src/HOL/Library/Euclidean_Space.thy
author huffman
Sat Feb 21 09:55:32 2009 -0800 (2009-02-21)
changeset 30039 7208c88df507
parent 29906 80369da39838
child 30040 e2cd1acda1ab
permissions -rw-r--r--
fix real_vector, real_algebra instances
     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   uses ("normarith.ML")
    12 begin
    13 
    14 text{* Some common special cases.*}
    15 
    16 lemma forall_1: "(\<forall>(i::'a::{order,one}). 1 <= i \<and> i <= 1 --> P i) \<longleftrightarrow> P 1"
    17   by (metis order_eq_iff)
    18 lemma forall_dimindex_1: "(\<forall>i \<in> {1..dimindex(UNIV:: 1 set)}. P i) \<longleftrightarrow> P 1"
    19   by (simp add: dimindex_def)
    20 
    21 lemma forall_2: "(\<forall>(i::nat). 1 <= i \<and> i <= 2 --> P i) \<longleftrightarrow> P 1 \<and> P 2"
    22 proof-
    23   have "\<And>i::nat. 1 <= i \<and> i <= 2 \<longleftrightarrow> i = 1 \<or> i = 2" by arith
    24   thus ?thesis by metis
    25 qed
    26 
    27 lemma forall_3: "(\<forall>(i::nat). 1 <= i \<and> i <= 3 --> P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
    28 proof-
    29   have "\<And>i::nat. 1 <= i \<and> i <= 3 \<longleftrightarrow> i = 1 \<or> i = 2 \<or> i = 3" by arith
    30   thus ?thesis by metis
    31 qed
    32 
    33 lemma setsum_singleton[simp]: "setsum f {x} = f x" by simp
    34 lemma setsum_1: "setsum f {(1::'a::{order,one})..1} = f 1" 
    35   by (simp add: atLeastAtMost_singleton)
    36 
    37 lemma setsum_2: "setsum f {1::nat..2} = f 1 + f 2" 
    38   by (simp add: nat_number  atLeastAtMostSuc_conv add_commute)
    39 
    40 lemma setsum_3: "setsum f {1::nat..3} = f 1 + f 2 + f 3" 
    41   by (simp add: nat_number  atLeastAtMostSuc_conv add_commute)
    42 
    43 subsection{* Basic componentwise operations on vectors. *}
    44 
    45 instantiation "^" :: (plus,type) plus
    46 begin
    47 definition  vector_add_def : "op + \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) + (y$i)))" 
    48 instance ..
    49 end
    50 
    51 instantiation "^" :: (times,type) times
    52 begin
    53   definition vector_mult_def : "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))" 
    54   instance ..
    55 end
    56 
    57 instantiation "^" :: (minus,type) minus begin
    58   definition vector_minus_def : "op - \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) - (y$i)))"
    59 instance ..
    60 end
    61 
    62 instantiation "^" :: (uminus,type) uminus begin
    63   definition vector_uminus_def : "uminus \<equiv> (\<lambda> x.  (\<chi> i. - (x$i)))"
    64 instance ..
    65 end
    66 instantiation "^" :: (zero,type) zero begin
    67   definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)" 
    68 instance ..
    69 end
    70 
    71 instantiation "^" :: (one,type) one begin
    72   definition vector_one_def : "1 \<equiv> (\<chi> i. 1)" 
    73 instance ..
    74 end
    75 
    76 instantiation "^" :: (ord,type) ord
    77  begin
    78 definition vector_less_eq_def:
    79   "less_eq (x :: 'a ^'b) y = (ALL i : {1 .. dimindex (UNIV :: 'b set)}.
    80   x$i <= y$i)"
    81 definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i : {1 ..
    82   dimindex (UNIV :: 'b set)}. x$i < y$i)"
    83  
    84 instance by (intro_classes)
    85 end
    86 
    87 instantiation "^" :: (scaleR, type) scaleR
    88 begin
    89 definition vector_scaleR_def: "scaleR = (\<lambda> r x.  (\<chi> i. scaleR r (x$i)))" 
    90 instance ..
    91 end
    92 
    93 text{* Also the scalar-vector multiplication. *}
    94 
    95 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'n" (infixr "*s" 75)
    96   where "c *s x = (\<chi> i. c * (x$i))"
    97 
    98 text{* Constant Vectors *}
    99 
   100 definition "vec x = (\<chi> i. x)"
   101 
   102 text{* Dot products. *}
   103 
   104 definition dot :: "'a::{comm_monoid_add, times} ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" (infix "\<bullet>" 70) where
   105   "x \<bullet> y = setsum (\<lambda>i. x$i * y$i) {1 .. dimindex (UNIV:: 'n set)}"
   106 lemma dot_1[simp]: "(x::'a::{comm_monoid_add, times}^1) \<bullet> y = (x$1) * (y$1)"
   107   by (simp add: dot_def dimindex_def)
   108 
   109 lemma dot_2[simp]: "(x::'a::{comm_monoid_add, times}^2) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2)"
   110   by (simp add: dot_def dimindex_def nat_number)
   111 
   112 lemma dot_3[simp]: "(x::'a::{comm_monoid_add, times}^3) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2) + (x$3) * (y$3)"
   113   by (simp add: dot_def dimindex_def nat_number)
   114 
   115 subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
   116 
   117 lemmas Cart_lambda_beta' = Cart_lambda_beta[rule_format]
   118 method_setup vector = {*
   119 let
   120   val ss1 = HOL_basic_ss addsimps [@{thm dot_def}, @{thm setsum_addf} RS sym, 
   121   @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib}, 
   122   @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
   123   val ss2 = @{simpset} addsimps 
   124              [@{thm vector_add_def}, @{thm vector_mult_def},  
   125               @{thm vector_minus_def}, @{thm vector_uminus_def}, 
   126               @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def}, 
   127               @{thm vector_scaleR_def},
   128               @{thm Cart_lambda_beta'}, @{thm vector_scalar_mult_def}]
   129  fun vector_arith_tac ths = 
   130    simp_tac ss1
   131    THEN' (fn i => rtac @{thm setsum_cong2} i
   132          ORELSE rtac @{thm setsum_0'} i 
   133          ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i)
   134    (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
   135    THEN' asm_full_simp_tac (ss2 addsimps ths)
   136  in
   137   Method.thms_args (Method.SIMPLE_METHOD' o vector_arith_tac)
   138 end
   139 *} "Lifts trivial vector statements to real arith statements"
   140 
   141 lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def)
   142 lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def)
   143 
   144 
   145 
   146 text{* Obvious "component-pushing". *}
   147 
   148 lemma vec_component: " i \<in> {1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (vec x :: 'a ^ 'n)$i = x" 
   149   by (vector vec_def) 
   150 
   151 lemma vector_add_component: 
   152   fixes x y :: "'a::{plus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
   153   shows "(x + y)$i = x$i + y$i"
   154   using i by vector
   155 
   156 lemma vector_minus_component: 
   157   fixes x y :: "'a::{minus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
   158   shows "(x - y)$i = x$i - y$i"
   159   using i  by vector
   160 
   161 lemma vector_mult_component: 
   162   fixes x y :: "'a::{times} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
   163   shows "(x * y)$i = x$i * y$i"
   164   using i by vector
   165 
   166 lemma vector_smult_component: 
   167   fixes y :: "'a::{times} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
   168   shows "(c *s y)$i = c * (y$i)"
   169   using i by vector
   170 
   171 lemma vector_uminus_component: 
   172   fixes x :: "'a::{uminus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
   173   shows "(- x)$i = - (x$i)"
   174   using i by vector
   175 
   176 lemma vector_scaleR_component:
   177   fixes x :: "'a::scaleR ^ 'n"
   178   assumes i: "i \<in> {1 .. dimindex(UNIV :: 'n set)}"
   179   shows "(scaleR r x)$i = scaleR r (x$i)"
   180   using i by vector
   181 
   182 lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
   183 
   184 lemmas vector_component =
   185   vec_component vector_add_component vector_mult_component
   186   vector_smult_component vector_minus_component vector_uminus_component
   187   vector_scaleR_component cond_component
   188 
   189 subsection {* Some frequently useful arithmetic lemmas over vectors. *}
   190 
   191 instance "^" :: (semigroup_add,type) semigroup_add 
   192   apply (intro_classes) by (vector add_assoc)
   193 
   194 
   195 instance "^" :: (monoid_add,type) monoid_add 
   196   apply (intro_classes) by vector+ 
   197 
   198 instance "^" :: (group_add,type) group_add 
   199   apply (intro_classes) by (vector algebra_simps)+ 
   200 
   201 instance "^" :: (ab_semigroup_add,type) ab_semigroup_add 
   202   apply (intro_classes) by (vector add_commute)
   203 
   204 instance "^" :: (comm_monoid_add,type) comm_monoid_add
   205   apply (intro_classes) by vector
   206 
   207 instance "^" :: (ab_group_add,type) ab_group_add 
   208   apply (intro_classes) by vector+
   209 
   210 instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add 
   211   apply (intro_classes)
   212   by (vector Cart_eq)+
   213 
   214 instance "^" :: (cancel_ab_semigroup_add,type) cancel_ab_semigroup_add
   215   apply (intro_classes)
   216   by (vector Cart_eq)
   217 
   218 instance "^" :: (real_vector, type) real_vector
   219   by default (vector scaleR_left_distrib scaleR_right_distrib)+
   220 
   221 instance "^" :: (semigroup_mult,type) semigroup_mult 
   222   apply (intro_classes) by (vector mult_assoc)
   223 
   224 instance "^" :: (monoid_mult,type) monoid_mult 
   225   apply (intro_classes) by vector+
   226 
   227 instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult 
   228   apply (intro_classes) by (vector mult_commute)
   229 
   230 instance "^" :: (ab_semigroup_idem_mult,type) ab_semigroup_idem_mult 
   231   apply (intro_classes) by (vector mult_idem)
   232 
   233 instance "^" :: (comm_monoid_mult,type) comm_monoid_mult 
   234   apply (intro_classes) by vector
   235 
   236 fun vector_power :: "('a::{one,times} ^'n) \<Rightarrow> nat \<Rightarrow> 'a^'n" where
   237   "vector_power x 0 = 1"
   238   | "vector_power x (Suc n) = x * vector_power x n"
   239 
   240 instantiation "^" :: (recpower,type) recpower 
   241 begin
   242   definition vec_power_def: "op ^ \<equiv> vector_power"
   243   instance 
   244   apply (intro_classes) by (simp_all add: vec_power_def) 
   245 end
   246 
   247 instance "^" :: (semiring,type) semiring
   248   apply (intro_classes) by (vector ring_simps)+
   249 
   250 instance "^" :: (semiring_0,type) semiring_0
   251   apply (intro_classes) by (vector ring_simps)+
   252 instance "^" :: (semiring_1,type) semiring_1
   253   apply (intro_classes) apply vector using dimindex_ge_1 by auto 
   254 instance "^" :: (comm_semiring,type) comm_semiring
   255   apply (intro_classes) by (vector ring_simps)+
   256 
   257 instance "^" :: (comm_semiring_0,type) comm_semiring_0 by (intro_classes) 
   258 instance "^" :: (cancel_comm_monoid_add, type) cancel_comm_monoid_add ..
   259 instance "^" :: (semiring_0_cancel,type) semiring_0_cancel by (intro_classes) 
   260 instance "^" :: (comm_semiring_0_cancel,type) comm_semiring_0_cancel by (intro_classes) 
   261 instance "^" :: (ring,type) ring by (intro_classes) 
   262 instance "^" :: (semiring_1_cancel,type) semiring_1_cancel by (intro_classes) 
   263 instance "^" :: (comm_semiring_1,type) comm_semiring_1 by (intro_classes)
   264 
   265 instance "^" :: (ring_1,type) ring_1 ..
   266 
   267 instance "^" :: (real_algebra,type) real_algebra
   268   apply intro_classes
   269   apply (simp_all add: vector_scaleR_def ring_simps)
   270   apply vector
   271   apply vector
   272   done
   273 
   274 instance "^" :: (real_algebra_1,type) real_algebra_1 ..
   275 
   276 lemma of_nat_index: 
   277   "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
   278   apply (induct n)
   279   apply vector
   280   apply vector
   281   done
   282 lemma zero_index[simp]: 
   283   "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (0 :: 'a::zero ^'n)$i = 0" by vector
   284 
   285 lemma one_index[simp]: 
   286   "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (1 :: 'a::one ^'n)$i = 1" by vector
   287 
   288 lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \<noteq> 0"
   289 proof-
   290   have "(1::'a) + of_nat n = 0 \<longleftrightarrow> of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp
   291   also have "\<dots> \<longleftrightarrow> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff) 
   292   finally show ?thesis by simp 
   293 qed
   294 
   295 instance "^" :: (semiring_char_0,type) semiring_char_0 
   296 proof (intro_classes) 
   297   fix m n ::nat
   298   show "(of_nat m :: 'a^'b) = of_nat n \<longleftrightarrow> m = n"
   299   proof(induct m arbitrary: n)
   300     case 0 thus ?case apply vector 
   301       apply (induct n,auto simp add: ring_simps)
   302       using dimindex_ge_1 apply auto
   303       apply vector
   304       by (auto simp add: of_nat_index one_plus_of_nat_neq_0)
   305   next
   306     case (Suc n m)
   307     thus ?case  apply vector
   308       apply (induct m, auto simp add: ring_simps of_nat_index zero_index)
   309       using dimindex_ge_1 apply simp apply blast
   310       apply (simp add: one_plus_of_nat_neq_0)
   311       using dimindex_ge_1 apply simp apply blast
   312       apply (simp add: vector_component one_index of_nat_index)
   313       apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)
   314       using  dimindex_ge_1 apply simp apply blast
   315       apply (simp add: vector_component one_index of_nat_index)
   316       apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)
   317       using dimindex_ge_1 apply simp apply blast
   318       apply (simp add: vector_component one_index of_nat_index)
   319       done
   320   qed
   321 qed
   322 
   323 instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes
   324 instance "^" :: (ring_char_0,type) ring_char_0 by intro_classes
   325 
   326 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"  
   327   by (vector mult_assoc)
   328 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" 
   329   by (vector ring_simps)
   330 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" 
   331   by (vector ring_simps)
   332 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
   333 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
   334 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y" 
   335   by (vector ring_simps)
   336 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
   337 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
   338 lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
   339 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
   340 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x" 
   341   by (vector ring_simps)
   342 
   343 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)" 
   344   apply (auto simp add: vec_def Cart_eq vec_component Cart_lambda_beta )
   345   using dimindex_ge_1 apply auto done
   346 
   347 subsection{* Properties of the dot product.  *}
   348 
   349 lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x" 
   350   by (vector mult_commute)
   351 lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \<bullet> z = (x \<bullet> z) + (y \<bullet> z)"
   352   by (vector ring_simps)
   353 lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)" 
   354   by (vector ring_simps)
   355 lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)" 
   356   by (vector ring_simps)
   357 lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)" 
   358   by (vector ring_simps)
   359 lemma dot_lmult: "(c *s x) \<bullet> y = (c::'a::ring) * (x \<bullet> y)" by (vector ring_simps)
   360 lemma dot_rmult: "x \<bullet> (c *s y) = (c::'a::comm_ring) * (x \<bullet> y)" by (vector ring_simps)
   361 lemma dot_lneg: "(-x) \<bullet> (y::'a::ring ^ 'n) = -(x \<bullet> y)" by vector
   362 lemma dot_rneg: "(x::'a::ring ^ 'n) \<bullet> (-y) = -(x \<bullet> y)" by vector
   363 lemma dot_lzero[simp]: "0 \<bullet> x = (0::'a::{comm_monoid_add, mult_zero})" by vector
   364 lemma dot_rzero[simp]: "x \<bullet> 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector
   365 lemma dot_pos_le[simp]: "(0::'a\<Colon>ordered_ring_strict) <= x \<bullet> x"
   366   by (simp add: dot_def setsum_nonneg)
   367 
   368 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)"
   369 using fS fp setsum_nonneg[OF fp]
   370 proof (induct set: finite)
   371   case empty thus ?case by simp
   372 next
   373   case (insert x F)
   374   from insert.prems have Fx: "f x \<ge> 0" and Fp: "\<forall> a \<in> F. f a \<ge> 0" by simp_all
   375   from insert.hyps Fp setsum_nonneg[OF Fp]
   376   have h: "setsum f F = 0 \<longleftrightarrow> (\<forall>a \<in>F. f a = 0)" by metis
   377   from sum_nonneg_eq_zero_iff[OF Fx  setsum_nonneg[OF Fp]] insert.hyps(1,2)
   378   show ?case by (simp add: h)
   379 qed
   380 
   381 lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) = 0"
   382 proof-
   383   {assume f: "finite (UNIV :: 'n set)"
   384     let ?S = "{Suc 0 .. card (UNIV :: 'n set)}"
   385     have fS: "finite ?S" using f by simp
   386     have fp: "\<forall> i\<in> ?S. x$i * x$i>= 0" by simp
   387     have ?thesis by (vector dimindex_def f setsum_squares_eq_0_iff[OF fS fp])}
   388   moreover
   389   {assume "\<not> finite (UNIV :: 'n set)" then have ?thesis by (vector dimindex_def)}
   390   ultimately show ?thesis by metis
   391 qed
   392 
   393 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] 
   394   by (auto simp add: le_less) 
   395 
   396 subsection {* Introduce norms, but defer many properties till we get square roots. *}
   397 text{* FIXME : This is ugly *}
   398 defs (overloaded) 
   399   real_of_real_def [code inline, simp]: "real == id"
   400 
   401 instantiation "^" :: ("{times, comm_monoid_add}", type) norm begin
   402 definition  real_vector_norm_def: "norm \<equiv> (\<lambda>x. sqrt (real (x \<bullet> x)))" 
   403 instance ..
   404 end
   405 
   406 
   407 subsection{* The collapse of the general concepts to dimention one. *}
   408 
   409 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
   410   by (vector dimindex_def)
   411 
   412 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
   413   apply auto
   414   apply (erule_tac x= "x$1" in allE)
   415   apply (simp only: vector_one[symmetric])
   416   done
   417 
   418 lemma norm_real: "norm(x::real ^ 1) = abs(x$1)" 
   419   by (simp add: real_vector_norm_def)
   420 
   421 text{* Metric *}
   422 
   423 definition dist:: "real ^ 'n \<Rightarrow> real ^ 'n \<Rightarrow> real" where 
   424   "dist x y = norm (x - y)"
   425 
   426 lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
   427   using dimindex_ge_1[of "UNIV :: 1 set"]
   428   by (auto simp add: norm_real dist_def vector_component Cart_lambda_beta[where ?'a = "1"] )
   429 
   430 subsection {* A connectedness or intermediate value lemma with several applications. *}
   431 
   432 lemma connected_real_lemma:
   433   fixes f :: "real \<Rightarrow> real ^ 'n"
   434   assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
   435   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"
   436   and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
   437   and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
   438   and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
   439   shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
   440 proof-
   441   let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
   442   have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa) 
   443   have Sub: "\<exists>y. isUb UNIV ?S y" 
   444     apply (rule exI[where x= b])
   445     using ab fb e12 by (auto simp add: isUb_def setle_def)  
   446   from reals_complete[OF Se Sub] obtain l where 
   447     l: "isLub UNIV ?S l"by blast
   448   have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
   449     apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)    
   450     by (metis linorder_linear)
   451   have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
   452     apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
   453     by (metis linorder_linear not_le)
   454     have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
   455     have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
   456     have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo
   457     {assume le2: "f l \<in> e2"
   458       from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
   459       hence lap: "l - a > 0" using alb by arith
   460       from e2[rule_format, OF le2] obtain e where 
   461 	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
   462       from dst[OF alb e(1)] obtain d where 
   463 	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
   464       have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1) 
   465 	apply ferrack by arith
   466       then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
   467       from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
   468       from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
   469       moreover
   470       have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
   471       ultimately have False using e12 alb d' by auto}
   472     moreover
   473     {assume le1: "f l \<in> e1"
   474     from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
   475       hence blp: "b - l > 0" using alb by arith
   476       from e1[rule_format, OF le1] obtain e where 
   477 	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
   478       from dst[OF alb e(1)] obtain d where 
   479 	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
   480       have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo 
   481       then obtain d' where d': "d' > 0" "d' < d" by metis
   482       from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
   483       hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
   484       with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
   485       with l d' have False 
   486 	by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
   487     ultimately show ?thesis using alb by metis
   488 qed
   489 
   490 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"} *}
   491 
   492 lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
   493 proof-
   494   have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith 
   495   thus ?thesis by (simp add: ring_simps power2_eq_square)
   496 qed
   497 
   498 lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
   499   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)
   500   apply (rule_tac x="s" in exI)
   501   apply auto
   502   apply (erule_tac x=y in allE)
   503   apply auto
   504   done
   505 
   506 lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
   507   using real_sqrt_le_iff[of x "y^2"] by simp
   508 
   509 lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
   510   using real_sqrt_le_mono[of "x^2" y] by simp
   511 
   512 lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
   513   using real_sqrt_less_mono[of "x^2" y] by simp
   514 
   515 lemma sqrt_even_pow2: assumes n: "even n" 
   516   shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
   517 proof-
   518   from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2 
   519     by (auto simp add: nat_number) 
   520   from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
   521     by (simp only: power_mult[symmetric] mult_commute)
   522   then show ?thesis  using m by simp 
   523 qed
   524 
   525 lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
   526   apply (cases "x = 0", simp_all)
   527   using sqrt_divide_self_eq[of x]
   528   apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps)
   529   done
   530 
   531 text{* Hence derive more interesting properties of the norm. *}
   532 
   533 lemma norm_0: "norm (0::real ^ 'n) = 0"
   534   by (simp add: real_vector_norm_def dot_eq_0)
   535 
   536 lemma norm_pos_le: "0 <= norm (x::real^'n)" 
   537   by (simp add: real_vector_norm_def dot_pos_le)
   538 lemma norm_neg: " norm(-x) = norm (x:: real ^ 'n)" 
   539   by (simp add: real_vector_norm_def dot_lneg dot_rneg)
   540 lemma norm_sub: "norm(x - y) = norm(y - (x::real ^ 'n))" 
   541   by (metis norm_neg minus_diff_eq)
   542 lemma norm_mul: "norm(a *s x) = abs(a) * norm x"
   543   by (simp add: real_vector_norm_def dot_lmult dot_rmult mult_assoc[symmetric] real_sqrt_mult)
   544 lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
   545   by (simp add: real_vector_norm_def)
   546 lemma norm_eq_0: "norm x = 0 \<longleftrightarrow> x = (0::real ^ 'n)"
   547   by (simp add: real_vector_norm_def dot_eq_0)
   548 lemma norm_pos_lt: "0 < norm x \<longleftrightarrow> x \<noteq> (0::real ^ 'n)"
   549   by (metis less_le real_vector_norm_def norm_pos_le norm_eq_0)
   550 lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
   551   by (simp add: real_vector_norm_def dot_pos_le)
   552 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_0)
   553 lemma norm_le_0: "norm x <= 0 \<longleftrightarrow> x = (0::real ^'n)"
   554   by (metis norm_eq_0 norm_pos_le order_antisym) 
   555 lemma vector_mul_eq_0: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
   556   by vector
   557 lemma vector_mul_lcancel: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
   558   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
   559 lemma vector_mul_rcancel: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
   560   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
   561 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
   562   by (metis vector_mul_lcancel)
   563 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
   564   by (metis vector_mul_rcancel)
   565 lemma norm_cauchy_schwarz: "x \<bullet> y <= norm x * norm y"
   566 proof-
   567   {assume "norm x = 0"
   568     hence ?thesis by (simp add: norm_eq_0 dot_lzero dot_rzero norm_0)}
   569   moreover
   570   {assume "norm y = 0" 
   571     hence ?thesis by (simp add: norm_eq_0 dot_lzero dot_rzero norm_0)}
   572   moreover
   573   {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
   574     let ?z = "norm y *s x - norm x *s y"
   575     from h have p: "norm x * norm y > 0" by (metis norm_pos_le le_less zero_compare_simps)
   576     from dot_pos_le[of ?z]
   577     have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"
   578       apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)
   579       by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym)
   580     hence "x\<bullet>y \<le> (norm x ^2 * norm y ^2) / (norm x * norm y)" using p
   581       by (simp add: field_simps)
   582     hence ?thesis using h by (simp add: power2_eq_square)}
   583   ultimately show ?thesis by metis
   584 qed
   585 
   586 lemma norm_abs[simp]: "abs (norm x) = norm (x::real ^'n)" 
   587   using norm_pos_le[of x] by (simp add: real_abs_def linorder_linear)
   588 
   589 lemma norm_cauchy_schwarz_abs: "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
   590   using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
   591   by (simp add: real_abs_def dot_rneg norm_neg)
   592 lemma norm_triangle: "norm(x + y) <= norm x + norm (y::real ^'n)"
   593   unfolding real_vector_norm_def
   594   apply (rule real_le_lsqrt)
   595   apply (auto simp add: dot_pos_le real_vector_norm_def[symmetric] norm_pos_le norm_pow_2[symmetric] intro: add_nonneg_nonneg)[1]
   596   apply (auto simp add: dot_pos_le real_vector_norm_def[symmetric] norm_pos_le norm_pow_2[symmetric] intro: add_nonneg_nonneg)[1]
   597   apply (simp add: dot_ladd dot_radd dot_sym )
   598     by (simp add: norm_pow_2[symmetric] power2_eq_square ring_simps norm_cauchy_schwarz)
   599 
   600 lemma norm_triangle_sub: "norm (x::real ^'n) <= norm(y) + norm(x - y)"
   601   using norm_triangle[of "y" "x - y"] by (simp add: ring_simps)
   602 lemma norm_triangle_le: "norm(x::real ^'n) + norm y <= e ==> norm(x + y) <= e"
   603   by (metis order_trans norm_triangle)
   604 lemma norm_triangle_lt: "norm(x::real ^'n) + norm(y) < e ==> norm(x + y) < e"
   605   by (metis basic_trans_rules(21) norm_triangle)
   606 
   607 lemma setsum_delta: 
   608   assumes fS: "finite S"
   609   shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
   610 proof-
   611   let ?f = "(\<lambda>k. if k=a then b k else 0)"
   612   {assume a: "a \<notin> S"
   613     hence "\<forall> k\<in> S. ?f k = 0" by simp
   614     hence ?thesis  using a by simp}
   615   moreover 
   616   {assume a: "a \<in> S"
   617     let ?A = "S - {a}"
   618     let ?B = "{a}"
   619     have eq: "S = ?A \<union> ?B" using a by blast 
   620     have dj: "?A \<inter> ?B = {}" by simp
   621     from fS have fAB: "finite ?A" "finite ?B" by auto  
   622     have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
   623       using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
   624       by simp
   625     then have ?thesis  using a by simp}
   626   ultimately show ?thesis by blast
   627 qed
   628   
   629 lemma component_le_norm: "i \<in> {1 .. dimindex(UNIV :: 'n set)} ==> \<bar>x$i\<bar> <= norm (x::real ^ 'n)"
   630 proof(simp add: real_vector_norm_def, rule real_le_rsqrt, clarsimp)
   631   assume i: "Suc 0 \<le> i" "i \<le> dimindex (UNIV :: 'n set)"
   632   let ?S = "{1 .. dimindex(UNIV :: 'n set)}"
   633   let ?f = "(\<lambda>k. if k = i then x$i ^2 else 0)"
   634   have fS: "finite ?S" by simp
   635   from i setsum_delta[OF fS, of i "\<lambda>k. x$i ^ 2"]
   636   have th: "x$i^2 = setsum ?f ?S" by simp
   637   let ?g = "\<lambda>k. x$k * x$k"
   638   {fix x assume x: "x \<in> ?S" have "?f x \<le> ?g x" by (simp add: power2_eq_square)}
   639   with setsum_mono[of ?S ?f ?g] 
   640   have "setsum ?f ?S \<le> setsum ?g ?S" by blast 
   641   then show "x$i ^2 \<le> x \<bullet> (x:: real ^ 'n)" unfolding dot_def th[symmetric] .
