 
  1625 lemma linear_vmul_component:
  1626   fixes f:: "'a::semiring_1^'m \<Rightarrow> 'a^'n"
  1627   assumes lf: "linear f" and k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
  1628   shows "linear (\<lambda>x. f x $ k *s v)"
  1629   using lf k
  1630   apply (auto simp add: linear_def )
  1631   by (vector ring_simps)+
  1632 
  1633 lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)"
  1634   unfolding linear_def
  1635   apply clarsimp
  1636   apply (erule allE[where x="0::'a"])
  1637   apply simp
  1638   done
  1639 
  1640 lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def)
  1641 
  1642 lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \<Rightarrow> _) ==> f (-x) = - f x"
  1643   unfolding vector_sneg_minus1
  1644   using linear_cmul[of f] by auto 
  1645 
  1646 lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def) 
  1647 
  1648 lemma linear_sub: "linear (f::'a::ring_1 ^'n \<Rightarrow> _) ==> f(x - y) = f x - f y"
  1649   by (simp add: diff_def linear_add linear_neg)
  1650 
  1651 lemma linear_setsum: 
  1652   fixes f:: "'a::semiring_1^'n \<Rightarrow> _"
  1653   assumes lf: "linear f" and fS: "finite S"
  1654   shows "f (setsum g S) = setsum (f o g) S"
  1655 proof (induct rule: finite_induct[OF fS])
  1656   case 1 thus ?case by (simp add: linear_0[OF lf])
  1657 next
  1658   case (2 x F)
  1659   have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
  1660     by simp
  1661   also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
  1662   also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
  1663   finally show ?case .
  1664 qed
  1665 
  1666 lemma linear_setsum_mul:
  1667   fixes f:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m"
  1668   assumes lf: "linear f" and fS: "finite S"
  1669   shows "f (setsum (\<lambda>i. c i *s v i) S) = setsum (\<lambda>i. c i *s f (v i)) S"
  1670   using linear_setsum[OF lf fS, of "\<lambda>i. c i *s v i" , unfolded o_def]
  1671   linear_cmul[OF lf] by simp 
  1672 
  1673 lemma linear_injective_0:
  1674   assumes lf: "linear (f:: 'a::ring_1 ^ 'n \<Rightarrow> _)"
  1675   shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
  1676 proof-
  1677   have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
  1678   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
  1679   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)" 
  1680     by (simp add: linear_sub[OF lf])
  1681   also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
  1682   finally show ?thesis .
  1683 qed
  1684 
  1685 lemma linear_bounded:
  1686   fixes f:: "real ^'m \<Rightarrow> real ^'n"
  1687   assumes lf: "linear f"
  1688   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
  1689 proof-
  1690   let ?S = "{1..dimindex(UNIV:: 'm set)}"
  1691   let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
  1692   have fS: "finite ?S" by simp
  1693   {fix x:: "real ^ 'm"
  1694     let ?g = "(\<lambda>i::nat. (x$i) *s (basis i) :: real ^ 'm)"
  1695     have "norm (f x) = norm (f (setsum (\<lambda>i. (x$i) *s (basis i)) ?S))"
  1696       by (simp only:  basis_expansion)
  1697     also have "\<dots> = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)"
  1698       using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf]
  1699       by auto
  1700     finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)" .
  1701     {fix i assume i: "i \<in> ?S"
  1702       from component_le_norm[OF i, of x]
  1703       have "norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x"
  1704       unfolding norm_mul
  1705       apply (simp only: mult_commute)
  1706       apply (rule mult_mono)
  1707       by (auto simp add: ring_simps norm_ge_zero) }
  1708     then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
  1709     from real_setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (f (basis i))", OF th]
  1710     have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
  1711   then show ?thesis by blast
  1712 qed
  1713 
  1714 lemma linear_bounded_pos:
  1715   fixes f:: "real ^'n \<Rightarrow> real ^ 'm"
  1716   assumes lf: "linear f"
  1717   shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
  1718 proof-
  1719   from linear_bounded[OF lf] obtain B where 
  1720     B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
  1721   let ?K = "\<bar>B\<bar> + 1"
  1722   have Kp: "?K > 0" by arith
  1723     {assume C: "B < 0"
  1724       have "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff)
  1725       with C have "B * norm (1:: real ^ 'n) < 0"
  1726 	by (simp add: zero_compare_simps)
  1727       with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp
  1728     }
  1729     then have Bp: "B \<ge> 0" by ferrack
  1730     {fix x::"real ^ 'n"
  1731       have "norm (f x) \<le> ?K *  norm x"
  1732       using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
  1733       apply (auto simp add: ring_simps split add: abs_split)
  1734       apply (erule order_trans, simp)
  1735       done
  1736   }
  1737   then show ?thesis using Kp by blast
  1738 qed
  1739 
  1740 subsection{* Bilinear functions. *}
  1741 
  1742 definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
  1743 
  1744 lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
  1745   by (simp add: bilinear_def linear_def)
  1746 lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
  1747   by (simp add: bilinear_def linear_def)
  1748 
  1749 lemma bilinear_lmul: "bilinear h ==> h (c *s x) y = c *s (h x y)"
  1750   by (simp add: bilinear_def linear_def)
  1751 
  1752 lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)"
  1753   by (simp add: bilinear_def linear_def)
  1754 
  1755 lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)"
  1756   by (simp only: vector_sneg_minus1 bilinear_lmul)
  1757 
  1758 lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y"
  1759   by (simp only: vector_sneg_minus1 bilinear_rmul)
  1760 
  1761 lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
  1762   using add_imp_eq[of x y 0] by auto
  1763     
  1764 lemma bilinear_lzero: 
  1765   fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h 0 x = 0"
  1766   using bilinear_ladd[OF bh, of 0 0 x] 
  1767     by (simp add: eq_add_iff ring_simps)
  1768 
  1769 lemma bilinear_rzero: 
  1770   fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h x 0 = 0"
  1771   using bilinear_radd[OF bh, of x 0 0 ] 
  1772     by (simp add: eq_add_iff ring_simps)
  1773 
  1774 lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ 'n)) z = h x z - h y z"
  1775   by (simp  add: diff_def bilinear_ladd bilinear_lneg)
  1776 
  1777 lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ 'n)) = h z x - h z y"
  1778   by (simp  add: diff_def bilinear_radd bilinear_rneg)
  1779 
  1780 lemma bilinear_setsum:
  1781   fixes h:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m \<Rightarrow> 'a ^ 'k"
  1782   assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
  1783   shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
  1784 proof- 
  1785   have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
  1786     apply (rule linear_setsum[unfolded o_def])
  1787     using bh fS by (auto simp add: bilinear_def)
  1788   also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
  1789     apply (rule setsum_cong, simp)
  1790     apply (rule linear_setsum[unfolded o_def])
  1791     using bh fT by (auto simp add: bilinear_def)
  1792   finally show ?thesis unfolding setsum_cartesian_product .
  1793 qed
  1794 
  1795 lemma bilinear_bounded:
  1796   fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^ 'k"
  1797   assumes bh: "bilinear h"
  1798   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
  1799 proof- 
  1800   let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
  1801   let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
  1802   let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
  1803   have fM: "finite ?M" and fN: "finite ?N" by simp_all
  1804   {fix x:: "real ^ 'm" and  y :: "real^'n"
  1805     have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$i) *s basis i) ?M) (setsum (\<lambda>i. (y$i) *s basis i) ?N))" unfolding basis_expansion ..
  1806     also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$i) *s basis i) ((y$j) *s basis j)) (?M \<times> ?N))"  unfolding bilinear_setsum[OF bh fM fN] ..
  1807     finally have th: "norm (h x y) = \<dots>" .
  1808     have "norm (h x y) \<le> ?B * norm x * norm y"
  1809       apply (simp add: setsum_left_distrib th)
  1810       apply (rule real_setsum_norm_le)
  1811       using fN fM
  1812       apply simp
  1813       apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)
  1814       apply (rule mult_mono)
  1815       apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
  1816       apply (rule mult_mono)
  1817       apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
  1818       done}
  1819   then show ?thesis by metis
  1820 qed
  1821 
  1822 lemma bilinear_bounded_pos:
  1823   fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^ 'k"
  1824   assumes bh: "bilinear h"
  1825   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
  1826 proof-
  1827   from bilinear_bounded[OF bh] obtain B where 
  1828     B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
  1829   let ?K = "\<bar>B\<bar> + 1"
  1830   have Kp: "?K > 0" by arith
  1831   have KB: "B < ?K" by arith
  1832   {fix x::"real ^'m" and y :: "real ^'n"
  1833     from KB Kp
  1834     have "B * norm x * norm y \<le> ?K * norm x * norm y"
  1835       apply - 
  1836       apply (rule mult_right_mono, rule mult_right_mono)
  1837       by (auto simp add: norm_ge_zero)
  1838     then have "norm (h x y) \<le> ?K * norm x * norm y"
  1839       using B[rule_format, of x y] by simp} 
  1840   with Kp show ?thesis by blast
  1841 qed
  1842 
  1843 subsection{* Adjoints. *}
  1844 
  1845 definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
  1846 
  1847 lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
  1848 
  1849 lemma adjoint_works_lemma:
  1850   fixes f:: "'a::ring_1 ^'n \<Rightarrow> 'a ^ 'm"
  1851   assumes lf: "linear f"
  1852   shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
  1853 proof-
  1854   let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
  1855   let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
  1856   have fN: "finite ?N" by simp
  1857   have fM: "finite ?M" by simp
  1858   {fix y:: "'a ^ 'm"
  1859     let ?w = "(\<chi> i. (f (basis i) \<bullet> y)) :: 'a ^ 'n"
  1860     {fix x
  1861       have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *s basis i) ?N) \<bullet> y"
  1862 	by (simp only: basis_expansion)
  1863       also have "\<dots> = (setsum (\<lambda>i. (x$i) *s f (basis i)) ?N) \<bullet> y"
  1864 	unfolding linear_setsum[OF lf fN] 
  1865 	by (simp add: linear_cmul[OF lf])
  1866       finally have "f x \<bullet> y = x \<bullet> ?w"
  1867 	apply (simp only: )
  1868 	apply (simp add: dot_def setsum_component Cart_lambda_beta setsum_left_distrib setsum_right_distrib vector_component setsum_commute[of _ ?M ?N] ring_simps del: One_nat_def)
  1869 	done}
  1870   }
  1871   then show ?thesis unfolding adjoint_def 
  1872     some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
  1873     using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
  1874     by metis
  1875 qed
  1876 
  1877 lemma adjoint_works:
  1878   fixes f:: "'a::ring_1 ^'n \<Rightarrow> 'a ^ 'm"
  1879   assumes lf: "linear f"
  1880   shows "x \<bullet> adjoint f y = f x \<bullet> y"
  1881   using adjoint_works_lemma[OF lf] by metis
  1882 
  1883 
  1884 lemma adjoint_linear:
  1885   fixes f :: "'a::comm_ring_1 ^'n \<Rightarrow> 'a ^ 'm"
  1886   assumes lf: "linear f"
  1887   shows "linear (adjoint f)"
  1888   by (simp add: linear_def vector_eq_ldot[symmetric] dot_radd dot_rmult adjoint_works[OF lf])
  1889 
  1890 lemma adjoint_clauses:
  1891   fixes f:: "'a::comm_ring_1 ^'n \<Rightarrow> 'a ^ 'm"
  1892   assumes lf: "linear f"
  1893   shows "x \<bullet> adjoint f y = f x \<bullet> y"
  1894   and "adjoint f y \<bullet> x = y \<bullet> f x"
  1895   by (simp_all add: adjoint_works[OF lf] dot_sym )
  1896 
  1897 lemma adjoint_adjoint:
  1898   fixes f:: "'a::comm_ring_1 ^ 'n \<Rightarrow> _"
  1899   assumes lf: "linear f"
  1900   shows "adjoint (adjoint f) = f"
  1901   apply (rule ext)
  1902   by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf])
  1903 
  1904 lemma adjoint_unique:
  1905   fixes f:: "'a::comm_ring_1 ^ 'n \<Rightarrow> 'a ^ 'm"
  1906   assumes lf: "linear f" and u: "\<forall>x y. f' x \<bullet> y = x \<bullet> f y"
  1907   shows "f' = adjoint f"
  1908   apply (rule ext)
  1909   using u
  1910   by (simp add: vector_eq_rdot[symmetric] adjoint_clauses[OF lf])
  1911 
  1912 text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
  1913 
  1914 consts generic_mult :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" (infixr "\<star>" 75)
  1915 
  1916 defs (overloaded) 
  1917 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)$j)) {1 .. dimindex (UNIV :: 'n set)}) ::'a ^ 'p ^'m"
  1918 
  1919 abbreviation 
  1920   matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)
  1921   where "m ** m' == m\<star> m'"
  1922 
  1923 defs (overloaded) 
  1924   matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) {1..dimindex(UNIV ::'n set)}) :: 'a^'m"
  1925 
  1926 abbreviation 
  1927   matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)
  1928   where 
  1929   "m *v v == m \<star> v"
  1930 
  1931 defs (overloaded) 
  1932   vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (x$i)) {1..dimindex(UNIV :: 'm set)}) :: 'a^'n"
  1933 
  1934 abbreviation 
  1935   vactor_matrix_mult' :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)
  1936   where 
  1937   "v v* m == v \<star> m"
  1938 
  1939 definition "(mat::'a::zero => 'a ^'n^'m) k = (\<chi> i j. if i = j then k else 0)"
  1940 definition "(transp::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
  1941 definition "(row::nat => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
  1942 definition "(column::nat =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
  1943 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> {1 .. dimindex(UNIV :: 'm set)}}"
  1944 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> {1 .. dimindex(UNIV :: 'n set)}}"
  1945 
  1946 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
  1947 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \<star> B) + (A \<star> C)"
  1948   by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps)
  1949 
  1950 lemma setsum_delta': 
  1951   assumes fS: "finite S" shows 
  1952   "setsum (\<lambda>k. if a = k then b k else 0) S = 
  1953      (if a\<in> S then b a else 0)"
  1954   using setsum_delta[OF fS, of a b, symmetric] 
  1955   by (auto intro: setsum_cong)
  1956 
  1957 lemma matrix_mul_lid: "mat 1 ** A = A"
  1958   apply (simp add: matrix_matrix_mult_def mat_def)
  1959   apply vector
  1960   by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite_atLeastAtMost]  mult_1_left mult_zero_left if_True)
  1961 
  1962 
  1963 lemma matrix_mul_rid: "A ** mat 1 = A"
  1964   apply (simp add: matrix_matrix_mult_def mat_def)
  1965   apply vector
  1966   by (auto simp only: cond_value_iff cond_application_beta setsum_delta[OF finite_atLeastAtMost]  mult_1_right mult_zero_right if_True cong: if_cong)
  1967 
  1968 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
  1969   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
  1970   apply (subst setsum_commute)
  1971   apply simp
  1972   done
  1973 
  1974 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
  1975   apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
  1976   apply (subst setsum_commute)
  1977   apply simp
  1978   done
  1979 
  1980 lemma matrix_vector_mul_lid: "mat 1 *v x = x"
  1981   apply (vector matrix_vector_mult_def mat_def)
  1982   by (simp add: cond_value_iff cond_application_beta 
  1983     setsum_delta' cong del: if_weak_cong)
  1984 
  1985 lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)"
  1986   by (simp add: matrix_matrix_mult_def transp_def Cart_eq Cart_lambda_beta mult_commute)
  1987 
  1988 lemma matrix_eq: "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
  1989   apply auto
  1990   apply (subst Cart_eq)
  1991   apply clarify
  1992   apply (clarsimp simp add: matrix_vector_mult_def basis_def cond_value_iff cond_application_beta Cart_eq Cart_lambda_beta cong del: if_weak_cong)
  1993   apply (erule_tac x="basis ia" in allE)
  1994   apply (erule_tac x="i" in ballE)
  1995   by (auto simp add: basis_def cond_value_iff cond_application_beta Cart_lambda_beta setsum_delta[OF finite_atLeastAtMost] cong del: if_weak_cong)
  1996 
  1997 lemma matrix_vector_mul_component: 
  1998   assumes k: "k \<in> {1.. dimindex (UNIV :: 'm set)}"
  1999   shows "((A::'a::semiring_1^'n'^'m) *v x)$k = (A$k) \<bullet> x"
  2000   using k
  2001   by (simp add: matrix_vector_mult_def Cart_lambda_beta dot_def)
  2002 
  2003 lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) v* A) \<bullet> y = x \<bullet> (A *v y)"
  2004   apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib Cart_lambda_beta mult_ac)
  2005   apply (subst setsum_commute)
  2006   by simp
  2007 
  2008 lemma transp_mat: "transp (mat n) = mat n"
  2009   by (vector transp_def mat_def)
  2010 
  2011 lemma transp_transp: "transp(transp A) = A"
  2012   by (vector transp_def)
  2013 
  2014 lemma row_transp: 
  2015   fixes A:: "'a::semiring_1^'n^'m"
  2016   assumes i: "i \<in> {1.. dimindex (UNIV :: 'n set)}"
  2017   shows "row i (transp A) = column i A"
  2018   using i 
  2019   by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)
  2020 
  2021 lemma column_transp:
  2022   fixes A:: "'a::semiring_1^'n^'m"
  2023   assumes i: "i \<in> {1.. dimindex (UNIV :: 'm set)}"
  2024   shows "column i (transp A) = row i A"
  2025   using i 
  2026   by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)
  2027 
  2028 lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A"
  2029 apply (auto simp add: rows_def columns_def row_transp intro: set_ext)
  2030 apply (rule_tac x=i in exI)
  2031 apply (auto simp add: row_transp)
  2032 done
  2033 
  2034 lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp)
  2035 
  2036 text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
  2037 
  2038 lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
  2039   by (simp add: matrix_vector_mult_def dot_def)
  2040 
  2041 lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) {1 .. dimindex(UNIV:: 'n set)}"
  2042   by (simp add: matrix_vector_mult_def Cart_eq setsum_component Cart_lambda_beta vector_component column_def mult_commute)
  2043 
  2044 lemma vector_componentwise:
  2045   "(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) {1..dimindex(UNIV :: 'n set)})"
  2046   apply (subst basis_expansion[symmetric])
  2047   by (vector Cart_eq Cart_lambda_beta setsum_component)
  2048 
  2049 lemma linear_componentwise:
  2050   fixes f:: "'a::ring_1 ^ 'm \<Rightarrow> 'a ^ 'n"
  2051   assumes lf: "linear f" and j: "j \<in> {1 .. dimindex (UNIV :: 'n set)}"
  2052   shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) {1 .. dimindex (UNIV :: 'm set)}" (is "?lhs = ?rhs")
  2053 proof-
  2054   let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
  2055   let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
  2056   have fM: "finite ?M" by simp
  2057   have "?rhs = (setsum (\<lambda>i.(x$i) *s f (basis i) ) ?M)$j"
  2058     unfolding vector_smult_component[OF j, symmetric]
  2059     unfolding setsum_component[OF j, of "(\<lambda>i.(x$i) *s f (basis i :: 'a^'m))" ?M]
  2060     ..
  2061   then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion ..
  2062 qed
  2063 
  2064 text{* Inverse matrices  (not necessarily square) *}
  2065 
  2066 definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
  2067 
  2068 definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
  2069         (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
  2070 
  2071 text{* Correspondence between matrices and linear operators. *}
  2072 
  2073 definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
  2074 where "matrix f = (\<chi> i j. (f(basis j))$i)"
  2075 
  2076 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::'a::comm_semiring_1 ^ 'n))"
  2077   by (simp add: linear_def matrix_vector_mult_def Cart_eq Cart_lambda_beta vector_component ring_simps setsum_right_distrib setsum_addf)
  2078 
  2079 lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n)"
  2080 apply (simp add: matrix_def matrix_vector_mult_def Cart_eq Cart_lambda_beta mult_commute del: One_nat_def)
  2081 apply clarify
  2082 apply (rule linear_componentwise[OF lf, symmetric])
  2083 apply simp
  2084 done
  2085 
  2086 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::'a::comm_ring_1 ^ 'n))" by (simp add: ext matrix_works)
  2087 
  2088 lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n)) = A"
  2089   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
  2090 
  2091 lemma matrix_compose: 
  2092   assumes lf: "linear (f::'a::comm_ring_1^'n \<Rightarrow> _)" and lg: "linear g" 
  2093   shows "matrix (g o f) = matrix g ** matrix f"
  2094   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
  2095   by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
  2096 
  2097 lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s ((transp A)$i)) {1..dimindex(UNIV:: 'n set)}"
  2098   by (simp add: matrix_vector_mult_def transp_def Cart_eq Cart_lambda_beta setsum_component vector_component mult_commute)
  2099 
  2100 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::'a::comm_ring_1^'n^'m) *v x) = (\<lambda>x. transp A *v x)"
  2101   apply (rule adjoint_unique[symmetric])
  2102   apply (rule matrix_vector_mul_linear)
  2103   apply (simp add: transp_def dot_def Cart_lambda_beta matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
  2104   apply (subst setsum_commute)
  2105   apply (auto simp add: mult_ac)
  2106   done
  2107 
  2108 lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n \<Rightarrow> 'a ^ 'm)"
  2109   shows "matrix(adjoint f) = transp(matrix f)"
  2110   apply (subst matrix_vector_mul[OF lf])
  2111   unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
  2112 
  2113 subsection{* Interlude: Some properties of real sets *}
  2114 
  2115 lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
  2116   shows "\<forall>n \<ge> m. d n < e m"
  2117   using prems apply auto
  2118   apply (erule_tac x="n" in allE)
  2119   apply (erule_tac x="n" in allE)
  2120   apply auto
  2121   done
  2122 
  2123 
  2124 lemma real_convex_bound_lt: 
  2125   assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"
  2126   and uv: "u + v = 1" 
  2127   shows "u * x + v * y < a"
  2128 proof-
  2129   have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
  2130   have "a = a * (u + v)" unfolding uv  by simp
  2131   hence th: "u * a + v * a = a" by (simp add: ring_simps)
  2132   from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_compare_simps)
  2133   from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_compare_simps)
  2134   from xa ya u v have "u * x + v * y < u * a + v * a"
  2135     apply (cases "u = 0", simp_all add: uv')
  2136     apply(rule mult_strict_left_mono)
  2137     using uv' apply simp_all
  2138     
  2139     apply (rule add_less_le_mono)
  2140     apply(rule mult_strict_left_mono)
  2141     apply simp_all
  2142     apply (rule mult_left_mono)
  2143     apply simp_all
  2144     done
  2145   thus ?thesis unfolding th .
