src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
author huffman
Thu Aug 11 09:11:15 2011 -0700 (2011-08-11)
changeset 44165 d26a45f3c835
parent 44140 2c10c35dd4be
child 44166 d12d89a66742
permissions -rw-r--r--
remove lemma stupid_ext
     1 
     2 header {*Instanciates the finite cartesian product of euclidean spaces as a euclidean space.*}
     3 
     4 theory Cartesian_Euclidean_Space
     5 imports Finite_Cartesian_Product Integration
     6 begin
     7 
     8 lemma delta_mult_idempotent:
     9   "(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)
    10 
    11 lemma setsum_Plus:
    12   "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
    13     (\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))"
    14   unfolding Plus_def
    15   by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)
    16 
    17 lemma setsum_UNIV_sum:
    18   fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
    19   shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
    20   apply (subst UNIV_Plus_UNIV [symmetric])
    21   apply (rule setsum_Plus [OF finite finite])
    22   done
    23 
    24 lemma setsum_mult_product:
    25   "setsum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
    26   unfolding sumr_group[of h B A, unfolded atLeast0LessThan, symmetric]
    27 proof (rule setsum_cong, simp, rule setsum_reindex_cong)
    28   fix i show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
    29   show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
    30   proof safe
    31     fix j assume "j \<in> {i * B..<i * B + B}"
    32     thus "j \<in> (\<lambda>j. j + i * B) ` {..<B}"
    33       by (auto intro!: image_eqI[of _ _ "j - i * B"])
    34   qed simp
    35 qed simp
    36 
    37 subsection{* Basic componentwise operations on vectors. *}
    38 
    39 instantiation vec :: (times, finite) times
    40 begin
    41   definition "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
    42   instance ..
    43 end
    44 
    45 instantiation vec :: (one, finite) one
    46 begin
    47   definition "1 \<equiv> (\<chi> i. 1)"
    48   instance ..
    49 end
    50 
    51 instantiation vec :: (ord, finite) ord
    52 begin
    53   definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"
    54   definition "x < y \<longleftrightarrow> (\<forall>i. x$i < y$i)"
    55   instance ..
    56 end
    57 
    58 text{* The ordering on one-dimensional vectors is linear. *}
    59 
    60 class cart_one = assumes UNIV_one: "card (UNIV \<Colon> 'a set) = Suc 0"
    61 begin
    62   subclass finite
    63   proof from UNIV_one show "finite (UNIV :: 'a set)"
    64       by (auto intro!: card_ge_0_finite) qed
    65 end
    66 
    67 instantiation vec :: (linorder,cart_one) linorder begin
    68 instance proof
    69   guess a B using UNIV_one[where 'a='b] unfolding card_Suc_eq apply- by(erule exE)+
    70   hence *:"UNIV = {a}" by auto
    71   have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P a" unfolding * by auto hence all:"\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a" by auto
    72   fix x y z::"'a^'b::cart_one" note * = less_eq_vec_def less_vec_def all vec_eq_iff
    73   show "x\<le>x" "(x < y) = (x \<le> y \<and> \<not> y \<le> x)" "x\<le>y \<or> y\<le>x" unfolding * by(auto simp only:field_simps)
    74   { assume "x\<le>y" "y\<le>z" thus "x\<le>z" unfolding * by(auto simp only:field_simps) }
    75   { assume "x\<le>y" "y\<le>x" thus "x=y" unfolding * by(auto simp only:field_simps) }
    76 qed end
    77 
    78 text{* Constant Vectors *} 
    79 
    80 definition "vec x = (\<chi> i. x)"
    81 
    82 text{* Also the scalar-vector multiplication. *}
    83 
    84 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
    85   where "c *s x = (\<chi> i. c * (x$i))"
    86 
    87 subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
    88 
    89 method_setup vector = {*
    90 let
    91   val ss1 = HOL_basic_ss addsimps [@{thm setsum_addf} RS sym,
    92   @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
    93   @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
    94   val ss2 = @{simpset} addsimps
    95              [@{thm plus_vec_def}, @{thm times_vec_def},
    96               @{thm minus_vec_def}, @{thm uminus_vec_def},
    97               @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
    98               @{thm scaleR_vec_def},
    99               @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}]
   100  fun vector_arith_tac ths =
   101    simp_tac ss1
   102    THEN' (fn i => rtac @{thm setsum_cong2} i
   103          ORELSE rtac @{thm setsum_0'} i
   104          ORELSE simp_tac (HOL_basic_ss addsimps [@{thm vec_eq_iff}]) i)
   105    (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
   106    THEN' asm_full_simp_tac (ss2 addsimps ths)
   107  in
   108   Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths)))
   109  end
   110 *} "lift trivial vector statements to real arith statements"
   111 
   112 lemma vec_0[simp]: "vec 0 = 0" by (vector zero_vec_def)
   113 lemma vec_1[simp]: "vec 1 = 1" by (vector one_vec_def)
   114 
   115 lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
   116 
   117 lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
   118 
   119 lemma vec_add: "vec(x + y) = vec x + vec y"  by (vector vec_def)
   120 lemma vec_sub: "vec(x - y) = vec x - vec y" by (vector vec_def)
   121 lemma vec_cmul: "vec(c * x) = c *s vec x " by (vector vec_def)
   122 lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def)
   123 
   124 lemma vec_setsum: assumes fS: "finite S"
   125   shows "vec(setsum f S) = setsum (vec o f) S"
   126   apply (induct rule: finite_induct[OF fS])
   127   apply (simp)
   128   apply (auto simp add: vec_add)
   129   done
   130 
   131 text{* Obvious "component-pushing". *}
   132 
   133 lemma vec_component [simp]: "vec x $ i = x"
   134   by (vector vec_def)
   135 
   136 lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
   137   by vector
   138 
   139 lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
   140   by vector
   141 
   142 lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
   143 
   144 lemmas vector_component =
   145   vec_component vector_add_component vector_mult_component
   146   vector_smult_component vector_minus_component vector_uminus_component
   147   vector_scaleR_component cond_component
   148 
   149 subsection {* Some frequently useful arithmetic lemmas over vectors. *}
   150 
   151 instance vec :: (semigroup_mult, finite) semigroup_mult
   152   by default (vector mult_assoc)
   153 
   154 instance vec :: (monoid_mult, finite) monoid_mult
   155   by default vector+
   156 
   157 instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
   158   by default (vector mult_commute)
   159 
   160 instance vec :: (ab_semigroup_idem_mult, finite) ab_semigroup_idem_mult