   642 qed    
   643 lemma norm_bound_component_le: "norm(x::real ^ 'n) <= e
   644                 ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> <= e"
   645   by (metis component_le_norm order_trans)
   646 
   647 lemma norm_bound_component_lt: "norm(x::real ^ 'n) < e
   648                 ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> < e"
   649   by (metis component_le_norm basic_trans_rules(21))
   650 
   651 lemma norm_le_l1: "norm (x:: real ^'n) <= setsum(\<lambda>i. \<bar>x$i\<bar>) {1..dimindex(UNIV::'n set)}"
   652 proof (simp add: real_vector_norm_def, rule real_le_lsqrt,simp add: dot_pos_le, simp add: setsum_mono, simp add: dot_def, induct "dimindex(UNIV::'n set)")
   653   case 0 thus ?case by simp
   654 next
   655   case (Suc n)
   656   have th: "2 * (\<bar>x$(Suc n)\<bar> * (\<Sum>i = Suc 0..n. \<bar>x$i\<bar>)) \<ge> 0" 
   657     apply simp
   658     apply (rule mult_nonneg_nonneg)
   659     by (simp_all add: setsum_abs_ge_zero)
   660   
   661   from Suc
   662   show ?case using th by (simp add: power2_eq_square ring_simps)
   663 qed
   664 
   665 lemma real_abs_norm: "\<bar> norm x\<bar> = norm (x :: real ^'n)" 
   666   by (simp add: norm_pos_le)
   667 lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n) - norm y\<bar> <= norm(x - y)"
   668   apply (simp add: abs_le_iff ring_simps)
   669   by (metis norm_triangle_sub norm_sub)
   670 lemma norm_le: "norm(x::real ^ 'n) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
   671   by (simp add: real_vector_norm_def)
   672 lemma norm_lt: "norm(x::real ^'n) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
   673   by (simp add: real_vector_norm_def)
   674 lemma norm_eq: "norm (x::real ^'n) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
   675   by (simp add: order_eq_iff norm_le)
   676 lemma norm_eq_1: "norm(x::real ^ 'n) = 1 \<longleftrightarrow> x \<bullet> x = 1"
   677   by (simp add: real_vector_norm_def)
   678 
   679 text{* Squaring equations and inequalities involving norms.  *}
   680 
   681 lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
   682   by (simp add: real_vector_norm_def  dot_pos_le )
   683 
   684 lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
   685 proof-
   686   have th: "\<And>x y::real. x^2 = y^2 \<longleftrightarrow> x = y \<or> x = -y" by algebra
   687   show ?thesis using norm_pos_le[of x]
   688   apply (simp add: dot_square_norm th)
   689   apply arith
   690   done
   691 qed
   692 
   693 lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
   694 proof-
   695   have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: ring_simps power2_eq_square)
   696   also have "\<dots> \<longleftrightarrow> \<bar>x\<bar> \<le> \<bar>y\<bar>" apply (simp add: zero_compare_simps real_abs_def not_less) by arith
   697 finally show ?thesis ..
   698 qed
   699 
   700 lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
   701   using norm_pos_le[of x]
   702   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
   703   apply arith
   704   done
   705 
   706 lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2" 
   707   using norm_pos_le[of x]
   708   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
   709   apply arith
   710   done
   711 
   712 lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
   713   by (metis not_le norm_ge_square)
   714 lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
   715   by (metis norm_le_square not_less)
   716 
   717 text{* Dot product in terms of the norm rather than conversely. *}
   718 
   719 lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
   720   by (simp add: norm_pow_2 dot_ladd dot_radd dot_sym)
   721 
   722 lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
   723   by (simp add: norm_pow_2 dot_ladd dot_radd dot_lsub dot_rsub dot_sym)
   724 
   725 
   726 text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
   727 
   728 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")
   729 proof
   730   assume "?lhs" then show ?rhs by simp
   731 next
   732   assume ?rhs
   733   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y\<bullet> y = 0" by simp
   734   hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" 
   735     by (simp add: dot_rsub dot_lsub dot_sym)
   736   then have "(x - y) \<bullet> (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub)
   737   then show "x = y" by (simp add: dot_eq_0)
   738 qed
   739 
   740 
   741 subsection{* General linear decision procedure for normed spaces. *}
   742 
   743 lemma norm_cmul_rule_thm: "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(c *s x)"
   744   apply (clarsimp simp add: norm_mul)
   745   apply (rule mult_mono1)
   746   apply simp_all
   747   done
   748 
   749 lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n) \<Longrightarrow> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)"
   750   apply (rule norm_triangle_le) by simp
   751 
   752 lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"
   753   by (simp add: ring_simps)
   754 
   755 lemma pth_1: "(x::real^'n) == 1 *s x" by (simp only: vector_smult_lid)
   756 lemma pth_2: "x - (y::real^'n) == x + -y" by (atomize (full)) simp
   757 lemma pth_3: "(-x::real^'n) == -1 *s x" by vector
   758 lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+
   759 lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector
   760 lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps)
   761 lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all 
   762 lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps) 
   763 lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z"
   764   "c *s x + (d *s x + z) == (c + d) *s x + z"
   765   "(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+
   766 lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector
   767 lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y" 
   768   "(c *s x + z) + d *s y == c *s x + (z + d *s y)"
   769   "c *s x + (d *s y + z) == c *s x + (d *s y + z)"
   770   "(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))"
   771   by ((atomize (full)), vector)+
   772 lemma pth_c: "(c::real) *s x + d *s y == d *s y + c *s x"
   773   "(c *s x + z) + d *s y == d *s y + (c *s x + z)"
   774   "c *s x + (d *s y + z) == d *s y + (c *s x + z)"
   775   "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+
   776 lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector
   777 
   778 lemma norm_imp_pos_and_ge: "norm (x::real ^ 'n) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
   779   by (atomize) (auto simp add: norm_pos_le)
   780 
   781 lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
   782 
   783 lemma norm_pths: 
   784   "(x::real ^'n) = y \<longleftrightarrow> norm (x - y) \<le> 0"
   785   "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
   786   using norm_pos_le[of "x - y"] by (auto simp add: norm_0 norm_eq_0)
   787 
   788 use "normarith.ML"
   789 
   790 method_setup norm = {* Method.ctxt_args (Method.SIMPLE_METHOD' o NormArith.norm_arith_tac)
   791 *} "Proves simple linear statements about vector norms"
   792 
   793 
   794 
   795 text{* Hence more metric properties. *}
   796 
   797 lemma dist_refl: "dist x x = 0" by norm
   798 
   799 lemma dist_sym: "dist x y = dist y x"by norm
   800 
   801 lemma dist_pos_le: "0 <= dist x y" by norm
   802 
   803 lemma dist_triangle: "dist x z <= dist x y + dist y z" by norm
   804 
   805 lemma dist_triangle_alt: "dist y z <= dist x y + dist x z" by norm
   806 
   807 lemma dist_eq_0: "dist x y = 0 \<longleftrightarrow> x = y" by norm
   808 
   809 lemma dist_pos_lt: "x \<noteq> y ==> 0 < dist x y" by norm 
   810 lemma dist_nz:  "x \<noteq> y \<longleftrightarrow> 0 < dist x y" by norm 
   811 
   812 lemma dist_triangle_le: "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e" by norm 
   813 
   814 lemma dist_triangle_lt: "dist x z + dist y z < e ==> dist x y < e" by norm 
   815 
   816 lemma dist_triangle_half_l: "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 ==> dist x1 x2 < e" by norm 
   817 
   818 lemma dist_triangle_half_r: "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 ==> dist x1 x2 < e" by norm 
   819 
   820 lemma dist_triangle_add: "dist (x + y) (x' + y') <= dist x x' + dist y y'"
   821   by norm 
   822 
   823 lemma dist_mul: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y" 
   824   unfolding dist_def vector_ssub_ldistrib[symmetric] norm_mul .. 
   825 
   826 lemma dist_triangle_add_half: " dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 ==> dist(x + y) (x' + y') < e" by norm 
   827 
   828 lemma dist_le_0: "dist x y <= 0 \<longleftrightarrow> x = y" by norm 
   829 
   830 instantiation "^" :: (monoid_add,type) monoid_add
   831 begin
   832   instance by (intro_classes)
   833 end
   834 
   835 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
   836   apply vector
   837   apply auto
   838   apply (cases "finite S")
   839   apply (rule finite_induct[of S])
   840   apply (auto simp add: vector_component zero_index)
   841   done
   842 
   843 lemma setsum_clauses: 
   844   shows "setsum f {} = 0"
   845   and "finite S \<Longrightarrow> setsum f (insert x S) =
   846                  (if x \<in> S then setsum f S else f x + setsum f S)"
   847   by (auto simp add: insert_absorb)
   848 
   849 lemma setsum_cmul: 
   850   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
   851   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
   852   by (simp add: setsum_eq Cart_eq Cart_lambda_beta vector_component setsum_right_distrib)
   853 
   854 lemma setsum_component: 
   855   fixes f:: " 'a \<Rightarrow> ('b::semiring_1) ^'n"
   856   assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
   857   shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
   858   using i by (simp add: setsum_eq Cart_lambda_beta)
   859 
   860   (* This needs finiteness assumption due to the definition of fold!!! *)
   861 
   862 lemma setsum_superset:
   863   assumes fb: "finite B" and ab: "A \<subseteq> B" 
   864   and f0: "\<forall>x \<in> B - A. f x = 0"
   865   shows "setsum f B = setsum f A"
   866 proof-
   867   from ab fb have fa: "finite A" by (metis finite_subset)
   868   from fb have fba: "finite (B - A)" by (metis finite_Diff)
   869   have d: "A \<inter> (B - A) = {}" by blast
   870   from ab have b: "B = A \<union> (B - A)" by blast
   871   from setsum_Un_disjoint[OF fa fba d, of f] b
   872     setsum_0'[OF f0]
   873   show "setsum f B = setsum f A" by simp
   874 qed
   875 
   876 lemma setsum_restrict_set:
   877   assumes fA: "finite A"
   878   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
   879 proof-
   880   from fA have fab: "finite (A \<inter> B)" by auto
   881   have aba: "A \<inter> B \<subseteq> A" by blast
   882   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
   883   from setsum_superset[OF fA aba, of ?g]
   884   show ?thesis by simp
   885 qed
   886 
   887 lemma setsum_cases:
   888   assumes fA: "finite A"
   889   shows "setsum (\<lambda>x. if x \<in> B then f x else g x) A =
   890          setsum f (A \<inter> B) + setsum g (A \<inter> - B)"
   891 proof-
   892   have a: "A = A \<inter> B \<union> A \<inter> -B" "(A \<inter> B) \<inter> (A \<inter> -B) = {}" 
   893     by blast+
   894   from fA 
   895   have f: "finite (A \<inter> B)" "finite (A \<inter> -B)" by auto
   896   let ?g = "\<lambda>x. if x \<in> B then f x else g x"
   897   from setsum_Un_disjoint[OF f a(2), of ?g] a(1)
   898   show ?thesis by simp
   899 qed
   900 
   901 lemma setsum_norm: 
   902   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   903   assumes fS: "finite S"
   904   shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
   905 proof(induct rule: finite_induct[OF fS])
   906   case 1 thus ?case by (simp add: norm_zero)
   907 next
   908   case (2 x S)
   909   from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
   910   also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
   911     using "2.hyps" by simp
   912   finally  show ?case  using "2.hyps" by simp
   913 qed
   914 
   915 lemma real_setsum_norm: 
   916   fixes f :: "'a \<Rightarrow> real ^'n"
   917   assumes fS: "finite S"
   918   shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
   919 proof(induct rule: finite_induct[OF fS])
   920   case 1 thus ?case by simp norm
   921 next
   922   case (2 x S)
   923   from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" apply (simp add: norm_triangle_ineq) by norm
   924   also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
   925     using "2.hyps" by simp
   926   finally  show ?case  using "2.hyps" by simp
   927 qed
   928 
   929 lemma setsum_norm_le: 
   930   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   931   assumes fS: "finite S"
   932   and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
   933   shows "norm (setsum f S) \<le> setsum g S"
   934 proof-
   935   from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S" 
   936     by - (rule setsum_mono, simp)
   937   then show ?thesis using setsum_norm[OF fS, of f] fg
   938     by arith
   939 qed
   940 
   941 lemma real_setsum_norm_le: 
   942   fixes f :: "'a \<Rightarrow> real ^ 'n"
   943   assumes fS: "finite S"
   944   and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
   945   shows "norm (setsum f S) \<le> setsum g S"
   946 proof-
   947   from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S" 
   948     by - (rule setsum_mono, simp)
   949   then show ?thesis using real_setsum_norm[OF fS, of f] fg
   950     by arith
   951 qed
   952 
   953 lemma setsum_norm_bound:
   954   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   955   assumes fS: "finite S"
   956   and K: "\<forall>x \<in> S. norm (f x) \<le> K"
   957   shows "norm (setsum f S) \<le> of_nat (card S) * K"
   958   using setsum_norm_le[OF fS K] setsum_constant[symmetric]
   959   by simp
   960 
   961 lemma real_setsum_norm_bound:
   962   fixes f :: "'a \<Rightarrow> real ^ 'n"
   963   assumes fS: "finite S"
   964   and K: "\<forall>x \<in> S. norm (f x) \<le> K"
   965   shows "norm (setsum f S) \<le> of_nat (card S) * K"
   966   using real_setsum_norm_le[OF fS K] setsum_constant[symmetric]
   967   by simp
   968 
   969 lemma setsum_vmul:
   970   fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector,semiring, mult_zero}"
   971   assumes fS: "finite S"
   972   shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
   973 proof(induct rule: finite_induct[OF fS])
   974   case 1 then show ?case by (simp add: vector_smult_lzero)
   975 next
   976   case (2 x F)
   977   from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v" 
   978     by simp
   979   also have "\<dots> = f x *s v + setsum f F *s v" 
   980     by (simp add: vector_sadd_rdistrib)
   981   also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp
   982   finally show ?case .
   983 qed
   984 
   985 (* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \<Rightarrow> real ^'n"]  ---
   986  Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *)
   987 
   988 lemma setsum_add_split: assumes mn: "(m::nat) \<le> n + 1"
   989   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
   990 proof-
   991   let ?A = "{m .. n}"
   992   let ?B = "{n + 1 .. n + p}"
   993   have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto 
   994   have d: "?A \<inter> ?B = {}" by auto
   995   from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto
   996 qed
   997 
   998 lemma setsum_reindex_nonzero: 
   999   assumes fS: "finite S"
  1000   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"
  1001   shows "setsum h (f ` S) = setsum (h o f) S"
  1002 using nz
  1003 proof(induct rule: finite_induct[OF fS])
  1004   case 1 thus ?case by simp
  1005 next
  1006   case (2 x F) 
  1007   {assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
  1008     then obtain y where y: "y \<in> F" "f x = f y" by auto 
  1009     from "2.hyps" y have xy: "x \<noteq> y" by auto
  1010     
  1011     from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
  1012     have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
  1013     also have "\<dots> = setsum (h o f) (insert x F)" 
  1014       using "2.hyps" "2.prems" h0  by auto 
  1015     finally have ?case .}
  1016   moreover
  1017   {assume fxF: "f x \<notin> f ` F"
  1018     have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" 
  1019       using fxF "2.hyps" by simp 
  1020     also have "\<dots> = setsum (h o f) (insert x F)"  
  1021       using "2.hyps" "2.prems" fxF
  1022       apply auto apply metis done
  1023     finally have ?case .}
  1024   ultimately show ?case by blast
  1025 qed
  1026 
  1027 lemma setsum_Un_nonzero:
  1028   assumes fS: "finite S" and fF: "finite F"
  1029   and f: "\<forall> x\<in> S \<inter> F . f x = (0::'a::ab_group_add)"
  1030   shows "setsum f (S \<union> F) = setsum f S + setsum f F"
  1031   using setsum_Un[OF fS fF, of f] setsum_0'[OF f] by simp
  1032 
  1033 lemma setsum_natinterval_left:
  1034   assumes mn: "(m::nat) <= n" 
  1035   shows "setsum f {m..n} = f m + setsum f {m + 1..n}"
  1036 proof-
  1037   from mn have "{m .. n} = insert m {m+1 .. n}" by auto
  1038   then show ?thesis by auto
  1039 qed
  1040 
  1041 lemma setsum_natinterval_difff: 
  1042   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
  1043   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
  1044           (if m <= n then f m - f(n + 1) else 0)"
  1045 by (induct n, auto simp add: ring_simps not_le le_Suc_eq)
  1046 
  1047 lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
  1048 
  1049 lemma setsum_setsum_restrict:
  1050   "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"
  1051   apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
  1052   by (rule setsum_commute)
  1053 
  1054 lemma setsum_image_gen: assumes fS: "finite S"
  1055   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
  1056 proof-
  1057   {fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto}
  1058   note th0 = this
  1059   have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S" 
  1060     apply (rule setsum_cong2) 
  1061     by (simp add: th0)
  1062   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
  1063     apply (rule setsum_setsum_restrict[OF fS])
  1064     by (rule finite_imageI[OF fS])
  1065   finally show ?thesis .
  1066 qed
  1067 
  1068     (* FIXME: Here too need stupid finiteness assumption on T!!! *)
  1069 lemma setsum_group:
  1070   assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
  1071   shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
  1072   
  1073 apply (subst setsum_image_gen[OF fS, of g f])
  1074 apply (rule setsum_superset[OF fT fST])
  1075 by (auto intro: setsum_0')
  1076 
  1077 (* FIXME: Change the name to fold_image\<dots> *)
  1078 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"
  1079   apply (induct set: finite)
  1080   apply simp by (auto simp add: fold_image_insert)
  1081 
  1082 lemma (in comm_monoid_mult) fold_union_nonzero:
  1083   assumes fS: "finite S" and fT: "finite T"
  1084   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
  1085   shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
  1086 proof-
  1087   have "fold_image op * f 1 (S \<inter> T) = 1" 
  1088     apply (rule fold_1')
  1089     using fS fT I0 by auto 
  1090   with fold_image_Un_Int[OF fS fT] show ?thesis by simp
  1091 qed
  1092 
  1093 lemma setsum_union_nonzero:  
  1094   assumes fS: "finite S" and fT: "finite T"
  1095   and I0: "\<forall>x \<in> S\<inter>T. f x = 0"
  1096   shows "setsum f (S \<union> T) = setsum f S  + setsum f T"
  1097   using fS fT
  1098   apply (simp add: setsum_def)
  1099   apply (rule comm_monoid_add.fold_union_nonzero)
  1100   using I0 by auto
  1101 
  1102 lemma setprod_union_nonzero:  
  1103   assumes fS: "finite S" and fT: "finite T"
  1104   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
  1105   shows "setprod f (S \<union> T) = setprod f S  * setprod f T"
  1106   using fS fT
  1107   apply (simp add: setprod_def)
  1108   apply (rule fold_union_nonzero)
  1109   using I0 by auto
  1110 
  1111 lemma setsum_unions_nonzero: 
  1112   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
  1113   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"
  1114   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
  1115   using fSS f0
  1116 proof(induct rule: finite_induct[OF fS])
  1117   case 1 thus ?case by simp
  1118 next
  1119   case (2 T F)
  1120   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F" 
  1121     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by (auto simp add: finite_insert)
  1122   from fTF have fUF: "finite (\<Union>F)" by (auto intro: finite_Union)
  1123   from "2.prems" TF fTF
  1124   show ?case 
  1125     by (auto simp add: H[symmetric] intro: setsum_union_nonzero[OF fTF(1) fUF, of f])
  1126 qed
  1127 
  1128   (* FIXME : Copied from Pocklington --- should be moved to Finite_Set!!!!!!!! *)
  1129 
  1130 
  1131 lemma (in comm_monoid_mult) fold_related: 
  1132   assumes Re: "R e e" 
  1133   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
  1134   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
  1135   shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
  1136   using fS by (rule finite_subset_induct) (insert assms, auto)
  1137 
  1138   (* FIXME: I think we can get rid of the finite assumption!! *)	
  1139 lemma (in comm_monoid_mult) 
  1140   fold_eq_general:
  1141   assumes fS: "finite S"
  1142   and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
  1143   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
  1144   shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
  1145 proof-
  1146   from h f12 have hS: "h ` S = S'" by auto
  1147   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
  1148     from f12 h H  have "x = y" by auto }
  1149   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
  1150   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
  1151   from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
  1152   also have "\<dots> = fold_image (op *) (f2 o h) e S" 
  1153     using fold_image_reindex[OF fS hinj, of f2 e] .
  1154   also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
  1155     by blast
  1156   finally show ?thesis ..
  1157 qed
  1158 
  1159 lemma (in comm_monoid_mult) fold_eq_general_inverses:
  1160   assumes fS: "finite S" 
  1161   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
  1162   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
  1163   shows "fold_image (op *) f e S = fold_image (op *) g e T"
  1164   using fold_eq_general[OF fS, of T h g f e] kh hk by metis
  1165 
  1166 lemma setsum_eq_general_reverses:
  1167   assumes fS: "finite S" and fT: "finite T"
  1168   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
  1169   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
  1170   shows "setsum f S = setsum g T"
  1171   apply (simp add: setsum_def fS fT)
  1172   apply (rule comm_monoid_add.fold_eq_general_inverses[OF fS])
  1173   apply (erule kh)
  1174   apply (erule hk)
  1175   done
  1176 
  1177 lemma vsum_norm_allsubsets_bound:
  1178   fixes f:: "'a \<Rightarrow> real ^'n"
  1179   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e" 
  1180   shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real (dimindex(UNIV :: 'n set)) *  e"
  1181 proof-
  1182   let ?d = "real (dimindex (UNIV ::'n set))"
  1183   let ?nf = "\<lambda>x. norm (f x)"
  1184   let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
  1185   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"
  1186     by (rule setsum_commute)
  1187   have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
  1188   have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
  1189     apply (rule setsum_mono)
  1190     by (rule norm_le_l1)
  1191   also have "\<dots> \<le> 2 * ?d * e"
  1192     unfolding th0 th1
  1193   proof(rule setsum_bounded)
  1194     fix i assume i: "i \<in> ?U"
  1195     let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
  1196     let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
  1197     have thp: "P = ?Pp \<union> ?Pn" by auto
  1198     have thp0: "?Pp \<inter> ?Pn ={}" by auto
  1199     have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
  1200     have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
  1201       using i component_le_norm[OF i, of "setsum (\<lambda>x. f x) ?Pp"]  fPs[OF PpP]
  1202       by (auto simp add: setsum_component intro: abs_le_D1)
  1203     have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
  1204       using i component_le_norm[OF i, of "setsum (\<lambda>x. - f x) ?Pn"]  fPs[OF PnP]
  1205       by (auto simp add: setsum_negf norm_neg setsum_component vector_component intro: abs_le_D1)
  1206     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" 
  1207       apply (subst thp)
  1208       apply (rule setsum_Un_nonzero) 
  1209       using fP thp0 by auto
  1210     also have "\<dots> \<le> 2*e" using Pne Ppe by arith
  1211     finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
  1212   qed
  1213   finally show ?thesis .