  2146 qed
  2147 
  2148 lemma real_convex_bound_le: 
  2149   assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"
  2150   and uv: "u + v = 1" 
  2151   shows "u * x + v * y \<le> a"
  2152 proof-
  2153   from xa ya u v have "u * x + v * y \<le> u * a + v * a" by (simp add: add_mono mult_left_mono)
  2154   also have "\<dots> \<le> (u + v) * a" by (simp add: ring_simps)
  2155   finally show ?thesis unfolding uv by simp
  2156 qed
  2157 
  2158 lemma infinite_enumerate: assumes fS: "infinite S"
  2159   shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
  2160 unfolding subseq_def
  2161 using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
  2162 
  2163 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)"
  2164 apply auto
  2165 apply (rule_tac x="d/2" in exI)
  2166 apply auto
  2167 done
  2168 
  2169 
  2170 lemma triangle_lemma: 
  2171   assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
  2172   shows "x <= y + z"
  2173 proof-
  2174   have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y  by (simp add: zero_compare_simps)
  2175   with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square ring_simps)
  2176   from y z have yz: "y + z \<ge> 0" by arith
  2177   from power2_le_imp_le[OF th yz] show ?thesis .
  2178 qed
  2179 
  2180 
  2181 lemma lambda_skolem: "(\<forall>i \<in> {1 .. dimindex(UNIV :: 'n set)}. \<exists>x. P i x) \<longleftrightarrow>
  2182    (\<exists>x::'a ^ 'n. \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")
  2183 proof-
  2184   let ?S = "{1 .. dimindex(UNIV :: 'n set)}"
  2185   {assume H: "?rhs"
  2186     then have ?lhs by auto}
  2187   moreover
  2188   {assume H: "?lhs"
  2189     then obtain f where f:"\<forall>i\<in> ?S. P i (f i)" unfolding Ball_def choice_iff by metis
  2190     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
  2191     {fix i assume i: "i \<in> ?S"
  2192       with f i have "P i (f i)" by metis
  2193       then have "P i (?x$i)" using Cart_lambda_beta[of f, rule_format, OF i] by auto 
  2194     }
  2195     hence "\<forall>i \<in> ?S. P i (?x$i)" by metis
  2196     hence ?rhs by metis }
  2197   ultimately show ?thesis by metis
  2198 qed 
  2199 
  2200 (* Supremum and infimum of real sets *)
  2201 
  2202 
  2203 definition rsup:: "real set \<Rightarrow> real" where
  2204   "rsup S = (SOME a. isLub UNIV S a)"
  2205 
  2206 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)
  2207 
  2208 lemma rsup: assumes Se: "S \<noteq> {}" and b: "\<exists>b. S *<= b"
  2209   shows "isLub UNIV S (rsup S)"
  2210 using Se b
  2211 unfolding rsup_def
  2212 apply clarify
  2213 apply (rule someI_ex)
  2214 apply (rule reals_complete)
  2215 by (auto simp add: isUb_def setle_def)
  2216 
  2217 lemma rsup_le: assumes Se: "S \<noteq> {}" and Sb: "S *<= b" shows "rsup S \<le> b"
  2218 proof-
  2219   from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def)
  2220   from rsup[OF Se] Sb have "isLub UNIV S (rsup S)"  by blast 
  2221   then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def)
  2222 qed
  2223 
  2224 lemma rsup_finite_Max: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2225   shows "rsup S = Max S"
  2226 using fS Se
  2227 proof-
  2228   let ?m = "Max S"
  2229   from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis
  2230   with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def)
  2231   from Max_in[OF fS Se] lub have mrS: "?m \<le> rsup S" 
  2232     by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)
  2233   moreover 
  2234   have "rsup S \<le> ?m" using Sm lub
  2235     by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
  2236   ultimately  show ?thesis by arith 
  2237 qed
  2238 
  2239 lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2240   shows "rsup S \<in> S"
  2241   using rsup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis
  2242 
  2243 lemma rsup_finite_Ub: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2244   shows "isUb S S (rsup S)"
  2245   using rsup_finite_Max[OF fS Se] rsup_finite_in[OF fS Se] Max_ge[OF fS]
  2246   unfolding isUb_def setle_def by metis
  2247 
  2248 lemma rsup_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2249   shows "a \<le> rsup S \<longleftrightarrow> (\<exists> x \<in> S. a \<le> x)"
  2250 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2251 
  2252 lemma rsup_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2253   shows "a \<ge> rsup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)"
  2254 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2255 
  2256 lemma rsup_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2257   shows "a < rsup S \<longleftrightarrow> (\<exists> x \<in> S. a < x)"
  2258 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2259 
  2260 lemma rsup_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2261   shows "a > rsup S \<longleftrightarrow> (\<forall> x \<in> S. a > x)"
  2262 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2263 
  2264 lemma rsup_unique: assumes b: "S *<= b" and S: "\<forall>b' < b. \<exists>x \<in> S. b' < x"
  2265   shows "rsup S = b"
  2266 using b S  
  2267 unfolding setle_def rsup_alt
  2268 apply -
  2269 apply (rule some_equality)
  2270 apply (metis  linorder_not_le order_eq_iff[symmetric])+
  2271 done
  2272 
  2273 lemma rsup_le_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. T *<= b) \<Longrightarrow> rsup S \<le> rsup T"
  2274   apply (rule rsup_le)
  2275   apply simp
  2276   using rsup[of T] by (auto simp add: isLub_def leastP_def setge_def setle_def isUb_def)
  2277 
  2278 lemma isUb_def': "isUb R S = (\<lambda>x. S *<= x \<and> x \<in> R)"
  2279   apply (rule ext)
  2280   by (metis isUb_def)
  2281 
  2282 lemma UNIV_trivial: "UNIV x" using UNIV_I[of x] by (metis mem_def)
  2283 lemma rsup_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
  2284   shows "a \<le> rsup S \<and> rsup S \<le> b"
  2285 proof-
  2286   from rsup[OF Se] u have lub: "isLub UNIV S (rsup S)" by blast
  2287   hence b: "rsup S \<le> b" using u by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
  2288   from Se obtain y where y: "y \<in> S" by blast
  2289   from lub l have "a \<le> rsup S" apply (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
  2290     apply (erule ballE[where x=y])
  2291     apply (erule ballE[where x=y])
  2292     apply arith
  2293     using y apply auto
  2294     done
  2295   with b show ?thesis by blast
  2296 qed
  2297 
  2298 lemma rsup_abs_le: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rsup S\<bar> \<le> a"
  2299   unfolding abs_le_interval_iff  using rsup_bounds[of S "-a" a]
  2300   by (auto simp add: setge_def setle_def)
  2301 
  2302 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"
  2303 proof-
  2304   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
  2305   show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th 
  2306     by  (auto simp add: setge_def setle_def)
  2307 qed
  2308 
  2309 definition rinf:: "real set \<Rightarrow> real" where
  2310   "rinf S = (SOME a. isGlb UNIV S a)"
  2311 
  2312 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)
  2313 
  2314 lemma reals_complete_Glb: assumes Se: "\<exists>x. x \<in> S" and lb: "\<exists> y. isLb UNIV S y"
  2315   shows "\<exists>(t::real). isGlb UNIV S t"
  2316 proof-
  2317   let ?M = "uminus ` S"
  2318   from lb have th: "\<exists>y. isUb UNIV ?M y" apply (auto simp add: isUb_def isLb_def setle_def setge_def)
  2319     by (rule_tac x="-y" in exI, auto)
  2320   from Se have Me: "\<exists>x. x \<in> ?M" by blast
  2321   from reals_complete[OF Me th] obtain t where t: "isLub UNIV ?M t" by blast
  2322   have "isGlb UNIV S (- t)" using t
  2323     apply (auto simp add: isLub_def isGlb_def leastP_def greatestP_def setle_def setge_def isUb_def isLb_def)
  2324     apply (erule_tac x="-y" in allE)
  2325     apply auto
  2326     done
  2327   then show ?thesis by metis
  2328 qed
  2329 
  2330 lemma rinf: assumes Se: "S \<noteq> {}" and b: "\<exists>b. b <=* S"
  2331   shows "isGlb UNIV S (rinf S)"
  2332 using Se b
  2333 unfolding rinf_def
  2334 apply clarify
  2335 apply (rule someI_ex)
  2336 apply (rule reals_complete_Glb)
  2337 apply (auto simp add: isLb_def setle_def setge_def)
  2338 done
  2339 
  2340 lemma rinf_ge: assumes Se: "S \<noteq> {}" and Sb: "b <=* S" shows "rinf S \<ge> b"
  2341 proof-
  2342   from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def)
  2343   from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)"  by blast 
  2344   then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def)
  2345 qed
  2346 
  2347 lemma rinf_finite_Min: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2348   shows "rinf S = Min S"
  2349 using fS Se
  2350 proof-
  2351   let ?m = "Min S"
  2352   from Min_le[OF fS] have Sm: "\<forall> x\<in> S. x \<ge> ?m" by metis
  2353   with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def)
  2354   from Min_in[OF fS Se] glb have mrS: "?m \<ge> rinf S" 
  2355     by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def)
  2356   moreover 
  2357   have "rinf S \<ge> ?m" using Sm glb
  2358     by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def)
  2359   ultimately  show ?thesis by arith 
  2360 qed
  2361 
  2362 lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2363   shows "rinf S \<in> S"
  2364   using rinf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis
  2365 
  2366 lemma rinf_finite_Lb: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2367   shows "isLb S S (rinf S)"
  2368   using rinf_finite_Min[OF fS Se] rinf_finite_in[OF fS Se] Min_le[OF fS]
  2369   unfolding isLb_def setge_def by metis
  2370 
  2371 lemma rinf_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2372   shows "a \<le> rinf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)"
  2373 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2374 
  2375 lemma rinf_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2376   shows "a \<ge> rinf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)"
  2377 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2378 
  2379 lemma rinf_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2380   shows "a < rinf S \<longleftrightarrow> (\<forall> x \<in> S. a < x)"
  2381 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2382 
  2383 lemma rinf_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2384   shows "a > rinf S \<longleftrightarrow> (\<exists> x \<in> S. a > x)"
  2385 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2386 
  2387 lemma rinf_unique: assumes b: "b <=* S" and S: "\<forall>b' > b. \<exists>x \<in> S. b' > x"
  2388   shows "rinf S = b"
  2389 using b S  
  2390 unfolding setge_def rinf_alt
  2391 apply -
  2392 apply (rule some_equality)
  2393 apply (metis  linorder_not_le order_eq_iff[symmetric])+
  2394 done
  2395 
  2396 lemma rinf_ge_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. b <=* T) \<Longrightarrow> rinf S >= rinf T"
  2397   apply (rule rinf_ge)
  2398   apply simp
  2399   using rinf[of T] by (auto simp add: isGlb_def greatestP_def setge_def setle_def isLb_def)
  2400 
  2401 lemma isLb_def': "isLb R S = (\<lambda>x. x <=* S \<and> x \<in> R)"
  2402   apply (rule ext)
  2403   by (metis isLb_def)
  2404 
  2405 lemma rinf_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
  2406   shows "a \<le> rinf S \<and> rinf S \<le> b"
  2407 proof-
  2408   from rinf[OF Se] l have lub: "isGlb UNIV S (rinf S)" by blast
  2409   hence b: "a \<le> rinf S" using l by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
  2410   from Se obtain y where y: "y \<in> S" by blast
  2411   from lub u have "b \<ge> rinf S" apply (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
  2412     apply (erule ballE[where x=y])
  2413     apply (erule ballE[where x=y])
  2414     apply arith
  2415     using y apply auto
  2416     done
  2417   with b show ?thesis by blast
  2418 qed
  2419 
  2420 lemma rinf_abs_ge: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rinf S\<bar> \<le> a"
  2421   unfolding abs_le_interval_iff  using rinf_bounds[of S "-a" a]
  2422   by (auto simp add: setge_def setle_def)
  2423 
  2424 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"
  2425 proof-
  2426   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
  2427   show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th 
  2428     by  (auto simp add: setge_def setle_def)
  2429 qed
  2430 
  2431 
  2432 
  2433 subsection{* Operator norm. *}
  2434 
  2435 definition "onorm f = rsup {norm (f x)| x. norm x = 1}"
  2436 
  2437 lemma norm_bound_generalize:
  2438   fixes f:: "real ^'n \<Rightarrow> real^'m"
  2439   assumes lf: "linear f"
  2440   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")
  2441 proof-
  2442   {assume H: ?rhs
  2443     {fix x :: "real^'n" assume x: "norm x = 1"
  2444       from H[rule_format, of x] x have "norm (f x) \<le> b" by simp}
  2445     then have ?lhs by blast }
  2446 
  2447   moreover
  2448   {assume H: ?lhs
  2449     from H[rule_format, of "basis 1"] 
  2450     have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"]
  2451       by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])
  2452     {fix x :: "real ^'n"
  2453       {assume "x = 0"
  2454 	then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}
  2455       moreover
  2456       {assume x0: "x \<noteq> 0"
  2457 	hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
  2458 	let ?c = "1/ norm x"
  2459 	have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul)
  2460 	with H have "norm (f(?c*s x)) \<le> b" by blast
  2461 	hence "?c * norm (f x) \<le> b" 
  2462 	  by (simp add: linear_cmul[OF lf] norm_mul)
  2463 	hence "norm (f x) \<le> b * norm x" 
  2464 	  using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
  2465       ultimately have "norm (f x) \<le> b * norm x" by blast}
  2466     then have ?rhs by blast}
  2467   ultimately show ?thesis by blast
  2468 qed
  2469 
  2470 lemma onorm:
  2471   fixes f:: "real ^'n \<Rightarrow> real ^'m"
  2472   assumes lf: "linear f"
  2473   shows "norm (f x) <= onorm f * norm x"
  2474   and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
  2475 proof-
  2476   {
  2477     let ?S = "{norm (f x) |x. norm x = 1}"
  2478     have Se: "?S \<noteq> {}" using  norm_basis_1 by auto
  2479     from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b" 
  2480       unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
  2481     {from rsup[OF Se b, unfolded onorm_def[symmetric]]
  2482       show "norm (f x) <= onorm f * norm x" 
  2483 	apply - 
  2484 	apply (rule spec[where x = x])
  2485 	unfolding norm_bound_generalize[OF lf, symmetric]
  2486 	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
  2487     {
  2488       show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"  
  2489 	using rsup[OF Se b, unfolded onorm_def[symmetric]]
  2490 	unfolding norm_bound_generalize[OF lf, symmetric]
  2491 	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
  2492   }
  2493 qed
  2494 
  2495 lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" shows "0 <= onorm f"
  2496   using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis 1"], unfolded norm_basis_1] by simp
  2497 
  2498 lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" 
  2499   shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
  2500   using onorm[OF lf]
  2501   apply (auto simp add: onorm_pos_le)
  2502   apply atomize
  2503   apply (erule allE[where x="0::real"])
  2504   using onorm_pos_le[OF lf]
  2505   apply arith
  2506   done
  2507 
  2508 lemma onorm_const: "onorm(\<lambda>x::real^'n. (y::real ^ 'm)) = norm y"
  2509 proof-
  2510   let ?f = "\<lambda>x::real^'n. (y::real ^ 'm)"
  2511   have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
  2512     by(auto intro: vector_choose_size set_ext)
  2513   show ?thesis
  2514     unfolding onorm_def th
  2515     apply (rule rsup_unique) by (simp_all  add: setle_def)
  2516 qed
  2517 
  2518 lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n \<Rightarrow> real ^'m)" 
  2519   shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"
  2520   unfolding onorm_eq_0[OF lf, symmetric]
  2521   using onorm_pos_le[OF lf] by arith
  2522 
  2523 lemma onorm_compose:
  2524   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and lg: "linear g"
  2525   shows "onorm (f o g) <= onorm f * onorm g"
  2526   apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
  2527   unfolding o_def
  2528   apply (subst mult_assoc)
  2529   apply (rule order_trans)
  2530   apply (rule onorm(1)[OF lf])
  2531   apply (rule mult_mono1)
  2532   apply (rule onorm(1)[OF lg])
  2533   apply (rule onorm_pos_le[OF lf])
  2534   done
  2535 
  2536 lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
  2537   shows "onorm (\<lambda>x. - f x) \<le> onorm f"
  2538   using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
  2539   unfolding norm_minus_cancel by metis
  2540 
  2541 lemma onorm_neg: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
  2542   shows "onorm (\<lambda>x. - f x) = onorm f"
  2543   using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]]
  2544   by simp
  2545 
  2546 lemma onorm_triangle:
  2547   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and lg: "linear g"
  2548   shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
  2549   apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
  2550   apply (rule order_trans)
  2551   apply (rule norm_triangle_ineq)
  2552   apply (simp add: distrib)
  2553   apply (rule add_mono)
  2554   apply (rule onorm(1)[OF lf])
  2555   apply (rule onorm(1)[OF lg])
  2556   done
  2557 
  2558 lemma onorm_triangle_le: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
  2559   \<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"
  2560   apply (rule order_trans)
  2561   apply (rule onorm_triangle)
  2562   apply assumption+
  2563   done
  2564 
  2565 lemma onorm_triangle_lt: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
  2566   ==> onorm(\<lambda>x. f x + g x) < e"
  2567   apply (rule order_le_less_trans)
  2568   apply (rule onorm_triangle)
  2569   by assumption+
  2570 
  2571 (* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)
  2572 
  2573 definition vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x = (\<chi> i. x)"
  2574 
  2575 definition dest_vec1:: "'a ^1 \<Rightarrow> 'a" 
  2576   where "dest_vec1 x = (x$1)"
  2577 
  2578 lemma vec1_component[simp]: "(vec1 x)$1 = x"
  2579   by (simp add: vec1_def)
  2580 
  2581 lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
  2582   by (simp_all add: vec1_def dest_vec1_def Cart_eq Cart_lambda_beta dimindex_def del: One_nat_def)
  2583 
  2584 lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))" by (metis vec1_dest_vec1)
  2585 
  2586 lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))" by (metis vec1_dest_vec1) 
  2587 
  2588 lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))"  by (metis vec1_dest_vec1)
  2589 
  2590 lemma exists_dest_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(dest_vec1 x))"by (metis vec1_dest_vec1)
  2591 
  2592 lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
  2593 
  2594 lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
  2595 
  2596 lemma vec1_in_image_vec1: "vec1 x \<in> (vec1 ` S) \<longleftrightarrow> x \<in> S" by auto
  2597 
  2598 lemma vec1_vec: "vec1 x = vec x" by (vector vec1_def)
  2599 
  2600 lemma vec1_add: "vec1(x + y) = vec1 x + vec1 y" by (vector vec1_def)
  2601 lemma vec1_sub: "vec1(x - y) = vec1 x - vec1 y" by (vector vec1_def)
  2602 lemma vec1_cmul: "vec1(c* x) = c *s vec1 x " by (vector vec1_def)
  2603 lemma vec1_neg: "vec1(- x) = - vec1 x " by (vector vec1_def)
  2604 
  2605 lemma vec1_setsum: assumes fS: "finite S"
  2606   shows "vec1(setsum f S) = setsum (vec1 o f) S"
  2607   apply (induct rule: finite_induct[OF fS])
  2608   apply (simp add: vec1_vec)
  2609   apply (auto simp add: vec1_add)
  2610   done
  2611 
  2612 lemma dest_vec1_lambda: "dest_vec1(\<chi> i. x i) = x 1"
  2613   by (simp add: dest_vec1_def)
  2614 
  2615 lemma dest_vec1_vec: "dest_vec1(vec x) = x"
  2616   by (simp add: vec1_vec[symmetric])
  2617 
  2618 lemma dest_vec1_add: "dest_vec1(x + y) = dest_vec1 x + dest_vec1 y"
  2619  by (metis vec1_dest_vec1 vec1_add)
  2620 
  2621 lemma dest_vec1_sub: "dest_vec1(x - y) = dest_vec1 x - dest_vec1 y"
  2622  by (metis vec1_dest_vec1 vec1_sub)
  2623 
  2624 lemma dest_vec1_cmul: "dest_vec1(c*sx) = c * dest_vec1 x"
  2625  by (metis vec1_dest_vec1 vec1_cmul)
  2626 
  2627 lemma dest_vec1_neg: "dest_vec1(- x) = - dest_vec1 x"
  2628  by (metis vec1_dest_vec1 vec1_neg)
  2629 
  2630 lemma dest_vec1_0[simp]: "dest_vec1 0 = 0" by (metis vec_0 dest_vec1_vec)
  2631 
  2632 lemma dest_vec1_sum: assumes fS: "finite S"
  2633   shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"
  2634   apply (induct rule: finite_induct[OF fS])
  2635   apply (simp add: dest_vec1_vec)
  2636   apply (auto simp add: dest_vec1_add)
  2637   done
  2638 
  2639 lemma norm_vec1: "norm(vec1 x) = abs(x)"
  2640   by (simp add: vec1_def norm_real)
  2641 
  2642 lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)"
  2643   by (simp only: dist_real vec1_component)
  2644 lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
  2645   by (metis vec1_dest_vec1 norm_vec1)
  2646 
  2647 lemma linear_vmul_dest_vec1: 
  2648   fixes f:: "'a::semiring_1^'n \<Rightarrow> 'a^1"
  2649   shows "linear f \<Longrightarrow> linear (\<lambda>x. dest_vec1(f x) *s v)"
  2650   unfolding dest_vec1_def
  2651   apply (rule linear_vmul_component)
  2652   by (auto simp add: dimindex_def)
  2653 
  2654 lemma linear_from_scalars:
  2655   assumes lf: "linear (f::'a::comm_ring_1 ^1 \<Rightarrow> 'a^'n)"
  2656   shows "f = (\<lambda>x. dest_vec1 x *s column 1 (matrix f))"
  2657   apply (rule ext)
  2658   apply (subst matrix_works[OF lf, symmetric])
  2659   apply (auto simp add: Cart_eq matrix_vector_mult_def dest_vec1_def column_def Cart_lambda_beta vector_component dimindex_def mult_commute del: One_nat_def )
  2660   done
  2661 
  2662 lemma linear_to_scalars: assumes lf: "linear (f::'a::comm_ring_1 ^'n \<Rightarrow> 'a^1)"
  2663   shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
  2664   apply (rule ext)
  2665   apply (subst matrix_works[OF lf, symmetric])
  2666   apply (auto simp add: Cart_eq matrix_vector_mult_def vec1_def row_def Cart_lambda_beta vector_component dimindex_def dot_def mult_commute)
  2667   done
  2668 
  2669 lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
  2670   by (simp add: dest_vec1_eq[symmetric])
  2671 
  2672 lemma setsum_scalars: assumes fS: "finite S"
  2673   shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)"
  2674   unfolding vec1_setsum[OF fS] by simp
  2675 
  2676 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"
  2677   apply (cases "dest_vec1 x \<le> dest_vec1 y")
  2678   apply simp
  2679   apply (subgoal_tac "dest_vec1 y \<le> dest_vec1 x")
  2680   apply (auto)
  2681   done
  2682 
  2683 text{* Pasting vectors. *}
  2684 
  2685 lemma linear_fstcart: "linear fstcart"
  2686   by (auto simp add: linear_def fstcart_def Cart_eq Cart_lambda_beta vector_component dimindex_finite_sum)
  2687 
  2688 lemma linear_sndcart: "linear sndcart"
  2689   by (auto simp add: linear_def sndcart_def Cart_eq Cart_lambda_beta vector_component dimindex_finite_sum)
  2690 
  2691 lemma fstcart_vec[simp]: "fstcart(vec x) = vec x"
  2692   by (vector fstcart_def vec_def dimindex_finite_sum)
  2693 
  2694 lemma fstcart_add[simp]:"fstcart(x + y) = fstcart (x::'a::{plus,times}^('b,'c) finite_sum) + fstcart y"
  2695   using linear_fstcart[unfolded linear_def] by blast
  2696 
  2697 lemma fstcart_cmul[simp]:"fstcart(c*s x) = c*s fstcart (x::'a::{plus,times}^('b,'c) finite_sum)"
  2698   using linear_fstcart[unfolded linear_def] by blast
  2699 
  2700 lemma fstcart_neg[simp]:"fstcart(- x) = - fstcart (x::'a::ring_1^('b,'c) finite_sum)"
  2701 unfolding vector_sneg_minus1 fstcart_cmul ..