   161   by default (vector mult_idem)
   162 
   163 instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
   164   by default vector
   165 
   166 instance vec :: (semiring, finite) semiring
   167   by default (vector field_simps)+
   168 
   169 instance vec :: (semiring_0, finite) semiring_0
   170   by default (vector field_simps)+
   171 instance vec :: (semiring_1, finite) semiring_1
   172   by default vector
   173 instance vec :: (comm_semiring, finite) comm_semiring
   174   by default (vector field_simps)+
   175 
   176 instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
   177 instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
   178 instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
   179 instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
   180 instance vec :: (ring, finite) ring ..
   181 instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
   182 instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
   183 
   184 instance vec :: (ring_1, finite) ring_1 ..
   185 
   186 instance vec :: (real_algebra, finite) real_algebra
   187   apply intro_classes
   188   apply (simp_all add: vec_eq_iff)
   189   done
   190 
   191 instance vec :: (real_algebra_1, finite) real_algebra_1 ..
   192 
   193 lemma of_nat_index:
   194   "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
   195   apply (induct n)
   196   apply vector
   197   apply vector
   198   done
   199 
   200 lemma one_index[simp]:
   201   "(1 :: 'a::one ^'n)$i = 1" by vector
   202 
   203 instance vec :: (semiring_char_0, finite) semiring_char_0
   204 proof
   205   fix m n :: nat
   206   show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
   207     by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
   208 qed
   209 
   210 instance vec :: (comm_ring_1, finite) comm_ring_1 ..
   211 instance vec :: (ring_char_0, finite) ring_char_0 ..
   212 
   213 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
   214   by (vector mult_assoc)
   215 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
   216   by (vector field_simps)
   217 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
   218   by (vector field_simps)
   219 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
   220 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
   221 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
   222   by (vector field_simps)
   223 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
   224 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
   225 lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
   226 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
   227 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
   228   by (vector field_simps)
   229 
   230 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
   231   by (simp add: vec_eq_iff)
   232 
   233 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
   234 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
   235   by vector
   236 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
   237   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
   238 lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
   239   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
   240 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
   241   by (metis vector_mul_lcancel)
   242 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
   243   by (metis vector_mul_rcancel)
   244 
   245 lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x"
   246   apply (simp add: norm_vec_def)
   247   apply (rule member_le_setL2, simp_all)
   248   done
   249 
   250 lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x$i\<bar> <= e"
   251   by (metis component_le_norm_cart order_trans)
   252 
   253 lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
   254   by (metis component_le_norm_cart basic_trans_rules(21))
   255 
   256 lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
   257   by (simp add: norm_vec_def setL2_le_setsum)
   258 
   259 lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
   260   unfolding scaleR_vec_def vector_scalar_mult_def by simp
   261 
   262 lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
   263   unfolding dist_norm scalar_mult_eq_scaleR
   264   unfolding scaleR_right_diff_distrib[symmetric] by simp
   265 
   266 lemma setsum_component [simp]:
   267   fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
   268   shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
   269   by (cases "finite S", induct S set: finite, simp_all)
   270 
   271 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
   272   by (simp add: vec_eq_iff)
   273 
   274 lemma setsum_cmul:
   275   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
   276   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
   277   by (simp add: vec_eq_iff setsum_right_distrib)
   278 
   279 (* TODO: use setsum_norm_allsubsets_bound *)
   280 lemma setsum_norm_allsubsets_bound_cart:
   281   fixes f:: "'a \<Rightarrow> real ^'n"
   282   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
   283   shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
   284 proof-
   285   let ?d = "real CARD('n)"
   286   let ?nf = "\<lambda>x. norm (f x)"
   287   let ?U = "UNIV :: 'n set"
   288   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"
   289     by (rule setsum_commute)
   290   have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
   291   have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
   292     apply (rule setsum_mono)    by (rule norm_le_l1_cart)
   293   also have "\<dots> \<le> 2 * ?d * e"
   294     unfolding th0 th1
   295   proof(rule setsum_bounded)
   296     fix i assume i: "i \<in> ?U"
   297     let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
   298     let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
   299     have thp: "P = ?Pp \<union> ?Pn" by auto
   300     have thp0: "?Pp \<inter> ?Pn ={}" by auto
   301     have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
   302     have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
   303       using component_le_norm_cart[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
   304       by (auto intro: abs_le_D1)
   305     have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
   306       using component_le_norm_cart[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
   307       by (auto simp add: setsum_negf intro: abs_le_D1)
   308     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"
   309       apply (subst thp)
   310       apply (rule setsum_Un_zero)
   311       using fP thp0 by auto
   312     also have "\<dots> \<le> 2*e" using Pne Ppe by arith
   313     finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
   314   qed
   315   finally show ?thesis .