  1214 qed
  1215 
  1216 lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> (y::'a::{comm_ring}^'n) = setsum (\<lambda>x. f x \<bullet> y) S "
  1217   by (induct rule: finite_induct, auto simp add: dot_lzero dot_ladd)
  1218 
  1219 lemma dot_rsum: "finite S \<Longrightarrow> (y::'a::{comm_ring}^'n) \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
  1220   by (induct rule: finite_induct, auto simp add: dot_rzero dot_radd)
  1221 
  1222 subsection{* Basis vectors in coordinate directions. *}
  1223 
  1224 
  1225 definition "basis k = (\<chi> i. if i = k then 1 else 0)"
  1226 
  1227 lemma delta_mult_idempotent: 
  1228   "(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)
  1229 
  1230 lemma norm_basis:
  1231   assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
  1232   shows "norm (basis k :: real ^'n) = 1"
  1233   using k 
  1234   apply (simp add: basis_def real_vector_norm_def dot_def)
  1235   apply (vector delta_mult_idempotent)
  1236   using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "k" "\<lambda>k. 1::real"]
  1237   apply auto
  1238   done
  1239 
  1240 lemma norm_basis_1: "norm(basis 1 :: real ^'n) = 1"
  1241   apply (simp add: basis_def real_vector_norm_def dot_def)
  1242   apply (vector delta_mult_idempotent)
  1243   using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "1" "\<lambda>k. 1::real"] dimindex_nonzero[of "UNIV :: 'n set"]
  1244   apply auto
  1245   done
  1246 
  1247 lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c"
  1248   apply (rule exI[where x="c *s basis 1"])
  1249   by (simp only: norm_mul norm_basis_1)
  1250 
  1251 lemma vector_choose_dist: assumes e: "0 <= e" 
  1252   shows "\<exists>(y::real^'n). dist x y = e"
  1253 proof-
  1254   from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"
  1255     by blast
  1256   then have "dist x (x - c) = e" by (simp add: dist_def)
  1257   then show ?thesis by blast
  1258 qed
  1259 
  1260 lemma basis_inj: "inj_on (basis :: nat \<Rightarrow> real ^'n) {1 .. dimindex (UNIV :: 'n set)}"
  1261   by (auto simp add: inj_on_def basis_def Cart_eq Cart_lambda_beta)
  1262 
  1263 lemma basis_component: "i \<in> {1 .. dimindex(UNIV:: 'n set)} ==> (basis k ::('a::semiring_1)^'n)$i = (if k=i then 1 else 0)"
  1264   by (simp add: basis_def Cart_lambda_beta)
  1265 
  1266 lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
  1267   by auto
  1268 
  1269 lemma basis_expansion:
  1270   "setsum (\<lambda>i. (x$i) *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
  1271   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)
  1272 
  1273 lemma basis_expansion_unique: 
  1274   "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$i)"
  1275   by (simp add: Cart_eq setsum_component vector_component basis_component setsum_delta cond_value_iff cong del: if_weak_cong)
  1276 
  1277 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
  1278   by auto
  1279 
  1280 lemma dot_basis:
  1281   assumes i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
  1282   shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i :: 'a^'n) = (x$i :: 'a::semiring_1)"
  1283   using i
  1284   by (auto simp add: dot_def basis_def Cart_lambda_beta cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)
  1285 
  1286 lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> i \<notin> {1..dimindex(UNIV ::'n set)}"
  1287   by (auto simp add: Cart_eq basis_component zero_index)
  1288 
  1289 lemma basis_nonzero: 
  1290   assumes k: "k \<in> {1 .. dimindex(UNIV ::'n set)}"
  1291   shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
  1292   using k by (simp add: basis_eq_0)
  1293 
  1294 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = (z::'a::semiring_1^'n)"
  1295   apply (auto simp add: Cart_eq dot_basis)
  1296   apply (erule_tac x="basis i" in allE)
  1297   apply (simp add: dot_basis)
  1298   apply (subgoal_tac "y = z")
  1299   apply simp
  1300   apply vector
  1301   done
  1302 
  1303 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = (y::'a::semiring_1^'n)"
  1304   apply (auto simp add: Cart_eq dot_basis)
  1305   apply (erule_tac x="basis i" in allE)
  1306   apply (simp add: dot_basis)
  1307   apply (subgoal_tac "x = y")
  1308   apply simp
  1309   apply vector
  1310   done
  1311 
  1312 subsection{* Orthogonality. *}
  1313 
  1314 definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
  1315 
  1316 lemma orthogonal_basis:
  1317   assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}" 
  1318   shows "orthogonal (basis i :: 'a^'n) x \<longleftrightarrow> x$i = (0::'a::ring_1)"
  1319   using i
  1320   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)
  1321 
  1322 lemma orthogonal_basis_basis:
  1323   assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}" 
  1324   and j: "j \<in> {1 .. dimindex(UNIV ::'n set)}" 
  1325   shows "orthogonal (basis i :: 'a::ring_1^'n) (basis j) \<longleftrightarrow> i \<noteq> j" 
  1326   unfolding orthogonal_basis[OF i] basis_component[OF i] by simp
  1327 
  1328   (* FIXME : Maybe some of these require less than comm_ring, but not all*)
  1329 lemma orthogonal_clauses:
  1330   "orthogonal a (0::'a::comm_ring ^'n)"
  1331   "orthogonal a x ==> orthogonal a (c *s x)"
  1332   "orthogonal a x ==> orthogonal a (-x)"
  1333   "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x + y)"
  1334   "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x - y)"
  1335   "orthogonal 0 a"
  1336   "orthogonal x a ==> orthogonal (c *s x) a"
  1337   "orthogonal x a ==> orthogonal (-x) a"
  1338   "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x + y) a"
  1339   "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x - y) a"
  1340   unfolding orthogonal_def dot_rneg dot_rmult dot_radd dot_rsub
  1341   dot_lzero dot_rzero dot_lneg dot_lmult dot_ladd dot_lsub
  1342   by simp_all
  1343 
  1344 lemma orthogonal_commute: "orthogonal (x::'a::{ab_semigroup_mult,comm_monoid_add} ^'n)y \<longleftrightarrow> orthogonal y x"
  1345   by (simp add: orthogonal_def dot_sym)
  1346 
  1347 subsection{* Explicit vector construction from lists. *}
  1348 
  1349 lemma Cart_lambda_beta_1[simp]: "(Cart_lambda g)$1 = g 1"
  1350   apply (rule Cart_lambda_beta[rule_format])
  1351   using dimindex_ge_1 apply auto done
  1352 
  1353 lemma Cart_lambda_beta_1'[simp]: "(Cart_lambda g)$(Suc 0) = g 1"
  1354   by (simp only: One_nat_def[symmetric] Cart_lambda_beta_1)
  1355 
  1356 definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"
  1357 
  1358 lemma vector_1: "(vector[x]) $1 = x"
  1359   using dimindex_ge_1
  1360   by (auto simp add: vector_def Cart_lambda_beta[rule_format])
  1361 lemma dimindex_2[simp]: "2 \<in> {1 .. dimindex (UNIV :: 2 set)}"
  1362   by (auto simp add: dimindex_def)
  1363 lemma dimindex_2'[simp]: "2 \<in> {Suc 0 .. dimindex (UNIV :: 2 set)}"
  1364   by (auto simp add: dimindex_def)
  1365 lemma dimindex_3[simp]: "2 \<in> {1 .. dimindex (UNIV :: 3 set)}" "3 \<in> {1 .. dimindex (UNIV :: 3 set)}"
  1366   by (auto simp add: dimindex_def)
  1367 
  1368 lemma dimindex_3'[simp]: "2 \<in> {Suc 0 .. dimindex (UNIV :: 3 set)}" "3 \<in> {Suc 0 .. dimindex (UNIV :: 3 set)}"
  1369   by (auto simp add: dimindex_def)
  1370 
  1371 lemma vector_2:
  1372  "(vector[x,y]) $1 = x"
  1373  "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
  1374   apply (simp add: vector_def)
  1375   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)"]
  1376   apply (simp only: vector_def )
  1377   apply auto
  1378   done
  1379 
  1380 lemma vector_3:
  1381  "(vector [x,y,z] ::('a::zero)^3)$1 = x"
  1382  "(vector [x,y,z] ::('a::zero)^3)$2 = y"
  1383  "(vector [x,y,z] ::('a::zero)^3)$3 = z"
  1384 apply (simp_all add: vector_def Cart_lambda_beta dimindex_3)
  1385   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)"]
  1386   by simp_all
  1387 
  1388 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
  1389   apply auto
  1390   apply (erule_tac x="v$1" in allE)
  1391   apply (subgoal_tac "vector [v$1] = v")
  1392   apply simp
  1393   by (vector vector_def dimindex_def)
  1394 
  1395 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
  1396   apply auto
  1397   apply (erule_tac x="v$1" in allE)
  1398   apply (erule_tac x="v$2" in allE)
  1399   apply (subgoal_tac "vector [v$1, v$2] = v")
  1400   apply simp
  1401   apply (vector vector_def dimindex_def)
  1402   apply auto
  1403   apply (subgoal_tac "i = 1 \<or> i =2", auto)
  1404   done
  1405 
  1406 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
  1407   apply auto
  1408   apply (erule_tac x="v$1" in allE)
  1409   apply (erule_tac x="v$2" in allE)
  1410   apply (erule_tac x="v$3" in allE)
  1411   apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
  1412   apply simp
  1413   apply (vector vector_def dimindex_def)
  1414   apply auto
  1415   apply (subgoal_tac "i = 1 \<or> i =2 \<or> i = 3", auto)
  1416   done
  1417 
  1418 subsection{* Linear functions. *}
  1419 
  1420 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)"
  1421 
  1422 lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. (c::'a::comm_semiring) *s f x)"
  1423   by (vector linear_def Cart_eq Cart_lambda_beta[rule_format] ring_simps)
  1424 
  1425 lemma linear_compose_neg: "linear (f :: 'a ^'n \<Rightarrow> 'a::comm_ring ^'m) ==> linear (\<lambda>x. -(f(x)))" by (vector linear_def Cart_eq)
  1426 
  1427 lemma linear_compose_add: "linear (f :: 'a ^'n \<Rightarrow> 'a::semiring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
  1428   by (vector linear_def Cart_eq ring_simps)
  1429 
  1430 lemma linear_compose_sub: "linear (f :: 'a ^'n \<Rightarrow> 'a::ring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
  1431   by (vector linear_def Cart_eq ring_simps)
  1432 
  1433 lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
  1434   by (simp add: linear_def)
  1435 
  1436 lemma linear_id: "linear id" by (simp add: linear_def id_def)
  1437 
  1438 lemma linear_zero: "linear (\<lambda>x. 0::'a::semiring_1 ^ 'n)" by (simp add: linear_def)
  1439 
  1440 lemma linear_compose_setsum:
  1441   assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a :: 'a::semiring_1 ^ 'n \<Rightarrow> 'a ^ 'm)"
  1442   shows "linear(\<lambda>x. setsum (\<lambda>a. f a x :: 'a::semiring_1 ^'m) S)"
  1443   using lS
  1444   apply (induct rule: finite_induct[OF fS])
  1445   by (auto simp add: linear_zero intro: linear_compose_add)
  1446 
  1447 lemma linear_vmul_component:
  1448   fixes f:: "'a::semiring_1^'m \<Rightarrow> 'a^'n"
  1449   assumes lf: "linear f" and k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
  1450   shows "linear (\<lambda>x. f x $ k *s v)"
  1451   using lf k
  1452   apply (auto simp add: linear_def )
  1453   by (vector ring_simps)+
  1454 
  1455 lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)"
  1456   unfolding linear_def
  1457   apply clarsimp
  1458   apply (erule allE[where x="0::'a"])
  1459   apply simp
  1460   done
  1461 
  1462 lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def)
  1463 
  1464 lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \<Rightarrow> _) ==> f (-x) = - f x"
  1465   unfolding vector_sneg_minus1
  1466   using linear_cmul[of f] by auto 
  1467 
  1468 lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def) 
  1469 
  1470 lemma linear_sub: "linear (f::'a::ring_1 ^'n \<Rightarrow> _) ==> f(x - y) = f x - f y"
  1471   by (simp add: diff_def linear_add linear_neg)
  1472 
  1473 lemma linear_setsum: 
  1474   fixes f:: "'a::semiring_1^'n \<Rightarrow> _"
  1475   assumes lf: "linear f" and fS: "finite S"
  1476   shows "f (setsum g S) = setsum (f o g) S"
  1477 proof (induct rule: finite_induct[OF fS])
  1478   case 1 thus ?case by (simp add: linear_0[OF lf])
  1479 next
  1480   case (2 x F)
  1481   have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
  1482     by simp
  1483   also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
  1484   also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
  1485   finally show ?case .
  1486 qed
  1487 
  1488 lemma linear_setsum_mul:
  1489   fixes f:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m"
  1490   assumes lf: "linear f" and fS: "finite S"
  1491   shows "f (setsum (\<lambda>i. c i *s v i) S) = setsum (\<lambda>i. c i *s f (v i)) S"
  1492   using linear_setsum[OF lf fS, of "\<lambda>i. c i *s v i" , unfolded o_def]
  1493   linear_cmul[OF lf] by simp 
  1494 
  1495 lemma linear_injective_0:
  1496   assumes lf: "linear (f:: 'a::ring_1 ^ 'n \<Rightarrow> _)"
  1497   shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
  1498 proof-
  1499   have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
  1500   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
  1501   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)" 
  1502     by (simp add: linear_sub[OF lf])
  1503   also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
  1504   finally show ?thesis .
  1505 qed
  1506 
  1507 lemma linear_bounded:
  1508   fixes f:: "real ^'m \<Rightarrow> real ^'n"
  1509   assumes lf: "linear f"
  1510   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
  1511 proof-
  1512   let ?S = "{1..dimindex(UNIV:: 'm set)}"
  1513   let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
  1514   have fS: "finite ?S" by simp
  1515   {fix x:: "real ^ 'm"
  1516     let ?g = "(\<lambda>i::nat. (x$i) *s (basis i) :: real ^ 'm)"
  1517     have "norm (f x) = norm (f (setsum (\<lambda>i. (x$i) *s (basis i)) ?S))"
  1518       by (simp only:  basis_expansion)
  1519     also have "\<dots> = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)"
  1520       using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf]
  1521       by auto
  1522     finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)" .
  1523     {fix i assume i: "i \<in> ?S"
  1524       from component_le_norm[OF i, of x]
  1525       have "norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x"
  1526       unfolding norm_mul
  1527       apply (simp only: mult_commute)
  1528       apply (rule mult_mono)
  1529       by (auto simp add: ring_simps norm_pos_le) }
  1530     then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
  1531     from real_setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (f (basis i))", OF th]
  1532     have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
  1533   then show ?thesis by blast
  1534 qed
  1535 
  1536 lemma linear_bounded_pos:
  1537   fixes f:: "real ^'n \<Rightarrow> real ^ 'm"
  1538   assumes lf: "linear f"
  1539   shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
  1540 proof-
  1541   from linear_bounded[OF lf] obtain B where 
  1542     B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
  1543   let ?K = "\<bar>B\<bar> + 1"
  1544   have Kp: "?K > 0" by arith
  1545     {assume C: "B < 0"
  1546       have "norm (1::real ^ 'n) > 0" by (simp add: norm_pos_lt)
  1547       with C have "B * norm (1:: real ^ 'n) < 0"
  1548 	by (simp add: zero_compare_simps)
  1549       with B[rule_format, of 1] norm_pos_le[of "f 1"] have False by simp
  1550     }
  1551     then have Bp: "B \<ge> 0" by ferrack
  1552     {fix x::"real ^ 'n"
  1553       have "norm (f x) \<le> ?K *  norm x"
  1554       using B[rule_format, of x] norm_pos_le[of x] norm_pos_le[of "f x"] Bp
  1555       by (auto simp add: ring_simps split add: abs_split)
  1556   }
  1557   then show ?thesis using Kp by blast
  1558 qed
  1559 
  1560 subsection{* Bilinear functions. *}
  1561 
  1562 definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
  1563 
  1564 lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
  1565   by (simp add: bilinear_def linear_def)
  1566 lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
  1567   by (simp add: bilinear_def linear_def)
  1568 
  1569 lemma bilinear_lmul: "bilinear h ==> h (c *s x) y = c *s (h x y)"
  1570   by (simp add: bilinear_def linear_def)
  1571 
  1572 lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)"
  1573   by (simp add: bilinear_def linear_def)
  1574 
  1575 lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)"
  1576   by (simp only: vector_sneg_minus1 bilinear_lmul)
  1577 
  1578 lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y"
  1579   by (simp only: vector_sneg_minus1 bilinear_rmul)
  1580 
  1581 lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
  1582   using add_imp_eq[of x y 0] by auto
  1583     
  1584 lemma bilinear_lzero: 
  1585   fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h 0 x = 0"
  1586   using bilinear_ladd[OF bh, of 0 0 x] 
  1587     by (simp add: eq_add_iff ring_simps)
  1588 
  1589 lemma bilinear_rzero: 
  1590   fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h x 0 = 0"
  1591   using bilinear_radd[OF bh, of x 0 0 ] 
  1592     by (simp add: eq_add_iff ring_simps)
  1593 
  1594 lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ 'n)) z = h x z - h y z"
  1595   by (simp  add: diff_def bilinear_ladd bilinear_lneg)
  1596 
  1597 lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ 'n)) = h z x - h z y"
  1598   by (simp  add: diff_def bilinear_radd bilinear_rneg)
  1599 
  1600 lemma bilinear_setsum:
  1601   fixes h:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m \<Rightarrow> 'a ^ 'k"
  1602   assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
  1603   shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
  1604 proof- 
  1605   have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
  1606     apply (rule linear_setsum[unfolded o_def])
  1607     using bh fS by (auto simp add: bilinear_def)
  1608   also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
  1609     apply (rule setsum_cong, simp)
  1610     apply (rule linear_setsum[unfolded o_def])
  1611     using bh fT by (auto simp add: bilinear_def)
  1612   finally show ?thesis unfolding setsum_cartesian_product .
  1613 qed
  1614 
  1615 lemma bilinear_bounded:
  1616   fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^ 'k"
  1617   assumes bh: "bilinear h"
  1618   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
  1619 proof- 
  1620   let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
  1621   let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
  1622   let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
  1623   have fM: "finite ?M" and fN: "finite ?N" by simp_all
  1624   {fix x:: "real ^ 'm" and  y :: "real^'n"
  1625     have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$i) *s basis i) ?M) (setsum (\<lambda>i. (y$i) *s basis i) ?N))" unfolding basis_expansion ..
  1626     also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$i) *s basis i) ((y$j) *s basis j)) (?M \<times> ?N))"  unfolding bilinear_setsum[OF bh fM fN] ..
  1627     finally have th: "norm (h x y) = \<dots>" .
  1628     have "norm (h x y) \<le> ?B * norm x * norm y"
  1629       apply (simp add: setsum_left_distrib th)
  1630       apply (rule real_setsum_norm_le)
  1631       using fN fM
  1632       apply simp
  1633       apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)
  1634       apply (rule mult_mono)
  1635       apply (auto simp add: norm_pos_le zero_le_mult_iff component_le_norm)
  1636       apply (rule mult_mono)
  1637       apply (auto simp add: norm_pos_le zero_le_mult_iff component_le_norm)
  1638       done}
  1639   then show ?thesis by metis
  1640 qed
  1641 
  1642 lemma bilinear_bounded_pos:
  1643   fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^ 'k"
  1644   assumes bh: "bilinear h"
  1645   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
  1646 proof-
  1647   from bilinear_bounded[OF bh] obtain B where 
  1648     B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
  1649   let ?K = "\<bar>B\<bar> + 1"
  1650   have Kp: "?K > 0" by arith
  1651   have KB: "B < ?K" by arith
  1652   {fix x::"real ^'m" and y :: "real ^'n"
  1653     from KB Kp
  1654     have "B * norm x * norm y \<le> ?K * norm x * norm y"
  1655       apply - 
  1656       apply (rule mult_right_mono, rule mult_right_mono)
  1657       by (auto simp add: norm_pos_le)
  1658     then have "norm (h x y) \<le> ?K * norm x * norm y"
  1659       using B[rule_format, of x y] by simp} 
  1660   with Kp show ?thesis by blast
  1661 qed
  1662 
  1663 subsection{* Adjoints. *}
  1664 
  1665 definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
  1666 
  1667 lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
  1668 
  1669 lemma adjoint_works_lemma:
  1670   fixes f:: "'a::ring_1 ^'n \<Rightarrow> 'a ^ 'm"
  1671   assumes lf: "linear f"
  1672   shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
  1673 proof-
  1674   let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
  1675   let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
  1676   have fN: "finite ?N" by simp
  1677   have fM: "finite ?M" by simp
  1678   {fix y:: "'a ^ 'm"
  1679     let ?w = "(\<chi> i. (f (basis i) \<bullet> y)) :: 'a ^ 'n"
  1680     {fix x
  1681       have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *s basis i) ?N) \<bullet> y"
  1682 	by (simp only: basis_expansion)
  1683       also have "\<dots> = (setsum (\<lambda>i. (x$i) *s f (basis i)) ?N) \<bullet> y"
  1684 	unfolding linear_setsum[OF lf fN] 
  1685 	by (simp add: linear_cmul[OF lf])
  1686       finally have "f x \<bullet> y = x \<bullet> ?w"
  1687 	apply (simp only: )
  1688 	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)
  1689 	done}
  1690   }
  1691   then show ?thesis unfolding adjoint_def 
  1692     some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
  1693     using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
  1694     by metis
  1695 qed
  1696 
  1697 lemma adjoint_works:
  1698   fixes f:: "'a::ring_1 ^'n \<Rightarrow> 'a ^ 'm"
  1699   assumes lf: "linear f"
  1700   shows "x \<bullet> adjoint f y = f x \<bullet> y"
  1701   using adjoint_works_lemma[OF lf] by metis
  1702 
  1703 
  1704 lemma adjoint_linear:
  1705   fixes f :: "'a::comm_ring_1 ^'n \<Rightarrow> 'a ^ 'm"
  1706   assumes lf: "linear f"
  1707   shows "linear (adjoint f)"
  1708   by (simp add: linear_def vector_eq_ldot[symmetric] dot_radd dot_rmult adjoint_works[OF lf])
  1709 
  1710 lemma adjoint_clauses:
  1711   fixes f:: "'a::comm_ring_1 ^'n \<Rightarrow> 'a ^ 'm"
  1712   assumes lf: "linear f"
  1713   shows "x \<bullet> adjoint f y = f x \<bullet> y"
  1714   and "adjoint f y \<bullet> x = y \<bullet> f x"
  1715   by (simp_all add: adjoint_works[OF lf] dot_sym )
  1716 
  1717 lemma adjoint_adjoint:
  1718   fixes f:: "'a::comm_ring_1 ^ 'n \<Rightarrow> _"
  1719   assumes lf: "linear f"
  1720   shows "adjoint (adjoint f) = f"
  1721   apply (rule ext)
  1722   by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf])
  1723 
  1724 lemma adjoint_unique:
  1725   fixes f:: "'a::comm_ring_1 ^ 'n \<Rightarrow> 'a ^ 'm"
  1726   assumes lf: "linear f" and u: "\<forall>x y. f' x \<bullet> y = x \<bullet> f y"
  1727   shows "f' = adjoint f"
  1728   apply (rule ext)
  1729   using u
  1730   by (simp add: vector_eq_rdot[symmetric] adjoint_clauses[OF lf])
  1731 
  1732 text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
  1733 
  1734 consts generic_mult :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" (infixr "\<star>" 75)
  1735 
  1736 defs (overloaded) 
  1737 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)$j)) {1 .. dimindex (UNIV :: 'n set)}) ::'a ^ 'p ^'m"
  1738 
  1739 abbreviation 
  1740   matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)
  1741   where "m ** m' == m\<star> m'"
  1742 
  1743 defs (overloaded) 
  1744   matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) {1..dimindex(UNIV ::'n set)}) :: 'a^'m"
  1745 
  1746 abbreviation 
  1747   matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)
  1748   where 
  1749   "m *v v == m \<star> v"
  1750 
  1751 defs (overloaded) 
  1752   vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (x$i)) {1..dimindex(UNIV :: 'm set)}) :: 'a^'n"
  1753 
  1754 abbreviation 
  1755   vactor_matrix_mult' :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)
  1756   where 
  1757   "v v* m == v \<star> m"
  1758 
  1759 definition "(mat::'a::zero => 'a ^'n^'m) k = (\<chi> i j. if i = j then k else 0)"
  1760 definition "(transp::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
  1761 definition "(row::nat => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
  1762 definition "(column::nat =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
  1763 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> {1 .. dimindex(UNIV :: 'm set)}}"
  1764 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> {1 .. dimindex(UNIV :: 'n set)}}"
  1765 
  1766 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
  1767 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \<star> B) + (A \<star> C)"
  1768   by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps)
  1769 
  1770 lemma setsum_delta': 
  1771   assumes fS: "finite S" shows 
  1772   "setsum (\<lambda>k. if a = k then b k else 0) S = 
  1773      (if a\<in> S then b a else 0)"
  1774   using setsum_delta[OF fS, of a b, symmetric] 
  1775   by (auto intro: setsum_cong)
  1776 
  1777 lemma matrix_mul_lid: "mat 1 ** A = A"
  1778   apply (simp add: matrix_matrix_mult_def mat_def)
  1779   apply vector
  1780   by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite_atLeastAtMost]  mult_1_left mult_zero_left if_True)
  1781 
  1782 
  1783 lemma matrix_mul_rid: "A ** mat 1 = A"
  1784   apply (simp add: matrix_matrix_mult_def mat_def)
  1785   apply vector
  1786   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)
  1787 
  1788 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
  1789   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
  1790   apply (subst setsum_commute)
  1791   apply simp
  1792   done
  1793 
  1794 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
  1795   apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
  1796   apply (subst setsum_commute)
  1797   apply simp
  1798   done
  1799 
  1800 lemma matrix_vector_mul_lid: "mat 1 *v x = x"
  1801   apply (vector matrix_vector_mult_def mat_def)
  1802   by (simp add: cond_value_iff cond_application_beta 
  1803     setsum_delta' cong del: if_weak_cong)
  1804 
  1805 lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)"
  1806   by (simp add: matrix_matrix_mult_def transp_def Cart_eq Cart_lambda_beta mult_commute)
  1807 
  1808 lemma matrix_eq: "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
  1809   apply auto
  1810   apply (subst Cart_eq)
  1811   apply clarify
  1812   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)
  1813   apply (erule_tac x="basis ia" in allE)
  1814   apply (erule_tac x="i" in ballE)
  1815   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)
  1816 
  1817 lemma matrix_vector_mul_component: 
  1818   assumes k: "k \<in> {1.. dimindex (UNIV :: 'm set)}"
  1819   shows "((A::'a::semiring_1^'n'^'m) *v x)$k = (A$k) \<bullet> x"
  1820   using k
  1821   by (simp add: matrix_vector_mult_def Cart_lambda_beta dot_def)
  1822 
  1823 lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) v* A) \<bullet> y = x \<bullet> (A *v y)"
  1824   apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib Cart_lambda_beta mult_ac)
  1825   apply (subst setsum_commute)
  1826   by simp
  1827 
  1828 lemma transp_mat: "transp (mat n) = mat n"
  1829   by (vector transp_def mat_def)
  1830 
  1831 lemma transp_transp: "transp(transp A) = A"
  1832   by (vector transp_def)
  1833 
  1834 lemma row_transp: 
  1835   fixes A:: "'a::semiring_1^'n^'m"
  1836   assumes i: "i \<in> {1.. dimindex (UNIV :: 'n set)}"
  1837   shows "row i (transp A) = column i A"
  1838   using i 
  1839   by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)
  1840 
  1841 lemma column_transp:
  1842   fixes A:: "'a::semiring_1^'n^'m"
  1843   assumes i: "i \<in> {1.. dimindex (UNIV :: 'm set)}"
  1844   shows "column i (transp A) = row i A"
  1845   using i 
  1846   by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)
  1847 
  1848 lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A"
  1849 apply (auto simp add: rows_def columns_def row_transp intro: set_ext)
  1850 apply (rule_tac x=i in exI)
  1851 apply (auto simp add: row_transp)
  1852 done
  1853 
  1854 lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp)
  1855 
  1856 text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
  1857 
  1858 lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
  1859   by (simp add: matrix_vector_mult_def dot_def)
  1860 
  1861 lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) {1 .. dimindex(UNIV:: 'n set)}"
  1862   by (simp add: matrix_vector_mult_def Cart_eq setsum_component Cart_lambda_beta vector_component column_def mult_commute)
  1863 
  1864 lemma vector_componentwise:
  1865   "(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) {1..dimindex(UNIV :: 'n set)})"
  1866   apply (subst basis_expansion[symmetric])
  1867   by (vector Cart_eq Cart_lambda_beta setsum_component)
  1868 
  1869 lemma linear_componentwise:
  1870   fixes f:: "'a::ring_1 ^ 'm \<Rightarrow> 'a ^ 'n"
  1871   assumes lf: "linear f" and j: "j \<in> {1 .. dimindex (UNIV :: 'n set)}"
  1872   shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) {1 .. dimindex (UNIV :: 'm set)}" (is "?lhs = ?rhs")
  1873 proof-
  1874   let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
  1875   let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
  1876   have fM: "finite ?M" by simp
  1877   have "?rhs = (setsum (\<lambda>i.(x$i) *s f (basis i) ) ?M)$j"
  1878     unfolding vector_smult_component[OF j, symmetric]
  1879     unfolding setsum_component[OF j, of "(\<lambda>i.(x$i) *s f (basis i :: 'a^'m))" ?M]
  1880     ..
  1881   then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion ..
  1882 qed
  1883 
  1884 text{* Inverse matrices  (not necessarily square) *}
  1885 
  1886 definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
  1887 
  1888 definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
  1889         (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
  1890 
  1891 text{* Correspondence between matrices and linear operators. *}
  1892 
  1893 definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
  1894 where "matrix f = (\<chi> i j. (f(basis j))$i)"
  1895 
  1896 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::'a::comm_semiring_1 ^ 'n))"
  1897   by (simp add: linear_def matrix_vector_mult_def Cart_eq Cart_lambda_beta vector_component ring_simps setsum_right_distrib setsum_addf)
  1898 
  1899 lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n)"
  1900 apply (simp add: matrix_def matrix_vector_mult_def Cart_eq Cart_lambda_beta mult_commute del: One_nat_def)
  1901 apply clarify
  1902 apply (rule linear_componentwise[OF lf, symmetric])
  1903 apply simp
  1904 done
  1905 
  1906 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::'a::comm_ring_1 ^ 'n))" by (simp add: ext matrix_works)
  1907 
  1908 lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n)) = A"
  1909   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
  1910 
  1911 lemma matrix_compose: 
  1912   assumes lf: "linear (f::'a::comm_ring_1^'n \<Rightarrow> _)" and lg: "linear g" 
  1913   shows "matrix (g o f) = matrix g ** matrix f"
  1914   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
  1915   by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
  1916 
  1917 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)}"
  1918   by (simp add: matrix_vector_mult_def transp_def Cart_eq Cart_lambda_beta setsum_component vector_component mult_commute)
  1919 
  1920 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::'a::comm_ring_1^'n^'m) *v x) = (\<lambda>x. transp A *v x)"
  1921   apply (rule adjoint_unique[symmetric])
  1922   apply (rule matrix_vector_mul_linear)
  1923   apply (simp add: transp_def dot_def Cart_lambda_beta matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
  1924   apply (subst setsum_commute)
  1925   apply (auto simp add: mult_ac)
  1926   done
  1927 
  1928 lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n \<Rightarrow> 'a ^ 'm)"
  1929   shows "matrix(adjoint f) = transp(matrix f)"
  1930   apply (subst matrix_vector_mul[OF lf])
  1931   unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
  1932 
  1933 subsection{* Interlude: Some properties of real sets *}
  1934 
  1935 lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
  1936   shows "\<forall>n \<ge> m. d n < e m"
  1937   using prems apply auto
  1938   apply (erule_tac x="n" in allE)
  1939   apply (erule_tac x="n" in allE)
  1940   apply auto
  1941   done
  1942 
  1943 
  1944 lemma real_convex_bound_lt: 
  1945   assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"
  1946   and uv: "u + v = 1" 
  1947   shows "u * x + v * y < a"
  1948 proof-
  1949   have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
  1950   have "a = a * (u + v)" unfolding uv  by simp
  1951   hence th: "u * a + v * a = a" by (simp add: ring_simps)
  1952   from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_compare_simps)
  1953   from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_compare_simps)
  1954   from xa ya u v have "u * x + v * y < u * a + v * a"
  1955     apply (cases "u = 0", simp_all add: uv')
  1956     apply(rule mult_strict_left_mono)
  1957     using uv' apply simp_all
  1958     
  1959     apply (rule add_less_le_mono)
  1960     apply(rule mult_strict_left_mono)
  1961     apply simp_all
  1962     apply (rule mult_left_mono)
  1963     apply simp_all
  1964     done
  1965   thus ?thesis unfolding th .
  1966 qed
  1967 
  1968 lemma real_convex_bound_le: 
  1969   assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"
  1970   and uv: "u + v = 1" 
  1971   shows "u * x + v * y \<le> a"
  1972 proof-
  1973   from xa ya u v have "u * x + v * y \<le> u * a + v * a" by (simp add: add_mono mult_left_mono)
  1974   also have "\<dots> \<le> (u + v) * a" by (simp add: ring_simps)
  1975   finally show ?thesis unfolding uv by simp
  1976 qed
  1977 
  1978 lemma infinite_enumerate: assumes fS: "infinite S"
  1979   shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
  1980 unfolding subseq_def
  1981 using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
  1982 
  1983 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)"
  1984 apply auto
  1985 apply (rule_tac x="d/2" in exI)
  1986 apply auto
  1987 done
  1988 
  1989 
  1990 lemma triangle_lemma: 
  1991   assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
  1992   shows "x <= y + z"
  1993 proof-
  1994   have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y  by (simp add: zero_compare_simps)
  1995   with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square ring_simps)
  1996   from y z have yz: "y + z \<ge> 0" by arith
  1997   from power2_le_imp_le[OF th yz] show ?thesis .