  2702 
  2703 lemma fstcart_sub[simp]:"fstcart(x - y) = fstcart (x::'a::ring_1^('b,'c) finite_sum) - fstcart y"
  2704   unfolding diff_def fstcart_add fstcart_neg  ..
  2705 
  2706 lemma fstcart_setsum:
  2707   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
  2708   assumes fS: "finite S"
  2709   shows "fstcart (setsum f S) = setsum (\<lambda>i. fstcart (f i)) S"
  2710   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
  2711 
  2712 lemma sndcart_vec[simp]: "sndcart(vec x) = vec x"
  2713   by (vector sndcart_def vec_def dimindex_finite_sum)
  2714 
  2715 lemma sndcart_add[simp]:"sndcart(x + y) = sndcart (x::'a::{plus,times}^('b,'c) finite_sum) + sndcart y"
  2716   using linear_sndcart[unfolded linear_def] by blast
  2717 
  2718 lemma sndcart_cmul[simp]:"sndcart(c*s x) = c*s sndcart (x::'a::{plus,times}^('b,'c) finite_sum)"
  2719   using linear_sndcart[unfolded linear_def] by blast
  2720 
  2721 lemma sndcart_neg[simp]:"sndcart(- x) = - sndcart (x::'a::ring_1^('b,'c) finite_sum)"
  2722 unfolding vector_sneg_minus1 sndcart_cmul ..
  2723 
  2724 lemma sndcart_sub[simp]:"sndcart(x - y) = sndcart (x::'a::ring_1^('b,'c) finite_sum) - sndcart y"
  2725   unfolding diff_def sndcart_add sndcart_neg  ..
  2726 
  2727 lemma sndcart_setsum:
  2728   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
  2729   assumes fS: "finite S"
  2730   shows "sndcart (setsum f S) = setsum (\<lambda>i. sndcart (f i)) S"
  2731   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
  2732 
  2733 lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x"
  2734   by (simp add: pastecart_eq fstcart_vec sndcart_vec fstcart_pastecart sndcart_pastecart)
  2735 
  2736 lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)"
  2737   by (simp add: pastecart_eq fstcart_add sndcart_add fstcart_pastecart sndcart_pastecart)
  2738 
  2739 lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1"
  2740   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
  2741 
  2742 lemma pastecart_neg[simp]: "pastecart (- (x::'a::ring_1^_)) (- y) = - pastecart x y"
  2743   unfolding vector_sneg_minus1 pastecart_cmul ..
  2744 
  2745 lemma pastecart_sub: "pastecart (x1::'a::ring_1^_) y1 - pastecart x2 y2 = pastecart (x1 - x2) (y1 - y2)"
  2746   by (simp add: diff_def pastecart_neg[symmetric] del: pastecart_neg)
  2747 
  2748 lemma pastecart_setsum:
  2749   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
  2750   assumes fS: "finite S"
  2751   shows "pastecart (setsum f S) (setsum g S) = setsum (\<lambda>i. pastecart (f i) (g i)) S"
  2752   by (simp  add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS] fstcart_pastecart sndcart_pastecart)
  2753 
  2754 lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n,'m) finite_sum)"
  2755 proof-
  2756   let ?n = "dimindex (UNIV :: 'n set)"
  2757   let ?m = "dimindex (UNIV :: 'm set)"
  2758   let ?N = "{1 .. ?n}"
  2759   let ?M = "{1 .. ?m}"
  2760   let ?NM = "{1 .. dimindex (UNIV :: ('n,'m) finite_sum set)}"
  2761   have th_0: "1 \<le> ?n +1" by simp
  2762   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
  2763     by (simp add: pastecart_fst_snd)
  2764   have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)" 
  2765     by (simp add: dot_def setsum_add_split[OF th_0, of _ ?m] pastecart_def dimindex_finite_sum Cart_lambda_beta setsum_nonneg zero_le_square del: One_nat_def)
  2766   then show ?thesis
  2767     unfolding th0 
  2768     unfolding real_vector_norm_def real_sqrt_le_iff id_def
  2769     by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta)
  2770 qed
  2771 
  2772 lemma dist_fstcart: "dist(fstcart (x::real^_)) (fstcart y) <= dist x y"
  2773   by (metis dist_def fstcart_sub[symmetric] norm_fstcart)
  2774 
  2775 lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n,'m) finite_sum)"
  2776 proof-
  2777   let ?n = "dimindex (UNIV :: 'n set)"
  2778   let ?m = "dimindex (UNIV :: 'm set)"
  2779   let ?N = "{1 .. ?n}"
  2780   let ?M = "{1 .. ?m}"
  2781   let ?nm = "dimindex (UNIV :: ('n,'m) finite_sum set)"
  2782   let ?NM = "{1 .. ?nm}"
  2783   have thnm[simp]: "?nm = ?n + ?m" by (simp add: dimindex_finite_sum)
  2784   have th_0: "1 \<le> ?n +1" by simp
  2785   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
  2786     by (simp add: pastecart_fst_snd)
  2787   let ?f = "\<lambda>n. n - ?n"
  2788   let ?S = "{?n+1 .. ?nm}"
  2789   have finj:"inj_on ?f ?S"
  2790     using dimindex_nonzero[of "UNIV :: 'n set"] dimindex_nonzero[of "UNIV :: 'm set"]
  2791     apply (simp add: Ball_def atLeastAtMost_iff inj_on_def dimindex_finite_sum del: One_nat_def)
  2792     by arith
  2793   have fS: "?f ` ?S = ?M" 
  2794     apply (rule set_ext)
  2795     apply (simp add: image_iff Bex_def) using dimindex_nonzero[of "UNIV :: 'n set"] dimindex_nonzero[of "UNIV :: 'm set"] by arith
  2796   have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)" 
  2797     by (simp add: dot_def setsum_add_split[OF th_0, of _ ?m] pastecart_def dimindex_finite_sum Cart_lambda_beta setsum_nonneg zero_le_square setsum_reindex[OF finj, unfolded fS] del: One_nat_def)    
  2798   then show ?thesis
  2799     unfolding th0 
  2800     unfolding real_vector_norm_def real_sqrt_le_iff id_def
  2801     by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta)
  2802 qed
  2803 
  2804 lemma dist_sndcart: "dist(sndcart (x::real^_)) (sndcart y) <= dist x y"
  2805   by (metis dist_def sndcart_sub[symmetric] norm_sndcart)
  2806 
  2807 lemma dot_pastecart: "(pastecart (x1::'a::{times,comm_monoid_add}^'n) (x2::'a::{times,comm_monoid_add}^'m)) \<bullet> (pastecart y1 y2) =  x1 \<bullet> y1 + x2 \<bullet> y2"
  2808 proof-
  2809   let ?n = "dimindex (UNIV :: 'n set)"
  2810   let ?m = "dimindex (UNIV :: 'm set)"
  2811   let ?N = "{1 .. ?n}"
  2812   let ?M = "{1 .. ?m}"
  2813   let ?nm = "dimindex (UNIV :: ('n,'m) finite_sum set)"
  2814   let ?NM = "{1 .. ?nm}"
  2815   have thnm: "?nm = ?n + ?m" by (simp add: dimindex_finite_sum)
  2816   have th_0: "1 \<le> ?n +1" by simp
  2817   have th_1: "\<And>i. i \<in> {?m + 1 .. ?nm} \<Longrightarrow> i - ?m \<in> ?N" apply (simp add: thnm) by arith
  2818   let ?f = "\<lambda>a b i. (a$i) * (b$i)"
  2819   let ?g = "?f (pastecart x1 x2) (pastecart y1 y2)"
  2820   let ?S = "{?n +1 .. ?nm}"
  2821   {fix i
  2822     assume i: "i \<in> ?N"
  2823     have "?g i = ?f x1 y1 i"
  2824       using i
  2825       apply (simp add: pastecart_def Cart_lambda_beta thnm) done
  2826   }
  2827   hence th2: "setsum ?g ?N = setsum (?f x1 y1) ?N"
  2828     apply -
  2829     apply (rule setsum_cong)
  2830     apply auto
  2831     done
  2832   {fix i
  2833     assume i: "i \<in> ?S"
  2834     have "?g i = ?f x2 y2 (i - ?n)"
  2835       using i
  2836       apply (simp add: pastecart_def Cart_lambda_beta thnm) done
  2837   }
  2838   hence th3: "setsum ?g ?S = setsum (\<lambda>i. ?f x2 y2 (i -?n)) ?S"
  2839     apply -
  2840     apply (rule setsum_cong)
  2841     apply auto
  2842     done
  2843   let ?r = "\<lambda>n. n - ?n"
  2844   have rinj: "inj_on ?r ?S" apply (simp add: inj_on_def Ball_def thnm) by arith
  2845   have rS: "?r ` ?S = ?M" apply (rule set_ext) 
  2846     apply (simp add: thnm image_iff Bex_def) by arith
  2847   have "pastecart x1 x2 \<bullet> (pastecart y1 y2) = setsum ?g ?NM" by (simp add: dot_def)
  2848   also have "\<dots> = setsum ?g ?N + setsum ?g ?S"
  2849     by (simp add: dot_def thnm setsum_add_split[OF th_0, of _ ?m] del: One_nat_def)
  2850   also have "\<dots> = setsum (?f x1 y1) ?N + setsum (?f x2 y2) ?M"
  2851     unfolding setsum_reindex[OF rinj, unfolded rS o_def] th2 th3 ..
  2852   finally 
  2853   show ?thesis by (simp add: dot_def)
  2854 qed
  2855 
  2856 lemma norm_pastecart: "norm(pastecart x y) <= norm(x :: real ^ _) + norm(y)"
  2857   unfolding real_vector_norm_def dot_pastecart real_sqrt_le_iff id_def
  2858   apply (rule power2_le_imp_le)
  2859   apply (simp add: real_sqrt_pow2[OF add_nonneg_nonneg[OF dot_pos_le[of x] dot_pos_le[of y]]])
  2860   apply (auto simp add: power2_eq_square ring_simps)
  2861   apply (simp add: power2_eq_square[symmetric])
  2862   apply (rule mult_nonneg_nonneg)
  2863   apply (simp_all add: real_sqrt_pow2[OF dot_pos_le])
  2864   apply (rule add_nonneg_nonneg)
  2865   apply (simp_all add: real_sqrt_pow2[OF dot_pos_le])
  2866   done
  2867 
  2868 subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
  2869 
  2870 definition hull :: "'a set set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) where
  2871   "S hull s = Inter {t. t \<in> S \<and> s \<subseteq> t}"
  2872 
  2873 lemma hull_same: "s \<in> S \<Longrightarrow> S hull s = s"
  2874   unfolding hull_def by auto
  2875 
  2876 lemma hull_in: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) \<in> S"
  2877 unfolding hull_def subset_iff by auto
  2878 
  2879 lemma hull_eq: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) = s \<longleftrightarrow> s \<in> S"
  2880 using hull_same[of s S] hull_in[of S s] by metis  
  2881 
  2882 
  2883 lemma hull_hull: "S hull (S hull s) = S hull s"
  2884   unfolding hull_def by blast
  2885 
  2886 lemma hull_subset: "s \<subseteq> (S hull s)"
  2887   unfolding hull_def by blast
  2888 
  2889 lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
  2890   unfolding hull_def by blast
  2891 
  2892 lemma hull_antimono: "S \<subseteq> T ==> (T hull s) \<subseteq> (S hull s)"
  2893   unfolding hull_def by blast
  2894 
  2895 lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> t \<in> S ==> (S hull s) \<subseteq> t"
  2896   unfolding hull_def by blast
  2897 
  2898 lemma subset_hull: "t \<in> S ==> S hull s \<subseteq> t \<longleftrightarrow>  s \<subseteq> t"
  2899   unfolding hull_def by blast
  2900 
  2901 lemma hull_unique: "s \<subseteq> t \<Longrightarrow> t \<in> S \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> t' \<in> S ==> t \<subseteq> t')
  2902            ==> (S hull s = t)"
  2903 unfolding hull_def by auto
  2904 
  2905 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"
  2906   using hull_minimal[of S "{x. P x}" Q]
  2907   by (auto simp add: subset_eq Collect_def mem_def)
  2908 
  2909 lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" by (metis hull_subset subset_eq)
  2910 
  2911 lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
  2912 unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
  2913 
  2914 lemma hull_union: assumes T: "\<And>T. T \<subseteq> S ==> Inter T \<in> S"
  2915   shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
  2916 apply rule
  2917 apply (rule hull_mono)
  2918 unfolding Un_subset_iff
  2919 apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
  2920 apply (rule hull_minimal)
  2921 apply (metis hull_union_subset)
  2922 apply (metis hull_in T)
  2923 done
  2924 
  2925 lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
  2926   unfolding hull_def by blast
  2927 
  2928 lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
  2929 by (metis hull_redundant_eq)
  2930 
  2931 text{* Archimedian properties and useful consequences. *}
  2932 
  2933 lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
  2934   using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto)
  2935 lemmas real_arch_lt = reals_Archimedean2
  2936 
  2937 lemmas real_arch = reals_Archimedean3
  2938 
  2939 lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
  2940   using reals_Archimedean
  2941   apply (auto simp add: field_simps inverse_positive_iff_positive)
  2942   apply (subgoal_tac "inverse (real n) > 0")
  2943   apply arith
  2944   apply simp
  2945   done
  2946 
  2947 lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
  2948 proof(induct n)
  2949   case 0 thus ?case by simp
  2950 next 
  2951   case (Suc n)
  2952   hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
  2953   from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
  2954   from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
  2955   also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric]) 
  2956     apply (simp add: ring_simps)
  2957     using mult_left_mono[OF p Suc.prems] by simp
  2958   finally show ?case  by (simp add: real_of_nat_Suc ring_simps)
  2959 qed
  2960 
  2961 lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
  2962 proof-
  2963   from x have x0: "x - 1 > 0" by arith
  2964   from real_arch[OF x0, rule_format, of y] 
  2965   obtain n::nat where n:"y < real n * (x - 1)" by metis
  2966   from x0 have x00: "x- 1 \<ge> 0" by arith
  2967   from real_pow_lbound[OF x00, of n] n 
  2968   have "y < x^n" by auto
  2969   then show ?thesis by metis
  2970 qed 
  2971 
  2972 lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
  2973   using real_arch_pow[of 2 x] by simp
  2974 
  2975 lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
  2976   shows "\<exists>n. x^n < y"
  2977 proof-
  2978   {assume x0: "x > 0" 
  2979     from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
  2980     from real_arch_pow[OF ix, of "1/y"]
  2981     obtain n where n: "1/y < (1/x)^n" by blast
  2982     then 
  2983     have ?thesis using y x0 by (auto simp add: field_simps power_divide) }
  2984   moreover 
  2985   {assume "\<not> x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)}
  2986   ultimately show ?thesis by metis
  2987 qed
  2988 
  2989 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)"
  2990   by (metis real_arch_inv)
  2991 
  2992 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"
  2993   apply (rule forall_pos_mono)
  2994   apply auto
  2995   apply (atomize)
  2996   apply (erule_tac x="n - 1" in allE)
  2997   apply auto
  2998   done
  2999 
  3000 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"
  3001   shows "x = 0"
  3002 proof-
  3003   {assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
  3004     from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x"  by blast
  3005     with xc[rule_format, of n] have "n = 0" by arith
  3006     with n c have False by simp}
  3007   then show ?thesis by blast
  3008 qed
  3009 
  3010 (* ------------------------------------------------------------------------- *)
  3011 (* Relate max and min to sup and inf.                                        *)
  3012 (* ------------------------------------------------------------------------- *)
  3013 
  3014 lemma real_max_rsup: "max x y = rsup {x,y}"
  3015 proof-
  3016   have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
  3017   from rsup_finite_le_iff[OF f, of "max x y"] have "rsup {x,y} \<le> max x y" by simp
  3018   moreover
  3019   have "max x y \<le> rsup {x,y}" using rsup_finite_ge_iff[OF f, of "max x y"]
  3020     by (simp add: linorder_linear)
  3021   ultimately show ?thesis by arith
  3022 qed 
  3023 
  3024 lemma real_min_rinf: "min x y = rinf {x,y}"
  3025 proof-
  3026   have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
  3027   from rinf_finite_le_iff[OF f, of "min x y"] have "rinf {x,y} \<le> min x y" 
  3028     by (simp add: linorder_linear)
  3029   moreover
  3030   have "min x y \<le> rinf {x,y}" using rinf_finite_ge_iff[OF f, of "min x y"]
  3031     by simp
  3032   ultimately show ?thesis by arith
  3033 qed 
  3034 
  3035 (* ------------------------------------------------------------------------- *)
  3036 (* Geometric progression.                                                    *)
  3037 (* ------------------------------------------------------------------------- *)
  3038 
  3039 lemma sum_gp_basic: "((1::'a::{field, recpower}) - x) * setsum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
  3040   (is "?lhs = ?rhs")
  3041 proof-
  3042   {assume x1: "x = 1" hence ?thesis by simp}
  3043   moreover
  3044   {assume x1: "x\<noteq>1"
  3045     hence x1': "x - 1 \<noteq> 0" "1 - x \<noteq> 0" "x - 1 = - (1 - x)" "- (1 - x) \<noteq> 0" by auto
  3046     from geometric_sum[OF x1, of "Suc n", unfolded x1']
  3047     have "(- (1 - x)) * setsum (\<lambda>i. x^i) {0 .. n} = - (1 - x^(Suc n))"
  3048       unfolding atLeastLessThanSuc_atLeastAtMost
  3049       using x1' apply (auto simp only: field_simps)
  3050       apply (simp add: ring_simps)
  3051       done
  3052     then have ?thesis by (simp add: ring_simps) }
  3053   ultimately show ?thesis by metis
  3054 qed
  3055 
  3056 lemma sum_gp_multiplied: assumes mn: "m <= n"
  3057   shows "((1::'a::{field, recpower}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
  3058   (is "?lhs = ?rhs")
  3059 proof-
  3060   let ?S = "{0..(n - m)}"
  3061   from mn have mn': "n - m \<ge> 0" by arith
  3062   let ?f = "op + m"
  3063   have i: "inj_on ?f ?S" unfolding inj_on_def by auto
  3064   have f: "?f ` ?S = {m..n}" 
  3065     using mn apply (auto simp add: image_iff Bex_def) by arith
  3066   have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)" 
  3067     by (rule ext, simp add: power_add power_mult)
  3068   from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
  3069   have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp
  3070   then show ?thesis unfolding sum_gp_basic using mn
  3071     by (simp add: ring_simps power_add[symmetric])
  3072 qed
  3073 
  3074 lemma sum_gp: "setsum (op ^ (x::'a::{field, recpower})) {m .. n} = 
  3075    (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m) 
  3076                     else (x^ m - x^ (Suc n)) / (1 - x))"
  3077 proof-
  3078   {assume nm: "n < m" hence ?thesis by simp}
  3079   moreover
  3080   {assume "\<not> n < m" hence nm: "m \<le> n" by arith