  1998 qed
  1999 
  2000 
  2001 lemma lambda_skolem: "(\<forall>i \<in> {1 .. dimindex(UNIV :: 'n set)}. \<exists>x. P i x) \<longleftrightarrow>
  2002    (\<exists>x::'a ^ 'n. \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")
  2003 proof-
  2004   let ?S = "{1 .. dimindex(UNIV :: 'n set)}"
  2005   {assume H: "?rhs"
  2006     then have ?lhs by auto}
  2007   moreover
  2008   {assume H: "?lhs"
  2009     then obtain f where f:"\<forall>i\<in> ?S. P i (f i)" unfolding Ball_def choice_iff by metis
  2010     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
  2011     {fix i assume i: "i \<in> ?S"
  2012       with f i have "P i (f i)" by metis
  2013       then have "P i (?x$i)" using Cart_lambda_beta[of f, rule_format, OF i] by auto 
  2014     }
  2015     hence "\<forall>i \<in> ?S. P i (?x$i)" by metis
  2016     hence ?rhs by metis }
  2017   ultimately show ?thesis by metis
  2018 qed 
  2019 
  2020 (* Supremum and infimum of real sets *)
  2021 
  2022 
  2023 definition rsup:: "real set \<Rightarrow> real" where
  2024   "rsup S = (SOME a. isLub UNIV S a)"
  2025 
  2026 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)
  2027 
  2028 lemma rsup: assumes Se: "S \<noteq> {}" and b: "\<exists>b. S *<= b"
  2029   shows "isLub UNIV S (rsup S)"
  2030 using Se b
  2031 unfolding rsup_def
  2032 apply clarify
  2033 apply (rule someI_ex)
  2034 apply (rule reals_complete)
  2035 by (auto simp add: isUb_def setle_def)
  2036 
  2037 lemma rsup_le: assumes Se: "S \<noteq> {}" and Sb: "S *<= b" shows "rsup S \<le> b"
  2038 proof-
  2039   from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def)
  2040   from rsup[OF Se] Sb have "isLub UNIV S (rsup S)"  by blast 
  2041   then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def)
  2042 qed
  2043 
  2044 lemma rsup_finite_Max: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2045   shows "rsup S = Max S"
  2046 using fS Se
  2047 proof-
  2048   let ?m = "Max S"
  2049   from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis
  2050   with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def)
  2051   from Max_in[OF fS Se] lub have mrS: "?m \<le> rsup S" 
  2052     by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)
  2053   moreover 
  2054   have "rsup S \<le> ?m" using Sm lub
  2055     by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
  2056   ultimately  show ?thesis by arith 
  2057 qed
  2058 
  2059 lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2060   shows "rsup S \<in> S"
  2061   using rsup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis
  2062 
  2063 lemma rsup_finite_Ub: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2064   shows "isUb S S (rsup S)"
  2065   using rsup_finite_Max[OF fS Se] rsup_finite_in[OF fS Se] Max_ge[OF fS]
  2066   unfolding isUb_def setle_def by metis
  2067 
  2068 lemma rsup_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2069   shows "a \<le> rsup S \<longleftrightarrow> (\<exists> x \<in> S. a \<le> x)"
  2070 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2071 
  2072 lemma rsup_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2073   shows "a \<ge> rsup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)"
  2074 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2075 
  2076 lemma rsup_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2077   shows "a < rsup S \<longleftrightarrow> (\<exists> x \<in> S. a < x)"
  2078 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2079 
  2080 lemma rsup_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2081   shows "a > rsup S \<longleftrightarrow> (\<forall> x \<in> S. a > x)"
  2082 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
  2083 
  2084 lemma rsup_unique: assumes b: "S *<= b" and S: "\<forall>b' < b. \<exists>x \<in> S. b' < x"
  2085   shows "rsup S = b"
  2086 using b S  
  2087 unfolding setle_def rsup_alt
  2088 apply -
  2089 apply (rule some_equality)
  2090 apply (metis  linorder_not_le order_eq_iff[symmetric])+
  2091 done
  2092 
  2093 lemma rsup_le_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. T *<= b) \<Longrightarrow> rsup S \<le> rsup T"
  2094   apply (rule rsup_le)
  2095   apply simp
  2096   using rsup[of T] by (auto simp add: isLub_def leastP_def setge_def setle_def isUb_def)
  2097 
  2098 lemma isUb_def': "isUb R S = (\<lambda>x. S *<= x \<and> x \<in> R)"
  2099   apply (rule ext)
  2100   by (metis isUb_def)
  2101 
  2102 lemma UNIV_trivial: "UNIV x" using UNIV_I[of x] by (metis mem_def)
  2103 lemma rsup_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
  2104   shows "a \<le> rsup S \<and> rsup S \<le> b"
  2105 proof-
  2106   from rsup[OF Se] u have lub: "isLub UNIV S (rsup S)" by blast
  2107   hence b: "rsup S \<le> b" using u by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
  2108   from Se obtain y where y: "y \<in> S" by blast
  2109   from lub l have "a \<le> rsup S" apply (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
  2110     apply (erule ballE[where x=y])
  2111     apply (erule ballE[where x=y])
  2112     apply arith
  2113     using y apply auto
  2114     done
  2115   with b show ?thesis by blast
  2116 qed
  2117 
  2118 lemma rsup_abs_le: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rsup S\<bar> \<le> a"
  2119   unfolding abs_le_interval_iff  using rsup_bounds[of S "-a" a]
  2120   by (auto simp add: setge_def setle_def)
  2121 
  2122 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"
  2123 proof-
  2124   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
  2125   show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th 
  2126     by  (auto simp add: setge_def setle_def)
  2127 qed
  2128 
  2129 definition rinf:: "real set \<Rightarrow> real" where
  2130   "rinf S = (SOME a. isGlb UNIV S a)"
  2131 
  2132 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)
  2133 
  2134 lemma reals_complete_Glb: assumes Se: "\<exists>x. x \<in> S" and lb: "\<exists> y. isLb UNIV S y"
  2135   shows "\<exists>(t::real). isGlb UNIV S t"
  2136 proof-
  2137   let ?M = "uminus ` S"
  2138   from lb have th: "\<exists>y. isUb UNIV ?M y" apply (auto simp add: isUb_def isLb_def setle_def setge_def)
  2139     by (rule_tac x="-y" in exI, auto)
  2140   from Se have Me: "\<exists>x. x \<in> ?M" by blast
  2141   from reals_complete[OF Me th] obtain t where t: "isLub UNIV ?M t" by blast
  2142   have "isGlb UNIV S (- t)" using t
  2143     apply (auto simp add: isLub_def isGlb_def leastP_def greatestP_def setle_def setge_def isUb_def isLb_def)
  2144     apply (erule_tac x="-y" in allE)
  2145     apply auto
  2146     done
  2147   then show ?thesis by metis
  2148 qed
  2149 
  2150 lemma rinf: assumes Se: "S \<noteq> {}" and b: "\<exists>b. b <=* S"
  2151   shows "isGlb UNIV S (rinf S)"
  2152 using Se b
  2153 unfolding rinf_def
  2154 apply clarify
  2155 apply (rule someI_ex)
  2156 apply (rule reals_complete_Glb)
  2157 apply (auto simp add: isLb_def setle_def setge_def)
  2158 done
  2159 
  2160 lemma rinf_ge: assumes Se: "S \<noteq> {}" and Sb: "b <=* S" shows "rinf S \<ge> b"
  2161 proof-
  2162   from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def)
  2163   from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)"  by blast 
  2164   then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def)
  2165 qed
  2166 
  2167 lemma rinf_finite_Min: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2168   shows "rinf S = Min S"
  2169 using fS Se
  2170 proof-
  2171   let ?m = "Min S"
  2172   from Min_le[OF fS] have Sm: "\<forall> x\<in> S. x \<ge> ?m" by metis
  2173   with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def)
  2174   from Min_in[OF fS Se] glb have mrS: "?m \<ge> rinf S" 
  2175     by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def)
  2176   moreover 
  2177   have "rinf S \<ge> ?m" using Sm glb
  2178     by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def)
  2179   ultimately  show ?thesis by arith 
  2180 qed
  2181 
  2182 lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2183   shows "rinf S \<in> S"
  2184   using rinf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis
  2185 
  2186 lemma rinf_finite_Lb: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2187   shows "isLb S S (rinf S)"
  2188   using rinf_finite_Min[OF fS Se] rinf_finite_in[OF fS Se] Min_le[OF fS]
  2189   unfolding isLb_def setge_def by metis
  2190 
  2191 lemma rinf_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2192   shows "a \<le> rinf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)"
  2193 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2194 
  2195 lemma rinf_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2196   shows "a \<ge> rinf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)"
  2197 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2198 
  2199 lemma rinf_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2200   shows "a < rinf S \<longleftrightarrow> (\<forall> x \<in> S. a < x)"
  2201 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2202 
  2203 lemma rinf_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
  2204   shows "a > rinf S \<longleftrightarrow> (\<exists> x \<in> S. a > x)"
  2205 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
  2206 
  2207 lemma rinf_unique: assumes b: "b <=* S" and S: "\<forall>b' > b. \<exists>x \<in> S. b' > x"
  2208   shows "rinf S = b"
  2209 using b S  
  2210 unfolding setge_def rinf_alt
  2211 apply -
  2212 apply (rule some_equality)
  2213 apply (metis  linorder_not_le order_eq_iff[symmetric])+
  2214 done
  2215 
  2216 lemma rinf_ge_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. b <=* T) \<Longrightarrow> rinf S >= rinf T"
  2217   apply (rule rinf_ge)
  2218   apply simp
  2219   using rinf[of T] by (auto simp add: isGlb_def greatestP_def setge_def setle_def isLb_def)
  2220 
  2221 lemma isLb_def': "isLb R S = (\<lambda>x. x <=* S \<and> x \<in> R)"
  2222   apply (rule ext)
  2223   by (metis isLb_def)
  2224 
  2225 lemma rinf_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
  2226   shows "a \<le> rinf S \<and> rinf S \<le> b"
  2227 proof-
  2228   from rinf[OF Se] l have lub: "isGlb UNIV S (rinf S)" by blast
  2229   hence b: "a \<le> rinf S" using l by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
  2230   from Se obtain y where y: "y \<in> S" by blast
  2231   from lub u have "b \<ge> rinf S" apply (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
  2232     apply (erule ballE[where x=y])
  2233     apply (erule ballE[where x=y])
  2234     apply arith
  2235     using y apply auto
  2236     done
  2237   with b show ?thesis by blast
  2238 qed
  2239 
  2240 lemma rinf_abs_ge: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rinf S\<bar> \<le> a"
  2241   unfolding abs_le_interval_iff  using rinf_bounds[of S "-a" a]
  2242   by (auto simp add: setge_def setle_def)
  2243 
  2244 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"
  2245 proof-
  2246   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
  2247   show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th 
  2248     by  (auto simp add: setge_def setle_def)
  2249 qed
  2250 
  2251 
  2252 
  2253 subsection{* Operator norm. *}
  2254 
  2255 definition "onorm f = rsup {norm (f x)| x. norm x = 1}"
  2256 
  2257 lemma norm_bound_generalize:
  2258   fixes f:: "real ^'n \<Rightarrow> real^'m"
  2259   assumes lf: "linear f"
  2260   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")
  2261 proof-
  2262   {assume H: ?rhs
  2263     {fix x :: "real^'n" assume x: "norm x = 1"
  2264       from H[rule_format, of x] x have "norm (f x) \<le> b" by simp}
  2265     then have ?lhs by blast }
  2266 
  2267   moreover
  2268   {assume H: ?lhs
  2269     from H[rule_format, of "basis 1"] 
  2270     have bp: "b \<ge> 0" using norm_pos_le[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"]
  2271       by (auto simp add: norm_basis) 
  2272     {fix x :: "real ^'n"
  2273       {assume "x = 0"
  2274 	then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] norm_0 bp)}
  2275       moreover
  2276       {assume x0: "x \<noteq> 0"
  2277 	hence n0: "norm x \<noteq> 0" by (metis norm_eq_0)
  2278 	let ?c = "1/ norm x"
  2279 	have "norm (?c*s x) = 1" by (simp add: n0 norm_mul)
  2280 	with H have "norm (f(?c*s x)) \<le> b" by blast
  2281 	hence "?c * norm (f x) \<le> b" 
  2282 	  by (simp add: linear_cmul[OF lf] norm_mul)
  2283 	hence "norm (f x) \<le> b * norm x" 
  2284 	  using n0 norm_pos_le[of x] by (auto simp add: field_simps)}
  2285       ultimately have "norm (f x) \<le> b * norm x" by blast}
  2286     then have ?rhs by blast}
  2287   ultimately show ?thesis by blast
  2288 qed
  2289 
  2290 lemma onorm:
  2291   fixes f:: "real ^'n \<Rightarrow> real ^'m"
  2292   assumes lf: "linear f"
  2293   shows "norm (f x) <= onorm f * norm x"
  2294   and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
  2295 proof-
  2296   {
  2297     let ?S = "{norm (f x) |x. norm x = 1}"
  2298     have Se: "?S \<noteq> {}" using  norm_basis_1 by auto
  2299     from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b" 
  2300       unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
  2301     {from rsup[OF Se b, unfolded onorm_def[symmetric]]
  2302       show "norm (f x) <= onorm f * norm x" 
  2303 	apply - 
  2304 	apply (rule spec[where x = x])
  2305 	unfolding norm_bound_generalize[OF lf, symmetric]
  2306 	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
  2307     {
  2308       show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"  
  2309 	using rsup[OF Se b, unfolded onorm_def[symmetric]]
  2310 	unfolding norm_bound_generalize[OF lf, symmetric]
  2311 	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
  2312   }
  2313 qed
  2314 
  2315 lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" shows "0 <= onorm f"
  2316   using order_trans[OF norm_pos_le onorm(1)[OF lf, of "basis 1"], unfolded norm_basis_1] by simp
  2317 
  2318 lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" 
  2319   shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
  2320   using onorm[OF lf]
  2321   apply (auto simp add: norm_0 onorm_pos_le norm_le_0)
  2322   apply atomize
  2323   apply (erule allE[where x="0::real"])
  2324   using onorm_pos_le[OF lf]
  2325   apply arith
  2326   done
  2327 
  2328 lemma onorm_const: "onorm(\<lambda>x::real^'n. (y::real ^ 'm)) = norm y"
  2329 proof-
  2330   let ?f = "\<lambda>x::real^'n. (y::real ^ 'm)"
  2331   have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
  2332     by(auto intro: vector_choose_size set_ext)
  2333   show ?thesis
  2334     unfolding onorm_def th
  2335     apply (rule rsup_unique) by (simp_all  add: setle_def)
  2336 qed
  2337 
  2338 lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n \<Rightarrow> real ^'m)" 
  2339   shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"
  2340   unfolding onorm_eq_0[OF lf, symmetric]
  2341   using onorm_pos_le[OF lf] by arith
  2342 
  2343 lemma onorm_compose:
  2344   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and lg: "linear g"
  2345   shows "onorm (f o g) <= onorm f * onorm g"
  2346   apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
  2347   unfolding o_def
  2348   apply (subst mult_assoc)
  2349   apply (rule order_trans)
  2350   apply (rule onorm(1)[OF lf])
  2351   apply (rule mult_mono1)
  2352   apply (rule onorm(1)[OF lg])
  2353   apply (rule onorm_pos_le[OF lf])
  2354   done
  2355 
  2356 lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
  2357   shows "onorm (\<lambda>x. - f x) \<le> onorm f"
  2358   using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
  2359   unfolding norm_neg by metis
  2360 
  2361 lemma onorm_neg: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
  2362   shows "onorm (\<lambda>x. - f x) = onorm f"
  2363   using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]]
  2364   by simp
  2365 
  2366 lemma onorm_triangle:
  2367   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and lg: "linear g"
  2368   shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
  2369   apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
  2370   apply (rule order_trans)
  2371   apply (rule norm_triangle)
  2372   apply (simp add: distrib)
  2373   apply (rule add_mono)
  2374   apply (rule onorm(1)[OF lf])
  2375   apply (rule onorm(1)[OF lg])
  2376   done
  2377 
  2378 lemma onorm_triangle_le: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
  2379   \<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"
  2380   apply (rule order_trans)
  2381   apply (rule onorm_triangle)
  2382   apply assumption+
  2383   done
  2384 
  2385 lemma onorm_triangle_lt: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
  2386   ==> onorm(\<lambda>x. f x + g x) < e"
  2387   apply (rule order_le_less_trans)
  2388   apply (rule onorm_triangle)
  2389   by assumption+
  2390 
  2391 (* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)
  2392 
  2393 definition vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x = (\<chi> i. x)"
  2394 
  2395 definition dest_vec1:: "'a ^1 \<Rightarrow> 'a" 
  2396   where "dest_vec1 x = (x$1)"
  2397 
  2398 lemma vec1_component[simp]: "(vec1 x)$1 = x"
  2399   by (simp add: vec1_def)
  2400 
  2401 lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
  2402   by (simp_all add: vec1_def dest_vec1_def Cart_eq Cart_lambda_beta dimindex_def del: One_nat_def)
  2403 
  2404 lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))" by (metis vec1_dest_vec1)
  2405 
  2406 lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))" by (metis vec1_dest_vec1) 
  2407 
  2408 lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))"  by (metis vec1_dest_vec1)
  2409 
  2410 lemma exists_dest_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(dest_vec1 x))"by (metis vec1_dest_vec1)
  2411 
  2412 lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
  2413 
  2414 lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
  2415 
  2416 lemma vec1_in_image_vec1: "vec1 x \<in> (vec1 ` S) \<longleftrightarrow> x \<in> S" by auto
  2417 
  2418 lemma vec1_vec: "vec1 x = vec x" by (vector vec1_def)
  2419 
  2420 lemma vec1_add: "vec1(x + y) = vec1 x + vec1 y" by (vector vec1_def)
  2421 lemma vec1_sub: "vec1(x - y) = vec1 x - vec1 y" by (vector vec1_def)
  2422 lemma vec1_cmul: "vec1(c* x) = c *s vec1 x " by (vector vec1_def)
  2423 lemma vec1_neg: "vec1(- x) = - vec1 x " by (vector vec1_def)
  2424 
  2425 lemma vec1_setsum: assumes fS: "finite S"
  2426   shows "vec1(setsum f S) = setsum (vec1 o f) S"
  2427   apply (induct rule: finite_induct[OF fS])
  2428   apply (simp add: vec1_vec)
  2429   apply (auto simp add: vec1_add)
  2430   done
  2431 
  2432 lemma dest_vec1_lambda: "dest_vec1(\<chi> i. x i) = x 1"
  2433   by (simp add: dest_vec1_def)
  2434 
  2435 lemma dest_vec1_vec: "dest_vec1(vec x) = x"
  2436   by (simp add: vec1_vec[symmetric])
  2437 
  2438 lemma dest_vec1_add: "dest_vec1(x + y) = dest_vec1 x + dest_vec1 y"
  2439  by (metis vec1_dest_vec1 vec1_add)
  2440 
  2441 lemma dest_vec1_sub: "dest_vec1(x - y) = dest_vec1 x - dest_vec1 y"
  2442  by (metis vec1_dest_vec1 vec1_sub)
  2443 
  2444 lemma dest_vec1_cmul: "dest_vec1(c*sx) = c * dest_vec1 x"
  2445  by (metis vec1_dest_vec1 vec1_cmul)
  2446 
  2447 lemma dest_vec1_neg: "dest_vec1(- x) = - dest_vec1 x"
  2448  by (metis vec1_dest_vec1 vec1_neg)
  2449 
  2450 lemma dest_vec1_0[simp]: "dest_vec1 0 = 0" by (metis vec_0 dest_vec1_vec)
  2451 
  2452 lemma dest_vec1_sum: assumes fS: "finite S"
  2453   shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"
  2454   apply (induct rule: finite_induct[OF fS])
  2455   apply (simp add: dest_vec1_vec)
  2456   apply (auto simp add: dest_vec1_add)
  2457   done
  2458 
  2459 lemma norm_vec1: "norm(vec1 x) = abs(x)"
  2460   by (simp add: vec1_def norm_real)
  2461 
  2462 lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)"
  2463   by (simp only: dist_real vec1_component)
  2464 lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
  2465   by (metis vec1_dest_vec1 norm_vec1)
  2466 
  2467 lemma linear_vmul_dest_vec1: 
  2468   fixes f:: "'a::semiring_1^'n \<Rightarrow> 'a^1"
  2469   shows "linear f \<Longrightarrow> linear (\<lambda>x. dest_vec1(f x) *s v)"
  2470   unfolding dest_vec1_def
  2471   apply (rule linear_vmul_component)
  2472   by (auto simp add: dimindex_def)
  2473 
  2474 lemma linear_from_scalars:
  2475   assumes lf: "linear (f::'a::comm_ring_1 ^1 \<Rightarrow> 'a^'n)"
  2476   shows "f = (\<lambda>x. dest_vec1 x *s column 1 (matrix f))"
  2477   apply (rule ext)
  2478   apply (subst matrix_works[OF lf, symmetric])
  2479   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 )
  2480   done
  2481 
  2482 lemma linear_to_scalars: assumes lf: "linear (f::'a::comm_ring_1 ^'n \<Rightarrow> 'a^1)"
  2483   shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
  2484   apply (rule ext)
  2485   apply (subst matrix_works[OF lf, symmetric])
  2486   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)
  2487   done
  2488 
  2489 lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
  2490   by (simp add: dest_vec1_eq[symmetric])
  2491 
  2492 lemma setsum_scalars: assumes fS: "finite S"
  2493   shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)"
  2494   unfolding vec1_setsum[OF fS] by simp
  2495 
  2496 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"
  2497   apply (cases "dest_vec1 x \<le> dest_vec1 y")
  2498   apply simp
  2499   apply (subgoal_tac "dest_vec1 y \<le> dest_vec1 x")
  2500   apply (auto)
  2501   done
  2502 
  2503 text{* Pasting vectors. *}
  2504 
  2505 lemma linear_fstcart: "linear fstcart"
  2506   by (auto simp add: linear_def fstcart_def Cart_eq Cart_lambda_beta vector_component dimindex_finite_sum)
  2507 
  2508 lemma linear_sndcart: "linear sndcart"
  2509   by (auto simp add: linear_def sndcart_def Cart_eq Cart_lambda_beta vector_component dimindex_finite_sum)
  2510 
  2511 lemma fstcart_vec[simp]: "fstcart(vec x) = vec x"
  2512   by (vector fstcart_def vec_def dimindex_finite_sum)
  2513 
  2514 lemma fstcart_add[simp]:"fstcart(x + y) = fstcart (x::'a::{plus,times}^('b,'c) finite_sum) + fstcart y"
  2515   using linear_fstcart[unfolded linear_def] by blast
  2516 
  2517 lemma fstcart_cmul[simp]:"fstcart(c*s x) = c*s fstcart (x::'a::{plus,times}^('b,'c) finite_sum)"
  2518   using linear_fstcart[unfolded linear_def] by blast
  2519 
  2520 lemma fstcart_neg[simp]:"fstcart(- x) = - fstcart (x::'a::ring_1^('b,'c) finite_sum)"
  2521 unfolding vector_sneg_minus1 fstcart_cmul ..
  2522 
  2523 lemma fstcart_sub[simp]:"fstcart(x - y) = fstcart (x::'a::ring_1^('b,'c) finite_sum) - fstcart y"
  2524   unfolding diff_def fstcart_add fstcart_neg  ..
  2525 
  2526 lemma fstcart_setsum:
  2527   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
  2528   assumes fS: "finite S"
  2529   shows "fstcart (setsum f S) = setsum (\<lambda>i. fstcart (f i)) S"
  2530   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
  2531 
  2532 lemma sndcart_vec[simp]: "sndcart(vec x) = vec x"
  2533   by (vector sndcart_def vec_def dimindex_finite_sum)
  2534 
  2535 lemma sndcart_add[simp]:"sndcart(x + y) = sndcart (x::'a::{plus,times}^('b,'c) finite_sum) + sndcart y"
  2536   using linear_sndcart[unfolded linear_def] by blast
  2537 
  2538 lemma sndcart_cmul[simp]:"sndcart(c*s x) = c*s sndcart (x::'a::{plus,times}^('b,'c) finite_sum)"
  2539   using linear_sndcart[unfolded linear_def] by blast
  2540 
  2541 lemma sndcart_neg[simp]:"sndcart(- x) = - sndcart (x::'a::ring_1^('b,'c) finite_sum)"
  2542 unfolding vector_sneg_minus1 sndcart_cmul ..
  2543 
  2544 lemma sndcart_sub[simp]:"sndcart(x - y) = sndcart (x::'a::ring_1^('b,'c) finite_sum) - sndcart y"
  2545   unfolding diff_def sndcart_add sndcart_neg  ..
  2546 
  2547 lemma sndcart_setsum:
  2548   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
  2549   assumes fS: "finite S"
  2550   shows "sndcart (setsum f S) = setsum (\<lambda>i. sndcart (f i)) S"
  2551   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
  2552 
  2553 lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x"
  2554   by (simp add: pastecart_eq fstcart_vec sndcart_vec fstcart_pastecart sndcart_pastecart)
  2555 
  2556 lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)"
  2557   by (simp add: pastecart_eq fstcart_add sndcart_add fstcart_pastecart sndcart_pastecart)
  2558 
  2559 lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1"
  2560   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
  2561 
  2562 lemma pastecart_neg[simp]: "pastecart (- (x::'a::ring_1^_)) (- y) = - pastecart x y"
  2563   unfolding vector_sneg_minus1 pastecart_cmul ..
  2564 
  2565 lemma pastecart_sub: "pastecart (x1::'a::ring_1^_) y1 - pastecart x2 y2 = pastecart (x1 - x2) (y1 - y2)"
  2566   by (simp add: diff_def pastecart_neg[symmetric] del: pastecart_neg)
  2567 
  2568 lemma pastecart_setsum:
  2569   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
  2570   assumes fS: "finite S"
  2571   shows "pastecart (setsum f S) (setsum g S) = setsum (\<lambda>i. pastecart (f i) (g i)) S"
  2572   by (simp  add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS] fstcart_pastecart sndcart_pastecart)
  2573 
  2574 lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n,'m) finite_sum)"
  2575 proof-
  2576   let ?n = "dimindex (UNIV :: 'n set)"
  2577   let ?m = "dimindex (UNIV :: 'm set)"
  2578   let ?N = "{1 .. ?n}"
  2579   let ?M = "{1 .. ?m}"
  2580   let ?NM = "{1 .. dimindex (UNIV :: ('n,'m) finite_sum set)}"
  2581   have th_0: "1 \<le> ?n +1" by simp
  2582   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
  2583     by (simp add: pastecart_fst_snd)
  2584   have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)" 
  2585     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)
  2586   then show ?thesis
  2587     unfolding th0 
  2588     unfolding real_vector_norm_def real_sqrt_le_iff real_of_real_def id_def
  2589     by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta)
  2590 qed
  2591 
  2592 lemma dist_fstcart: "dist(fstcart (x::real^_)) (fstcart y) <= dist x y"
  2593   by (metis dist_def fstcart_sub[symmetric] norm_fstcart)
  2594 
  2595 lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n,'m) finite_sum)"
  2596 proof-
  2597   let ?n = "dimindex (UNIV :: 'n set)"
  2598   let ?m = "dimindex (UNIV :: 'm set)"
  2599   let ?N = "{1 .. ?n}"
  2600   let ?M = "{1 .. ?m}"
  2601   let ?nm = "dimindex (UNIV :: ('n,'m) finite_sum set)"
  2602   let ?NM = "{1 .. ?nm}"
  2603   have thnm[simp]: "?nm = ?n + ?m" by (simp add: dimindex_finite_sum)
  2604   have th_0: "1 \<le> ?n +1" by simp
  2605   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
  2606     by (simp add: pastecart_fst_snd)
  2607   let ?f = "\<lambda>n. n - ?n"
  2608   let ?S = "{?n+1 .. ?nm}"
  2609   have finj:"inj_on ?f ?S"
  2610     using dimindex_nonzero[of "UNIV :: 'n set"] dimindex_nonzero[of "UNIV :: 'm set"]
  2611     apply (simp add: Ball_def atLeastAtMost_iff inj_on_def dimindex_finite_sum del: One_nat_def)
  2612     by arith
  2613   have fS: "?f ` ?S = ?M" 
  2614     apply (rule set_ext)
  2615     apply (simp add: image_iff Bex_def) using dimindex_nonzero[of "UNIV :: 'n set"] dimindex_nonzero[of "UNIV :: 'm set"] by arith
  2616   have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)" 
  2617     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)    
  2618   then show ?thesis
  2619     unfolding th0 
  2620     unfolding real_vector_norm_def real_sqrt_le_iff real_of_real_def id_def
  2621     by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta)
  2622 qed
  2623 
  2624 lemma dist_sndcart: "dist(sndcart (x::real^_)) (sndcart y) <= dist x y"
  2625   by (metis dist_def sndcart_sub[symmetric] norm_sndcart)
  2626 
  2627 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"
  2628 proof-
  2629   let ?n = "dimindex (UNIV :: 'n set)"
  2630   let ?m = "dimindex (UNIV :: 'm set)"
  2631   let ?N = "{1 .. ?n}"
  2632   let ?M = "{1 .. ?m}"
  2633   let ?nm = "dimindex (UNIV :: ('n,'m) finite_sum set)"
  2634   let ?NM = "{1 .. ?nm}"
  2635   have thnm: "?nm = ?n + ?m" by (simp add: dimindex_finite_sum)
  2636   have th_0: "1 \<le> ?n +1" by simp
  2637   have th_1: "\<And>i. i \<in> {?m + 1 .. ?nm} \<Longrightarrow> i - ?m \<in> ?N" apply (simp add: thnm) by arith
  2638   let ?f = "\<lambda>a b i. (a$i) * (b$i)"
  2639   let ?g = "?f (pastecart x1 x2) (pastecart y1 y2)"
  2640   let ?S = "{?n +1 .. ?nm}"
  2641   {fix i
  2642     assume i: "i \<in> ?N"
  2643     have "?g i = ?f x1 y1 i"
  2644       using i
  2645       apply (simp add: pastecart_def Cart_lambda_beta thnm) done
  2646   }
  2647   hence th2: "setsum ?g ?N = setsum (?f x1 y1) ?N"
  2648     apply -
  2649     apply (rule setsum_cong)
  2650     apply auto
  2651     done
  2652   {fix i
  2653     assume i: "i \<in> ?S"
  2654     have "?g i = ?f x2 y2 (i - ?n)"
  2655       using i
  2656       apply (simp add: pastecart_def Cart_lambda_beta thnm) done
  2657   }
  2658   hence th3: "setsum ?g ?S = setsum (\<lambda>i. ?f x2 y2 (i -?n)) ?S"
  2659     apply -
  2660     apply (rule setsum_cong)
  2661     apply auto
  2662     done
  2663   let ?r = "\<lambda>n. n - ?n"
  2664   have rinj: "inj_on ?r ?S" apply (simp add: inj_on_def Ball_def thnm) by arith
  2665   have rS: "?r ` ?S = ?M" apply (rule set_ext) 
  2666     apply (simp add: thnm image_iff Bex_def) by arith
  2667   have "pastecart x1 x2 \<bullet> (pastecart y1 y2) = setsum ?g ?NM" by (simp add: dot_def)
  2668   also have "\<dots> = setsum ?g ?N + setsum ?g ?S"
  2669     by (simp add: dot_def thnm setsum_add_split[OF th_0, of _ ?m] del: One_nat_def)
  2670   also have "\<dots> = setsum (?f x1 y1) ?N + setsum (?f x2 y2) ?M"
  2671     unfolding setsum_reindex[OF rinj, unfolded rS o_def] th2 th3 ..
  2672   finally 
  2673   show ?thesis by (simp add: dot_def)
  2674 qed
  2675 
  2676 lemma norm_pastecart: "norm(pastecart x y) <= norm(x :: real ^ _) + norm(y)"
  2677   unfolding real_vector_norm_def dot_pastecart real_sqrt_le_iff real_of_real_def id_def
  2678   apply (rule power2_le_imp_le)
  2679   apply (simp add: real_sqrt_pow2[OF add_nonneg_nonneg[OF dot_pos_le[of x] dot_pos_le[of y]]])
  2680   apply (auto simp add: power2_eq_square ring_simps)
  2681   apply (simp add: power2_eq_square[symmetric])
  2682   apply (rule mult_nonneg_nonneg)
  2683   apply (simp_all add: real_sqrt_pow2[OF dot_pos_le])
  2684   apply (rule add_nonneg_nonneg)
  2685   apply (simp_all add: real_sqrt_pow2[OF dot_pos_le])
  2686   done
  2687 
  2688 subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
  2689 
  2690 definition hull :: "'a set set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) where
  2691   "S hull s = Inter {t. t \<in> S \<and> s \<subseteq> t}"
  2692 
  2693 lemma hull_same: "s \<in> S \<Longrightarrow> S hull s = s"
  2694   unfolding hull_def by auto
  2695 
  2696 lemma hull_in: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) \<in> S"
  2697 unfolding hull_def subset_iff by auto
  2698 
  2699 lemma hull_eq: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) = s \<longleftrightarrow> s \<in> S"
  2700 using hull_same[of s S] hull_in[of S s] by metis  
  2701 
  2702 
  2703 lemma hull_hull: "S hull (S hull s) = S hull s"
  2704   unfolding hull_def by blast
  2705 
  2706 lemma hull_subset: "s \<subseteq> (S hull s)"
  2707   unfolding hull_def by blast
  2708 
  2709 lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
  2710   unfolding hull_def by blast
  2711 
  2712 lemma hull_antimono: "S \<subseteq> T ==> (T hull s) \<subseteq> (S hull s)"
  2713   unfolding hull_def by blast
  2714 
  2715 lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> t \<in> S ==> (S hull s) \<subseteq> t"
  2716   unfolding hull_def by blast
  2717 
  2718 lemma subset_hull: "t \<in> S ==> S hull s \<subseteq> t \<longleftrightarrow>  s \<subseteq> t"
  2719   unfolding hull_def by blast
  2720 
  2721 lemma hull_unique: "s \<subseteq> t \<Longrightarrow> t \<in> S \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> t' \<in> S ==> t \<subseteq> t')
  2722            ==> (S hull s = t)"
  2723 unfolding hull_def by auto
  2724 
  2725 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"
  2726   using hull_minimal[of S "{x. P x}" Q]
  2727   by (auto simp add: subset_eq Collect_def mem_def)
  2728 
  2729 lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" by (metis hull_subset subset_eq)
  2730 
  2731 lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
  2732 unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
  2733 
  2734 lemma hull_union: assumes T: "\<And>T. T \<subseteq> S ==> Inter T \<in> S"
  2735   shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
  2736 apply rule
  2737 apply (rule hull_mono)
  2738 unfolding Un_subset_iff
  2739 apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
  2740 apply (rule hull_minimal)
  2741 apply (metis hull_union_subset)
  2742 apply (metis hull_in T)
  2743 done
  2744 
  2745 lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
  2746   unfolding hull_def by blast
  2747 
  2748 lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
  2749 by (metis hull_redundant_eq)
  2750 
  2751 text{* Archimedian properties and useful consequences. *}
  2752 
  2753 lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
  2754   using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto)
  2755 lemmas real_arch_lt = reals_Archimedean2
  2756 
  2757 lemmas real_arch = reals_Archimedean3
  2758 
  2759 lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
  2760   using reals_Archimedean
  2761   apply (auto simp add: field_simps inverse_positive_iff_positive)
  2762   apply (subgoal_tac "inverse (real n) > 0")
  2763   apply arith
  2764   apply simp
  2765   done
  2766 
  2767 lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
  2768 proof(induct n)
  2769   case 0 thus ?case by simp
  2770 next 
  2771   case (Suc n)
  2772   hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
  2773   from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
  2774   from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
  2775   also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric]) 
  2776     apply (simp add: ring_simps)
  2777     using mult_left_mono[OF p Suc.prems] by simp
  2778   finally show ?case  by (simp add: real_of_nat_Suc ring_simps)
  2779 qed
  2780 
  2781 lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
  2782 proof-
  2783   from x have x0: "x - 1 > 0" by arith
  2784   from real_arch[OF x0, rule_format, of y] 
  2785   obtain n::nat where n:"y < real n * (x - 1)" by metis
  2786   from x0 have x00: "x- 1 \<ge> 0" by arith
  2787   from real_pow_lbound[OF x00, of n] n 
  2788   have "y < x^n" by auto
  2789   then show ?thesis by metis
  2790 qed 
  2791 
  2792 lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
  2793   using real_arch_pow[of 2 x] by simp
  2794 
  2795 lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
  2796   shows "\<exists>n. x^n < y"
  2797 proof-
  2798   {assume x0: "x > 0" 
  2799     from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
  2800     from real_arch_pow[OF ix, of "1/y"]
  2801     obtain n where n: "1/y < (1/x)^n" by blast
  2802     then 
  2803     have ?thesis using y x0 by (auto simp add: field_simps power_divide) }
  2804   moreover 
  2805   {assume "\<not> x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)}
  2806   ultimately show ?thesis by metis
  2807 qed
  2808 
  2809 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)"
  2810   by (metis real_arch_inv)
  2811 
  2812 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"
  2813   apply (rule forall_pos_mono)
  2814   apply auto
  2815   apply (atomize)
  2816   apply (erule_tac x="n - 1" in allE)
  2817   apply auto
  2818   done
  2819 
  2820 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"
  2821   shows "x = 0"
  2822 proof-
  2823   {assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
  2824     from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x"  by blast
  2825     with xc[rule_format, of n] have "n = 0" by arith
  2826     with n c have False by simp}
  2827   then show ?thesis by blast
  2828 qed
  2829 
  2830 (* ------------------------------------------------------------------------- *)
  2831 (* Relate max and min to sup and inf.                                        *)
  2832 (* ------------------------------------------------------------------------- *)
  2833 
  2834 lemma real_max_rsup: "max x y = rsup {x,y}"
  2835 proof-
  2836   have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
  2837   from rsup_finite_le_iff[OF f, of "max x y"] have "rsup {x,y} \<le> max x y" by simp
  2838   moreover
  2839   have "max x y \<le> rsup {x,y}" using rsup_finite_ge_iff[OF f, of "max x y"]
  2840     by (simp add: linorder_linear)
  2841   ultimately show ?thesis by arith
  2842 qed 
  2843 
  2844 lemma real_min_rinf: "min x y = rinf {x,y}"
  2845 proof-
  2846   have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
  2847   from rinf_finite_le_iff[OF f, of "min x y"] have "rinf {x,y} \<le> min x y" 
  2848     by (simp add: linorder_linear)
  2849   moreover
  2850   have "min x y \<le> rinf {x,y}" using rinf_finite_ge_iff[OF f, of "min x y"]
  2851     by simp
  2852   ultimately show ?thesis by arith
  2853 qed 
  2854 
  2855 (* ------------------------------------------------------------------------- *)
  2856 (* Geometric progression.                                                    *)
  2857 (* ------------------------------------------------------------------------- *)
  2858 
  2859 lemma sum_gp_basic: "((1::'a::{field, recpower}) - x) * setsum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
  2860   (is "?lhs = ?rhs")
  2861 proof-
  2862   {assume x1: "x = 1" hence ?thesis by simp}
  2863   moreover
  2864   {assume x1: "x\<noteq>1"
  2865     hence x1': "x - 1 \<noteq> 0" "1 - x \<noteq> 0" "x - 1 = - (1 - x)" "- (1 - x) \<noteq> 0" by auto
  2866     from geometric_sum[OF x1, of "Suc n", unfolded x1']
  2867     have "(- (1 - x)) * setsum (\<lambda>i. x^i) {0 .. n} = - (1 - x^(Suc n))"
  2868       unfolding atLeastLessThanSuc_atLeastAtMost
  2869       using x1' apply (auto simp only: field_simps)
  2870       apply (simp add: ring_simps)
  2871       done
  2872     then have ?thesis by (simp add: ring_simps) }
  2873   ultimately show ?thesis by metis
  2874 qed
  2875 
  2876 lemma sum_gp_multiplied: assumes mn: "m <= n"
  2877   shows "((1::'a::{field, recpower}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
  2878   (is "?lhs = ?rhs")
  2879 proof-
  2880   let ?S = "{0..(n - m)}"
  2881   from mn have mn': "n - m \<ge> 0" by arith
  2882   let ?f = "op + m"
  2883   have i: "inj_on ?f ?S" unfolding inj_on_def by auto
  2884   have f: "?f ` ?S = {m..n}" 
  2885     using mn apply (auto simp add: image_iff Bex_def) by arith
  2886   have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)" 
  2887     by (rule ext, simp add: power_add power_mult)
  2888   from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
  2889   have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp
  2890   then show ?thesis unfolding sum_gp_basic using mn
  2891     by (simp add: ring_simps power_add[symmetric])
  2892 qed
  2893 
  2894 lemma sum_gp: "setsum (op ^ (x::'a::{field, recpower})) {m .. n} = 
  2895    (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m) 
  2896                     else (x^ m - x^ (Suc n)) / (1 - x))"
  2897 proof-
  2898   {assume nm: "n < m" hence ?thesis by simp}
  2899   moreover
  2900   {assume "\<not> n < m" hence nm: "m \<le> n" by arith
  2901     {assume x: "x = 1"  hence ?thesis by simp}
  2902     moreover
  2903     {assume x: "x \<noteq> 1" hence nz: "1 - x \<noteq> 0" by simp
  2904       from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)}
  2905     ultimately have ?thesis by metis
  2906   }
  2907   ultimately show ?thesis by metis
  2908 qed
  2909 
  2910 lemma sum_gp_offset: "setsum (op ^ (x::'a::{field,recpower})) {m .. m+n} = 
  2911   (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
  2912   unfolding sum_gp[of x m "m + n"] power_Suc
  2913   by (simp add: ring_simps power_add)
  2914 
  2915 
  2916 subsection{* A bit of linear algebra. *}
  2917 
  2918 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 )"
  2919 definition "span S = (subspace hull S)"
  2920 definition "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
  2921 abbreviation "independent s == ~(dependent s)"
  2922 
  2923 (* Closure properties of subspaces.                                          *)
  2924 
  2925 lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
  2926 
  2927 lemma subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
  2928 
  2929 lemma subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S" 
  2930   by (metis subspace_def)
  2931 
  2932 lemma subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *s x \<in> S"
  2933   by (metis subspace_def)
  2934 
  2935 lemma subspace_neg: "subspace S \<Longrightarrow> (x::'a::ring_1^'n) \<in> S \<Longrightarrow> - x \<in> S"
  2936   by (metis vector_sneg_minus1 subspace_mul)
  2937 
  2938 lemma subspace_sub: "subspace S \<Longrightarrow> (x::'a::ring_1^'n) \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
  2939   by (metis diff_def subspace_add subspace_neg)
  2940 
  2941 lemma subspace_setsum:
  2942   assumes sA: "subspace A" and fB: "finite B"
  2943   and f: "\<forall>x\<in> B. f x \<in> A"
  2944   shows "setsum f B \<in> A"
  2945   using  fB f sA
  2946   apply(induct rule: finite_induct[OF fB])
  2947   by (simp add: subspace_def sA, auto simp add: sA subspace_add) 
  2948 
  2949 lemma subspace_linear_image: 
  2950   assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and sS: "subspace S" 
  2951   shows "subspace(f ` S)"
  2952   using lf sS linear_0[OF lf]
  2953   unfolding linear_def subspace_def
  2954   apply (auto simp add: image_iff)
  2955   apply (rule_tac x="x + y" in bexI, auto)
  2956   apply (rule_tac x="c*s x" in bexI, auto)
  2957   done
  2958 
  2959 lemma subspace_linear_preimage: "linear (f::'a::semiring_1^'n \<Rightarrow> _) ==> subspace S ==> subspace {x. f x \<in> S}"
  2960   by (auto simp add: subspace_def linear_def linear_0[of f])
  2961 
  2962 lemma subspace_trivial: "subspace {0::'a::semiring_1 ^_}"
  2963   by (simp add: subspace_def)
  2964 
  2965 lemma subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
  2966   by (simp add: subspace_def)
  2967 
  2968 
  2969 lemma span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
  2970   by (metis span_def hull_mono)
  2971 
  2972 lemma subspace_span: "subspace(span S)"
  2973   unfolding span_def
  2974   apply (rule hull_in[unfolded mem_def])
  2975   apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
  2976   apply auto
  2977   apply (erule_tac x="X" in ballE)
  2978   apply (simp add: mem_def)
  2979   apply blast
  2980   apply (erule_tac x="X" in ballE)
  2981   apply (erule_tac x="X" in ballE)
  2982   apply (erule_tac x="X" in ballE)
  2983   apply (clarsimp simp add: mem_def)
  2984   apply simp
  2985   apply simp
  2986   apply simp
  2987   apply (erule_tac x="X" in ballE)
  2988   apply (erule_tac x="X" in ballE)
  2989   apply (simp add: mem_def)
  2990   apply simp
  2991   apply simp
  2992   done
  2993 
  2994 lemma span_clauses:
  2995   "a \<in> S ==> a \<in> span S"
  2996   "0 \<in> span S"
  2997   "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
  2998   "x \<in> span S \<Longrightarrow> c *s x \<in> span S"
  2999   by (metis span_def hull_subset subset_eq subspace_span subspace_def)+
  3000 
  3001 lemma span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
  3002   and P: "subspace P" and x: "x \<in> span S" shows "P x"
  3003 proof-
  3004   from SP have SP': "S \<subseteq> P" by (simp add: mem_def subset_eq)
  3005   from P have P': "P \<in> subspace" by (simp add: mem_def)
  3006   from x hull_minimal[OF SP' P', unfolded span_def[symmetric]]
  3007   show "P x" by (metis mem_def subset_eq) 
  3008 qed
  3009 
  3010 lemma span_empty: "span {} = {(0::'a::semiring_0 ^ 'n)}"
  3011   apply (simp add: span_def)
  3012   apply (rule hull_unique)
  3013   apply (auto simp add: mem_def subspace_def)
  3014   unfolding mem_def[of "0::'a^'n", symmetric]
  3015   apply simp
  3016   done
  3017 
  3018 lemma independent_empty: "independent {}"
  3019   by (simp add: dependent_def)
  3020 
  3021 lemma independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
  3022   apply (clarsimp simp add: dependent_def span_mono)
  3023   apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
  3024   apply force
  3025   apply (rule span_mono)
  3026   apply auto
  3027   done
  3028 
  3029 lemma span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
  3030   by (metis order_antisym span_def hull_minimal mem_def)
  3031 
  3032 lemma span_induct': assumes SP: "\<forall>x \<in> S. P x"
  3033   and P: "subspace P" shows "\<forall>x \<in> span S. P x"
  3034   using span_induct SP P by blast
  3035 
  3036 inductive span_induct_alt_help for S:: "'a::semiring_1^'n \<Rightarrow> bool"
  3037   where 
  3038   span_induct_alt_help_0: "span_induct_alt_help S 0"
  3039   | 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)"
  3040 
  3041 lemma span_induct_alt': 
  3042   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"
  3043 proof-
  3044   {fix x:: "'a^'n" assume x: "span_induct_alt_help S x"
  3045     have "h x"
  3046       apply (rule span_induct_alt_help.induct[OF x])
  3047       apply (rule h0)
  3048       apply (rule hS, assumption, assumption)
  3049       done}
  3050   note th0 = this
  3051   {fix x assume x: "x \<in> span S"
  3052     
  3053     have "span_induct_alt_help S x"
  3054       proof(rule span_induct[where x=x and S=S])
  3055 	show "x \<in> span S" using x .
  3056       next
  3057 	fix x assume xS : "x \<in> S"
  3058 	  from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
  3059 	  show "span_induct_alt_help S x" by simp
  3060 	next
  3061 	have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0)
  3062 	moreover
  3063 	{fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y"
  3064 	  from h 
  3065 	  have "span_induct_alt_help S (x + y)"
  3066 	    apply (induct rule: span_induct_alt_help.induct)
  3067 	    apply simp
  3068 	    unfolding add_assoc
  3069 	    apply (rule span_induct_alt_help_S)
  3070 	    apply assumption
  3071 	    apply simp
  3072 	    done}
  3073 	moreover
  3074 	{fix c x assume xt: "span_induct_alt_help S x"
  3075 	  then have "span_induct_alt_help S (c*s x)" 
  3076 	    apply (induct rule: span_induct_alt_help.induct)
  3077 	    apply (simp add: span_induct_alt_help_0)
  3078 	    apply (simp add: vector_smult_assoc vector_add_ldistrib)
  3079 	    apply (rule span_induct_alt_help_S)
  3080 	    apply assumption
  3081 	    apply simp
  3082 	    done
  3083 	}
  3084 	ultimately show "subspace (span_induct_alt_help S)" 
  3085 	  unfolding subspace_def mem_def Ball_def by blast
  3086       qed}
  3087   with th0 show ?thesis by blast
  3088 qed 
  3089 
  3090 lemma span_induct_alt: 
  3091   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"
  3092   shows "h x"
  3093 using span_induct_alt'[of h S] h0 hS x by blast
  3094 
  3095 (* Individual closure properties. *)
  3096 
  3097 lemma span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses)
  3098 
  3099 lemma span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
  3100 
  3101 lemma span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
  3102   by (metis subspace_add subspace_span)
  3103 
  3104 lemma span_mul: "x \<in> span S ==> (c *s x) \<in> span S"
  3105   by (metis subspace_span subspace_mul)
  3106 
  3107 lemma span_neg: "x \<in> span S ==> - (x::'a::ring_1^'n) \<in> span S"
  3108   by (metis subspace_neg subspace_span)
  3109 
  3110 lemma span_sub: "(x::'a::ring_1^'n) \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
  3111   by (metis subspace_span subspace_sub)
  3112 
  3113 lemma span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
  3114   apply (rule subspace_setsum)
  3115   by (metis subspace_span subspace_setsum)+
  3116 
  3117 lemma span_add_eq: "(x::'a::ring_1^'n) \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
  3118   apply (auto simp only: span_add span_sub)
  3119   apply (subgoal_tac "(x + y) - x \<in> span S", simp)
  3120   by (simp only: span_add span_sub)
  3121 
  3122 (* Mapping under linear image. *)
  3123 
  3124 lemma span_linear_image: assumes lf: "linear (f::'a::semiring_1 ^ 'n => _)"
  3125   shows "span (f ` S) = f ` (span S)"
  3126 proof-
  3127   {fix x
  3128     assume x: "x \<in> span (f ` S)"
  3129     have "x \<in> f ` span S"
  3130       apply (rule span_induct[where x=x and S = "f ` S"])
  3131       apply (clarsimp simp add: image_iff)
  3132       apply (frule span_superset)
  3133       apply blast
  3134       apply (simp only: mem_def)
  3135       apply (rule subspace_linear_image[OF lf])
  3136       apply (rule subspace_span)
  3137       apply (rule x)
  3138       done}
  3139   moreover 
  3140   {fix x assume x: "x \<in> span S"
  3141     have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_ext) 
  3142       unfolding mem_def Collect_def ..
  3143     have "f x \<in> span (f ` S)"
  3144       apply (rule span_induct[where S=S])
  3145       apply (rule span_superset)
  3146       apply simp
  3147       apply (subst th0)
  3148       apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"])
  3149       apply (rule x)
  3150       done}
  3151   ultimately show ?thesis by blast
  3152 qed
  3153 
  3154 (* The key breakdown property. *)
  3155 
  3156 lemma span_breakdown:
  3157   assumes bS: "(b::'a::ring_1 ^ 'n) \<in> S" and aS: "a \<in> span S"
  3158   shows "\<exists>k. a - k*s b \<in> span (S - {b})" (is "?P a")
  3159 proof-
  3160   {fix x assume xS: "x \<in> S"
  3161     {assume ab: "x = b"
  3162       then have "?P x"
  3163 	apply simp
  3164 	apply (rule exI[where x="1"], simp)
  3165 	by (rule span_0)}
  3166     moreover
  3167     {assume ab: "x \<noteq> b" 
  3168       then have "?P x"  using xS
  3169 	apply -
  3170 	apply (rule exI[where x=0])
  3171 	apply (rule span_superset)
  3172 	by simp}
  3173     ultimately have "?P x" by blast}
  3174   moreover have "subspace ?P" 
  3175     unfolding subspace_def 
  3176     apply auto
  3177     apply (simp add: mem_def)
  3178     apply (rule exI[where x=0])
  3179     using span_0[of "S - {b}"]
  3180     apply (simp add: mem_def)
  3181     apply (clarsimp simp add: mem_def)
  3182     apply (rule_tac x="k + ka" in exI)
  3183     apply (subgoal_tac "x + y - (k + ka) *s b = (x - k*s b) + (y - ka *s b)")
  3184     apply (simp only: )
  3185     apply (rule span_add[unfolded mem_def])
  3186     apply assumption+
  3187     apply (vector ring_simps)
  3188     apply (clarsimp simp add: mem_def)
  3189     apply (rule_tac x= "c*k" in exI)
  3190     apply (subgoal_tac "c *s x - (c * k) *s b = c*s (x - k*s b)")
  3191     apply (simp only: )
  3192     apply (rule span_mul[unfolded mem_def])
  3193     apply assumption
  3194     by (vector ring_simps)
  3195   ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis 
  3196 qed
  3197 
  3198 lemma span_breakdown_eq:
  3199   "(x::'a::ring_1^'n) \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *s a) \<in> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
  3200 proof-
  3201   {assume x: "x \<in> span (insert a S)"
  3202     from x span_breakdown[of "a" "insert a S" "x"]
  3203     have ?rhs apply clarsimp
  3204       apply (rule_tac x= "k" in exI)
  3205       apply (rule set_rev_mp[of _ "span (S - {a})" _])
  3206       apply assumption
  3207       apply (rule span_mono)      
  3208       apply blast
  3209       done}
  3210   moreover
  3211   { fix k assume k: "x - k *s a \<in> span S"
  3212     have eq: "x = (x - k *s a) + k *s a" by vector
  3213     have "(x - k *s a) + k *s a \<in> span (insert a S)"
  3214       apply (rule span_add)
  3215       apply (rule set_rev_mp[of _ "span S" _])
  3216       apply (rule k)
  3217       apply (rule span_mono)      
  3218       apply blast
  3219       apply (rule span_mul)
  3220       apply (rule span_superset)
  3221       apply blast
  3222       done
  3223     then have ?lhs using eq by metis}
  3224   ultimately show ?thesis by blast
  3225 qed
  3226 
  3227 (* Hence some "reversal" results.*)
  3228 
  3229 lemma in_span_insert:
  3230   assumes a: "(a::'a::field^'n) \<in> span (insert b S)" and na: "a \<notin> span S"
  3231   shows "b \<in> span (insert a S)"
  3232 proof-
  3233   from span_breakdown[of b "insert b S" a, OF insertI1 a]
  3234   obtain k where k: "a - k*s b \<in> span (S - {b})" by auto
  3235   {assume k0: "k = 0"
  3236     with k have "a \<in> span S"
  3237       apply (simp)
  3238       apply (rule set_rev_mp)
  3239       apply assumption
  3240       apply (rule span_mono)
  3241       apply blast
  3242       done
  3243     with na  have ?thesis by blast}
  3244   moreover
  3245   {assume k0: "k \<noteq> 0" 
  3246     have eq: "b = (1/k) *s a - ((1/k) *s a - b)" by vector
  3247     from k0 have eq': "(1/k) *s (a - k*s b) = (1/k) *s a - b"
  3248       by (vector field_simps)
  3249     from k have "(1/k) *s (a - k*s b) \<in> span (S - {b})"
  3250       by (rule span_mul)
  3251     hence th: "(1/k) *s a - b \<in> span (S - {b})"
  3252       unfolding eq' .