  3081     {assume x: "x = 1"  hence ?thesis by simp}
  3082     moreover
  3083     {assume x: "x \<noteq> 1" hence nz: "1 - x \<noteq> 0" by simp
  3084       from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)}
  3085     ultimately have ?thesis by metis
  3086   }
  3087   ultimately show ?thesis by metis
  3088 qed
  3089 
  3090 lemma sum_gp_offset: "setsum (op ^ (x::'a::{field,recpower})) {m .. m+n} = 
  3091   (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
  3092   unfolding sum_gp[of x m "m + n"] power_Suc
  3093   by (simp add: ring_simps power_add)
  3094 
  3095 
  3096 subsection{* A bit of linear algebra. *}
  3097 
  3098 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 )"
  3099 definition "span S = (subspace hull S)"
  3100 definition "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
  3101 abbreviation "independent s == ~(dependent s)"
  3102 
  3103 (* Closure properties of subspaces.                                          *)
  3104 
  3105 lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
  3106 
  3107 lemma subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
  3108 
  3109 lemma subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S" 
  3110   by (metis subspace_def)
  3111 
  3112 lemma subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *s x \<in> S"
  3113   by (metis subspace_def)
  3114 
  3115 lemma subspace_neg: "subspace S \<Longrightarrow> (x::'a::ring_1^'n) \<in> S \<Longrightarrow> - x \<in> S"
  3116   by (metis vector_sneg_minus1 subspace_mul)
  3117 
  3118 lemma subspace_sub: "subspace S \<Longrightarrow> (x::'a::ring_1^'n) \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
  3119   by (metis diff_def subspace_add subspace_neg)
  3120 
  3121 lemma subspace_setsum:
  3122   assumes sA: "subspace A" and fB: "finite B"
  3123   and f: "\<forall>x\<in> B. f x \<in> A"
  3124   shows "setsum f B \<in> A"
  3125   using  fB f sA
  3126   apply(induct rule: finite_induct[OF fB])
  3127   by (simp add: subspace_def sA, auto simp add: sA subspace_add) 
  3128 
  3129 lemma subspace_linear_image: 
  3130   assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and sS: "subspace S" 
  3131   shows "subspace(f ` S)"
  3132   using lf sS linear_0[OF lf]
  3133   unfolding linear_def subspace_def
  3134   apply (auto simp add: image_iff)
  3135   apply (rule_tac x="x + y" in bexI, auto)
  3136   apply (rule_tac x="c*s x" in bexI, auto)
  3137   done
  3138 
  3139 lemma subspace_linear_preimage: "linear (f::'a::semiring_1^'n \<Rightarrow> _) ==> subspace S ==> subspace {x. f x \<in> S}"
  3140   by (auto simp add: subspace_def linear_def linear_0[of f])
  3141 
  3142 lemma subspace_trivial: "subspace {0::'a::semiring_1 ^_}"
  3143   by (simp add: subspace_def)
  3144 
  3145 lemma subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
  3146   by (simp add: subspace_def)
  3147 
  3148 
  3149 lemma span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
  3150   by (metis span_def hull_mono)
  3151 
  3152 lemma subspace_span: "subspace(span S)"
  3153   unfolding span_def
  3154   apply (rule hull_in[unfolded mem_def])
  3155   apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
  3156   apply auto
  3157   apply (erule_tac x="X" in ballE)
  3158   apply (simp add: mem_def)
  3159   apply blast
  3160   apply (erule_tac x="X" in ballE)
  3161   apply (erule_tac x="X" in ballE)
  3162   apply (erule_tac x="X" in ballE)
  3163   apply (clarsimp simp add: mem_def)
  3164   apply simp
  3165   apply simp
  3166   apply simp
  3167   apply (erule_tac x="X" in ballE)
  3168   apply (erule_tac x="X" in ballE)
  3169   apply (simp add: mem_def)
  3170   apply simp
  3171   apply simp
  3172   done
  3173 
  3174 lemma span_clauses:
  3175   "a \<in> S ==> a \<in> span S"
  3176   "0 \<in> span S"
  3177   "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
  3178   "x \<in> span S \<Longrightarrow> c *s x \<in> span S"
  3179   by (metis span_def hull_subset subset_eq subspace_span subspace_def)+
  3180 
  3181 lemma span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
  3182   and P: "subspace P" and x: "x \<in> span S" shows "P x"
  3183 proof-
  3184   from SP have SP': "S \<subseteq> P" by (simp add: mem_def subset_eq)
  3185   from P have P': "P \<in> subspace" by (simp add: mem_def)
  3186   from x hull_minimal[OF SP' P', unfolded span_def[symmetric]]
  3187   show "P x" by (metis mem_def subset_eq) 
  3188 qed
  3189 
  3190 lemma span_empty: "span {} = {(0::'a::semiring_0 ^ 'n)}"
  3191   apply (simp add: span_def)
  3192   apply (rule hull_unique)
  3193   apply (auto simp add: mem_def subspace_def)
  3194   unfolding mem_def[of "0::'a^'n", symmetric]
  3195   apply simp
  3196   done
  3197 
  3198 lemma independent_empty: "independent {}"
  3199   by (simp add: dependent_def)
  3200 
  3201 lemma independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
  3202   apply (clarsimp simp add: dependent_def span_mono)
  3203   apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
  3204   apply force
  3205   apply (rule span_mono)
  3206   apply auto
  3207   done
  3208 
  3209 lemma span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
  3210   by (metis order_antisym span_def hull_minimal mem_def)
  3211 
  3212 lemma span_induct': assumes SP: "\<forall>x \<in> S. P x"
  3213   and P: "subspace P" shows "\<forall>x \<in> span S. P x"
  3214   using span_induct SP P by blast
  3215 
  3216 inductive span_induct_alt_help for S:: "'a::semiring_1^'n \<Rightarrow> bool"
  3217   where 
  3218   span_induct_alt_help_0: "span_induct_alt_help S 0"
  3219   | 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)"
  3220 
  3221 lemma span_induct_alt': 
  3222   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"
  3223 proof-
  3224   {fix x:: "'a^'n" assume x: "span_induct_alt_help S x"
  3225     have "h x"
  3226       apply (rule span_induct_alt_help.induct[OF x])
  3227       apply (rule h0)
  3228       apply (rule hS, assumption, assumption)
  3229       done}
  3230   note th0 = this
  3231   {fix x assume x: "x \<in> span S"
  3232     
  3233     have "span_induct_alt_help S x"
  3234       proof(rule span_induct[where x=x and S=S])
  3235 	show "x \<in> span S" using x .
  3236       next
  3237 	fix x assume xS : "x \<in> S"
  3238 	  from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
  3239 	  show "span_induct_alt_help S x" by simp
  3240 	next
  3241 	have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0)
  3242 	moreover
  3243 	{fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y"
  3244 	  from h 
  3245 	  have "span_induct_alt_help S (x + y)"
  3246 	    apply (induct rule: span_induct_alt_help.induct)
  3247 	    apply simp
  3248 	    unfolding add_assoc
  3249 	    apply (rule span_induct_alt_help_S)
  3250 	    apply assumption
  3251 	    apply simp
  3252 	    done}
  3253 	moreover
  3254 	{fix c x assume xt: "span_induct_alt_help S x"
  3255 	  then have "span_induct_alt_help S (c*s x)" 
  3256 	    apply (induct rule: span_induct_alt_help.induct)
  3257 	    apply (simp add: span_induct_alt_help_0)
  3258 	    apply (simp add: vector_smult_assoc vector_add_ldistrib)
  3259 	    apply (rule span_induct_alt_help_S)
  3260 	    apply assumption
  3261 	    apply simp
  3262 	    done
  3263 	}
  3264 	ultimately show "subspace (span_induct_alt_help S)" 
  3265 	  unfolding subspace_def mem_def Ball_def by blast
  3266       qed}
  3267   with th0 show ?thesis by blast
  3268 qed 
  3269 
  3270 lemma span_induct_alt: 
  3271   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"
  3272   shows "h x"
  3273 using span_induct_alt'[of h S] h0 hS x by blast
  3274 
  3275 (* Individual closure properties. *)
  3276 
  3277 lemma span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses)
  3278 
  3279 lemma span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
  3280 
  3281 lemma span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
  3282   by (metis subspace_add subspace_span)
  3283 
  3284 lemma span_mul: "x \<in> span S ==> (c *s x) \<in> span S"
  3285   by (metis subspace_span subspace_mul)
  3286 
  3287 lemma span_neg: "x \<in> span S ==> - (x::'a::ring_1^'n) \<in> span S"
  3288   by (metis subspace_neg subspace_span)
  3289 
  3290 lemma span_sub: "(x::'a::ring_1^'n) \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
  3291   by (metis subspace_span subspace_sub)
  3292 
  3293 lemma span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
  3294   apply (rule subspace_setsum)
  3295   by (metis subspace_span subspace_setsum)+
  3296 
  3297 lemma span_add_eq: "(x::'a::ring_1^'n) \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
  3298   apply (auto simp only: span_add span_sub)
  3299   apply (subgoal_tac "(x + y) - x \<in> span S", simp)
  3300   by (simp only: span_add span_sub)
  3301 
  3302 (* Mapping under linear image. *)
  3303 
  3304 lemma span_linear_image: assumes lf: "linear (f::'a::semiring_1 ^ 'n => _)"
  3305   shows "span (f ` S) = f ` (span S)"
  3306 proof-
  3307   {fix x
  3308     assume x: "x \<in> span (f ` S)"
  3309     have "x \<in> f ` span S"
  3310       apply (rule span_induct[where x=x and S = "f ` S"])
  3311       apply (clarsimp simp add: image_iff)
  3312       apply (frule span_superset)
  3313       apply blast
  3314       apply (simp only: mem_def)
  3315       apply (rule subspace_linear_image[OF lf])
  3316       apply (rule subspace_span)
  3317       apply (rule x)
  3318       done}
  3319   moreover 
  3320   {fix x assume x: "x \<in> span S"
  3321     have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_ext) 
  3322       unfolding mem_def Collect_def ..
  3323     have "f x \<in> span (f ` S)"
  3324       apply (rule span_induct[where S=S])
  3325       apply (rule span_superset)
  3326       apply simp
  3327       apply (subst th0)
  3328       apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"])
  3329       apply (rule x)
  3330       done}
  3331   ultimately show ?thesis by blast
  3332 qed
  3333 
  3334 (* The key breakdown property. *)
  3335 
  3336 lemma span_breakdown:
  3337   assumes bS: "(b::'a::ring_1 ^ 'n) \<in> S" and aS: "a \<in> span S"
  3338   shows "\<exists>k. a - k*s b \<in> span (S - {b})" (is "?P a")
  3339 proof-
  3340   {fix x assume xS: "x \<in> S"
  3341     {assume ab: "x = b"
  3342       then have "?P x"
  3343 	apply simp
  3344 	apply (rule exI[where x="1"], simp)
  3345 	by (rule span_0)}
  3346     moreover
  3347     {assume ab: "x \<noteq> b" 
  3348       then have "?P x"  using xS
  3349 	apply -
  3350 	apply (rule exI[where x=0])
  3351 	apply (rule span_superset)
  3352 	by simp}
  3353     ultimately have "?P x" by blast}
  3354   moreover have "subspace ?P" 
  3355     unfolding subspace_def 
  3356     apply auto
  3357     apply (simp add: mem_def)
  3358     apply (rule exI[where x=0])
  3359     using span_0[of "S - {b}"]
  3360     apply (simp add: mem_def)
  3361     apply (clarsimp simp add: mem_def)
  3362     apply (rule_tac x="k + ka" in exI)
  3363     apply (subgoal_tac "x + y - (k + ka) *s b = (x - k*s b) + (y - ka *s b)")
  3364     apply (simp only: )
  3365     apply (rule span_add[unfolded mem_def])
  3366     apply assumption+
  3367     apply (vector ring_simps)
  3368     apply (clarsimp simp add: mem_def)
  3369     apply (rule_tac x= "c*k" in exI)
  3370     apply (subgoal_tac "c *s x - (c * k) *s b = c*s (x - k*s b)")
  3371     apply (simp only: )
  3372     apply (rule span_mul[unfolded mem_def])
  3373     apply assumption
  3374     by (vector ring_simps)
  3375   ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis 
  3376 qed
  3377 
  3378 lemma span_breakdown_eq:
  3379   "(x::'a::ring_1^'n) \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *s a) \<in> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
  3380 proof-
  3381   {assume x: "x \<in> span (insert a S)"
  3382     from x span_breakdown[of "a" "insert a S" "x"]
  3383     have ?rhs apply clarsimp
  3384       apply (rule_tac x= "k" in exI)
  3385       apply (rule set_rev_mp[of _ "span (S - {a})" _])
  3386       apply assumption
  3387       apply (rule span_mono)      
  3388       apply blast
  3389       done}
  3390   moreover
  3391   { fix k assume k: "x - k *s a \<in> span S"
  3392     have eq: "x = (x - k *s a) + k *s a" by vector
  3393     have "(x - k *s a) + k *s a \<in> span (insert a S)"
  3394       apply (rule span_add)
  3395       apply (rule set_rev_mp[of _ "span S" _])
  3396       apply (rule k)
  3397       apply (rule span_mono)      
  3398       apply blast
  3399       apply (rule span_mul)
  3400       apply (rule span_superset)
  3401       apply blast
  3402       done
  3403     then have ?lhs using eq by metis}
  3404   ultimately show ?thesis by blast
  3405 qed
  3406 
  3407 (* Hence some "reversal" results.*)
  3408 
  3409 lemma in_span_insert:
  3410   assumes a: "(a::'a::field^'n) \<in> span (insert b S)" and na: "a \<notin> span S"
  3411   shows "b \<in> span (insert a S)"
  3412 proof-
  3413   from span_breakdown[of b "insert b S" a, OF insertI1 a]
  3414   obtain k where k: "a - k*s b \<in> span (S - {b})" by auto
  3415   {assume k0: "k = 0"
  3416     with k have "a \<in> span S"
  3417       apply (simp)
  3418       apply (rule set_rev_mp)
  3419       apply assumption
  3420       apply (rule span_mono)
  3421       apply blast
  3422       done
  3423     with na  have ?thesis by blast}
  3424   moreover
  3425   {assume k0: "k \<noteq> 0" 
  3426     have eq: "b = (1/k) *s a - ((1/k) *s a - b)" by vector
  3427     from k0 have eq': "(1/k) *s (a - k*s b) = (1/k) *s a - b"
  3428       by (vector field_simps)
  3429     from k have "(1/k) *s (a - k*s b) \<in> span (S - {b})"
  3430       by (rule span_mul)
  3431     hence th: "(1/k) *s a - b \<in> span (S - {b})"
  3432       unfolding eq' .
  3433 
  3434     from k
  3435     have ?thesis
  3436       apply (subst eq)
  3437       apply (rule span_sub)
  3438       apply (rule span_mul)
  3439       apply (rule span_superset)
  3440       apply blast
  3441       apply (rule set_rev_mp)
  3442       apply (rule th)
  3443       apply (rule span_mono)
  3444       using na by blast}
  3445   ultimately show ?thesis by blast
  3446 qed
  3447 
  3448 lemma in_span_delete: 
  3449   assumes a: "(a::'a::field^'n) \<in> span S" 
  3450   and na: "a \<notin> span (S-{b})"
  3451   shows "b \<in> span (insert a (S - {b}))"
  3452   apply (rule in_span_insert)
  3453   apply (rule set_rev_mp)
  3454   apply (rule a)
  3455   apply (rule span_mono)
  3456   apply blast
  3457   apply (rule na)
  3458   done
  3459 
  3460 (* Transitivity property. *)
  3461 
  3462 lemma span_trans:
  3463   assumes x: "(x::'a::ring_1^'n) \<in> span S" and y: "y \<in> span (insert x S)"
  3464   shows "y \<in> span S"
  3465 proof-
  3466   from span_breakdown[of x "insert x S" y, OF insertI1 y]
  3467   obtain k where k: "y -k*s x \<in> span (S - {x})" by auto
  3468   have eq: "y = (y - k *s x) + k *s x" by vector
  3469   show ?thesis 
  3470     apply (subst eq)
  3471     apply (rule span_add)
  3472     apply (rule set_rev_mp)
  3473     apply (rule k)
  3474     apply (rule span_mono)
  3475     apply blast
  3476     apply (rule span_mul)
  3477     by (rule x)
  3478 qed
  3479 
  3480 (* ------------------------------------------------------------------------- *)
  3481 (* An explicit expansion is sometimes needed.                                *)
  3482 (* ------------------------------------------------------------------------- *)
  3483 
  3484 lemma span_explicit:
  3485   "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}"
  3486   (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
  3487 proof-
  3488   {fix x assume x: "x \<in> ?E"
  3489     then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *s v) S = x"
  3490       by blast
  3491     have "x \<in> span P"
  3492       unfolding u[symmetric]
  3493       apply (rule span_setsum[OF fS])
  3494       using span_mono[OF SP]
  3495       by (auto intro: span_superset span_mul)}
  3496   moreover
  3497   have "\<forall>x \<in> span P. x \<in> ?E"
  3498     unfolding mem_def Collect_def
  3499   proof(rule span_induct_alt')
  3500     show "?h 0"
  3501       apply (rule exI[where x="{}"]) by simp
  3502   next
  3503     fix c x y
  3504     assume x: "x \<in> P" and hy: "?h y"
  3505     from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P" 
  3506       and u: "setsum (\<lambda>v. u v *s v) S = y" by blast
  3507     let ?S = "insert x S"
  3508     let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c)
  3509                   else u y"
  3510     from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
  3511     {assume xS: "x \<in> S"
  3512       have S1: "S = (S - {x}) \<union> {x}" 
  3513 	and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
  3514       have "setsum (\<lambda>v. ?u v *s v) ?S =(\<Sum>v\<in>S - {x}. u v *s v) + (u x + c) *s x"
  3515 	using xS 
  3516 	by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]] 
  3517 	  setsum_clauses(2)[OF fS] cong del: if_weak_cong)
  3518       also have "\<dots> = (\<Sum>v\<in>S. u v *s v) + c *s x"
  3519 	apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
  3520 	by (vector ring_simps)
  3521       also have "\<dots> = c*s x + y"
  3522 	by (simp add: add_commute u)
  3523       finally have "setsum (\<lambda>v. ?u v *s v) ?S = c*s x + y" .