  3253 
  3254     from k
  3255     have ?thesis
  3256       apply (subst eq)
  3257       apply (rule span_sub)
  3258       apply (rule span_mul)
  3259       apply (rule span_superset)
  3260       apply blast
  3261       apply (rule set_rev_mp)
  3262       apply (rule th)
  3263       apply (rule span_mono)
  3264       using na by blast}
  3265   ultimately show ?thesis by blast
  3266 qed
  3267 
  3268 lemma in_span_delete: 
  3269   assumes a: "(a::'a::field^'n) \<in> span S" 
  3270   and na: "a \<notin> span (S-{b})"
  3271   shows "b \<in> span (insert a (S - {b}))"
  3272   apply (rule in_span_insert)
  3273   apply (rule set_rev_mp)
  3274   apply (rule a)
  3275   apply (rule span_mono)
  3276   apply blast
  3277   apply (rule na)
  3278   done
  3279 
  3280 (* Transitivity property. *)
  3281 
  3282 lemma span_trans:
  3283   assumes x: "(x::'a::ring_1^'n) \<in> span S" and y: "y \<in> span (insert x S)"
  3284   shows "y \<in> span S"
  3285 proof-
  3286   from span_breakdown[of x "insert x S" y, OF insertI1 y]
  3287   obtain k where k: "y -k*s x \<in> span (S - {x})" by auto
  3288   have eq: "y = (y - k *s x) + k *s x" by vector
  3289   show ?thesis 
  3290     apply (subst eq)
  3291     apply (rule span_add)
  3292     apply (rule set_rev_mp)
  3293     apply (rule k)
  3294     apply (rule span_mono)
  3295     apply blast
  3296     apply (rule span_mul)
  3297     by (rule x)
  3298 qed
  3299 
  3300 (* ------------------------------------------------------------------------- *)
  3301 (* An explicit expansion is sometimes needed.                                *)
  3302 (* ------------------------------------------------------------------------- *)
  3303 
  3304 lemma span_explicit:
  3305   "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}"
  3306   (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
  3307 proof-
  3308   {fix x assume x: "x \<in> ?E"
  3309     then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *s v) S = x"
  3310       by blast
  3311     have "x \<in> span P"
  3312       unfolding u[symmetric]
  3313       apply (rule span_setsum[OF fS])
  3314       using span_mono[OF SP]
  3315       by (auto intro: span_superset span_mul)}
  3316   moreover
  3317   have "\<forall>x \<in> span P. x \<in> ?E"
  3318     unfolding mem_def Collect_def
  3319   proof(rule span_induct_alt')
  3320     show "?h 0"
  3321       apply (rule exI[where x="{}"]) by simp
  3322   next
  3323     fix c x y
  3324     assume x: "x \<in> P" and hy: "?h y"
  3325     from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P" 
  3326       and u: "setsum (\<lambda>v. u v *s v) S = y" by blast
  3327     let ?S = "insert x S"
  3328     let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c)
  3329                   else u y"
  3330     from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
  3331     {assume xS: "x \<in> S"
  3332       have S1: "S = (S - {x}) \<union> {x}" 
  3333 	and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
  3334       have "setsum (\<lambda>v. ?u v *s v) ?S =(\<Sum>v\<in>S - {x}. u v *s v) + (u x + c) *s x"
  3335 	using xS 
  3336 	by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]] 
  3337 	  setsum_clauses(2)[OF fS] cong del: if_weak_cong)
  3338       also have "\<dots> = (\<Sum>v\<in>S. u v *s v) + c *s x"
  3339 	apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
  3340 	by (vector ring_simps)
  3341       also have "\<dots> = c*s x + y"
  3342 	by (simp add: add_commute u)
  3343       finally have "setsum (\<lambda>v. ?u v *s v) ?S = c*s x + y" .
  3344     then have "?Q ?S ?u (c*s x + y)" using th0 by blast}
  3345   moreover 
  3346   {assume xS: "x \<notin> S"
  3347     have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *s v) = y"
  3348       unfolding u[symmetric]
  3349       apply (rule setsum_cong2)
  3350       using xS by auto
  3351     have "?Q ?S ?u (c*s x + y)" using fS xS th0
  3352       by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)}
  3353   ultimately have "?Q ?S ?u (c*s x + y)"
  3354     by (cases "x \<in> S", simp, simp)
  3355     then show "?h (c*s x + y)" 
  3356       apply -
  3357       apply (rule exI[where x="?S"])
  3358       apply (rule exI[where x="?u"]) by metis
  3359   qed
  3360   ultimately show ?thesis by blast
  3361 qed
  3362 
  3363 lemma dependent_explicit:
  3364   "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")
  3365 proof-
  3366   {assume dP: "dependent P"
  3367     then obtain a S u where aP: "a \<in> P" and fS: "finite S" 
  3368       and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *s v) S = a" 
  3369       unfolding dependent_def span_explicit by blast
  3370     let ?S = "insert a S" 
  3371     let ?u = "\<lambda>y. if y = a then - 1 else u y" 
  3372     let ?v = a
  3373     from aP SP have aS: "a \<notin> S" by blast
  3374     from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
  3375     have s0: "setsum (\<lambda>v. ?u v *s v) ?S = 0"
  3376       using fS aS
  3377       apply (simp add: vector_smult_lneg vector_smult_lid setsum_clauses ring_simps )
  3378       apply (subst (2) ua[symmetric])
  3379       apply (rule setsum_cong2)
  3380       by auto
  3381     with th0 have ?rhs
  3382       apply -
  3383       apply (rule exI[where x= "?S"])
  3384       apply (rule exI[where x= "?u"])
  3385       by clarsimp}
  3386   moreover
  3387   {fix S u v assume fS: "finite S" 
  3388       and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0" 
  3389     and u: "setsum (\<lambda>v. u v *s v) S = 0"
  3390     let ?a = v 
  3391     let ?S = "S - {v}"
  3392     let ?u = "\<lambda>i. (- u i) / u v"
  3393     have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"       using fS SP vS by auto 
  3394     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"
  3395       using fS vS uv 
  3396       by (simp add: setsum_diff1 vector_smult_lneg divide_inverse 
  3397 	vector_smult_assoc field_simps)
  3398     also have "\<dots> = ?a"
  3399       unfolding setsum_cmul u
  3400       using uv by (simp add: vector_smult_lneg)
  3401     finally  have "setsum (\<lambda>v. ?u v *s v) ?S = ?a" .
  3402     with th0 have ?lhs
  3403       unfolding dependent_def span_explicit
  3404       apply -
  3405       apply (rule bexI[where x= "?a"])
  3406       apply simp_all
  3407       apply (rule exI[where x= "?S"])
  3408       by auto}
  3409   ultimately show ?thesis by blast
  3410 qed
  3411 
  3412 
  3413 lemma span_finite:
  3414   assumes fS: "finite S"
  3415   shows "span S = {(y::'a::semiring_1^'n). \<exists>u. setsum (\<lambda>v. u v *s v) S = y}"
  3416   (is "_ = ?rhs")
  3417 proof-
  3418   {fix y assume y: "y \<in> span S"
  3419     from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and 
  3420       u: "setsum (\<lambda>v. u v *s v) S' = y" unfolding span_explicit by blast
  3421     let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
  3422     from setsum_restrict_set[OF fS, of "\<lambda>v. u v *s v" S', symmetric] SS'
  3423     have "setsum (\<lambda>v. ?u v *s v) S = setsum (\<lambda>v. u v *s v) S'"
  3424       unfolding cond_value_iff cond_application_beta
  3425       apply (simp add: cond_value_iff cong del: if_weak_cong)
  3426       apply (rule setsum_cong)
  3427       apply auto
  3428       done
  3429     hence "setsum (\<lambda>v. ?u v *s v) S = y" by (metis u)
  3430     hence "y \<in> ?rhs" by auto}
  3431   moreover 
  3432   {fix y u assume u: "setsum (\<lambda>v. u v *s v) S = y"
  3433     then have "y \<in> span S" using fS unfolding span_explicit by auto}
  3434   ultimately show ?thesis by blast
  3435 qed
  3436 
  3437 
  3438 (* Standard bases are a spanning set, and obviously finite.                  *)
  3439 
  3440 lemma span_stdbasis:"span {basis i :: 'a::ring_1^'n | i. i \<in> {1 .. dimindex(UNIV :: 'n set)}} = UNIV"
  3441 apply (rule set_ext)
  3442 apply auto
  3443 apply (subst basis_expansion[symmetric])
  3444 apply (rule span_setsum)
  3445 apply simp
  3446 apply auto
  3447 apply (rule span_mul)
  3448 apply (rule span_superset)
  3449 apply (auto simp add: Collect_def mem_def)
  3450 done
  3451 
  3452   
  3453 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")
  3454 proof-
  3455   have eq: "?S = basis ` {1 .. ?n}" by blast
  3456   show ?thesis unfolding eq
  3457     apply (rule hassize_image_inj[OF basis_inj])
  3458     by (simp add: hassize_def)
  3459 qed
  3460 
  3461 lemma finite_stdbasis: "finite {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV:: 'n set)}}"
  3462   using has_size_stdbasis[unfolded hassize_def]
  3463   ..
  3464 
  3465 lemma card_stdbasis: "card {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}} = dimindex(UNIV :: 'n set)"
  3466   using has_size_stdbasis[unfolded hassize_def]
  3467   ..
  3468 
  3469 lemma independent_stdbasis_lemma:
  3470   assumes x: "(x::'a::semiring_1 ^ 'n) \<in> span (basis ` S)"
  3471   and i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
  3472   and iS: "i \<notin> S"
  3473   shows "(x$i) = 0"
  3474 proof-
  3475   let ?n = "dimindex (UNIV :: 'n set)"
  3476   let ?U = "{1 .. ?n}"
  3477   let ?B = "basis ` S"
  3478   let ?P = "\<lambda>(x::'a^'n). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0"
  3479  {fix x::"'a^'n" assume xS: "x\<in> ?B"
  3480    from xS have "?P x" by (auto simp add: basis_component)}
  3481  moreover
  3482  have "subspace ?P" 
  3483    by (auto simp add: subspace_def Collect_def mem_def zero_index vector_component)
  3484  ultimately show ?thesis
  3485    using x span_induct[of ?B ?P x] i iS by blast 
  3486 qed
  3487 
  3488 lemma independent_stdbasis: "independent {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
  3489 proof-
  3490   let ?n = "dimindex (UNIV :: 'n set)"
  3491   let ?I = "{1 .. ?n}"
  3492   let ?b = "basis :: nat \<Rightarrow> real ^'n"
  3493   let ?B = "?b ` ?I"
  3494   have eq: "{?b i|i. i \<in> ?I} = ?B"
  3495     by auto
  3496   {assume d: "dependent ?B"
  3497     then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"
  3498       unfolding dependent_def by auto
  3499     have eq1: "?B - {?b k} = ?B - ?b ` {k}"  by simp
  3500     have eq2: "?B - {?b k} = ?b ` (?I - {k})"
  3501       unfolding eq1
  3502       apply (rule inj_on_image_set_diff[symmetric])
  3503       apply (rule basis_inj) using k(1) by auto
  3504     from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
  3505     from independent_stdbasis_lemma[OF th0 k(1), simplified]
  3506     have False by (simp add: basis_component[OF k(1), of k])}
  3507   then show ?thesis unfolding eq dependent_def ..
  3508 qed
  3509 
  3510 (* This is useful for building a basis step-by-step.                         *)
  3511 
  3512 lemma independent_insert:
  3513   "independent(insert (a::'a::field ^'n) S) \<longleftrightarrow>
  3514       (if a \<in> S then independent S
  3515                 else independent S \<and> a \<notin> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
  3516 proof-
  3517   {assume aS: "a \<in> S"
  3518     hence ?thesis using insert_absorb[OF aS] by simp}
  3519   moreover
  3520   {assume aS: "a \<notin> S"
  3521     {assume i: ?lhs
  3522       then have ?rhs using aS
  3523 	apply simp
  3524 	apply (rule conjI)
  3525 	apply (rule independent_mono)
  3526 	apply assumption
  3527 	apply blast
  3528 	by (simp add: dependent_def)}
  3529     moreover 
  3530     {assume i: ?rhs
  3531       have ?lhs using i aS
  3532 	apply simp
  3533 	apply (auto simp add: dependent_def)
  3534 	apply (case_tac "aa = a", auto)
  3535 	apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
  3536 	apply simp
  3537 	apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
  3538 	apply (subgoal_tac "insert aa (S - {aa}) = S")
  3539 	apply simp
  3540 	apply blast
  3541 	apply (rule in_span_insert)
  3542 	apply assumption
  3543 	apply blast
  3544 	apply blast
  3545 	done}
  3546     ultimately have ?thesis by blast}
  3547   ultimately show ?thesis by blast
  3548 qed
  3549 
  3550 (* The degenerate case of the Exchange Lemma.  *)
  3551 
  3552 lemma mem_delete: "x \<in> (A - {a}) \<longleftrightarrow> x \<noteq> a \<and> x \<in> A"
  3553   by blast
  3554 
  3555 lemma span_span: "span (span A) = span A"
  3556   unfolding span_def hull_hull ..
  3557 
  3558 lemma span_inc: "S \<subseteq> span S"
  3559   by (metis subset_eq span_superset)
  3560 
  3561 lemma spanning_subset_independent:
  3562   assumes BA: "B \<subseteq> A" and iA: "independent (A::('a::field ^'n) set)" 
  3563   and AsB: "A \<subseteq> span B"
  3564   shows "A = B"
  3565 proof
  3566   from BA show "B \<subseteq> A" .
  3567 next
  3568   from span_mono[OF BA] span_mono[OF AsB]
  3569   have sAB: "span A = span B" unfolding span_span by blast
  3570 
  3571   {fix x assume x: "x \<in> A"
  3572     from iA have th0: "x \<notin> span (A - {x})"
  3573       unfolding dependent_def using x by blast
  3574     from x have xsA: "x \<in> span A" by (blast intro: span_superset)
  3575     have "A - {x} \<subseteq> A" by blast
  3576     hence th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
  3577     {assume xB: "x \<notin> B"
  3578       from xB BA have "B \<subseteq> A -{x}" by blast
  3579       hence "span B \<subseteq> span (A - {x})" by (metis span_mono)
  3580       with th1 th0 sAB have "x \<notin> span A" by blast
  3581       with x have False by (metis span_superset)}
  3582     then have "x \<in> B" by blast}
  3583   then show "A \<subseteq> B" by blast
  3584 qed
  3585 
  3586 (* The general case of the Exchange Lemma, the key to what follows.  *)
  3587 
  3588 lemma exchange_lemma:
  3589   assumes f:"finite (t:: ('a::field^'n) set)" and i: "independent s"
  3590   and sp:"s \<subseteq> span t" 
  3591   shows "\<exists>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
  3592 using f i sp
  3593 proof(induct c\<equiv>"card(t - s)" arbitrary: s t rule: nat_less_induct)
  3594   fix n:: nat and s t :: "('a ^'n) set"
  3595   assume H: " \<forall>m<n. \<forall>(x:: ('a ^'n) set) xa.
  3596                 finite xa \<longrightarrow>
  3597                 independent x \<longrightarrow>
  3598                 x \<subseteq> span xa \<longrightarrow>
  3599                 m = card (xa - x) \<longrightarrow>
  3600                 (\<exists>t'. (t' hassize card xa) \<and>
  3601                       x \<subseteq> t' \<and> t' \<subseteq> x \<union> xa \<and> x \<subseteq> span t')"
  3602     and ft: "finite t" and s: "independent s" and sp: "s \<subseteq> span t"
  3603     and n: "n = card (t - s)"
  3604   let ?P = "\<lambda>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
  3605   let ?ths = "\<exists>t'. ?P t'" 
  3606   {assume st: "s \<subseteq> t" 
  3607     from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t]) 
  3608       by (auto simp add: hassize_def intro: span_superset)}
  3609   moreover
  3610   {assume st: "t \<subseteq> s"
  3611     
  3612     from spanning_subset_independent[OF st s sp] 
  3613       st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t]) 
  3614       by (auto simp add: hassize_def intro: span_superset)}
  3615   moreover
  3616   {assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
  3617     from st(2) obtain b where b: "b \<in> t" "b \<notin> s" by blast
  3618       from b have "t - {b} - s \<subset> t - s" by blast
  3619       then have cardlt: "card (t - {b} - s) < n" using n ft
  3620  	by (auto intro: psubset_card_mono)
  3621       from b ft have ct0: "card t \<noteq> 0" by auto
  3622     {assume stb: "s \<subseteq> span(t -{b})"
  3623       from ft have ftb: "finite (t -{b})" by auto
  3624       from H[rule_format, OF cardlt ftb s stb] 
  3625       obtain u where u: "u hassize card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" by blast
  3626       let ?w = "insert b u"
  3627       have th0: "s \<subseteq> insert b u" using u by blast
  3628       from u(3) b have "u \<subseteq> s \<union> t" by blast 
  3629       then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
  3630       have bu: "b \<notin> u" using b u by blast
  3631       from u(1) have fu: "finite u" by (simp add: hassize_def)
  3632       from u(1) ft b have "u hassize (card t - 1)" by auto
  3633       then 
  3634       have th2: "insert b u hassize card t" 
  3635 	using  card_insert_disjoint[OF fu bu] ct0 by (auto simp add: hassize_def)
  3636       from u(4) have "s \<subseteq> span u" .
  3637       also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
  3638       finally have th3: "s \<subseteq> span (insert b u)" .      from th0 th1 th2 th3 have th: "?P ?w"  by blast
  3639       from th have ?ths by blast}
  3640     moreover
  3641     {assume stb: "\<not> s \<subseteq> span(t -{b})" 
  3642       from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
  3643       have ab: "a \<noteq> b" using a b by blast
  3644       have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
  3645       have mlt: "card ((insert a (t - {b})) - s) < n" 
  3646 	using cardlt ft n  a b by auto
  3647       have ft': "finite (insert a (t - {b}))" using ft by auto
  3648       {fix x assume xs: "x \<in> s"
  3649 	have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
  3650 	from b(1) have "b \<in> span t" by (simp add: span_superset)
  3651 	have bs: "b \<in> span (insert a (t - {b}))"
  3652 	  by (metis in_span_delete a sp mem_def subset_eq)
  3653 	from xs sp have "x \<in> span t" by blast
  3654 	with span_mono[OF t]
  3655 	have x: "x \<in> span (insert b (insert a (t - {b})))" ..
  3656 	from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))"  .}
  3657       then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
  3658       
  3659       from H[rule_format, OF mlt ft' s sp' refl] obtain u where 
  3660 	u: "u hassize card (insert a (t -{b}))" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
  3661 	"s \<subseteq> span u" by blast
  3662       from u a b ft at ct0 have "?P u" by (auto simp add: hassize_def)
  3663       then have ?ths by blast }
  3664     ultimately have ?ths by blast
  3665   }
  3666   ultimately 
  3667   show ?ths  by blast
  3668 qed
  3669 
  3670 (* This implies corresponding size bounds.                                   *)
  3671 
  3672 lemma independent_span_bound:
  3673   assumes f: "finite t" and i: "independent (s::('a::field^'n) set)" and sp:"s \<subseteq> span t"
  3674   shows "finite s \<and> card s \<le> card t"
  3675   by (metis exchange_lemma[OF f i sp] hassize_def finite_subset card_mono)
  3676 
  3677 lemma finite_Atleast_Atmost[simp]: "finite {f x |x. x\<in> {(i::'a::finite_intvl_succ) .. j}}"
  3678 proof-
  3679   have eq: "{f x |x. x\<in> {i .. j}} = f ` {i .. j}" by auto
  3680   show ?thesis unfolding eq 
  3681     apply (rule finite_imageI)
  3682     apply (rule finite_intvl)
  3683     done
  3684 qed
  3685 
  3686 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> {(i::nat) .. j}}"
  3687 proof-
  3688   have eq: "{f x |x. x\<in> {i .. j}} = f ` {i .. j}" by auto
  3689   show ?thesis unfolding eq 
  3690     apply (rule finite_imageI)
  3691     apply (rule finite_atLeastAtMost)
  3692     done
  3693 qed
  3694 
  3695 
  3696 lemma independent_bound:
  3697   fixes S:: "(real^'n) set"
  3698   shows "independent S \<Longrightarrow> finite S \<and> card S <= dimindex(UNIV :: 'n set)"
  3699   apply (subst card_stdbasis[symmetric])
  3700   apply (rule independent_span_bound)
  3701   apply (rule finite_Atleast_Atmost_nat)
  3702   apply assumption
  3703   unfolding span_stdbasis 
  3704   apply (rule subset_UNIV)
  3705   done
  3706 
  3707 lemma dependent_biggerset: "(finite (S::(real ^'n) set) ==> card S > dimindex(UNIV:: 'n set)) ==> dependent S"
  3708   by (metis independent_bound not_less)
  3709 
  3710 (* Hence we can create a maximal independent subset.                         *)
  3711 
  3712 lemma maximal_independent_subset_extend:
  3713   assumes sv: "(S::(real^'n) set) \<subseteq> V" and iS: "independent S"
  3714   shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
  3715   using sv iS
  3716 proof(induct d\<equiv> "dimindex (UNIV :: 'n set) - card S" arbitrary: S rule: nat_less_induct)
  3717   fix n and S:: "(real^'n) set"
  3718   assume H: "\<forall>m<n. \<forall>S \<subseteq> V. independent S \<longrightarrow> m = dimindex (UNIV::'n set) - card S \<longrightarrow>
  3719               (\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B)"
  3720     and sv: "S \<subseteq> V" and i: "independent S" and n: "n = dimindex (UNIV :: 'n set) - card S"
  3721   let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
  3722   let ?ths = "\<exists>x. ?P x"
  3723   let ?d = "dimindex (UNIV :: 'n set)"
  3724   {assume "V \<subseteq> span S"
  3725     then have ?ths  using sv i by blast }
  3726   moreover
  3727   {assume VS: "\<not> V \<subseteq> span S"
  3728     from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
  3729     from a have aS: "a \<notin> S" by (auto simp add: span_superset)
  3730     have th0: "insert a S \<subseteq> V" using a sv by blast
  3731     from independent_insert[of a S]  i a 
  3732     have th1: "independent (insert a S)" by auto
  3733     have mlt: "?d - card (insert a S) < n" 
  3734       using aS a n independent_bound[OF th1] dimindex_ge_1[of "UNIV :: 'n set"] 
  3735       by auto 
  3736       
  3737     from H[rule_format, OF mlt th0 th1 refl] 
  3738     obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B" 
  3739       by blast
  3740     from B have "?P B" by auto
  3741     then have ?ths by blast}
  3742   ultimately show ?ths by blast
  3743 qed
  3744 
  3745 lemma maximal_independent_subset:
  3746   "\<exists>(B:: (real ^'n) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
  3747   by (metis maximal_independent_subset_extend[of "{}:: (real ^'n) set"] empty_subsetI independent_empty)
  3748 
  3749 (* Notion of dimension.                                                      *)
  3750 
  3751 definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n))"
  3752 
  3753 lemma basis_exists:  "\<exists>B. (B :: (real ^'n) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize dim V)" 
  3754 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)"]
  3755 unfolding hassize_def
  3756 using maximal_independent_subset[of V] independent_bound
  3757 by auto
  3758 
  3759 (* Consequences of independence or spanning for cardinality.                 *)
  3760 
  3761 lemma independent_card_le_dim: "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B \<le> dim V"
  3762 by (metis basis_exists[of V] independent_span_bound[where ?'a=real] hassize_def subset_trans)
  3763 
  3764 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"
  3765   by (metis basis_exists[of V] independent_span_bound hassize_def subset_trans)
  3766 
  3767 lemma basis_card_eq_dim:
  3768   "B \<subseteq> (V:: (real ^'n) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
  3769   by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_mono)
  3770 
  3771 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"
  3772   by (metis basis_card_eq_dim hassize_def)
  3773 
  3774 (* More lemmas about dimension.                                              *)
  3775 
  3776 lemma dim_univ: "dim (UNIV :: (real^'n) set) = dimindex (UNIV :: 'n set)"
  3777   apply (rule dim_unique[of "{basis i |i. i\<in> {1 .. dimindex (UNIV :: 'n set)}}"])
  3778   by (auto simp only: span_stdbasis has_size_stdbasis independent_stdbasis)
  3779 
  3780 lemma dim_subset:
  3781   "(S:: (real ^'n) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
  3782   using basis_exists[of T] basis_exists[of S]
  3783   by (metis independent_span_bound[where ?'a = real and ?'n = 'n] subset_eq hassize_def)
  3784 
  3785 lemma dim_subset_univ: "dim (S:: (real^'n) set) \<le> dimindex (UNIV :: 'n set)"
  3786   by (metis dim_subset subset_UNIV dim_univ)
  3787 
  3788 (* Converses to those.                                                       *)
  3789 
  3790 lemma card_ge_dim_independent:
  3791   assumes BV:"(B::(real ^'n) set) \<subseteq> V" and iB:"independent B" and dVB:"dim V \<le> card B"
  3792   shows "V \<subseteq> span B"
  3793 proof-
  3794   {fix a assume aV: "a \<in> V"
  3795     {assume aB: "a \<notin> span B"
  3796       then have iaB: "independent (insert a B)" using iB aV  BV by (simp add: independent_insert)
  3797       from aV BV have th0: "insert a B \<subseteq> V" by blast
  3798       from aB have "a \<notin>B" by (auto simp add: span_superset)
  3799       with independent_card_le_dim[OF th0 iaB] dVB  have False by auto}
  3800     then have "a \<in> span B"  by blast}
  3801   then show ?thesis by blast
  3802 qed
  3803 
  3804 lemma card_le_dim_spanning:
  3805   assumes BV: "(B:: (real ^'n) set) \<subseteq> V" and VB: "V \<subseteq> span B" 
  3806   and fB: "finite B" and dVB: "dim V \<ge> card B"
  3807   shows "independent B"
  3808 proof-
  3809   {fix a assume a: "a \<in> B" "a \<in> span (B -{a})"
  3810     from a fB have c0: "card B \<noteq> 0" by auto
  3811     from a fB have cb: "card (B -{a}) = card B - 1" by auto
  3812     from BV a have th0: "B -{a} \<subseteq> V" by blast
  3813     {fix x assume x: "x \<in> V"
  3814       from a have eq: "insert a (B -{a}) = B" by blast
  3815       from x VB have x': "x \<in> span B" by blast 
  3816       from span_trans[OF a(2), unfolded eq, OF x']
  3817       have "x \<in> span (B -{a})" . }
  3818     then have th1: "V \<subseteq> span (B -{a})" by blast 
  3819     have th2: "finite (B -{a})" using fB by auto
  3820     from span_card_ge_dim[OF th0 th1 th2]
  3821     have c: "dim V \<le> card (B -{a})" .
  3822     from c c0 dVB cb have False by simp}
  3823   then show ?thesis unfolding dependent_def by blast
  3824 qed
  3825 
  3826 lemma card_eq_dim: "(B:: (real ^'n) set) \<subseteq> V \<Longrightarrow> B hassize dim V \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
  3827   by (metis hassize_def order_eq_iff card_le_dim_spanning 
  3828     card_ge_dim_independent)
  3829 
  3830 (* ------------------------------------------------------------------------- *)
  3831 (* More general size bound lemmas.                                           *)
  3832 (* ------------------------------------------------------------------------- *)
  3833 
  3834 lemma independent_bound_general:
  3835   "independent (S:: (real^'n) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
  3836   by (metis independent_card_le_dim independent_bound subset_refl)
  3837 
  3838 lemma dependent_biggerset_general: "(finite (S:: (real^'n) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
  3839   using independent_bound_general[of S] by (metis linorder_not_le) 
  3840 
  3841 lemma dim_span: "dim (span (S:: (real ^'n) set)) = dim S"
  3842 proof-
  3843   have th0: "dim S \<le> dim (span S)" 
  3844     by (auto simp add: subset_eq intro: dim_subset span_superset)
  3845   from basis_exists[of S] 
  3846   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
  3847   from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
  3848   have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc) 
  3849   have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span) 
  3850   from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis 
  3851     using fB(2)  by arith
  3852 qed
  3853 
  3854 lemma subset_le_dim: "(S:: (real ^'n) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
  3855   by (metis dim_span dim_subset)
  3856 
  3857 lemma span_eq_dim: "span (S:: (real ^'n) set) = span T ==> dim S = dim T"
  3858   by (metis dim_span)
  3859 
  3860 lemma spans_image:
  3861   assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and VB: "V \<subseteq> span B"
  3862   shows "f ` V \<subseteq> span (f ` B)"
  3863   unfolding span_linear_image[OF lf]
  3864   by (metis VB image_mono)
  3865 
  3866 lemma dim_image_le: assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S:: (real ^'n) set)"
  3867 proof-
  3868   from basis_exists[of S] obtain B where 
  3869     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
  3870   from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
  3871   have "dim (f ` S) \<le> card (f ` B)"
  3872     apply (rule span_card_ge_dim)
  3873     using lf B fB by (auto simp add: span_linear_image spans_image subset_image_iff)
  3874   also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
  3875   finally show ?thesis .
  3876 qed
  3877 
  3878 (* Relation between bases and injectivity/surjectivity of map.               *)
  3879 
  3880 lemma spanning_surjective_image:
  3881   assumes us: "UNIV \<subseteq> span (S:: ('a::semiring_1 ^'n) set)" 
  3882   and lf: "linear f" and sf: "surj f"
  3883   shows "UNIV \<subseteq> span (f ` S)"
  3884 proof-
  3885   have "UNIV \<subseteq> f ` UNIV" using sf by (auto simp add: surj_def)
  3886   also have " \<dots> \<subseteq> span (f ` S)" using spans_image[OF lf us] .
  3887 finally show ?thesis .
  3888 qed
  3889 
  3890 lemma independent_injective_image:
  3891   assumes iS: "independent (S::('a::semiring_1^'n) set)" and lf: "linear f" and fi: "inj f"
  3892   shows "independent (f ` S)"
  3893 proof-
  3894   {fix a assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
  3895     have eq: "f ` S - {f a} = f ` (S - {a})" using fi
  3896       by (auto simp add: inj_on_def)
  3897     from a have "f a \<in> f ` span (S -{a})"
  3898       unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
  3899     hence "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
  3900     with a(1) iS  have False by (simp add: dependent_def) }
  3901   then show ?thesis unfolding dependent_def by blast
  3902 qed 
  3903 
  3904 (* ------------------------------------------------------------------------- *)
  3905 (* Picking an orthogonal replacement for a spanning set.                     *)
  3906 (* ------------------------------------------------------------------------- *)
  3907     (* FIXME : Move to some general theory ?*)
  3908 definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
  3909 
  3910 lemma vector_sub_project_orthogonal: "(b::'a::ordered_field^'n) \<bullet> (x - ((b \<bullet> x) / (b\<bullet>b)) *s b) = 0"
  3911   apply (cases "b = 0", simp)
  3912   apply (simp add: dot_rsub dot_rmult)
  3913   unfolding times_divide_eq_right[symmetric]
  3914   by (simp add: field_simps dot_eq_0)
  3915 
  3916 lemma basis_orthogonal:
  3917   fixes B :: "(real ^'n) set"
  3918   assumes fB: "finite B"
  3919   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
  3920   (is " \<exists>C. ?P B C")
  3921 proof(induct rule: finite_induct[OF fB])
  3922   case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
  3923 next
  3924   case (2 a B)
  3925   note fB = `finite B` and aB = `a \<notin> B` 
  3926   from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C` 
  3927   obtain C where C: "finite C" "card C \<le> card B" 
  3928     "span C = span B" "pairwise orthogonal C" by blast
  3929   let ?a = "a - setsum (\<lambda>x. (x\<bullet>a / (x\<bullet>x)) *s x) C"
  3930   let ?C = "insert ?a C"
  3931   from C(1) have fC: "finite ?C" by simp
  3932   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" by (simp add: card_insert_if)
  3933   {fix x k 
  3934     have th0: "\<And>(a::'b::comm_ring) b c. a - (b - c) = c + (a - b)" by (simp add: ring_simps)
  3935     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"
  3936       apply (simp only: vector_ssub_ldistrib th0)
  3937       apply (rule span_add_eq)
  3938       apply (rule span_mul)
  3939       apply (rule span_setsum[OF C(1)])
  3940       apply clarify
  3941       apply (rule span_mul)
  3942       by (rule span_superset)}
  3943   then have SC: "span ?C = span (insert a B)"
  3944     unfolding expand_set_eq span_breakdown_eq C(3)[symmetric] by auto
  3945   thm pairwise_def 
  3946   {fix x y assume xC: "x \<in> ?C" and yC: "y \<in> ?C" and xy: "x \<noteq> y"
  3947     {assume xa: "x = ?a" and ya: "y = ?a" 
  3948       have "orthogonal x y" using xa ya xy by blast}
  3949     moreover
  3950     {assume xa: "x = ?a" and ya: "y \<noteq> ?a" "y \<in> C" 
  3951       from ya have Cy: "C = insert y (C - {y})" by blast
  3952       have fth: "finite (C - {y})" using C by simp
  3953       have "orthogonal x y"
  3954 	using xa ya
  3955 	unfolding orthogonal_def xa dot_lsub dot_rsub diff_eq_0_iff_eq
  3956 	apply simp 
  3957 	apply (subst Cy)
  3958 	using C(1) fth
  3959 	apply (simp only: setsum_clauses)
  3960 	apply (auto simp add: dot_ladd dot_lmult dot_eq_0 dot_sym[of y a] dot_lsum[OF fth])
  3961 	apply (rule setsum_0')
  3962 	apply clarsimp
  3963 	apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
  3964 	by auto}
  3965     moreover
  3966     {assume xa: "x \<noteq> ?a" "x \<in> C" and ya: "y = ?a" 
  3967       from xa have Cx: "C = insert x (C - {x})" by blast
  3968       have fth: "finite (C - {x})" using C by simp
  3969       have "orthogonal x y"
  3970 	using xa ya
  3971 	unfolding orthogonal_def ya dot_rsub dot_lsub diff_eq_0_iff_eq
  3972 	apply simp 
  3973 	apply (subst Cx)
  3974 	using C(1) fth
  3975 	apply (simp only: setsum_clauses)
  3976 	apply (subst dot_sym[of x])
  3977 	apply (auto simp add: dot_radd dot_rmult dot_eq_0 dot_sym[of x a] dot_rsum[OF fth])
  3978 	apply (rule setsum_0')
  3979 	apply clarsimp
  3980 	apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
  3981 	by auto}
  3982     moreover
  3983     {assume xa: "x \<in> C" and ya: "y \<in> C" 
  3984       have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast}
  3985     ultimately have "orthogonal x y" using xC yC by blast}
  3986   then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast
  3987   from fC cC SC CPO have "?P (insert a B) ?C" by blast
  3988   then show ?case by blast 
  3989 qed
  3990 
  3991 lemma orthogonal_basis_exists:
  3992   fixes V :: "(real ^'n) set"
  3993   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (B hassize dim V) \<and> pairwise orthogonal B"
  3994 proof-
  3995   from basis_exists[of V] obtain B where B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "B hassize dim V" by blast
  3996   from B have fB: "finite B" "card B = dim V" by (simp_all add: hassize_def)
  3997   from basis_orthogonal[OF fB(1)] obtain C where 
  3998     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
  3999   from C B 
  4000   have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans) 
  4001   from span_mono[OF B(3)]  C have SVC: "span V \<subseteq> span C" by (simp add: span_span)
  4002   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
  4003   have iC: "independent C" by (simp add: dim_span) 
  4004   from C fB have "card C \<le> dim V" by simp
  4005   moreover have "dim V \<le> card C" using span_card_ge_dim[OF CSV SVC C(1)]
  4006     by (simp add: dim_span)
  4007   ultimately have CdV: "C hassize dim V" unfolding hassize_def using C(1) by simp
  4008   from C B CSV CdV iC show ?thesis by auto 
  4009 qed
  4010 
  4011 lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
  4012   by (metis set_eq_subset span_mono span_span span_inc)
  4013 
  4014 (* ------------------------------------------------------------------------- *)
  4015 (* Low-dimensional subset is in a hyperplane (weak orthogonal complement).   *)
  4016 (* ------------------------------------------------------------------------- *)
  4017 
  4018 lemma span_not_univ_orthogonal:
  4019   assumes sU: "span S \<noteq> UNIV"
  4020   shows "\<exists>(a:: real ^'n). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
  4021 proof-
  4022   from sU obtain a where a: "a \<notin> span S" by blast
  4023   from orthogonal_basis_exists obtain B where 
  4024     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "B hassize dim S" "pairwise orthogonal B" 
  4025     by blast
  4026   from B have fB: "finite B" "card B = dim S" by (simp_all add: hassize_def)
  4027   from span_mono[OF B(2)] span_mono[OF B(3)]
  4028   have sSB: "span S = span B" by (simp add: span_span)
  4029   let ?a = "a - setsum (\<lambda>b. (a\<bullet>b / (b\<bullet>b)) *s b) B"
  4030   have "setsum (\<lambda>b. (a\<bullet>b / (b\<bullet>b)) *s b) B \<in> span S"
  4031     unfolding sSB
  4032     apply (rule span_setsum[OF fB(1)])
  4033     apply clarsimp
  4034     apply (rule span_mul)
  4035     by (rule span_superset)
  4036   with a have a0:"?a  \<noteq> 0" by auto
  4037   have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
  4038   proof(rule span_induct')
  4039     show "subspace (\<lambda>x. ?a \<bullet> x = 0)"
  4040       by (auto simp add: subspace_def mem_def dot_radd dot_rmult) 
  4041   next
  4042     {fix x assume x: "x \<in> B"
  4043       from x have B': "B = insert x (B - {x})" by blast
  4044       have fth: "finite (B - {x})" using fB by simp
  4045       have "?a \<bullet> x = 0" 
  4046 	apply (subst B') using fB fth
  4047 	unfolding setsum_clauses(2)[OF fth]
  4048 	apply simp
  4049 	apply (clarsimp simp add: dot_lsub dot_ladd dot_lmult dot_lsum dot_eq_0)
  4050 	apply (rule setsum_0', rule ballI)
  4051 	unfolding dot_sym
  4052 	by (auto simp add: x field_simps dot_eq_0 intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
  4053     then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
  4054   qed
  4055   with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
  4056 qed
  4057 
  4058 lemma span_not_univ_subset_hyperplane: 
  4059   assumes SU: "span S \<noteq> (UNIV ::(real^'n) set)"
  4060   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
  4061   using span_not_univ_orthogonal[OF SU] by auto
  4062 
  4063 lemma lowdim_subset_hyperplane:
  4064   assumes d: "dim S < dimindex (UNIV :: 'n set)"
  4065   shows "\<exists>(a::real ^'n). a  \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
  4066 proof-
  4067   {assume "span S = UNIV"
  4068     hence "dim (span S) = dim (UNIV :: (real ^'n) set)" by simp
  4069     hence "dim S = dimindex (UNIV :: 'n set)" by (simp add: dim_span dim_univ)
  4070     with d have False by arith}
  4071   hence th: "span S \<noteq> UNIV" by blast
  4072   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
  4073 qed
  4074 
  4075 (* We can extend a linear basis-basis injection to the whole set.            *)
  4076 
  4077 lemma linear_indep_image_lemma:
  4078   assumes lf: "linear f" and fB: "finite B" 
  4079   and ifB: "independent (f ` B)"
  4080   and fi: "inj_on f B" and xsB: "x \<in> span B" 
  4081   and fx: "f (x::'a::field^'n) = 0"
  4082   shows "x = 0"
  4083   using fB ifB fi xsB fx
  4084 proof(induct arbitrary: x rule: finite_induct[OF fB])
  4085   case 1 thus ?case by (auto simp add:  span_empty)
  4086 next
  4087   case (2 a b x)
  4088   have fb: "finite b" using "2.prems" by simp
  4089   have th0: "f ` b \<subseteq> f ` (insert a b)"
  4090     apply (rule image_mono) by blast 
  4091   from independent_mono[ OF "2.prems"(2) th0]
  4092   have ifb: "independent (f ` b)"  .
  4093   have fib: "inj_on f b" 
  4094     apply (rule subset_inj_on [OF "2.prems"(3)]) 
  4095     by blast
  4096   from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
  4097   obtain k where k: "x - k*s a \<in> span (b -{a})" by blast
  4098   have "f (x - k*s a) \<in> span (f ` b)"
  4099     unfolding span_linear_image[OF lf]
  4100     apply (rule imageI)
  4101     using k span_mono[of "b-{a}" b] by blast
  4102   hence "f x - k*s f a \<in> span (f ` b)"
  4103     by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
  4104   hence th: "-k *s f a \<in> span (f ` b)" 
  4105     using "2.prems"(5) by (simp add: vector_smult_lneg)
  4106   {assume k0: "k = 0" 
  4107     from k0 k have "x \<in> span (b -{a})" by simp
  4108     then have "x \<in> span b" using span_mono[of "b-{a}" b]
  4109       by blast}
  4110   moreover
  4111   {assume k0: "k \<noteq> 0"
  4112     from span_mul[OF th, of "- 1/ k"] k0
  4113     have th1: "f a \<in> span (f ` b)" 
  4114       by (auto simp add: vector_smult_assoc)
  4115     from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
  4116     have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
  4117     from "2.prems"(2)[unfolded dependent_def bex_simps(10), rule_format, of "f a"]
  4118     have "f a \<notin> span (f ` b)" using tha
  4119       using "2.hyps"(2)
  4120       "2.prems"(3) by auto
  4121     with th1 have False by blast
  4122     then have "x \<in> span b" by blast}
  4123   ultimately have xsb: "x \<in> span b" by blast
  4124   from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)]
  4125   show "x = 0" .
  4126 qed
  4127 
  4128 (* We can extend a linear mapping from basis.                                *)
  4129 
  4130 lemma linear_independent_extend_lemma:
  4131   assumes fi: "finite B" and ib: "independent B"
  4132   shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g ((x::'a::field^'n) + y) = g x + g y) 
  4133            \<and> (\<forall>x\<in> span B. \<forall>c. g (c*s x) = c *s g x)
  4134            \<and> (\<forall>x\<in> B. g x = f x)"
  4135 using ib fi
  4136 proof(induct rule: finite_induct[OF fi])
  4137   case 1 thus ?case by (auto simp add: span_empty) 
  4138 next
  4139   case (2 a b)
  4140   from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
  4141     by (simp_all add: independent_insert)
  4142   from "2.hyps"(3)[OF ibf] obtain g where 
  4143     g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
  4144     "\<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
  4145   let ?h = "\<lambda>z. SOME k. (z - k *s a) \<in> span b"
  4146   {fix z assume z: "z \<in> span (insert a b)"
  4147     have th0: "z - ?h z *s a \<in> span b"
  4148       apply (rule someI_ex)
  4149       unfolding span_breakdown_eq[symmetric]
  4150       using z .
  4151     {fix k assume k: "z - k *s a \<in> span b"
  4152       have eq: "z - ?h z *s a - (z - k*s a) = (k - ?h z) *s a" 
  4153 	by (simp add: ring_simps vector_sadd_rdistrib[symmetric])
  4154       from span_sub[OF th0 k]
  4155       have khz: "(k - ?h z) *s a \<in> span b" by (simp add: eq)
  4156       {assume "k \<noteq> ?h z" hence k0: "k - ?h z \<noteq> 0" by simp
  4157 	from k0 span_mul[OF khz, of "1 /(k - ?h z)"] 
  4158 	have "a \<in> span b" by (simp add: vector_smult_assoc)
  4159 	with "2.prems"(1) "2.hyps"(2) have False
  4160 	  by (auto simp add: dependent_def)}
  4161       then have "k = ?h z" by blast}
  4162     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}
  4163   note h = this
  4164   let ?g = "\<lambda>z. ?h z *s f a + g (z - ?h z *s a)"
  4165   {fix x y assume x: "x \<in> span (insert a b)" and y: "y \<in> span (insert a b)"
  4166     have tha: "\<And>(x::'a^'n) y a k l. (x + y) - (k + l) *s a = (x - k *s a) + (y - l *s a)" 
  4167       by (vector ring_simps)
  4168     have addh: "?h (x + y) = ?h x + ?h y"
  4169       apply (rule conjunct2[OF h, rule_format, symmetric])
  4170       apply (rule span_add[OF x y])
  4171       unfolding tha
  4172       by (metis span_add x y conjunct1[OF h, rule_format])
  4173     have "?g (x + y) = ?g x + ?g y" 
  4174       unfolding addh tha
  4175       g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
  4176       by (simp add: vector_sadd_rdistrib)}
  4177   moreover
  4178   {fix x:: "'a^'n" and c:: 'a  assume x: "x \<in> span (insert a b)"
  4179     have tha: "\<And>(x::'a^'n) c k a. c *s x - (c * k) *s a = c *s (x - k *s a)" 
  4180       by (vector ring_simps)
  4181     have hc: "?h (c *s x) = c * ?h x" 
  4182       apply (rule conjunct2[OF h, rule_format, symmetric])
  4183       apply (metis span_mul x)
  4184       by (metis tha span_mul x conjunct1[OF h])
  4185     have "?g (c *s x) = c*s ?g x" 
  4186       unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
  4187       by (vector ring_simps)}
  4188   moreover
  4189   {fix x assume x: "x \<in> (insert a b)"
  4190     {assume xa: "x = a"
  4191       have ha1: "1 = ?h a"
  4192 	apply (rule conjunct2[OF h, rule_format])
  4193 	apply (metis span_superset insertI1)
  4194 	using conjunct1[OF h, OF span_superset, OF insertI1]
  4195 	by (auto simp add: span_0)
  4196 
  4197       from xa ha1[symmetric] have "?g x = f x" 
  4198 	apply simp
  4199 	using g(2)[rule_format, OF span_0, of 0]
  4200 	by simp}
  4201     moreover
  4202     {assume xb: "x \<in> b"
  4203       have h0: "0 = ?h x"
  4204 	apply (rule conjunct2[OF h, rule_format])
  4205 	apply (metis  span_superset insertI1 xb x)
  4206 	apply simp
  4207 	apply (metis span_superset xb)
  4208 	done
  4209       have "?g x = f x"
  4210 	by (simp add: h0[symmetric] g(3)[rule_format, OF xb])}
  4211     ultimately have "?g x = f x" using x by blast }
  4212   ultimately show ?case apply - apply (rule exI[where x="?g"]) by blast
  4213 qed
  4214 
  4215 lemma linear_independent_extend:
  4216   assumes iB: "independent (B:: (real ^'n) set)"
  4217   shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
  4218 proof-
  4219   from maximal_independent_subset_extend[of B "UNIV"] iB
  4220   obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
  4221   
  4222   from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
  4223   obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) 
  4224            \<and> (\<forall>x\<in> span C. \<forall>c. g (c*s x) = c *s g x)
  4225            \<and> (\<forall>x\<in> C. g x = f x)" by blast
  4226   from g show ?thesis unfolding linear_def using C 
  4227     apply clarsimp by blast
  4228 qed
  4229 
  4230 (* Can construct an isomorphism between spaces of same dimension.            *)
  4231 
  4232 lemma card_le_inj: assumes fA: "finite A" and fB: "finite B"
  4233   and c: "card A \<le> card B" shows "(\<exists>f. f ` A \<subseteq> B \<and> inj_on f A)"
  4234 using fB c
  4235 proof(induct arbitrary: B rule: finite_induct[OF fA])
  4236   case 1 thus ?case by simp
  4237 next
  4238   case (2 x s t) 
  4239   thus ?case
  4240   proof(induct rule: finite_induct[OF "2.prems"(1)])
  4241     case 1    then show ?case by simp
  4242   next
  4243     case (2 y t)
  4244     from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
  4245     from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where
  4246       f: "f ` s \<subseteq> t \<and> inj_on f s" by blast
  4247     from f "2.prems"(2) "2.hyps"(2) show ?case
  4248       apply -
  4249       apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
  4250       by (auto simp add: inj_on_def)
  4251   qed
  4252 qed
  4253 
  4254 lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and 
  4255   c: "card A = card B"
  4256   shows "A = B"
  4257 proof-
  4258   from fB AB have fA: "finite A" by (auto intro: finite_subset)
  4259   from fA fB have fBA: "finite (B - A)" by auto
  4260   have e: "A \<inter> (B - A) = {}" by blast
  4261   have eq: "A \<union> (B - A) = B" using AB by blast
  4262   from card_Un_disjoint[OF fA fBA e, unfolded eq c]
  4263   have "card (B - A) = 0" by arith
  4264   hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
  4265   with AB show "A = B" by blast  
  4266 qed
  4267 
  4268 lemma subspace_isomorphism:
  4269   assumes s: "subspace (S:: (real ^'n) set)" and t: "subspace T" 
  4270   and d: "dim S = dim T"
  4271   shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
  4272 proof-
  4273   from basis_exists[of S] obtain B where 
  4274     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
  4275   from basis_exists[of T] obtain C where 
  4276     C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "C hassize dim T" by blast
  4277   from B(4) C(4) card_le_inj[of B C] d obtain f where
  4278     f: "f ` B \<subseteq> C" "inj_on f B" unfolding hassize_def by auto 
  4279   from linear_independent_extend[OF B(2)] obtain g where
  4280     g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
  4281   from B(4) have fB: "finite B" by (simp add: hassize_def)
  4282   from C(4) have fC: "finite C" by (simp add: hassize_def)
  4283   from inj_on_iff_eq_card[OF fB, of f] f(2) 
  4284   have "card (f ` B) = card B" by simp
  4285   with B(4) C(4) have ceq: "card (f ` B) = card C" using d 
  4286     by (simp add: hassize_def)
  4287   have "g ` B = f ` B" using g(2)
  4288     by (auto simp add: image_iff)
  4289   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
  4290   finally have gBC: "g ` B = C" .
  4291   have gi: "inj_on g B" using f(2) g(2)
  4292     by (auto simp add: inj_on_def)
  4293   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
  4294   {fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
  4295     from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
  4296     from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
  4297     have th1: "x - y \<in> span B" using x' y' by (metis span_sub) 
  4298     have "x=y" using g0[OF th1 th0] by simp }
  4299   then have giS: "inj_on g S" 
  4300     unfolding inj_on_def by blast
  4301   from span_subspace[OF B(1,3) s]
  4302   have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
  4303   also have "\<dots> = span C" unfolding gBC ..
  4304   also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
  4305   finally have gS: "g ` S = T" .
  4306   from g(1) gS giS show ?thesis by blast
  4307 qed
  4308 
  4309 (* linear functions are equal on a subspace if they are on a spanning set.   *)
  4310 
  4311 lemma subspace_kernel:
  4312   assumes lf: "linear (f::'a::semiring_1 ^'n \<Rightarrow> _)"
  4313   shows "subspace {x. f x = 0}"
  4314 apply (simp add: subspace_def)
  4315 by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
  4316 
  4317 lemma linear_eq_0_span:
  4318   assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
  4319   shows "\<forall>x \<in> span B. f x = (0::'a::semiring_1 ^'n)"
  4320 proof
  4321   fix x assume x: "x \<in> span B"
  4322   let ?P = "\<lambda>x. f x = 0"
  4323   from subspace_kernel[OF lf] have "subspace ?P" unfolding Collect_def .
  4324   with x f0 span_induct[of B "?P" x] show "f x = 0" by blast
  4325 qed
  4326 
  4327 lemma linear_eq_0:
  4328   assumes lf: "linear f" and SB: "S \<subseteq> span B" and f0: "\<forall>x\<in>B. f x = 0" 
  4329   shows "\<forall>x \<in> S. f x = (0::'a::semiring_1^'n)"
  4330   by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
  4331 
  4332 lemma linear_eq:
  4333   assumes lf: "linear (f::'a::ring_1^'n \<Rightarrow> _)" and lg: "linear g" and S: "S \<subseteq> span B"
  4334   and fg: "\<forall> x\<in> B. f x = g x" 
  4335   shows "\<forall>x\<in> S. f x = g x"
  4336 proof-
  4337   let ?h = "\<lambda>x. f x - g x"
  4338   from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
  4339   from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
  4340   show ?thesis by simp
  4341 qed    
  4342 
  4343 lemma linear_eq_stdbasis:
  4344   assumes lf: "linear (f::'a::ring_1^'m \<Rightarrow> 'a^'n)" and lg: "linear g"
  4345   and fg: "\<forall>i \<in> {1 .. dimindex(UNIV :: 'm set)}. f (basis i) = g(basis i)"
  4346   shows "f = g"
  4347 proof-
  4348   let ?U = "UNIV :: 'm set"
  4349   let ?I = "{basis i:: 'a^'m|i. i \<in> {1 .. dimindex ?U}}" 
  4350   {fix x assume x: "x \<in> (UNIV :: ('a^'m) set)"
  4351     from equalityD2[OF span_stdbasis]
  4352     have IU: " (UNIV :: ('a^'m) set) \<subseteq> span ?I" by blast
  4353     from linear_eq[OF lf lg IU] fg x
  4354     have "f x = g x" unfolding Collect_def  Ball_def mem_def by metis}
  4355   then show ?thesis by (auto intro: ext)
  4356 qed
  4357 
  4358 (* Similar results for bilinear functions.                                   *)
  4359 
  4360 lemma bilinear_eq:
  4361   assumes bf: "bilinear (f:: 'a::ring^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)" 
  4362   and bg: "bilinear g"
  4363   and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
  4364   and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
  4365   shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
  4366 proof-
  4367   let ?P = "\<lambda>x. \<forall>y\<in> span C. f x y = g x y"
  4368   from bf bg have sp: "subspace ?P" 
  4369     unfolding bilinear_def linear_def subspace_def bf bg  
  4370     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])
  4371 
  4372   have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y" 
  4373     apply -
  4374     apply (rule ballI)
  4375     apply (rule span_induct[of B ?P]) 
  4376     defer
  4377     apply (rule sp)
  4378     apply assumption
  4379     apply (clarsimp simp add: Ball_def)
  4380     apply (rule_tac P="\<lambda>y. f xa y = g xa y" and S=C in span_induct)
  4381     using fg 
  4382     apply (auto simp add: subspace_def)
  4383     using bf bg unfolding bilinear_def linear_def
  4384     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])
  4385   then show ?thesis using SB TC by (auto intro: ext)
  4386 qed
  4387 
  4388 lemma bilinear_eq_stdbasis:
  4389   assumes bf: "bilinear (f:: 'a::ring_1^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)" 
  4390   and bg: "bilinear g"
  4391   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)"
  4392   shows "f = g"
  4393 proof-
  4394   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
  4395   from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext)
  4396 qed
  4397 
  4398 (* Detailed theorems about left and right invertibility in general case.     *)
  4399 
  4400 lemma left_invertible_transp:
  4401   "(\<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)"
  4402   by (metis matrix_transp_mul transp_mat transp_transp)
  4403 
  4404 lemma right_invertible_transp:
  4405   "(\<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)"
  4406   by (metis matrix_transp_mul transp_mat transp_transp)
  4407 
  4408 lemma linear_injective_left_inverse:
  4409   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and fi: "inj f"
  4410   shows "\<exists>g. linear g \<and> g o f = id"
  4411 proof-
  4412   from linear_independent_extend[OF independent_injective_image, OF independent_stdbasis, OF lf fi]
  4413   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
  4414   from h(2) 
  4415   have th: "\<forall>i\<in>{1..dimindex (UNIV::'n set)}. (h \<circ> f) (basis i) = id (basis i)"
  4416     using inv_o_cancel[OF fi, unfolded stupid_ext[symmetric] id_def o_def]
  4417     apply auto
  4418     apply (erule_tac x="basis i" in allE)
  4419     by auto
  4420   
  4421   from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
  4422   have "h o f = id" .
  4423   then show ?thesis using h(1) by blast  
  4424 qed
  4425 
  4426 lemma linear_surjective_right_inverse:
  4427   assumes lf: "linear (f:: real ^'m \<Rightarrow> real ^'n)" and sf: "surj f"
  4428   shows "\<exists>g. linear g \<and> f o g = id"
  4429 proof-
  4430   from linear_independent_extend[OF independent_stdbasis]
  4431   obtain h:: "real ^'n \<Rightarrow> real ^'m" where 
  4432     h: "linear h" "\<forall> x\<in> {basis i| i. i\<in> {1 .. dimindex (UNIV :: 'n set)}}. h x = inv f x" by blast
  4433   from h(2) 
  4434   have th: "\<forall>i\<in>{1..dimindex (UNIV::'n set)}. (f o h) (basis i) = id (basis i)"
  4435     using sf
  4436     apply (auto simp add: surj_iff o_def stupid_ext[symmetric])
  4437     apply (erule_tac x="basis i" in allE)
  4438     by auto
  4439   
  4440   from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
  4441   have "f o h = id" .
  4442   then show ?thesis using h(1) by blast  
  4443 qed
  4444 
  4445 lemma matrix_left_invertible_injective:
  4446 "(\<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)"
  4447 proof-
  4448   {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
  4449     from xy have "B*v (A *v x) = B *v (A*v y)" by simp
  4450     hence "x = y"
  4451       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .}
  4452   moreover
  4453   {assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
  4454     hence i: "inj (op *v A)" unfolding inj_on_def by auto 
  4455     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
  4456     obtain g where g: "linear g" "g o op *v A = id" by blast
  4457     have "matrix g ** A = mat 1"
  4458       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
  4459       using g(2) by (simp add: o_def id_def stupid_ext)
  4460     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast}
  4461   ultimately show ?thesis by blast
  4462 qed
  4463 
  4464 lemma matrix_left_invertible_ker:
  4465   "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
  4466   unfolding matrix_left_invertible_injective
  4467   using linear_injective_0[OF matrix_vector_mul_linear, of A]
  4468   by (simp add: inj_on_def)
  4469 
  4470 lemma matrix_right_invertible_surjective:
  4471 "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
  4472 proof-
  4473   {fix B :: "real ^'m^'n"  assume AB: "A ** B = mat 1"
  4474     {fix x :: "real ^ 'm" 
  4475       have "A *v (B *v x) = x"
  4476 	by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)}
  4477     hence "surj (op *v A)" unfolding surj_def by metis }
  4478   moreover
  4479   {assume sf: "surj (op *v A)"
  4480     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
  4481     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id" 
  4482       by blast
  4483 
  4484     have "A ** (matrix g) = mat 1"
  4485       unfolding matrix_eq  matrix_vector_mul_lid 
  4486 	matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] 
  4487       using g(2) unfolding o_def stupid_ext[symmetric] id_def
  4488       .