  3524     then have "?Q ?S ?u (c*s x + y)" using th0 by blast}
  3525   moreover 
  3526   {assume xS: "x \<notin> S"
  3527     have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *s v) = y"
  3528       unfolding u[symmetric]
  3529       apply (rule setsum_cong2)
  3530       using xS by auto
  3531     have "?Q ?S ?u (c*s x + y)" using fS xS th0
  3532       by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)}
  3533   ultimately have "?Q ?S ?u (c*s x + y)"
  3534     by (cases "x \<in> S", simp, simp)
  3535     then show "?h (c*s x + y)" 
  3536       apply -
  3537       apply (rule exI[where x="?S"])
  3538       apply (rule exI[where x="?u"]) by metis
  3539   qed
  3540   ultimately show ?thesis by blast
  3541 qed
  3542 
  3543 lemma dependent_explicit:
  3544   "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")
  3545 proof-
  3546   {assume dP: "dependent P"
  3547     then obtain a S u where aP: "a \<in> P" and fS: "finite S" 
  3548       and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *s v) S = a" 
  3549       unfolding dependent_def span_explicit by blast
  3550     let ?S = "insert a S" 
  3551     let ?u = "\<lambda>y. if y = a then - 1 else u y" 
  3552     let ?v = a
  3553     from aP SP have aS: "a \<notin> S" by blast
  3554     from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
  3555     have s0: "setsum (\<lambda>v. ?u v *s v) ?S = 0"
  3556       using fS aS
  3557       apply (simp add: vector_smult_lneg vector_smult_lid setsum_clauses ring_simps )
  3558       apply (subst (2) ua[symmetric])
  3559       apply (rule setsum_cong2)
  3560       by auto
  3561     with th0 have ?rhs
  3562       apply -
  3563       apply (rule exI[where x= "?S"])
  3564       apply (rule exI[where x= "?u"])
  3565       by clarsimp}
  3566   moreover
  3567   {fix S u v assume fS: "finite S" 
  3568       and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0" 
  3569     and u: "setsum (\<lambda>v. u v *s v) S = 0"
  3570     let ?a = v 
  3571     let ?S = "S - {v}"
  3572     let ?u = "\<lambda>i. (- u i) / u v"
  3573     have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"       using fS SP vS by auto 
  3574     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"
  3575       using fS vS uv 
  3576       by (simp add: setsum_diff1 vector_smult_lneg divide_inverse 
  3577 	vector_smult_assoc field_simps)
  3578     also have "\<dots> = ?a"
  3579       unfolding setsum_cmul u
  3580       using uv by (simp add: vector_smult_lneg)
  3581     finally  have "setsum (\<lambda>v. ?u v *s v) ?S = ?a" .
  3582     with th0 have ?lhs
  3583       unfolding dependent_def span_explicit
  3584       apply -
  3585       apply (rule bexI[where x= "?a"])
  3586       apply simp_all
  3587       apply (rule exI[where x= "?S"])
  3588       by auto}
  3589   ultimately show ?thesis by blast
  3590 qed
  3591 
  3592 
  3593 lemma span_finite:
  3594   assumes fS: "finite S"
  3595   shows "span S = {(y::'a::semiring_1^'n). \<exists>u. setsum (\<lambda>v. u v *s v) S = y}"
  3596   (is "_ = ?rhs")
  3597 proof-
  3598   {fix y assume y: "y \<in> span S"
  3599     from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and 
  3600       u: "setsum (\<lambda>v. u v *s v) S' = y" unfolding span_explicit by blast
  3601     let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
  3602     from setsum_restrict_set[OF fS, of "\<lambda>v. u v *s v" S', symmetric] SS'
  3603     have "setsum (\<lambda>v. ?u v *s v) S = setsum (\<lambda>v. u v *s v) S'"
  3604       unfolding cond_value_iff cond_application_beta
  3605       apply (simp add: cond_value_iff cong del: if_weak_cong)
  3606       apply (rule setsum_cong)
  3607       apply auto
  3608       done
  3609     hence "setsum (\<lambda>v. ?u v *s v) S = y" by (metis u)
  3610     hence "y \<in> ?rhs" by auto}
  3611   moreover 
  3612   {fix y u assume u: "setsum (\<lambda>v. u v *s v) S = y"
  3613     then have "y \<in> span S" using fS unfolding span_explicit by auto}
  3614   ultimately show ?thesis by blast
  3615 qed
  3616 
  3617 
  3618 (* Standard bases are a spanning set, and obviously finite.                  *)
  3619 
  3620 lemma span_stdbasis:"span {basis i :: 'a::ring_1^'n | i. i \<in> {1 .. dimindex(UNIV :: 'n set)}} = UNIV"
  3621 apply (rule set_ext)
  3622 apply auto
  3623 apply (subst basis_expansion[symmetric])
  3624 apply (rule span_setsum)
  3625 apply simp
  3626 apply auto
  3627 apply (rule span_mul)
  3628 apply (rule span_superset)
  3629 apply (auto simp add: Collect_def mem_def)
  3630 done
  3631 
  3632   
  3633 lemma has_size_stdbasis: "{basis i ::real ^'n | i. i \<in> {1 .. dimindex (UNIV :: 'n set)}} hassize (dimindex(UNIV :: 'n set))" (is "?S hassize ?n")
  3634 proof-
  3635   have eq: "?S = basis ` {1 .. ?n}" by blast
  3636   show ?thesis unfolding eq
  3637     apply (rule hassize_image_inj[OF basis_inj])
  3638     by (simp add: hassize_def)
  3639 qed
  3640 
  3641 lemma finite_stdbasis: "finite {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV:: 'n set)}}"
  3642   using has_size_stdbasis[unfolded hassize_def]
  3643   ..
  3644 
  3645 lemma card_stdbasis: "card {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}} = dimindex(UNIV :: 'n set)"
  3646   using has_size_stdbasis[unfolded hassize_def]
  3647   ..
  3648 
  3649 lemma independent_stdbasis_lemma:
  3650   assumes x: "(x::'a::semiring_1 ^ 'n) \<in> span (basis ` S)"
  3651   and i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
  3652   and iS: "i \<notin> S"
  3653   shows "(x$i) = 0"
  3654 proof-
  3655   let ?n = "dimindex (UNIV :: 'n set)"
  3656   let ?U = "{1 .. ?n}"
  3657   let ?B = "basis ` S"
  3658   let ?P = "\<lambda>(x::'a^'n). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0"
  3659  {fix x::"'a^'n" assume xS: "x\<in> ?B"
  3660    from xS have "?P x" by (auto simp add: basis_component)}
  3661  moreover
  3662  have "subspace ?P" 
  3663    by (auto simp add: subspace_def Collect_def mem_def zero_index vector_component)
  3664  ultimately show ?thesis
  3665    using x span_induct[of ?B ?P x] i iS by blast 
  3666 qed
  3667 
  3668 lemma independent_stdbasis: "independent {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
  3669 proof-
  3670   let ?n = "dimindex (UNIV :: 'n set)"
  3671   let ?I = "{1 .. ?n}"
  3672   let ?b = "basis :: nat \<Rightarrow> real ^'n"
  3673   let ?B = "?b ` ?I"
  3674   have eq: "{?b i|i. i \<in> ?I} = ?B"
  3675     by auto
  3676   {assume d: "dependent ?B"
  3677     then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"
  3678       unfolding dependent_def by auto
  3679     have eq1: "?B - {?b k} = ?B - ?b ` {k}"  by simp
  3680     have eq2: "?B - {?b k} = ?b ` (?I - {k})"
  3681       unfolding eq1
  3682       apply (rule inj_on_image_set_diff[symmetric])
  3683       apply (rule basis_inj) using k(1) by auto
  3684     from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
  3685     from independent_stdbasis_lemma[OF th0 k(1), simplified]
  3686     have False by (simp add: basis_component[OF k(1), of k])}
  3687   then show ?thesis unfolding eq dependent_def ..
  3688 qed
  3689 
  3690 (* This is useful for building a basis step-by-step.                         *)
  3691 
  3692 lemma independent_insert:
  3693   "independent(insert (a::'a::field ^'n) S) \<longleftrightarrow>
  3694       (if a \<in> S then independent S
  3695                 else independent S \<and> a \<notin> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
  3696 proof-
  3697   {assume aS: "a \<in> S"
  3698     hence ?thesis using insert_absorb[OF aS] by simp}
  3699   moreover
  3700   {assume aS: "a \<notin> S"
  3701     {assume i: ?lhs
  3702       then have ?rhs using aS
  3703 	apply simp
  3704 	apply (rule conjI)
  3705 	apply (rule independent_mono)
  3706 	apply assumption
  3707 	apply blast
  3708 	by (simp add: dependent_def)}
  3709     moreover 
  3710     {assume i: ?rhs
  3711       have ?lhs using i aS
  3712 	apply simp
  3713 	apply (auto simp add: dependent_def)
  3714 	apply (case_tac "aa = a", auto)
  3715 	apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
  3716 	apply simp
  3717 	apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
  3718 	apply (subgoal_tac "insert aa (S - {aa}) = S")
  3719 	apply simp
  3720 	apply blast
  3721 	apply (rule in_span_insert)
  3722 	apply assumption
  3723 	apply blast
  3724 	apply blast
  3725 	done}
  3726     ultimately have ?thesis by blast}
  3727   ultimately show ?thesis by blast
  3728 qed
  3729 
  3730 (* The degenerate case of the Exchange Lemma.  *)
  3731 
  3732 lemma mem_delete: "x \<in> (A - {a}) \<longleftrightarrow> x \<noteq> a \<and> x \<in> A"
  3733   by blast
  3734 
  3735 lemma span_span: "span (span A) = span A"
  3736   unfolding span_def hull_hull ..
  3737 
  3738 lemma span_inc: "S \<subseteq> span S"
  3739   by (metis subset_eq span_superset)
  3740 
  3741 lemma spanning_subset_independent:
  3742   assumes BA: "B \<subseteq> A" and iA: "independent (A::('a::field ^'n) set)" 
  3743   and AsB: "A \<subseteq> span B"
  3744   shows "A = B"
  3745 proof
  3746   from BA show "B \<subseteq> A" .
  3747 next
  3748   from span_mono[OF BA] span_mono[OF AsB]
  3749   have sAB: "span A = span B" unfolding span_span by blast
  3750 
  3751   {fix x assume x: "x \<in> A"
  3752     from iA have th0: "x \<notin> span (A - {x})"
  3753       unfolding dependent_def using x by blast
  3754     from x have xsA: "x \<in> span A" by (blast intro: span_superset)
  3755     have "A - {x} \<subseteq> A" by blast
  3756     hence th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
  3757     {assume xB: "x \<notin> B"
  3758       from xB BA have "B \<subseteq> A -{x}" by blast
  3759       hence "span B \<subseteq> span (A - {x})" by (metis span_mono)
  3760       with th1 th0 sAB have "x \<notin> span A" by blast
  3761       with x have False by (metis span_superset)}
  3762     then have "x \<in> B" by blast}
  3763   then show "A \<subseteq> B" by blast
  3764 qed
  3765 
  3766 (* The general case of the Exchange Lemma, the key to what follows.  *)
  3767 
  3768 lemma exchange_lemma:
  3769   assumes f:"finite (t:: ('a::field^'n) set)" and i: "independent s"
  3770   and sp:"s \<subseteq> span t" 
  3771   shows "\<exists>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
  3772 using f i sp
  3773 proof(induct c\<equiv>"card(t - s)" arbitrary: s t rule: nat_less_induct)
  3774   fix n:: nat and s t :: "('a ^'n) set"
  3775   assume H: " \<forall>m<n. \<forall>(x:: ('a ^'n) set) xa.
  3776                 finite xa \<longrightarrow>
  3777                 independent x \<longrightarrow>
  3778                 x \<subseteq> span xa \<longrightarrow>
  3779                 m = card (xa - x) \<longrightarrow>
  3780                 (\<exists>t'. (t' hassize card xa) \<and>
  3781                       x \<subseteq> t' \<and> t' \<subseteq> x \<union> xa \<and> x \<subseteq> span t')"
  3782     and ft: "finite t" and s: "independent s" and sp: "s \<subseteq> span t"
  3783     and n: "n = card (t - s)"
  3784   let ?P = "\<lambda>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
  3785   let ?ths = "\<exists>t'. ?P t'" 
  3786   {assume st: "s \<subseteq> t" 
  3787     from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t]) 
  3788       by (auto simp add: hassize_def intro: span_superset)}
  3789   moreover
  3790   {assume st: "t \<subseteq> s"
  3791     
  3792     from spanning_subset_independent[OF st s sp] 
  3793       st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t]) 
  3794       by (auto simp add: hassize_def intro: span_superset)}
  3795   moreover
  3796   {assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
  3797     from st(2) obtain b where b: "b \<in> t" "b \<notin> s" by blast
  3798       from b have "t - {b} - s \<subset> t - s" by blast
  3799       then have cardlt: "card (t - {b} - s) < n" using n ft
  3800  	by (auto intro: psubset_card_mono)
  3801       from b ft have ct0: "card t \<noteq> 0" by auto
  3802     {assume stb: "s \<subseteq> span(t -{b})"
  3803       from ft have ftb: "finite (t -{b})" by auto
  3804       from H[rule_format, OF cardlt ftb s stb] 
  3805       obtain u where u: "u hassize card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" by blast
  3806       let ?w = "insert b u"
  3807       have th0: "s \<subseteq> insert b u" using u by blast
  3808       from u(3) b have "u \<subseteq> s \<union> t" by blast 
  3809       then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
  3810       have bu: "b \<notin> u" using b u by blast
  3811       from u(1) have fu: "finite u" by (simp add: hassize_def)
  3812       from u(1) ft b have "u hassize (card t - 1)" by auto
  3813       then 
  3814       have th2: "insert b u hassize card t" 
  3815 	using  card_insert_disjoint[OF fu bu] ct0 by (auto simp add: hassize_def)
  3816       from u(4) have "s \<subseteq> span u" .
  3817       also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
  3818       finally have th3: "s \<subseteq> span (insert b u)" .      from th0 th1 th2 th3 have th: "?P ?w"  by blast
  3819       from th have ?ths by blast}
  3820     moreover
  3821     {assume stb: "\<not> s \<subseteq> span(t -{b})" 
  3822       from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
  3823       have ab: "a \<noteq> b" using a b by blast
  3824       have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
  3825       have mlt: "card ((insert a (t - {b})) - s) < n" 
  3826 	using cardlt ft n  a b by auto
  3827       have ft': "finite (insert a (t - {b}))" using ft by auto
  3828       {fix x assume xs: "x \<in> s"
  3829 	have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
  3830 	from b(1) have "b \<in> span t" by (simp add: span_superset)
  3831 	have bs: "b \<in> span (insert a (t - {b}))"
  3832 	  by (metis in_span_delete a sp mem_def subset_eq)
  3833 	from xs sp have "x \<in> span t" by blast
  3834 	with span_mono[OF t]
  3835 	have x: "x \<in> span (insert b (insert a (t - {b})))" ..
  3836 	from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))"  .}
  3837       then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
  3838       
  3839       from H[rule_format, OF mlt ft' s sp' refl] obtain u where 
  3840 	u: "u hassize card (insert a (t -{b}))" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
  3841 	"s \<subseteq> span u" by blast
  3842       from u a b ft at ct0 have "?P u" by (auto simp add: hassize_def)
  3843       then have ?ths by blast }
  3844     ultimately have ?ths by blast
  3845   }
  3846   ultimately 
  3847   show ?ths  by blast
  3848 qed
  3849 
  3850 (* This implies corresponding size bounds.                                   *)
  3851 
  3852 lemma independent_span_bound:
  3853   assumes f: "finite t" and i: "independent (s::('a::field^'n) set)" and sp:"s \<subseteq> span t"
  3854   shows "finite s \<and> card s \<le> card t"
  3855   by (metis exchange_lemma[OF f i sp] hassize_def finite_subset card_mono)
  3856 
  3857 lemma finite_Atleast_Atmost[simp]: "finite {f x |x. x\<in> {(i::'a::finite_intvl_succ) .. j}}"
  3858 proof-
  3859   have eq: "{f x |x. x\<in> {i .. j}} = f ` {i .. j}" by auto
  3860   show ?thesis unfolding eq 
  3861     apply (rule finite_imageI)
  3862     apply (rule finite_intvl)
  3863     done
  3864 qed
  3865 
  3866 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> {(i::nat) .. j}}"
  3867 proof-
  3868   have eq: "{f x |x. x\<in> {i .. j}} = f ` {i .. j}" by auto
  3869   show ?thesis unfolding eq 
  3870     apply (rule finite_imageI)
  3871     apply (rule finite_atLeastAtMost)
  3872     done
  3873 qed
  3874 
  3875 
  3876 lemma independent_bound:
  3877   fixes S:: "(real^'n) set"
  3878   shows "independent S \<Longrightarrow> finite S \<and> card S <= dimindex(UNIV :: 'n set)"
  3879   apply (subst card_stdbasis[symmetric])
  3880   apply (rule independent_span_bound)
  3881   apply (rule finite_Atleast_Atmost_nat)
  3882   apply assumption
  3883   unfolding span_stdbasis 
  3884   apply (rule subset_UNIV)
  3885   done
  3886 
  3887 lemma dependent_biggerset: "(finite (S::(real ^'n) set) ==> card S > dimindex(UNIV:: 'n set)) ==> dependent S"
  3888   by (metis independent_bound not_less)
  3889 
  3890 (* Hence we can create a maximal independent subset.                         *)
  3891 
  3892 lemma maximal_independent_subset_extend:
  3893   assumes sv: "(S::(real^'n) set) \<subseteq> V" and iS: "independent S"
  3894   shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
  3895   using sv iS
  3896 proof(induct d\<equiv> "dimindex (UNIV :: 'n set) - card S" arbitrary: S rule: nat_less_induct)
  3897   fix n and S:: "(real^'n) set"
  3898   assume H: "\<forall>m<n. \<forall>S \<subseteq> V. independent S \<longrightarrow> m = dimindex (UNIV::'n set) - card S \<longrightarrow>
  3899               (\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B)"
  3900     and sv: "S \<subseteq> V" and i: "independent S" and n: "n = dimindex (UNIV :: 'n set) - card S"
  3901   let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
  3902   let ?ths = "\<exists>x. ?P x"
  3903   let ?d = "dimindex (UNIV :: 'n set)"
  3904   {assume "V \<subseteq> span S"
  3905     then have ?ths  using sv i by blast }
  3906   moreover
  3907   {assume VS: "\<not> V \<subseteq> span S"
  3908     from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
  3909     from a have aS: "a \<notin> S" by (auto simp add: span_superset)
  3910     have th0: "insert a S \<subseteq> V" using a sv by blast
  3911     from independent_insert[of a S]  i a 
  3912     have th1: "independent (insert a S)" by auto
  3913     have mlt: "?d - card (insert a S) < n" 
  3914       using aS a n independent_bound[OF th1] dimindex_ge_1[of "UNIV :: 'n set"] 
  3915       by auto 
  3916       
  3917     from H[rule_format, OF mlt th0 th1 refl] 
  3918     obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B" 
  3919       by blast
  3920     from B have "?P B" by auto
  3921     then have ?ths by blast}
  3922   ultimately show ?ths by blast
  3923 qed
  3924 
  3925 lemma maximal_independent_subset:
  3926   "\<exists>(B:: (real ^'n) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
  3927   by (metis maximal_independent_subset_extend[of "{}:: (real ^'n) set"] empty_subsetI independent_empty)
  3928 
  3929 (* Notion of dimension.                                                      *)
  3930 
  3931 definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n))"
  3932 
  3933 lemma basis_exists:  "\<exists>B. (B :: (real ^'n) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize dim V)" 
  3934 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)"]
  3935 unfolding hassize_def
  3936 using maximal_independent_subset[of V] independent_bound
  3937 by auto
  3938 
  3939 (* Consequences of independence or spanning for cardinality.                 *)
  3940 
  3941 lemma independent_card_le_dim: "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B \<le> dim V"
  3942 by (metis basis_exists[of V] independent_span_bound[where ?'a=real] hassize_def subset_trans)
  3943 
  3944 lemma span_card_ge_dim:  "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
  3945   by (metis basis_exists[of V] independent_span_bound hassize_def subset_trans)
  3946 
  3947 lemma basis_card_eq_dim:
  3948   "B \<subseteq> (V:: (real ^'n) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
  3949   by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_mono)
  3950 
  3951 lemma dim_unique: "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> B hassize n \<Longrightarrow> dim V = n"
  3952   by (metis basis_card_eq_dim hassize_def)
  3953 
  3954 (* More lemmas about dimension.                                              *)
  3955 
  3956 lemma dim_univ: "dim (UNIV :: (real^'n) set) = dimindex (UNIV :: 'n set)"
  3957   apply (rule dim_unique[of "{basis i |i. i\<in> {1 .. dimindex (UNIV :: 'n set)}}"])
  3958   by (auto simp only: span_stdbasis has_size_stdbasis independent_stdbasis)
  3959 
  3960 lemma dim_subset:
  3961   "(S:: (real ^'n) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
  3962   using basis_exists[of T] basis_exists[of S]
  3963   by (metis independent_span_bound[where ?'a = real and ?'n = 'n] subset_eq hassize_def)
  3964 
  3965 lemma dim_subset_univ: "dim (S:: (real^'n) set) \<le> dimindex (UNIV :: 'n set)"
  3966   by (metis dim_subset subset_UNIV dim_univ)
  3967 
  3968 (* Converses to those.                                                       *)
  3969 
  3970 lemma card_ge_dim_independent:
  3971   assumes BV:"(B::(real ^'n) set) \<subseteq> V" and iB:"independent B" and dVB:"dim V \<le> card B"
  3972   shows "V \<subseteq> span B"
  3973 proof-
  3974   {fix a assume aV: "a \<in> V"
  3975     {assume aB: "a \<notin> span B"
  3976       then have iaB: "independent (insert a B)" using iB aV  BV by (simp add: independent_insert)
  3977       from aV BV have th0: "insert a B \<subseteq> V" by blast
  3978       from aB have "a \<notin>B" by (auto simp add: span_superset)
  3979       with independent_card_le_dim[OF th0 iaB] dVB  have False by auto}
  3980     then have "a \<in> span B"  by blast}
  3981   then show ?thesis by blast
  3982 qed
  3983 
  3984 lemma card_le_dim_spanning:
  3985   assumes BV: "(B:: (real ^'n) set) \<subseteq> V" and VB: "V \<subseteq> span B" 
  3986   and fB: "finite B" and dVB: "dim V \<ge> card B"
  3987   shows "independent B"
  3988 proof-
  3989   {fix a assume a: "a \<in> B" "a \<in> span (B -{a})"
  3990     from a fB have c0: "card B \<noteq> 0" by auto
  3991     from a fB have cb: "card (B -{a}) = card B - 1" by auto
  3992     from BV a have th0: "B -{a} \<subseteq> V" by blast
  3993     {fix x assume x: "x \<in> V"
  3994       from a have eq: "insert a (B -{a}) = B" by blast
  3995       from x VB have x': "x \<in> span B" by blast 
  3996       from span_trans[OF a(2), unfolded eq, OF x']
  3997       have "x \<in> span (B -{a})" . }
  3998     then have th1: "V \<subseteq> span (B -{a})" by blast 
  3999     have th2: "finite (B -{a})" using fB by auto
  4000     from span_card_ge_dim[OF th0 th1 th2]
  4001     have c: "dim V \<le> card (B -{a})" .