  4489     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
  4490   }
  4491   ultimately show ?thesis unfolding surj_def by blast
  4492 qed    
  4493 
  4494 lemma matrix_left_invertible_independent_columns:
  4495   fixes A :: "real^'n^'m"
  4496   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))"
  4497    (is "?lhs \<longleftrightarrow> ?rhs")
  4498 proof-
  4499   let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
  4500   {assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
  4501     {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" 
  4502       and i: "i \<in> ?U"
  4503       let ?x = "\<chi> i. c i"
  4504       have th0:"A *v ?x = 0"
  4505 	using c
  4506 	unfolding matrix_mult_vsum Cart_eq
  4507 	by (auto simp add: vector_component zero_index setsum_component Cart_lambda_beta)
  4508       from k[rule_format, OF th0] i
  4509       have "c i = 0" by (vector Cart_eq)}
  4510     hence ?rhs by blast}
  4511   moreover
  4512   {assume H: ?rhs
  4513     {fix x assume x: "A *v x = 0" 
  4514       let ?c = "\<lambda>i. ((x$i ):: real)"
  4515       from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
  4516       have "x = 0" by vector}}
  4517   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast 
  4518 qed
  4519 
  4520 lemma matrix_right_invertible_independent_rows:
  4521   fixes A :: "real^'n^'m"
  4522   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))"
  4523   unfolding left_invertible_transp[symmetric]
  4524     matrix_left_invertible_independent_columns
  4525   by (simp add: column_transp)
  4526 
  4527 lemma matrix_right_invertible_span_columns:
  4528   "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")
  4529 proof-
  4530   let ?U = "{1 .. dimindex (UNIV :: 'm set)}"
  4531   have fU: "finite ?U" by simp
  4532   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
  4533     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
  4534     apply (subst eq_commute) ..    
  4535   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
  4536   {assume h: ?lhs
  4537     {fix x:: "real ^'n" 
  4538 	from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m"
  4539 	  where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
  4540 	have "x \<in> span (columns A)"  
  4541 	  unfolding y[symmetric]
  4542 	  apply (rule span_setsum[OF fU])
  4543 	  apply clarify
  4544 	  apply (rule span_mul)
  4545 	  apply (rule span_superset)
  4546 	  unfolding columns_def
  4547 	  by blast}
  4548     then have ?rhs unfolding rhseq by blast}
  4549   moreover
  4550   {assume h:?rhs
  4551     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
  4552     {fix y have "?P y" 
  4553       proof(rule span_induct_alt[of ?P "columns A"])
  4554 	show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
  4555 	  apply (rule exI[where x=0])
  4556 	  by (simp add: zero_index vector_smult_lzero)
  4557       next
  4558 	fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
  4559 	from y1 obtain i where i: "i \<in> ?U" "y1 = column i A" 
  4560 	  unfolding columns_def by blast
  4561 	from y2 obtain x:: "real ^'m" where 
  4562 	  x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
  4563 	let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
  4564 	show "?P (c*s y1 + y2)"
  4565 	  proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric]Cart_lambda_beta setsum_component cond_value_iff right_distrib cond_application_beta vector_component cong del: if_weak_cong, simp only: One_nat_def[symmetric])
  4566 	    fix j 
  4567 	    have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
  4568            else (x$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)
  4569 	      by (simp add: ring_simps)
  4570 	    have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
  4571            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)$j))) ?U"
  4572 	      apply (rule setsum_cong[OF refl])
  4573 	      using th by blast
  4574 	    also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
  4575 	      by (simp add: setsum_addf)
  4576 	    also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
  4577 	      unfolding setsum_delta[OF fU]
  4578 	      using i(1) by simp 
  4579 	    finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
  4580            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
  4581 	  qed
  4582 	next
  4583 	  show "y \<in> span (columns A)" unfolding h by blast
  4584 	qed}
  4585     then have ?lhs unfolding lhseq ..}
  4586   ultimately show ?thesis by blast
  4587 qed
  4588 
  4589 lemma matrix_left_invertible_span_rows:
  4590   "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
  4591   unfolding right_invertible_transp[symmetric]
  4592   unfolding columns_transp[symmetric]
  4593   unfolding matrix_right_invertible_span_columns
  4594  ..
  4595 
  4596 (* An injective map real^'n->real^'n is also surjective.                       *)
  4597 
  4598 lemma linear_injective_imp_surjective:
  4599   assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and fi: "inj f" 
  4600   shows "surj f"
  4601 proof-
  4602   let ?U = "UNIV :: (real ^'n) set"
  4603   from basis_exists[of ?U] obtain B 
  4604     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U" 
  4605     by blast
  4606   from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
  4607   have th: "?U \<subseteq> span (f ` B)"
  4608     apply (rule card_ge_dim_independent)
  4609     apply blast
  4610     apply (rule independent_injective_image[OF B(2) lf fi])
  4611     apply (rule order_eq_refl)
  4612     apply (rule sym)
  4613     unfolding d
  4614     apply (rule card_image)
  4615     apply (rule subset_inj_on[OF fi])
  4616     by blast
  4617   from th show ?thesis
  4618     unfolding span_linear_image[OF lf] surj_def
  4619     using B(3) by blast
  4620 qed
  4621 
  4622 (* And vice versa.                                                           *)
  4623 
  4624 lemma surjective_iff_injective_gen: 
  4625   assumes fS: "finite S" and fT: "finite T" and c: "card S = card T"
  4626   and ST: "f ` S \<subseteq> T"
  4627   shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs")
  4628 proof-
  4629   {assume h: "?lhs"
  4630     {fix x y assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
  4631       from x fS have S0: "card S \<noteq> 0" by auto
  4632       {assume xy: "x \<noteq> y"
  4633 	have th: "card S \<le> card (f ` (S - {y}))"
  4634 	  unfolding c
  4635 	  apply (rule card_mono)
  4636 	  apply (rule finite_imageI)
  4637 	  using fS apply simp
  4638 	  using h xy x y f unfolding subset_eq image_iff
  4639 	  apply auto
  4640 	  apply (case_tac "xa = f x")
  4641 	  apply (rule bexI[where x=x])
  4642 	  apply auto
  4643 	  done
  4644 	also have " \<dots> \<le> card (S -{y})"
  4645 	  apply (rule card_image_le)
  4646 	  using fS by simp
  4647 	also have "\<dots> \<le> card S - 1" using y fS by simp
  4648 	finally have False  using S0 by arith }
  4649       then have "x = y" by blast}
  4650     then have ?rhs unfolding inj_on_def by blast}
  4651   moreover
  4652   {assume h: ?rhs
  4653     have "f ` S = T"
  4654       apply (rule card_subset_eq[OF fT ST])
  4655       unfolding card_image[OF h] using c .
  4656     then have ?lhs by blast}
  4657   ultimately show ?thesis by blast
  4658 qed
  4659 
  4660 lemma linear_surjective_imp_injective:
  4661   assumes lf: "linear (f::real ^'n => real ^'n)" and sf: "surj f" 
  4662   shows "inj f"
  4663 proof-
  4664   let ?U = "UNIV :: (real ^'n) set"
  4665   from basis_exists[of ?U] obtain B 
  4666     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U" 
  4667     by blast
  4668   {fix x assume x: "x \<in> span B" and fx: "f x = 0"
  4669     from B(4) have fB: "finite B" by (simp add: hassize_def)
  4670     from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
  4671     have fBi: "independent (f ` B)" 
  4672       apply (rule card_le_dim_spanning[of "f ` B" ?U])
  4673       apply blast
  4674       using sf B(3)
  4675       unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
  4676       apply blast
  4677       using fB apply (blast intro: finite_imageI)
  4678       unfolding d
  4679       apply (rule card_image_le)
  4680       apply (rule fB)
  4681       done
  4682     have th0: "dim ?U \<le> card (f ` B)"
  4683       apply (rule span_card_ge_dim)
  4684       apply blast
  4685       unfolding span_linear_image[OF lf]
  4686       apply (rule subset_trans[where B = "f ` UNIV"])
  4687       using sf unfolding surj_def apply blast
  4688       apply (rule image_mono)
  4689       apply (rule B(3))
  4690       apply (metis finite_imageI fB)
  4691       done
  4692 
  4693     moreover have "card (f ` B) \<le> card B"
  4694       by (rule card_image_le, rule fB)
  4695     ultimately have th1: "card B = card (f ` B)" unfolding d by arith
  4696     have fiB: "inj_on f B" 
  4697       unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast
  4698     from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
  4699     have "x = 0" by blast}
  4700   note th = this
  4701   from th show ?thesis unfolding linear_injective_0[OF lf] 
  4702     using B(3) by blast
  4703 qed
  4704 
  4705 (* Hence either is enough for isomorphism.                                   *)
  4706 
  4707 lemma left_right_inverse_eq:
  4708   assumes fg: "f o g = id" and gh: "g o h = id"
  4709   shows "f = h" 
  4710 proof-
  4711   have "f = f o (g o h)" unfolding gh by simp
  4712   also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
  4713   finally show "f = h" unfolding fg by simp
  4714 qed
  4715 
  4716 lemma isomorphism_expand:
  4717   "f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
  4718   by (simp add: expand_fun_eq o_def id_def)
  4719 
  4720 lemma linear_injective_isomorphism:
  4721   assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'n)" and fi: "inj f"
  4722   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
  4723 unfolding isomorphism_expand[symmetric]
  4724 using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi]
  4725 by (metis left_right_inverse_eq)
  4726 
  4727 lemma linear_surjective_isomorphism:
  4728   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and sf: "surj f"
  4729   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
  4730 unfolding isomorphism_expand[symmetric]
  4731 using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
  4732 by (metis left_right_inverse_eq)
  4733 
  4734 (* Left and right inverses are the same for R^N->R^N.                        *)
  4735 
  4736 lemma linear_inverse_left:
  4737   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and lf': "linear f'"
  4738   shows "f o f' = id \<longleftrightarrow> f' o f = id"
  4739 proof-
  4740   {fix f f':: "real ^'n \<Rightarrow> real ^'n"
  4741     assume lf: "linear f" "linear f'" and f: "f o f' = id"
  4742     from f have sf: "surj f"
  4743       
  4744       apply (auto simp add: o_def stupid_ext[symmetric] id_def surj_def)
  4745       by metis
  4746     from linear_surjective_isomorphism[OF lf(1) sf] lf f
  4747     have "f' o f = id" unfolding stupid_ext[symmetric] o_def id_def
  4748       by metis}
  4749   then show ?thesis using lf lf' by metis
  4750 qed
  4751 
  4752 (* Moreover, a one-sided inverse is automatically linear.                    *)
  4753 
  4754 lemma left_inverse_linear:
  4755   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and gf: "g o f = id" 
  4756   shows "linear g"
  4757 proof-
  4758   from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def stupid_ext[symmetric])
  4759     by metis
  4760   from linear_injective_isomorphism[OF lf fi] 
  4761   obtain h:: "real ^'n \<Rightarrow> real ^'n" where 
  4762     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
  4763   have "h = g" apply (rule ext) using gf h(2,3)
  4764     apply (simp add: o_def id_def stupid_ext[symmetric])
  4765     by metis
  4766   with h(1) show ?thesis by blast
  4767 qed
  4768 
  4769 lemma right_inverse_linear:
  4770   assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and gf: "f o g = id" 
  4771   shows "linear g"
  4772 proof-
  4773   from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def stupid_ext[symmetric])
  4774     by metis
  4775   from linear_surjective_isomorphism[OF lf fi] 
  4776   obtain h:: "real ^'n \<Rightarrow> real ^'n" where 
  4777     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
  4778   have "h = g" apply (rule ext) using gf h(2,3)
  4779     apply (simp add: o_def id_def stupid_ext[symmetric])
  4780     by metis
  4781   with h(1) show ?thesis by blast
  4782 qed
  4783 
  4784 (* The same result in terms of square matrices.                              *)
  4785 
  4786 lemma matrix_left_right_inverse:
  4787   fixes A A' :: "real ^'n^'n" 
  4788   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
  4789 proof-
  4790   {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"
  4791     have sA: "surj (op *v A)"
  4792       unfolding surj_def
  4793       apply clarify
  4794       apply (rule_tac x="(A' *v y)" in exI)
  4795       by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
  4796     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
  4797     obtain f' :: "real ^'n \<Rightarrow> real ^'n"
  4798       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
  4799     have th: "matrix f' ** A = mat 1" 
  4800       by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
  4801     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
  4802     hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
  4803     hence "matrix f' ** A = A' ** A" by simp
  4804     hence "A' ** A = mat 1" by (simp add: th)}
  4805   then show ?thesis by blast
  4806 qed
  4807 
  4808 (* Considering an n-element vector as an n-by-1 or 1-by-n matrix.            *)
  4809 
  4810 definition "rowvector v = (\<chi> i j. (v$j))"
  4811 
  4812 definition "columnvector v = (\<chi> i j. (v$i))"
  4813 
  4814 lemma transp_columnvector:
  4815  "transp(columnvector v) = rowvector v"
  4816   by (simp add: transp_def rowvector_def columnvector_def Cart_eq Cart_lambda_beta)
  4817 
  4818 lemma transp_rowvector: "transp(rowvector v) = columnvector v"
  4819   by (simp add: transp_def columnvector_def rowvector_def Cart_eq Cart_lambda_beta)
  4820 
  4821 lemma dot_rowvector_columnvector:
  4822   "columnvector (A *v v) = A ** columnvector v"
  4823   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
  4824 
  4825 lemma dot_matrix_product: "(x::'a::semiring_1^'n) \<bullet> y = (((rowvector x ::'a^'n^1) ** (columnvector y :: 'a^1^'n))$1)$1"
  4826   apply (vector matrix_matrix_mult_def rowvector_def columnvector_def dot_def)
  4827   by (simp add: Cart_lambda_beta)
  4828 
  4829 lemma dot_matrix_vector_mul:
  4830   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
  4831   shows "(A *v x) \<bullet> (B *v y) =
  4832       (((rowvector x :: real^'n^1) ** ((transp A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
  4833 unfolding dot_matrix_product transp_columnvector[symmetric]
  4834   dot_rowvector_columnvector matrix_transp_mul matrix_mul_assoc ..
  4835 
  4836 (* Infinity norm.                                                            *)
  4837 
  4838 definition "infnorm (x::real^'n) = rsup {abs(x$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
  4839 
  4840 lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> {1 .. dimindex (UNIV :: 'n set)}"
  4841   using dimindex_ge_1 by auto
  4842 
  4843 lemma infnorm_set_image:
  4844   "{abs(x$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}} =
  4845   (\<lambda>i. abs(x$i)) ` {1 .. dimindex(UNIV :: 'n set)}" by blast
  4846 
  4847 lemma infnorm_set_lemma:
  4848   shows "finite {abs((x::'a::abs ^'n)$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
  4849   and "{abs(x$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}} \<noteq> {}"
  4850   unfolding infnorm_set_image
  4851   using dimindex_ge_1[of "UNIV :: 'n set"]
  4852   by (auto intro: finite_imageI)
  4853 
  4854 lemma infnorm_pos_le: "0 \<le> infnorm x"
  4855   unfolding infnorm_def
  4856   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
  4857   unfolding infnorm_set_image
  4858   using dimindex_ge_1
  4859   by auto
  4860 
  4861 lemma infnorm_triangle: "infnorm ((x::real^'n) + y) \<le> infnorm x + infnorm y"
  4862 proof-
  4863   have th: "\<And>x y (z::real). x - y <= z \<longleftrightarrow> x - z <= y" by arith
  4864   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
  4865   have th2: "\<And>x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith
  4866   show ?thesis 
  4867   unfolding infnorm_def
  4868   unfolding rsup_finite_le_iff[ OF infnorm_set_lemma]
  4869   apply (subst diff_le_eq[symmetric])
  4870   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
  4871   unfolding infnorm_set_image bex_simps 
  4872   apply (subst th)
  4873   unfolding th1 
  4874   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
  4875   
  4876   unfolding infnorm_set_image ball_simps bex_simps 
  4877   apply (simp add: vector_add_component)
  4878   apply (metis numseg_dimindex_nonempty th2)
  4879   done
  4880 qed
  4881 
  4882 lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::real ^'n) = 0"
  4883 proof-
  4884   have "infnorm x <= 0 \<longleftrightarrow> x = 0"
  4885     unfolding infnorm_def
  4886     unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
  4887     unfolding infnorm_set_image ball_simps
  4888     by vector
  4889   then show ?thesis using infnorm_pos_le[of x] by simp
  4890 qed
  4891 
  4892 lemma infnorm_0: "infnorm 0 = 0"
  4893   by (simp add: infnorm_eq_0)
  4894 
  4895 lemma infnorm_neg: "infnorm (- x) = infnorm x"
  4896   unfolding infnorm_def
  4897   apply (rule cong[of "rsup" "rsup"])
  4898   apply blast
  4899   apply (rule set_ext)
  4900   apply (auto simp add: vector_component abs_minus_cancel)
  4901   apply (rule_tac x="i" in exI)
  4902   apply (simp add: vector_component)
  4903   done
  4904 
  4905 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)" 
  4906 proof-
  4907   have "y - x = - (x - y)" by simp
  4908   then show ?thesis  by (metis infnorm_neg)
  4909 qed
  4910 
  4911 lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
  4912 proof-
  4913   have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
  4914     by arith
  4915   from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
  4916   have ths: "infnorm x \<le> infnorm (x - y) + infnorm y" 
  4917     "infnorm y \<le> infnorm (x - y) + infnorm x"
  4918     by (simp_all add: ring_simps infnorm_neg diff_def[symmetric])
  4919   from th[OF ths]  show ?thesis .
  4920 qed
  4921 
  4922 lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
  4923   using infnorm_pos_le[of x] by arith
  4924 
  4925 lemma component_le_infnorm: assumes i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
  4926   shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
  4927 proof-
  4928   let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
  4929   let ?S = "{\<bar>x$i\<bar> |i. i\<in> ?U}"
  4930   have fS: "finite ?S" unfolding image_Collect[symmetric]
  4931     apply (rule finite_imageI) unfolding Collect_def mem_def by simp  
  4932   have S0: "?S \<noteq> {}" using numseg_dimindex_nonempty by blast
  4933   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
  4934   from rsup_finite_in[OF fS S0] rsup_finite_Ub[OF fS S0] i
  4935   show ?thesis unfolding infnorm_def isUb_def setle_def 
  4936     unfolding infnorm_set_image ball_simps by auto
  4937 qed
  4938 
  4939 lemma infnorm_mul_lemma: "infnorm(a *s x) <= \<bar>a\<bar> * infnorm x"
  4940   apply (subst infnorm_def)
  4941   unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
  4942   unfolding infnorm_set_image ball_simps
  4943   apply (simp add: abs_mult vector_component del: One_nat_def)
  4944   apply (rule ballI)
  4945   apply (drule component_le_infnorm[of _ x])
  4946   apply (rule mult_mono)
  4947   apply auto
  4948   done
  4949 
  4950 lemma infnorm_mul: "infnorm(a *s x) = abs a * infnorm x"
  4951 proof-
  4952   {assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) }
  4953   moreover
  4954   {assume a0: "a \<noteq> 0"
  4955     from a0 have th: "(1/a) *s (a *s x) = x"
  4956       by (simp add: vector_smult_assoc)
  4957     from a0 have ap: "\<bar>a\<bar> > 0" by arith
  4958     from infnorm_mul_lemma[of "1/a" "a *s x"]
  4959     have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*s x)"
  4960       unfolding th by simp
  4961     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)
  4962     then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*s x)" 
  4963       using ap by (simp add: field_simps)
  4964     with infnorm_mul_lemma[of a x] have ?thesis by arith }
  4965   ultimately show ?thesis by blast
  4966 qed
  4967 
  4968 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
  4969   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
  4970 
  4971 (* Prove that it differs only up to a bound from Euclidean norm.             *)
  4972 
  4973 lemma infnorm_le_norm: "infnorm x \<le> norm x"
  4974   unfolding infnorm_def rsup_finite_le_iff[OF infnorm_set_lemma] 
  4975   unfolding infnorm_set_image  ball_simps
  4976   by (metis component_le_norm)
  4977 lemma card_enum: "card {1 .. n} = n" by auto
  4978 lemma norm_le_infnorm: "norm(x) <= sqrt(real (dimindex(UNIV ::'n set))) * infnorm(x::real ^'n)"
  4979 proof-
  4980   let ?d = "dimindex(UNIV ::'n set)"
  4981   have d: "?d = card {1 .. ?d}" by auto
  4982   have "real ?d \<ge> 0" by simp
  4983   hence d2: "(sqrt (real ?d))^2 = real ?d"
  4984     by (auto intro: real_sqrt_pow2)
  4985   have th: "sqrt (real ?d) * infnorm x \<ge> 0"
  4986     by (simp add: dimindex_ge_1 zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le)
  4987   have th1: "x\<bullet>x \<le> (sqrt (real ?d) * infnorm x)^2"
  4988     unfolding power_mult_distrib d2 
  4989     apply (subst d)
  4990     apply (subst power2_abs[symmetric])
  4991     unfolding real_of_nat_def dot_def power2_eq_square[symmetric]
  4992     apply (subst power2_abs[symmetric])
  4993     apply (rule setsum_bounded)
  4994     apply (rule power_mono)
  4995     unfolding abs_of_nonneg[OF infnorm_pos_le] 
  4996     unfolding infnorm_def  rsup_finite_ge_iff[OF infnorm_set_lemma]
  4997     unfolding infnorm_set_image bex_simps
  4998     apply blast
  4999     by (rule abs_ge_zero)
  5000   from real_le_lsqrt[OF dot_pos_le th th1]
  5001   show ?thesis unfolding real_vector_norm_def  real_of_real_def id_def . 
  5002 qed
  5003 
  5004 (* Equality in Cauchy-Schwarz and triangle inequalities.                     *)
  5005 
  5006 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")
  5007 proof-
  5008   {assume h: "x = 0"
  5009     hence ?thesis by (simp add: norm_0)}
  5010   moreover
  5011   {assume h: "y = 0"
  5012     hence ?thesis by (simp add: norm_0)}
  5013   moreover
  5014   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
  5015     from dot_eq_0[of "norm y *s x - norm x *s y"]
  5016     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)"
  5017       using x y
  5018       unfolding dot_rsub dot_lsub dot_lmult dot_rmult
  5019       unfolding norm_pow_2[symmetric] power2_eq_square diff_eq_0_iff_eq apply (simp add: dot_sym)
  5020       apply (simp add: ring_simps)
  5021       apply metis
  5022       done
  5023     also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
  5024       by (simp add: ring_simps dot_sym)
  5025     also have "\<dots> \<longleftrightarrow> ?lhs" using x y
  5026       apply (simp add: norm_eq_0)
  5027       by metis
  5028     finally have ?thesis by blast}
  5029   ultimately show ?thesis by blast
  5030 qed
  5031 
  5032 lemma norm_cauchy_schwarz_abs_eq: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
  5033                 norm x *s y = norm y *s x \<or> norm(x) *s y = - norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
  5034 proof-
  5035   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
  5036   have "?rhs \<longleftrightarrow> norm x *s y = norm y *s x \<or> norm (- x) *s y = norm y *s (- x)"
  5037     apply (simp add: norm_neg) by vector
  5038   also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or>
  5039      (-x) \<bullet> y = norm x * norm y)"
  5040     unfolding norm_cauchy_schwarz_eq[symmetric]
  5041     unfolding norm_neg
  5042       norm_mul by blast
  5043   also have "\<dots> \<longleftrightarrow> ?lhs"
  5044     unfolding th[OF mult_nonneg_nonneg, OF norm_pos_le[of x] norm_pos_le[of y]] dot_lneg
  5045     by arith
  5046   finally show ?thesis ..
  5047 qed
  5048 
  5049 lemma norm_triangle_eq: "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *s y = norm y *s x"
  5050 proof-
  5051   {assume x: "x =0 \<or> y =0"
  5052     hence ?thesis by (cases "x=0", simp_all add: norm_0)}
  5053   moreover
  5054   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
  5055     hence "norm x \<noteq> 0" "norm y \<noteq> 0"
  5056       by (simp_all add: norm_eq_0)
  5057     hence n: "norm x > 0" "norm y > 0" 
  5058       using norm_pos_le[of x] norm_pos_le[of y]
  5059       by arith+
  5060     have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
  5061     have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
  5062       apply (rule th) using n norm_pos_le[of "x + y"]
  5063       by arith
  5064     also have "\<dots> \<longleftrightarrow> norm x *s y = norm y *s x"
  5065       unfolding norm_cauchy_schwarz_eq[symmetric]
  5066       unfolding norm_pow_2 dot_ladd dot_radd
  5067       by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym ring_simps)
  5068     finally have ?thesis .}
  5069   ultimately show ?thesis by blast
  5070 qed
  5071 
  5072 (* Collinearity.*)
  5073 
  5074 definition "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *s u)"
  5075 
  5076 lemma collinear_empty:  "collinear {}" by (simp add: collinear_def)
  5077 
  5078 lemma collinear_sing: "collinear {(x::'a::ring_1^'n)}" 
  5079   apply (simp add: collinear_def)
  5080   apply (rule exI[where x=0])
  5081   by simp
  5082 
  5083 lemma collinear_2: "collinear {(x::'a::ring_1^'n),y}"
  5084   apply (simp add: collinear_def)
  5085   apply (rule exI[where x="x - y"])
  5086   apply auto
  5087   apply (rule exI[where x=0], simp)
  5088   apply (rule exI[where x=1], simp)
  5089   apply (rule exI[where x="- 1"], simp add: vector_sneg_minus1[symmetric])
  5090   apply (rule exI[where x=0], simp)
  5091   done
  5092 
  5093 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")
  5094 proof-
  5095   {assume "x=0 \<or> y = 0" hence ?thesis 
  5096       by (cases "x = 0", simp_all add: collinear_2 insert_commute)}
  5097   moreover
  5098   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
  5099     {assume h: "?lhs"
  5100       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
  5101       from u[rule_format, of x 0] u[rule_format, of y 0]
  5102       obtain cx and cy where 
  5103 	cx: "x = cx*s u" and cy: "y = cy*s u"
  5104 	by auto
  5105       from cx x have cx0: "cx \<noteq> 0" by auto
  5106       from cy y have cy0: "cy \<noteq> 0" by auto
  5107       let ?d = "cy / cx"
  5108       from cx cy cx0 have "y = ?d *s x" 
  5109 	by (simp add: vector_smult_assoc)
  5110       hence ?rhs using x y by blast}
  5111     moreover
  5112     {assume h: "?rhs"
  5113       then obtain c where c: "y = c*s x" using x y by blast
  5114       have ?lhs unfolding collinear_def c
  5115 	apply (rule exI[where x=x])
  5116 	apply auto
  5117 	apply (rule exI[where x=0], simp)
  5118 	apply (rule exI[where x="- 1"], simp only: vector_smult_lneg vector_smult_lid)
  5119 	apply (rule exI[where x= "-c"], simp only: vector_smult_lneg)
  5120 	apply (rule exI[where x=1], simp)
  5121 	apply (rule exI[where x=0], simp)
  5122 	apply (rule exI[where x="1 - c"], simp add: vector_smult_lneg vector_sub_rdistrib)
  5123 	apply (rule exI[where x="c - 1"], simp add: vector_smult_lneg vector_sub_rdistrib)
  5124 	apply (rule exI[where x=0], simp)
  5125 	done}
  5126     ultimately have ?thesis by blast}
  5127   ultimately show ?thesis by blast
  5128 qed
  5129 
  5130 lemma norm_cauchy_schwarz_equal: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {(0::real^'n),x,y}"
  5131 unfolding norm_cauchy_schwarz_abs_eq
  5132 apply (cases "x=0", simp_all add: collinear_2 norm_0)
  5133 apply (cases "y=0", simp_all add: collinear_2 norm_0 insert_commute)
  5134 unfolding collinear_lemma
  5135 apply simp
  5136 apply (subgoal_tac "norm x \<noteq> 0")
  5137 apply (subgoal_tac "norm y \<noteq> 0")
  5138 apply (rule iffI)
  5139 apply (cases "norm x *s y = norm y *s x")
  5140 apply (rule exI[where x="(1/norm x) * norm y"])
  5141 apply (drule sym)
  5142 unfolding vector_smult_assoc[symmetric]
  5143 apply (simp add: vector_smult_assoc field_simps)
  5144 apply (rule exI[where x="(1/norm x) * - norm y"])
  5145 apply clarify
  5146 apply (drule sym)
  5147 unfolding vector_smult_assoc[symmetric]
  5148 apply (simp add: vector_smult_assoc field_simps)
  5149 apply (erule exE)
  5150 apply (erule ssubst)
  5151 unfolding vector_smult_assoc
  5152 unfolding norm_mul
  5153 apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
  5154 apply (case_tac "c <= 0", simp add: ring_simps)
  5155 apply (simp add: ring_simps)
  5156 apply (case_tac "c <= 0", simp add: ring_simps)
  5157 apply (simp add: ring_simps)
  5158 apply (simp add: norm_eq_0)
  5159 apply (simp add: norm_eq_0)
  5160 done
  5161 
  5162 end