  4002     from c c0 dVB cb have False by simp}
  4003   then show ?thesis unfolding dependent_def by blast
  4004 qed
  4005 
  4006 lemma card_eq_dim: "(B:: (real ^'n) set) \<subseteq> V \<Longrightarrow> B hassize dim V \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
  4007   by (metis hassize_def order_eq_iff card_le_dim_spanning 
  4008     card_ge_dim_independent)
  4009 
  4010 (* ------------------------------------------------------------------------- *)
  4011 (* More general size bound lemmas.                                           *)
  4012 (* ------------------------------------------------------------------------- *)
  4013 
  4014 lemma independent_bound_general:
  4015   "independent (S:: (real^'n) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
  4016   by (metis independent_card_le_dim independent_bound subset_refl)
  4017 
  4018 lemma dependent_biggerset_general: "(finite (S:: (real^'n) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
  4019   using independent_bound_general[of S] by (metis linorder_not_le) 
  4020 
  4021 lemma dim_span: "dim (span (S:: (real ^'n) set)) = dim S"
  4022 proof-
  4023   have th0: "dim S \<le> dim (span S)" 
  4024     by (auto simp add: subset_eq intro: dim_subset span_superset)
  4025   from basis_exists[of S] 
  4026   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
  4027   from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
  4028   have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc) 
  4029   have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span) 
  4030   from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis 
  4031     using fB(2)  by arith
  4032 qed
  4033 
  4034 lemma subset_le_dim: "(S:: (real ^'n) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
  4035   by (metis dim_span dim_subset)
  4036 
  4037 lemma span_eq_dim: "span (S:: (real ^'n) set) = span T ==> dim S = dim T"
  4038   by (metis dim_span)
  4039 
  4040 lemma spans_image:
  4041   assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and VB: "V \<subseteq> span B"
  4042   shows "f ` V \<subseteq> span (f ` B)"
  4043   unfolding span_linear_image[OF lf]
  4044   by (metis VB image_mono)
  4045 
  4046 lemma dim_image_le: assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S:: (real ^'n) set)"
  4047 proof-
  4048   from basis_exists[of S] obtain B where 
  4049     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
  4050   from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
  4051   have "dim (f ` S) \<le> card (f ` B)"
  4052     apply (rule span_card_ge_dim)
  4053     using lf B fB by (auto simp add: span_linear_image spans_image subset_image_iff)
  4054   also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
  4055   finally show ?thesis .
  4056 qed
  4057 
  4058 (* Relation between bases and injectivity/surjectivity of map.               *)
  4059 
  4060 lemma spanning_surjective_image:
  4061   assumes us: "UNIV \<subseteq> span (S:: ('a::semiring_1 ^'n) set)" 
  4062   and lf: "linear f" and sf: "surj f"
  4063   shows "UNIV \<subseteq> span (f ` S)"
  4064 proof-
  4065   have "UNIV \<subseteq> f ` UNIV" using sf by (auto simp add: surj_def)
  4066   also have " \<dots> \<subseteq> span (f ` S)" using spans_image[OF lf us] .
  4067 finally show ?thesis .
  4068 qed
  4069 
  4070 lemma independent_injective_image:
  4071   assumes iS: "independent (S::('a::semiring_1^'n) set)" and lf: "linear f" and fi: "inj f"
  4072   shows "independent (f ` S)"
  4073 proof-
  4074   {fix a assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
  4075     have eq: "f ` S - {f a} = f ` (S - {a})" using fi
  4076       by (auto simp add: inj_on_def)
  4077     from a have "f a \<in> f ` span (S -{a})"
  4078       unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
  4079     hence "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
  4080     with a(1) iS  have False by (simp add: dependent_def) }
  4081   then show ?thesis unfolding dependent_def by blast
  4082 qed 
  4083 
  4084 (* ------------------------------------------------------------------------- *)
  4085 (* Picking an orthogonal replacement for a spanning set.                     *)
  4086 (* ------------------------------------------------------------------------- *)
  4087     (* FIXME : Move to some general theory ?*)
  4088 definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
  4089 
  4090 lemma vector_sub_project_orthogonal: "(b::'a::ordered_field^'n) \<bullet> (x - ((b \<bullet> x) / (b\<bullet>b)) *s b) = 0"
  4091   apply (cases "b = 0", simp)
  4092   apply (simp add: dot_rsub dot_rmult)
  4093   unfolding times_divide_eq_right[symmetric]
  4094   by (simp add: field_simps dot_eq_0)
  4095 
  4096 lemma basis_orthogonal:
  4097   fixes B :: "(real ^'n) set"
  4098   assumes fB: "finite B"
  4099   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
  4100   (is " \<exists>C. ?P B C")
  4101 proof(induct rule: finite_induct[OF fB])
  4102   case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
  4103 next
  4104   case (2 a B)
  4105   note fB = `finite B` and aB = `a \<notin> B` 
  4106   from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C` 
  4107   obtain C where C: "finite C" "card C \<le> card B" 
  4108     "span C = span B" "pairwise orthogonal C" by blast
  4109   let ?a = "a - setsum (\<lambda>x. (x\<bullet>a / (x\<bullet>x)) *s x) C"
  4110   let ?C = "insert ?a C"
  4111   from C(1) have fC: "finite ?C" by simp
  4112   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" by (simp add: card_insert_if)
  4113   {fix x k 
  4114     have th0: "\<And>(a::'b::comm_ring) b c. a - (b - c) = c + (a - b)" by (simp add: ring_simps)
  4115     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"
  4116       apply (simp only: vector_ssub_ldistrib th0)
  4117       apply (rule span_add_eq)
  4118       apply (rule span_mul)
  4119       apply (rule span_setsum[OF C(1)])
  4120       apply clarify
  4121       apply (rule span_mul)
  4122       by (rule span_superset)}
  4123   then have SC: "span ?C = span (insert a B)"
  4124     unfolding expand_set_eq span_breakdown_eq C(3)[symmetric] by auto
  4125   thm pairwise_def 
  4126   {fix x y assume xC: "x \<in> ?C" and yC: "y \<in> ?C" and xy: "x \<noteq> y"
  4127     {assume xa: "x = ?a" and ya: "y = ?a" 
  4128       have "orthogonal x y" using xa ya xy by blast}
  4129     moreover
  4130     {assume xa: "x = ?a" and ya: "y \<noteq> ?a" "y \<in> C" 
  4131       from ya have Cy: "C = insert y (C - {y})" by blast
  4132       have fth: "finite (C - {y})" using C by simp
  4133       have "orthogonal x y"
  4134 	using xa ya
  4135 	unfolding orthogonal_def xa dot_lsub dot_rsub diff_eq_0_iff_eq
  4136 	apply simp 
  4137 	apply (subst Cy)
  4138 	using C(1) fth
  4139 	apply (simp only: setsum_clauses)
  4140 	apply (auto simp add: dot_ladd dot_lmult dot_eq_0 dot_sym[of y a] dot_lsum[OF fth])
  4141 	apply (rule setsum_0')
  4142 	apply clarsimp
  4143 	apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
  4144 	by auto}
  4145     moreover
  4146     {assume xa: "x \<noteq> ?a" "x \<in> C" and ya: "y = ?a" 
  4147       from xa have Cx: "C = insert x (C - {x})" by blast
  4148       have fth: "finite (C - {x})" using C by simp
  4149       have "orthogonal x y"
  4150 	using xa ya
  4151 	unfolding orthogonal_def ya dot_rsub dot_lsub diff_eq_0_iff_eq
  4152 	apply simp 
  4153 	apply (subst Cx)
  4154 	using C(1) fth
  4155 	apply (simp only: setsum_clauses)
  4156 	apply (subst dot_sym[of x])
  4157 	apply (auto simp add: dot_radd dot_rmult dot_eq_0 dot_sym[of x a] dot_rsum[OF fth])
  4158 	apply (rule setsum_0')
  4159 	apply clarsimp
  4160 	apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
  4161 	by auto}
  4162     moreover
  4163     {assume xa: "x \<in> C" and ya: "y \<in> C" 
  4164       have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast}
  4165     ultimately have "orthogonal x y" using xC yC by blast}
  4166   then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast
  4167   from fC cC SC CPO have "?P (insert a B) ?C" by blast
  4168   then show ?case by blast 
  4169 qed
  4170 
  4171 lemma orthogonal_basis_exists:
  4172   fixes V :: "(real ^'n) set"
  4173   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (B hassize dim V) \<and> pairwise orthogonal B"
  4174 proof-
  4175   from basis_exists[of V] obtain B where B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "B hassize dim V" by blast
  4176   from B have fB: "finite B" "card B = dim V" by (simp_all add: hassize_def)
  4177   from basis_orthogonal[OF fB(1)] obtain C where 
  4178     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
  4179   from C B 
  4180   have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans) 
  4181   from span_mono[OF B(3)]  C have SVC: "span V \<subseteq> span C" by (simp add: span_span)
  4182   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
  4183   have iC: "independent C" by (simp add: dim_span) 
  4184   from C fB have "card C \<le> dim V" by simp
  4185   moreover have "dim V \<le> card C" using span_card_ge_dim[OF CSV SVC C(1)]
  4186     by (simp add: dim_span)
  4187   ultimately have CdV: "C hassize dim V" unfolding hassize_def using C(1) by simp
  4188   from C B CSV CdV iC show ?thesis by auto 
  4189 qed
  4190 
  4191 lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
  4192   by (metis set_eq_subset span_mono span_span span_inc)
  4193 
  4194 (* ------------------------------------------------------------------------- *)
  4195 (* Low-dimensional subset is in a hyperplane (weak orthogonal complement).   *)
  4196 (* ------------------------------------------------------------------------- *)
  4197 
  4198 lemma span_not_univ_orthogonal:
  4199   assumes sU: "span S \<noteq> UNIV"
  4200   shows "\<exists>(a:: real ^'n). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
  4201 proof-
  4202   from sU obtain a where a: "a \<notin> span S" by blast
  4203   from orthogonal_basis_exists obtain B where 
  4204     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "B hassize dim S" "pairwise orthogonal B" 
  4205     by blast
  4206   from B have fB: "finite B" "card B = dim S" by (simp_all add: hassize_def)
  4207   from span_mono[OF B(2)] span_mono[OF B(3)]
  4208   have sSB: "span S = span B" by (simp add: span_span)
  4209   let ?a = "a - setsum (\<lambda>b. (a\<bullet>b / (b\<bullet>b)) *s b) B"
  4210   have "setsum (\<lambda>b. (a\<bullet>b / (b\<bullet>b)) *s b) B \<in> span S"
  4211     unfolding sSB
  4212     apply (rule span_setsum[OF fB(1)])
  4213     apply clarsimp
  4214     apply (rule span_mul)
  4215     by (rule span_superset)
  4216   with a have a0:"?a  \<noteq> 0" by auto
  4217   have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
  4218   proof(rule span_induct')
  4219     show "subspace (\<lambda>x. ?a \<bullet> x = 0)"
  4220       by (auto simp add: subspace_def mem_def dot_radd dot_rmult) 
  4221   next
  4222     {fix x assume x: "x \<in> B"
  4223       from x have B': "B = insert x (B - {x})" by blast
  4224       have fth: "finite (B - {x})" using fB by simp
  4225       have "?a \<bullet> x = 0" 
  4226 	apply (subst B') using fB fth
  4227 	unfolding setsum_clauses(2)[OF fth]
  4228 	apply simp
  4229 	apply (clarsimp simp add: dot_lsub dot_ladd dot_lmult dot_lsum dot_eq_0)
  4230 	apply (rule setsum_0', rule ballI)
  4231 	unfolding dot_sym
  4232 	by (auto simp add: x field_simps dot_eq_0 intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
  4233     then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
  4234   qed
  4235   with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
  4236 qed
  4237 
  4238 lemma span_not_univ_subset_hyperplane: 
  4239   assumes SU: "span S \<noteq> (UNIV ::(real^'n) set)"
  4240   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
  4241   using span_not_univ_orthogonal[OF SU] by auto
  4242 
  4243 lemma lowdim_subset_hyperplane:
  4244   assumes d: "dim S < dimindex (UNIV :: 'n set)"
  4245   shows "\<exists>(a::real ^'n). a  \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
  4246 proof-
  4247   {assume "span S = UNIV"
  4248     hence "dim (span S) = dim (UNIV :: (real ^'n) set)" by simp
  4249     hence "dim S = dimindex (UNIV :: 'n set)" by (simp add: dim_span dim_univ)
  4250     with d have False by arith}
  4251   hence th: "span S \<noteq> UNIV" by blast
  4252   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
  4253 qed
  4254 
  4255 (* We can extend a linear basis-basis injection to the whole set.            *)
  4256 
  4257 lemma linear_indep_image_lemma:
  4258   assumes lf: "linear f" and fB: "finite B" 
  4259   and ifB: "independent (f ` B)"
  4260   and fi: "inj_on f B" and xsB: "x \<in> span B" 
  4261   and fx: "f (x::'a::field^'n) = 0"
  4262   shows "x = 0"
  4263   using fB ifB fi xsB fx
  4264 proof(induct arbitrary: x rule: finite_induct[OF fB])
  4265   case 1 thus ?case by (auto simp add:  span_empty)
  4266 next
  4267   case (2 a b x)
  4268   have fb: "finite b" using "2.prems" by simp
  4269   have th0: "f ` b \<subseteq> f ` (insert a b)"
  4270     apply (rule image_mono) by blast 
  4271   from independent_mono[ OF "2.prems"(2) th0]
  4272   have ifb: "independent (f ` b)"  .
  4273   have fib: "inj_on f b" 
  4274     apply (rule subset_inj_on [OF "2.prems"(3)]) 
  4275     by blast
  4276   from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
  4277   obtain k where k: "x - k*s a \<in> span (b -{a})" by blast
  4278   have "f (x - k*s a) \<in> span (f ` b)"
  4279     unfolding span_linear_image[OF lf]
  4280     apply (rule imageI)
  4281     using k span_mono[of "b-{a}" b] by blast
  4282   hence "f x - k*s f a \<in> span (f ` b)"
  4283     by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
  4284   hence th: "-k *s f a \<in> span (f ` b)" 
  4285     using "2.prems"(5) by (simp add: vector_smult_lneg)
  4286   {assume k0: "k = 0" 
  4287     from k0 k have "x \<in> span (b -{a})" by simp
  4288     then have "x \<in> span b" using span_mono[of "b-{a}" b]
  4289       by blast}
  4290   moreover
  4291   {assume k0: "k \<noteq> 0"
  4292     from span_mul[OF th, of "- 1/ k"] k0
  4293     have th1: "f a \<in> span (f ` b)" 
  4294       by (auto simp add: vector_smult_assoc)
  4295     from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
  4296     have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
  4297     from "2.prems"(2)[unfolded dependent_def bex_simps(10), rule_format, of "f a"]
  4298     have "f a \<notin> span (f ` b)" using tha
  4299       using "2.hyps"(2)
  4300       "2.prems"(3) by auto
  4301     with th1 have False by blast
  4302     then have "x \<in> span b" by blast}
  4303   ultimately have xsb: "x \<in> span b" by blast
  4304   from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)]
  4305   show "x = 0" .
  4306 qed
  4307 
  4308 (* We can extend a linear mapping from basis.                                *)
  4309 
  4310 lemma linear_independent_extend_lemma:
  4311   assumes fi: "finite B" and ib: "independent B"
  4312   shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g ((x::'a::field^'n) + y) = g x + g y) 
  4313            \<and> (\<forall>x\<in> span B. \<forall>c. g (c*s x) = c *s g x)
  4314            \<and> (\<forall>x\<in> B. g x = f x)"
  4315 using ib fi
  4316 proof(induct rule: finite_induct[OF fi])
  4317   case 1 thus ?case by (auto simp add: span_empty) 
  4318 next
  4319   case (2 a b)
  4320   from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
  4321     by (simp_all add: independent_insert)
  4322   from "2.hyps"(3)[OF ibf] obtain g where 
  4323     g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
  4324     "\<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
  4325   let ?h = "\<lambda>z. SOME k. (z - k *s a) \<in> span b"
  4326   {fix z assume z: "z \<in> span (insert a b)"
  4327     have th0: "z - ?h z *s a \<in> span b"
  4328       apply (rule someI_ex)
  4329       unfolding span_breakdown_eq[symmetric]
  4330       using z .
  4331     {fix k assume k: "z - k *s a \<in> span b"
  4332       have eq: "z - ?h z *s a - (z - k*s a) = (k - ?h z) *s a" 
  4333 	by (simp add: ring_simps vector_sadd_rdistrib[symmetric])
  4334       from span_sub[OF th0 k]
  4335       have khz: "(k - ?h z) *s a \<in> span b" by (simp add: eq)
  4336       {assume "k \<noteq> ?h z" hence k0: "k - ?h z \<noteq> 0" by simp
  4337 	from k0 span_mul[OF khz, of "1 /(k - ?h z)"] 
  4338 	have "a \<in> span b" by (simp add: vector_smult_assoc)
  4339 	with "2.prems"(1) "2.hyps"(2) have False
  4340 	  by (auto simp add: dependent_def)}
  4341       then have "k = ?h z" by blast}
  4342     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}
  4343   note h = this
  4344   let ?g = "\<lambda>z. ?h z *s f a + g (z - ?h z *s a)"
  4345   {fix x y assume x: "x \<in> span (insert a b)" and y: "y \<in> span (insert a b)"
  4346     have tha: "\<And>(x::'a^'n) y a k l. (x + y) - (k + l) *s a = (x - k *s a) + (y - l *s a)" 
  4347       by (vector ring_simps)
  4348     have addh: "?h (x + y) = ?h x + ?h y"
  4349       apply (rule conjunct2[OF h, rule_format, symmetric])
  4350       apply (rule span_add[OF x y])
  4351       unfolding tha
  4352       by (metis span_add x y conjunct1[OF h, rule_format])
  4353     have "?g (x + y) = ?g x + ?g y" 
  4354       unfolding addh tha
  4355       g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
  4356       by (simp add: vector_sadd_rdistrib)}
  4357   moreover
  4358   {fix x:: "'a^'n" and c:: 'a  assume x: "x \<in> span (insert a b)"
  4359     have tha: "\<And>(x::'a^'n) c k a. c *s x - (c * k) *s a = c *s (x - k *s a)" 
  4360       by (vector ring_simps)
  4361     have hc: "?h (c *s x) = c * ?h x" 
  4362       apply (rule conjunct2[OF h, rule_format, symmetric])
  4363       apply (metis span_mul x)
  4364       by (metis tha span_mul x conjunct1[OF h])
  4365     have "?g (c *s x) = c*s ?g x" 
  4366       unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
  4367       by (vector ring_simps)}
  4368   moreover
  4369   {fix x assume x: "x \<in> (insert a b)"
  4370     {assume xa: "x = a"
  4371       have ha1: "1 = ?h a"
  4372 	apply (rule conjunct2[OF h, rule_format])
  4373 	apply (metis span_superset insertI1)
  4374 	using conjunct1[OF h, OF span_superset, OF insertI1]
  4375 	by (auto simp add: span_0)
  4376 
  4377       from xa ha1[symmetric] have "?g x = f x" 
  4378 	apply simp
  4379 	using g(2)[rule_format, OF span_0, of 0]
  4380 	by simp}
  4381     moreover
  4382     {assume xb: "x \<in> b"
  4383       have h0: "0 = ?h x"
  4384 	apply (rule conjunct2[OF h, rule_format])
  4385 	apply (metis  span_superset insertI1 xb x)
  4386 	apply simp
  4387 	apply (metis span_superset xb)
  4388 	done
  4389       have "?g x = f x"
  4390 	by (simp add: h0[symmetric] g(3)[rule_format, OF xb])}
  4391     ultimately have "?g x = f x" using x by blast }
  4392   ultimately show ?case apply - apply (rule exI[where x="?g"]) by blast
  4393 qed
  4394 
  4395 lemma linear_independent_extend:
  4396   assumes iB: "independent (B:: (real ^'n) set)"
  4397   shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
  4398 proof-
  4399   from maximal_independent_subset_extend[of B "UNIV"] iB
  4400   obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
  4401   
  4402   from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
  4403   obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) 
  4404            \<and> (\<forall>x\<in> span C. \<forall>c. g (c*s x) = c *s g x)
  4405            \<and> (\<forall>x\<in> C. g x = f x)" by blast
  4406   from g show ?thesis unfolding linear_def using C 
  4407     apply clarsimp by blast
  4408 qed
  4409 
  4410 (* Can construct an isomorphism between spaces of same dimension.            *)
  4411 
  4412 lemma card_le_inj: assumes fA: "finite A" and fB: "finite B"
  4413   and c: "card A \<le> card B" shows "(\<exists>f. f ` A \<subseteq> B \<and> inj_on f A)"
  4414 using fB c
  4415 proof(induct arbitrary: B rule: finite_induct[OF fA])
  4416   case 1 thus ?case by simp
  4417 next
  4418   case (2 x s t) 
  4419   thus ?case
  4420   proof(induct rule: finite_induct[OF "2.prems"(1)])
  4421     case 1    then show ?case by simp
  4422   next
  4423     case (2 y t)
  4424     from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
  4425     from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where
  4426       f: "f ` s \<subseteq> t \<and> inj_on f s" by blast
  4427     from f "2.prems"(2) "2.hyps"(2) show ?case
  4428       apply -
  4429       apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
  4430       by (auto simp add: inj_on_def)
  4431   qed
  4432 qed
  4433 
  4434 lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and 
  4435   c: "card A = card B"
  4436   shows "A = B"
  4437 proof-
  4438   from fB AB have fA: "finite A" by (auto intro: finite_subset)
  4439   from fA fB have fBA: "finite (B - A)" by auto
  4440   have e: "A \<inter> (B - A) = {}" by blast
  4441   have eq: "A \<union> (B - A) = B" using AB by blast
  4442   from card_Un_disjoint[OF fA fBA e, unfolded eq c]
  4443   have "card (B - A) = 0" by arith
  4444   hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
  4445   with AB show "A = B" by blast  
  4446 qed
  4447 
  4448 lemma subspace_isomorphism:
  4449   assumes s: "subspace (S:: (real ^'n) set)" and t: "subspace T" 
  4450   and d: "dim S = dim T"
  4451   shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
  4452 proof-
  4453   from basis_exists[of S] obtain B where 
  4454     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
  4455   from basis_exists[of T] obtain C where 
  4456     C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "C hassize dim T" by blast
  4457   from B(4) C(4) card_le_inj[of B C] d obtain f where
  4458     f: "f ` B \<subseteq> C" "inj_on f B" unfolding hassize_def by auto 
  4459   from linear_independent_extend[OF B(2)] obtain g where
  4460     g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
  4461   from B(4) have fB: "finite B" by (simp add: hassize_def)
  4462   from C(4) have fC: "finite C" by (simp add: hassize_def)
  4463   from inj_on_iff_eq_card[OF fB, of f] f(2) 
  4464   have "card (f ` B) = card B" by simp
  4465   with B(4) C(4) have ceq: "card (f ` B) = card C" using d 
  4466     by (simp add: hassize_def)
  4467   have "g ` B = f ` B" using g(2)
  4468     by (auto simp add: image_iff)
  4469   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
  4470   finally have gBC: "g ` B = C" .
  4471   have gi: "inj_on g B" using f(2) g(2)
  4472     by (auto simp add: inj_on_def)
  4473   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
  4474   {fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
  4475     from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
  4476     from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
  4477     have th1: "x - y \<in> span B" using x' y' by (metis span_sub) 
  4478     have "x=y" using g0[OF th1 th0] by simp }
  4479   then have giS: "inj_on g S" 
  4480     unfolding inj_on_def by blast
  4481   from span_subspace[OF B(1,3) s]
  4482   have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
  4483   also have "\<dots> = span C" unfolding gBC ..
  4484   also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
  4485   finally have gS: "g ` S = T" .
  4486   from g(1) gS giS show ?thesis by blast
  4487 qed
  4488 
  4489 (* linear functions are equal on a subspace if they are on a spanning set.   *)
  4490 
  4491 lemma subspace_kernel:
  4492   assumes lf: "linear (f::'a::semiring_1 ^'n \<Rightarrow> _)"
  4493   shows "subspace {x. f x = 0}"
  4494 apply (simp add: subspace_def)
  4495 by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
  4496 
  4497 lemma linear_eq_0_span:
  4498   assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
  4499   shows "\<forall>x \<in> span B. f x = (0::'a::semiring_1 ^'n)"
  4500 proof
  4501   fix x assume x: "x \<in> span B"
  4502   let ?P = "\<lambda>x. f x = 0"
  4503   from subspace_kernel[OF lf] have "subspace ?P" unfolding Collect_def .
  4504   with x f0 span_induct[of B "?P" x] show "f x = 0" by blast
  4505 qed
  4506 
  4507 lemma linear_eq_0:
  4508   assumes lf: "linear f" and SB: "S \<subseteq> span B" and f0: "\<forall>x\<in>B. f x = 0" 
  4509   shows "\<forall>x \<in> S. f x = (0::'a::semiring_1^'n)"
  4510   by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
  4511 
  4512 lemma linear_eq:
  4513   assumes lf: "linear (f::'a::ring_1^'n \<Rightarrow> _)" and lg: "linear g" and S: "S \<subseteq> span B"
  4514   and fg: "\<forall> x\<in> B. f x = g x" 
  4515   shows "\<forall>x\<in> S. f x = g x"
  4516 proof-
  4517   let ?h = "\<lambda>x. f x - g x"
  4518   from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
  4519   from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
  4520   show ?thesis by simp
  4521 qed    
  4522 
  4523 lemma linear_eq_stdbasis:
  4524   assumes lf: "linear (f::'a::ring_1^'m \<Rightarrow> 'a^'n)" and lg: "linear g"
  4525   and fg: "\<forall>i \<in> {1 .. dimindex(UNIV :: 'm set)}. f (basis i) = g(basis i)"
  4526   shows "f = g"
  4527 proof-
  4528   let ?U = "UNIV :: 'm set"
  4529   let ?I = "{basis i:: 'a^'m|i. i \<in> {1 .. dimindex ?U}}" 
  4530   {fix x assume x: "x \<in> (UNIV :: ('a^'m) set)"
  4531     from equalityD2[OF span_stdbasis]
  4532     have IU: " (UNIV :: ('a^'m) set) \<subseteq> span ?I" by blast
  4533     from linear_eq[OF lf lg IU] fg x
  4534     have "f x = g x" unfolding Collect_def  Ball_def mem_def by metis}
  4535   then show ?thesis by (auto intro: ext)
  4536 qed
  4537 
  4538 (* Similar results for bilinear functions.                                   *)
  4539 
  4540 lemma bilinear_eq:
  4541   assumes bf: "bilinear (f:: 'a::ring^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)" 
  4542   and bg: "bilinear g"
  4543   and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
  4544   and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
  4545   shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
  4546 proof-
  4547   let ?P = "\<lambda>x. \<forall>y\<in> span C. f x y = g x y"
  4548   from bf bg have sp: "subspace ?P" 
  4549     unfolding bilinear_def linear_def subspace_def bf bg  
  4550     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])
  4551 
  4552   have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y" 
  4553     apply -
  4554     apply (rule ballI)
  4555     apply (rule span_induct[of B ?P]) 
  4556     defer
  4557     apply (rule sp)
  4558     apply assumption
  4559     apply (clarsimp simp add: Ball_def)
  4560     apply (rule_tac P="\<lambda>y. f xa y = g xa y" and S=C in span_induct)
  4561     using fg 
  4562     apply (auto simp add: subspace_def)
  4563     using bf bg unfolding bilinear_def linear_def
  4564     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])
  4565   then show ?thesis using SB TC by (auto intro: ext)
  4566 qed
  4567 
  4568 lemma bilinear_eq_stdbasis:
  4569   assumes bf: "bilinear (f:: 'a::ring_1^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)" 
  4570   and bg: "bilinear g"
  4571   and fg: "\<forall>i\<in> {1 .. dimindex (UNIV :: 'm set)}. \<forall>j\<in>  {1 .. dimindex (UNIV :: 'n set)}. f (basis i) (basis j) = g (basis i) (basis j)"
  4572   shows "f = g"
  4573 proof-
  4574   from fg have th: "\<forall>x \<in> {basis i| i. i\<in> {1 .. dimindex (UNIV :: 'm set)}}. \<forall>y\<in>  {basis j |j. j \<in> {1 .. dimindex (UNIV :: 'n set)}}. f x y = g x y" by blast
  4575   from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext)
  4576 qed
  4577 
  4578 (* Detailed theorems about left and right invertibility in general case.     *)
  4579 
  4580 lemma left_invertible_transp:
  4581   "(\<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)"
  4582   by (metis matrix_transp_mul transp_mat transp_transp)
  4583 
  4584 lemma right_invertible_transp:
  4585   "(\<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)"
  4586   by (metis matrix_transp_mul transp_mat transp_transp)
  4587 
  4588 lemma linear_injective_left_inverse:
  4589   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and fi: "inj f"
  4590   shows "\<exists>g. linear g \<and> g o f = id"
  4591 proof-
  4592   from linear_independent_extend[OF independent_injective_image, OF independent_stdbasis, OF lf fi]
  4593   obtain h:: "real ^'m \<Rightarrow> real ^'n" where h: "linear h" " \<forall>x \<in> f ` {basis i|i. i \<in> {1 .. dimindex (UNIV::'n set)}}. h x = inv f x" by blast
  4594   from h(2) 
  4595   have th: "\<forall>i\<in>{1..dimindex (UNIV::'n set)}. (h \<circ> f) (basis i) = id (basis i)"
  4596     using inv_o_cancel[OF fi, unfolded stupid_ext[symmetric] id_def o_def]
  4597     apply auto
  4598     apply (erule_tac x="basis i" in allE)
  4599     by auto
  4600   
  4601   from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
  4602   have "h o f = id" .
  4603   then show ?thesis using h(1) by blast  
  4604 qed
  4605 
  4606 lemma linear_surjective_right_inverse:
  4607   assumes lf: "linear (f:: real ^'m \<Rightarrow> real ^'n)" and sf: "surj f"
  4608   shows "\<exists>g. linear g \<and> f o g = id"
  4609 proof-
  4610   from linear_independent_extend[OF independent_stdbasis]
  4611   obtain h:: "real ^'n \<Rightarrow> real ^'m" where 
  4612     h: "linear h" "\<forall> x\<in> {basis i| i. i\<in> {1 .. dimindex (UNIV :: 'n set)}}. h x = inv f x" by blast
  4613   from h(2) 
  4614   have th: "\<forall>i\<in>{1..dimindex (UNIV::'n set)}. (f o h) (basis i) = id (basis i)"
  4615     using sf
  4616     apply (auto simp add: surj_iff o_def stupid_ext[symmetric])
  4617     apply (erule_tac x="basis i" in allE)
  4618     by auto
  4619   
  4620   from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
  4621   have "f o h = id" .
  4622   then show ?thesis using h(1) by blast  
  4623 qed
  4624 
  4625 lemma matrix_left_invertible_injective:
  4626 "(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
  4627 proof-
  4628   {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
  4629     from xy have "B*v (A *v x) = B *v (A*v y)" by simp
  4630     hence "x = y"
  4631       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .}
  4632   moreover
  4633   {assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
  4634     hence i: "inj (op *v A)" unfolding inj_on_def by auto 
  4635     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
  4636     obtain g where g: "linear g" "g o op *v A = id" by blast
  4637     have "matrix g ** A = mat 1"
  4638       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
  4639       using g(2) by (simp add: o_def id_def stupid_ext)
  4640     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast}
  4641   ultimately show ?thesis by blast
  4642 qed
  4643 
  4644 lemma matrix_left_invertible_ker:
  4645   "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
  4646   unfolding matrix_left_invertible_injective
  4647   using linear_injective_0[OF matrix_vector_mul_linear, of A]
  4648   by (simp add: inj_on_def)
  4649 
  4650 lemma matrix_right_invertible_surjective:
  4651 "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
  4652 proof-
  4653   {fix B :: "real ^'m^'n"  assume AB: "A ** B = mat 1"
  4654     {fix x :: "real ^ 'm" 
  4655       have "A *v (B *v x) = x"
  4656 	by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)}
  4657     hence "surj (op *v A)" unfolding surj_def by metis }
  4658   moreover
  4659   {assume sf: "surj (op *v A)"
  4660     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
  4661     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id" 
  4662       by blast
  4663 
  4664     have "A ** (matrix g) = mat 1"
  4665       unfolding matrix_eq  matrix_vector_mul_lid 
  4666 	matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] 
  4667       using g(2) unfolding o_def stupid_ext[symmetric] id_def
  4668       .
  4669     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
  4670   }
  4671   ultimately show ?thesis unfolding surj_def by blast
  4672 qed    
  4673 
  4674 lemma matrix_left_invertible_independent_columns:
  4675   fixes A :: "real^'n^'m"
  4676   shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s column i A) {1 .. dimindex(UNIV :: 'n set)} = 0 \<longrightarrow> (\<forall>i\<in> {1 .. dimindex (UNIV :: 'n set)}. c i = 0))"
  4677    (is "?lhs \<longleftrightarrow> ?rhs")
  4678 proof-
  4679   let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
  4680   {assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
  4681     {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" 
  4682       and i: "i \<in> ?U"
  4683       let ?x = "\<chi> i. c i"
  4684       have th0:"A *v ?x = 0"
  4685 	using c
  4686 	unfolding matrix_mult_vsum Cart_eq
  4687 	by (auto simp add: vector_component zero_index setsum_component Cart_lambda_beta)
  4688       from k[rule_format, OF th0] i
  4689       have "c i = 0" by (vector Cart_eq)}
  4690     hence ?rhs by blast}
  4691   moreover
  4692   {assume H: ?rhs
  4693     {fix x assume x: "A *v x = 0" 
  4694       let ?c = "\<lambda>i. ((x$i ):: real)"
  4695       from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
  4696       have "x = 0" by vector}}
  4697   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast 
  4698 qed
  4699 
  4700 lemma matrix_right_invertible_independent_rows:
  4701   fixes A :: "real^'n^'m"
  4702   shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s row i A) {1 .. dimindex(UNIV :: 'm set)} = 0 \<longrightarrow> (\<forall>i\<in> {1 .. dimindex (UNIV :: 'm set)}. c i = 0))"
  4703   unfolding left_invertible_transp[symmetric]
  4704     matrix_left_invertible_independent_columns
  4705   by (simp add: column_transp)
  4706 
  4707 lemma matrix_right_invertible_span_columns:
  4708   "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")
  4709 proof-
  4710   let ?U = "{1 .. dimindex (UNIV :: 'm set)}"
  4711   have fU: "finite ?U" by simp
  4712   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
  4713     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
  4714     apply (subst eq_commute) ..    
  4715   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
  4716   {assume h: ?lhs
  4717     {fix x:: "real ^'n" 
  4718 	from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m"
  4719 	  where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
  4720 	have "x \<in> span (columns A)"  
  4721 	  unfolding y[symmetric]
  4722 	  apply (rule span_setsum[OF fU])
  4723 	  apply clarify
  4724 	  apply (rule span_mul)
  4725 	  apply (rule span_superset)
  4726 	  unfolding columns_def
  4727 	  by blast}
  4728     then have ?rhs unfolding rhseq by blast}
  4729   moreover
  4730   {assume h:?rhs
  4731     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
  4732     {fix y have "?P y" 
  4733       proof(rule span_induct_alt[of ?P "columns A"])
  4734 	show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
  4735 	  apply (rule exI[where x=0])
  4736 	  by (simp add: zero_index vector_smult_lzero)
  4737       next
  4738 	fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
  4739 	from y1 obtain i where i: "i \<in> ?U" "y1 = column i A" 
  4740 	  unfolding columns_def by blast
  4741 	from y2 obtain x:: "real ^'m" where 
  4742 	  x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
  4743 	let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
  4744 	show "?P (c*s y1 + y2)"
  4745 	  proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric]Cart_lambda_beta setsum_component cond_value_iff right_distrib cond_application_beta vector_component cong del: if_weak_cong, simp only: One_nat_def[symmetric])
  4746 	    fix j 
  4747 	    have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
  4748            else (x$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)
  4749 	      by (simp add: ring_simps)
  4750 	    have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
  4751            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)$j))) ?U"
  4752 	      apply (rule setsum_cong[OF refl])
  4753 	      using th by blast
  4754 	    also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
  4755 	      by (simp add: setsum_addf)
  4756 	    also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
  4757 	      unfolding setsum_delta[OF fU]
  4758 	      using i(1) by simp 
  4759 	    finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
  4760            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
  4761 	  qed
  4762 	next
  4763 	  show "y \<in> span (columns A)" unfolding h by blast
  4764 	qed}
  4765     then have ?lhs unfolding lhseq ..}
  4766   ultimately show ?thesis by blast
  4767 qed
  4768 
  4769 lemma matrix_left_invertible_span_rows:
  4770   "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
  4771   unfolding right_invertible_transp[symmetric]
  4772   unfolding columns_transp[symmetric]
  4773   unfolding matrix_right_invertible_span_columns
  4774  ..
  4775 
  4776 (* An injective map real^'n->real^'n is also surjective.                       *)
  4777 
  4778 lemma linear_injective_imp_surjective:
  4779   assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and fi: "inj f" 
  4780   shows "surj f"
  4781 proof-
  4782   let ?U = "UNIV :: (real ^'n) set"
  4783   from basis_exists[of ?U] obtain B 
  4784     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U" 
  4785     by blast
  4786   from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
  4787   have th: "?U \<subseteq> span (f ` B)"
  4788     apply (rule card_ge_dim_independent)
  4789     apply blast
  4790     apply (rule independent_injective_image[OF B(2) lf fi])
  4791     apply (rule order_eq_refl)
  4792     apply (rule sym)
  4793     unfolding d
  4794     apply (rule card_image)
  4795     apply (rule subset_inj_on[OF fi])
  4796     by blast
  4797   from th show ?thesis
  4798     unfolding span_linear_image[OF lf] surj_def
  4799     using B(3) by blast
  4800 qed
  4801 
  4802 (* And vice versa.                                                           *)
  4803 
  4804 lemma surjective_iff_injective_gen: 
  4805   assumes fS: "finite S" and fT: "finite T" and c: "card S = card T"
  4806   and ST: "f ` S \<subseteq> T"
  4807   shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs")
  4808 proof-
  4809   {assume h: "?lhs"
  4810     {fix x y assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
  4811       from x fS have S0: "card S \<noteq> 0" by auto
  4812       {assume xy: "x \<noteq> y"
  4813 	have th: "card S \<le> card (f ` (S - {y}))"
  4814 	  unfolding c
  4815 	  apply (rule card_mono)
  4816 	  apply (rule finite_imageI)
  4817 	  using fS apply simp
  4818 	  using h xy x y f unfolding subset_eq image_iff
  4819 	  apply auto
  4820 	  apply (case_tac "xa = f x")
  4821 	  apply (rule bexI[where x=x])
  4822 	  apply auto
  4823 	  done
  4824 	also have " \<dots> \<le> card (S -{y})"
  4825 	  apply (rule card_image_le)
  4826 	  using fS by simp
  4827 	also have "\<dots> \<le> card S - 1" using y fS by simp
  4828 	finally have False  using S0 by arith }
  4829       then have "x = y" by blast}
  4830     then have ?rhs unfolding inj_on_def by blast}
  4831   moreover
  4832   {assume h: ?rhs
  4833     have "f ` S = T"
  4834       apply (rule card_subset_eq[OF fT ST])
  4835       unfolding card_image[OF h] using c .
  4836     then have ?lhs by blast}
  4837   ultimately show ?thesis by blast
  4838 qed
  4839 
  4840 lemma linear_surjective_imp_injective:
  4841   assumes lf: "linear (f::real ^'n => real ^'n)" and sf: "surj f" 
  4842   shows "inj f"
  4843 proof-
  4844   let ?U = "UNIV :: (real ^'n) set"
  4845   from basis_exists[of ?U] obtain B 
  4846     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U" 
  4847     by blast
  4848   {fix x assume x: "x \<in> span B" and fx: "f x = 0"
  4849     from B(4) have fB: "finite B" by (simp add: hassize_def)
  4850     from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
  4851     have fBi: "independent (f ` B)" 
  4852       apply (rule card_le_dim_spanning[of "f ` B" ?U])
  4853       apply blast
  4854       using sf B(3)
  4855       unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
  4856       apply blast
  4857       using fB apply (blast intro: finite_imageI)
  4858       unfolding d
  4859       apply (rule card_image_le)
  4860       apply (rule fB)
  4861       done
  4862     have th0: "dim ?U \<le> card (f ` B)"
  4863       apply (rule span_card_ge_dim)
  4864       apply blast
  4865       unfolding span_linear_image[OF lf]
  4866       apply (rule subset_trans[where B = "f ` UNIV"])
  4867       using sf unfolding surj_def apply blast
  4868       apply (rule image_mono)
  4869       apply (rule B(3))
  4870       apply (metis finite_imageI fB)
  4871       done
  4872 
  4873     moreover have "card (f ` B) \<le> card B"
  4874       by (rule card_image_le, rule fB)
  4875     ultimately have th1: "card B = card (f ` B)" unfolding d by arith
  4876     have fiB: "inj_on f B" 
  4877       unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast
  4878     from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
  4879     have "x = 0" by blast}
  4880   note th = this
  4881   from th show ?thesis unfolding linear_injective_0[OF lf] 
  4882     using B(3) by blast
  4883 qed
  4884 
  4885 (* Hence either is enough for isomorphism.                                   *)
  4886 
  4887 lemma left_right_inverse_eq:
  4888   assumes fg: "f o g = id" and gh: "g o h = id"
  4889   shows "f = h" 
  4890 proof-
  4891   have "f = f o (g o h)" unfolding gh by simp
  4892   also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
  4893   finally show "f = h" unfolding fg by simp
  4894 qed
  4895 
  4896 lemma isomorphism_expand:
  4897   "f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
  4898   by (simp add: expand_fun_eq o_def id_def)
  4899 
  4900 lemma linear_injective_isomorphism:
  4901   assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'n)" and fi: "inj f"
  4902   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
  4903 unfolding isomorphism_expand[symmetric]
  4904 using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi]
  4905 by (metis left_right_inverse_eq)
  4906 
  4907 lemma linear_surjective_isomorphism:
  4908   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and sf: "surj f"
  4909   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
  4910 unfolding isomorphism_expand[symmetric]
  4911 using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
  4912 by (metis left_right_inverse_eq)
  4913 
  4914 (* Left and right inverses are the same for R^N->R^N.                        *)
  4915 
  4916 lemma linear_inverse_left:
  4917   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and lf': "linear f'"
  4918   shows "f o f' = id \<longleftrightarrow> f' o f = id"
  4919 proof-
  4920   {fix f f':: "real ^'n \<Rightarrow> real ^'n"
  4921     assume lf: "linear f" "linear f'" and f: "f o f' = id"
  4922     from f have sf: "surj f"
  4923       
  4924       apply (auto simp add: o_def stupid_ext[symmetric] id_def surj_def)
  4925       by metis
  4926     from linear_surjective_isomorphism[OF lf(1) sf] lf f
  4927     have "f' o f = id" unfolding stupid_ext[symmetric] o_def id_def
  4928       by metis}
  4929   then show ?thesis using lf lf' by metis
  4930 qed
  4931 
  4932 (* Moreover, a one-sided inverse is automatically linear.                    *)
  4933 
  4934 lemma left_inverse_linear:
  4935   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and gf: "g o f = id" 
  4936   shows "linear g"
  4937 proof-
  4938   from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def stupid_ext[symmetric])
  4939     by metis
  4940   from linear_injective_isomorphism[OF lf fi] 
  4941   obtain h:: "real ^'n \<Rightarrow> real ^'n" where 
  4942     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
  4943   have "h = g" apply (rule ext) using gf h(2,3)
  4944     apply (simp add: o_def id_def stupid_ext[symmetric])
  4945     by metis
  4946   with h(1) show ?thesis by blast
  4947 qed
  4948 
  4949 lemma right_inverse_linear:
  4950   assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and gf: "f o g = id" 
  4951   shows "linear g"
  4952 proof-
  4953   from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def stupid_ext[symmetric])
  4954     by metis
  4955   from linear_surjective_isomorphism[OF lf fi] 
  4956   obtain h:: "real ^'n \<Rightarrow> real ^'n" where 
  4957     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
  4958   have "h = g" apply (rule ext) using gf h(2,3)
  4959     apply (simp add: o_def id_def stupid_ext[symmetric])
  4960     by metis
  4961   with h(1) show ?thesis by blast
  4962 qed
  4963 
  4964 (* The same result in terms of square matrices.                              *)
  4965 
  4966 lemma matrix_left_right_inverse:
  4967   fixes A A' :: "real ^'n^'n" 
  4968   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
  4969 proof-
  4970   {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"
  4971     have sA: "surj (op *v A)"
  4972       unfolding surj_def
  4973       apply clarify
  4974       apply (rule_tac x="(A' *v y)" in exI)
  4975       by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
  4976     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
  4977     obtain f' :: "real ^'n \<Rightarrow> real ^'n"
  4978       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
  4979     have th: "matrix f' ** A = mat 1" 
  4980       by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
  4981     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
  4982     hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
  4983     hence "matrix f' ** A = A' ** A" by simp
  4984     hence "A' ** A = mat 1" by (simp add: th)}
  4985   then show ?thesis by blast
  4986 qed
  4987 
  4988 (* Considering an n-element vector as an n-by-1 or 1-by-n matrix.            *)
  4989 
  4990 definition "rowvector v = (\<chi> i j. (v$j))"
  4991 
  4992 definition "columnvector v = (\<chi> i j. (v$i))"
  4993 
  4994 lemma transp_columnvector:
  4995  "transp(columnvector v) = rowvector v"
  4996   by (simp add: transp_def rowvector_def columnvector_def Cart_eq Cart_lambda_beta)
  4997 
  4998 lemma transp_rowvector: "transp(rowvector v) = columnvector v"
  4999   by (simp add: transp_def columnvector_def rowvector_def Cart_eq Cart_lambda_beta)
  5000 
  5001 lemma dot_rowvector_columnvector:
  5002   "columnvector (A *v v) = A ** columnvector v"
  5003   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
  5004 
  5005 lemma dot_matrix_product: "(x::'a::semiring_1^'n) \<bullet> y = (((rowvector x ::'a^'n^1) ** (columnvector y :: 'a^1^'n))$1)$1"
  5006   apply (vector matrix_matrix_mult_def rowvector_def columnvector_def dot_def)
  5007   by (simp add: Cart_lambda_beta)
  5008 
  5009 lemma dot_matrix_vector_mul:
  5010   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
  5011   shows "(A *v x) \<bullet> (B *v y) =
  5012       (((rowvector x :: real^'n^1) ** ((transp A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
  5013 unfolding dot_matrix_product transp_columnvector[symmetric]
  5014   dot_rowvector_columnvector matrix_transp_mul matrix_mul_assoc ..
  5015 
  5016 (* Infinity norm.                                                            *)
  5017 
  5018 definition "infnorm (x::real^'n) = rsup {abs(x$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
  5019 
  5020 lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> {1 .. dimindex (UNIV :: 'n set)}"
  5021   using dimindex_ge_1 by auto
  5022 
  5023 lemma infnorm_set_image:
  5024   "{abs(x$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}} =
  5025   (\<lambda>i. abs(x$i)) ` {1 .. dimindex(UNIV :: 'n set)}" by blast
  5026 
  5027 lemma infnorm_set_lemma:
  5028   shows "finite {abs((x::'a::abs ^'n)$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
  5029   and "{abs(x$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}} \<noteq> {}"
  5030   unfolding infnorm_set_image
  5031   using dimindex_ge_1[of "UNIV :: 'n set"]
  5032   by (auto intro: finite_imageI)
  5033 
  5034 lemma infnorm_pos_le: "0 \<le> infnorm x"
  5035   unfolding infnorm_def
  5036   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
  5037   unfolding infnorm_set_image
  5038   using dimindex_ge_1
  5039   by auto
  5040 
  5041 lemma infnorm_triangle: "infnorm ((x::real^'n) + y) \<le> infnorm x + infnorm y"
  5042 proof-
  5043   have th: "\<And>x y (z::real). x - y <= z \<longleftrightarrow> x - z <= y" by arith
  5044   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
  5045   have th2: "\<And>x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith
  5046   show ?thesis 
  5047   unfolding infnorm_def
  5048   unfolding rsup_finite_le_iff[ OF infnorm_set_lemma]
  5049   apply (subst diff_le_eq[symmetric])
  5050   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
  5051   unfolding infnorm_set_image bex_simps 
  5052   apply (subst th)
  5053   unfolding th1 
  5054   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
  5055   
  5056   unfolding infnorm_set_image ball_simps bex_simps 
  5057   apply (simp add: vector_add_component)
  5058   apply (metis numseg_dimindex_nonempty th2)
  5059   done
  5060 qed
  5061 
  5062 lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::real ^'n) = 0"
  5063 proof-
  5064   have "infnorm x <= 0 \<longleftrightarrow> x = 0"
  5065     unfolding infnorm_def
  5066     unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
  5067     unfolding infnorm_set_image ball_simps
  5068     by vector
  5069   then show ?thesis using infnorm_pos_le[of x] by simp
  5070 qed
  5071 
  5072 lemma infnorm_0: "infnorm 0 = 0"
  5073   by (simp add: infnorm_eq_0)
  5074 
  5075 lemma infnorm_neg: "infnorm (- x) = infnorm x"
  5076   unfolding infnorm_def
  5077   apply (rule cong[of "rsup" "rsup"])
  5078   apply blast
  5079   apply (rule set_ext)
  5080   apply (auto simp add: vector_component abs_minus_cancel)
  5081   apply (rule_tac x="i" in exI)
  5082   apply (simp add: vector_component)
  5083   done
  5084 
  5085 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)" 
  5086 proof-
  5087   have "y - x = - (x - y)" by simp
  5088   then show ?thesis  by (metis infnorm_neg)
  5089 qed
  5090 
  5091 lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
  5092 proof-
  5093   have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
  5094     by arith
  5095   from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
  5096   have ths: "infnorm x \<le> infnorm (x - y) + infnorm y" 
  5097     "infnorm y \<le> infnorm (x - y) + infnorm x"
  5098     by (simp_all add: ring_simps infnorm_neg diff_def[symmetric])
  5099   from th[OF ths]  show ?thesis .
  5100 qed
  5101 
  5102 lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
  5103   using infnorm_pos_le[of x] by arith
  5104 
  5105 lemma component_le_infnorm: assumes i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
  5106   shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
  5107 proof-
  5108   let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
  5109   let ?S = "{\<bar>x$i\<bar> |i. i\<in> ?U}"
  5110   have fS: "finite ?S" unfolding image_Collect[symmetric]
  5111     apply (rule finite_imageI) unfolding Collect_def mem_def by simp  
  5112   have S0: "?S \<noteq> {}" using numseg_dimindex_nonempty by blast
  5113   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
  5114   from rsup_finite_in[OF fS S0] rsup_finite_Ub[OF fS S0] i
  5115   show ?thesis unfolding infnorm_def isUb_def setle_def 
  5116     unfolding infnorm_set_image ball_simps by auto
  5117 qed
  5118 
  5119 lemma infnorm_mul_lemma: "infnorm(a *s x) <= \<bar>a\<bar> * infnorm x"
  5120   apply (subst infnorm_def)
  5121   unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
  5122   unfolding infnorm_set_image ball_simps
  5123   apply (simp add: abs_mult vector_component del: One_nat_def)
  5124   apply (rule ballI)
  5125   apply (drule component_le_infnorm[of _ x])
  5126   apply (rule mult_mono)
  5127   apply auto
  5128   done
  5129 
  5130 lemma infnorm_mul: "infnorm(a *s x) = abs a * infnorm x"
  5131 proof-
  5132   {assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) }
  5133   moreover
  5134   {assume a0: "a \<noteq> 0"
  5135     from a0 have th: "(1/a) *s (a *s x) = x"
  5136       by (simp add: vector_smult_assoc)
  5137     from a0 have ap: "\<bar>a\<bar> > 0" by arith
  5138     from infnorm_mul_lemma[of "1/a" "a *s x"]
  5139     have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*s x)"
  5140       unfolding th by simp
  5141     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)
  5142     then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*s x)" 
  5143       using ap by (simp add: field_simps)
  5144     with infnorm_mul_lemma[of a x] have ?thesis by arith }
  5145   ultimately show ?thesis by blast
  5146 qed
  5147 
  5148 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
  5149   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
  5150 
  5151 (* Prove that it differs only up to a bound from Euclidean norm.             *)
  5152 
  5153 lemma infnorm_le_norm: "infnorm x \<le> norm x"
  5154   unfolding infnorm_def rsup_finite_le_iff[OF infnorm_set_lemma] 
  5155   unfolding infnorm_set_image  ball_simps
  5156   by (metis component_le_norm)
  5157 lemma card_enum: "card {1 .. n} = n" by auto
  5158 lemma norm_le_infnorm: "norm(x) <= sqrt(real (dimindex(UNIV ::'n set))) * infnorm(x::real ^'n)"
  5159 proof-
  5160   let ?d = "dimindex(UNIV ::'n set)"
  5161   have d: "?d = card {1 .. ?d}" by auto
  5162   have "real ?d \<ge> 0" by simp
  5163   hence d2: "(sqrt (real ?d))^2 = real ?d"
  5164     by (auto intro: real_sqrt_pow2)
  5165   have th: "sqrt (real ?d) * infnorm x \<ge> 0"
  5166     by (simp add: dimindex_ge_1 zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le)
  5167   have th1: "x\<bullet>x \<le> (sqrt (real ?d) * infnorm x)^2"
  5168     unfolding power_mult_distrib d2 
  5169     apply (subst d)
  5170     apply (subst power2_abs[symmetric])
  5171     unfolding real_of_nat_def dot_def power2_eq_square[symmetric]
  5172     apply (subst power2_abs[symmetric])
  5173     apply (rule setsum_bounded)
  5174     apply (rule power_mono)
  5175     unfolding abs_of_nonneg[OF infnorm_pos_le] 
  5176     unfolding infnorm_def  rsup_finite_ge_iff[OF infnorm_set_lemma]
  5177     unfolding infnorm_set_image bex_simps
  5178     apply blast
  5179     by (rule abs_ge_zero)
  5180   from real_le_lsqrt[OF dot_pos_le th th1]
  5181   show ?thesis unfolding real_vector_norm_def id_def . 
  5182 qed
  5183 
  5184 (* Equality in Cauchy-Schwarz and triangle inequalities.                     *)
  5185 
  5186 lemma norm_cauchy_schwarz_eq: "(x::real ^'n) \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *s y = norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
  5187 proof-
  5188   {assume h: "x = 0"
  5189     hence ?thesis by simp}
  5190   moreover
  5191   {assume h: "y = 0"
  5192     hence ?thesis by simp}
  5193   moreover
  5194   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
  5195     from dot_eq_0[of "norm y *s x - norm x *s y"]
  5196     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)"
  5197       using x y
  5198       unfolding dot_rsub dot_lsub dot_lmult dot_rmult
  5199       unfolding norm_pow_2[symmetric] power2_eq_square diff_eq_0_iff_eq apply (simp add: dot_sym)
  5200       apply (simp add: ring_simps)
  5201       apply metis
  5202       done
  5203     also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
  5204       by (simp add: ring_simps dot_sym)
  5205     also have "\<dots> \<longleftrightarrow> ?lhs" using x y
  5206       apply simp
  5207       by metis
  5208     finally have ?thesis by blast}
  5209   ultimately show ?thesis by blast
  5210 qed
  5211 
  5212 lemma norm_cauchy_schwarz_abs_eq: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
  5213                 norm x *s y = norm y *s x \<or> norm(x) *s y = - norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
  5214 proof-
  5215   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
  5216   have "?rhs \<longleftrightarrow> norm x *s y = norm y *s x \<or> norm (- x) *s y = norm y *s (- x)"
  5217     apply simp by vector
  5218   also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or>
  5219      (-x) \<bullet> y = norm x * norm y)"
  5220     unfolding norm_cauchy_schwarz_eq[symmetric]
  5221     unfolding norm_minus_cancel
  5222       norm_mul by blast
  5223   also have "\<dots> \<longleftrightarrow> ?lhs"
  5224     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] dot_lneg
  5225     by arith
  5226   finally show ?thesis ..
  5227 qed
  5228 
  5229 lemma norm_triangle_eq: "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *s y = norm y *s x"
  5230 proof-
  5231   {assume x: "x =0 \<or> y =0"
  5232     hence ?thesis by (cases "x=0", simp_all)}
  5233   moreover
  5234   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
  5235     hence "norm x \<noteq> 0" "norm y \<noteq> 0"
  5236       by simp_all
  5237     hence n: "norm x > 0" "norm y > 0" 
  5238       using norm_ge_zero[of x] norm_ge_zero[of y]
  5239       by arith+
  5240     have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
  5241     have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
  5242       apply (rule th) using n norm_ge_zero[of "x + y"]
  5243       by arith
  5244     also have "\<dots> \<longleftrightarrow> norm x *s y = norm y *s x"
  5245       unfolding norm_cauchy_schwarz_eq[symmetric]
  5246       unfolding norm_pow_2 dot_ladd dot_radd
  5247       by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym ring_simps)
  5248     finally have ?thesis .}
  5249   ultimately show ?thesis by blast
  5250 qed
  5251 
  5252 (* Collinearity.*)
  5253 
  5254 definition "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *s u)"
  5255 
  5256 lemma collinear_empty:  "collinear {}" by (simp add: collinear_def)
  5257 
  5258 lemma collinear_sing: "collinear {(x::'a::ring_1^'n)}" 
  5259   apply (simp add: collinear_def)
  5260   apply (rule exI[where x=0])
  5261   by simp
  5262 
  5263 lemma collinear_2: "collinear {(x::'a::ring_1^'n),y}"
  5264   apply (simp add: collinear_def)
  5265   apply (rule exI[where x="x - y"])
  5266   apply auto
  5267   apply (rule exI[where x=0], simp)
  5268   apply (rule exI[where x=1], simp)
  5269   apply (rule exI[where x="- 1"], simp add: vector_sneg_minus1[symmetric])
  5270   apply (rule exI[where x=0], simp)
  5271   done
  5272 
  5273 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")
  5274 proof-
  5275   {assume "x=0 \<or> y = 0" hence ?thesis 
  5276       by (cases "x = 0", simp_all add: collinear_2 insert_commute)}
  5277   moreover
  5278   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
  5279     {assume h: "?lhs"
  5280       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
  5281       from u[rule_format, of x 0] u[rule_format, of y 0]
  5282       obtain cx and cy where 
  5283 	cx: "x = cx*s u" and cy: "y = cy*s u"
  5284 	by auto
  5285       from cx x have cx0: "cx \<noteq> 0" by auto
  5286       from cy y have cy0: "cy \<noteq> 0" by auto
  5287       let ?d = "cy / cx"
  5288       from cx cy cx0 have "y = ?d *s x" 
  5289 	by (simp add: vector_smult_assoc)
  5290       hence ?rhs using x y by blast}
  5291     moreover
  5292     {assume h: "?rhs"
  5293       then obtain c where c: "y = c*s x" using x y by blast
  5294       have ?lhs unfolding collinear_def c
  5295 	apply (rule exI[where x=x])
  5296 	apply auto
  5297 	apply (rule exI[where x=0], simp)
  5298 	apply (rule exI[where x="- 1"], simp only: vector_smult_lneg vector_smult_lid)
  5299 	apply (rule exI[where x= "-c"], simp only: vector_smult_lneg)
  5300 	apply (rule exI[where x=1], simp)
  5301 	apply (rule exI[where x=0], simp)
  5302 	apply (rule exI[where x="1 - c"], simp add: vector_smult_lneg vector_sub_rdistrib)
  5303 	apply (rule exI[where x="c - 1"], simp add: vector_smult_lneg vector_sub_rdistrib)
  5304 	apply (rule exI[where x=0], simp)
  5305 	done}
  5306     ultimately have ?thesis by blast}
  5307   ultimately show ?thesis by blast
  5308 qed
  5309 
  5310 lemma norm_cauchy_schwarz_equal: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {(0::real^'n),x,y}"
  5311 unfolding norm_cauchy_schwarz_abs_eq
  5312 apply (cases "x=0", simp_all add: collinear_2)
  5313 apply (cases "y=0", simp_all add: collinear_2 insert_commute)
  5314 unfolding collinear_lemma
  5315 apply simp
  5316 apply (subgoal_tac "norm x \<noteq> 0")
  5317 apply (subgoal_tac "norm y \<noteq> 0")
  5318 apply (rule iffI)
  5319 apply (cases "norm x *s y = norm y *s x")
  5320 apply (rule exI[where x="(1/norm x) * norm y"])
  5321 apply (drule sym)
  5322 unfolding vector_smult_assoc[symmetric]
  5323 apply (simp add: vector_smult_assoc field_simps)
  5324 apply (rule exI[where x="(1/norm x) * - norm y"])
  5325 apply clarify
  5326 apply (drule sym)
  5327 unfolding vector_smult_assoc[symmetric]
  5328 apply (simp add: vector_smult_assoc field_simps)
  5329 apply (erule exE)
  5330 apply (erule ssubst)
  5331 unfolding vector_smult_assoc
  5332 unfolding norm_mul
  5333 apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
  5334 apply (case_tac "c <= 0", simp add: ring_simps)
  5335 apply (simp add: ring_simps)
  5336 apply (case_tac "c <= 0", simp add: ring_simps)
  5337 apply (simp add: ring_simps)
  5338 apply simp
  5339 apply simp
  5340 done
  5341 
  5342 end