   316 qed
   317 
   318 lemma if_distr: "(if P then f else g) $ i = (if P then f $ i else g $ i)" by simp
   319 
   320 lemma split_dimensions'[consumes 1]:
   321   assumes "k < DIM('a::euclidean_space^'b)"
   322   obtains i j where "i < CARD('b::finite)" and "j < DIM('a::euclidean_space)" and "k = j + i * DIM('a::euclidean_space)"
   323 using split_times_into_modulo[OF assms[simplified]] .
   324 
   325 lemma cart_euclidean_bound[intro]:
   326   assumes j:"j < DIM('a::euclidean_space)"
   327   shows "j + \<pi>' (i::'b::finite) * DIM('a) < CARD('b) * DIM('a::euclidean_space)"
   328   using linear_less_than_times[OF pi'_range j, of i] .
   329 
   330 lemma (in euclidean_space) forall_CARD_DIM:
   331   "(\<forall>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<forall>(i::'b::finite) j. j<DIM('a) \<longrightarrow> P (j + \<pi>' i * DIM('a)))"
   332    (is "?l \<longleftrightarrow> ?r")
   333 proof (safe elim!: split_times_into_modulo)
   334   fix i :: 'b and j assume "j < DIM('a)"
   335   note linear_less_than_times[OF pi'_range[of i] this]
   336   moreover assume "?l"
   337   ultimately show "P (j + \<pi>' i * DIM('a))" by auto
   338 next
   339   fix i j assume "i < CARD('b)" "j < DIM('a)" and "?r"
   340   from `?r`[rule_format, OF `j < DIM('a)`, of "\<pi> i"] `i < CARD('b)`
   341   show "P (j + i * DIM('a))" by simp
   342 qed
   343 
   344 lemma (in euclidean_space) exists_CARD_DIM:
   345   "(\<exists>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<exists>i::'b::finite. \<exists>j<DIM('a). P (j + \<pi>' i * DIM('a)))"
   346   using forall_CARD_DIM[where 'b='b, of "\<lambda>x. \<not> P x"] by blast
   347 
   348 lemma forall_CARD:
   349   "(\<forall>i<CARD('b). P i) \<longleftrightarrow> (\<forall>i::'b::finite. P (\<pi>' i))"
   350   using forall_CARD_DIM[where 'a=real, of P] by simp
   351 
   352 lemma exists_CARD:
   353   "(\<exists>i<CARD('b). P i) \<longleftrightarrow> (\<exists>i::'b::finite. P (\<pi>' i))"
   354   using exists_CARD_DIM[where 'a=real, of P] by simp
   355 
   356 lemmas cart_simps = forall_CARD_DIM exists_CARD_DIM forall_CARD exists_CARD
   357 
   358 lemma cart_euclidean_nth[simp]:
   359   fixes x :: "('a::euclidean_space, 'b::finite) vec"
   360   assumes j:"j < DIM('a)"
   361   shows "x $$ (j + \<pi>' i * DIM('a)) = x $ i $$ j"
   362   unfolding euclidean_component_def inner_vec_def basis_eq_pi'[OF j] if_distrib cond_application_beta
   363   by (simp add: setsum_cases)
   364 
   365 lemma real_euclidean_nth:
   366   fixes x :: "real^'n"
   367   shows "x $$ \<pi>' i = (x $ i :: real)"
   368   using cart_euclidean_nth[where 'a=real, of 0 x i] by simp
   369 
   370 lemmas nth_conv_component = real_euclidean_nth[symmetric]
   371 
   372 lemma mult_split_eq:
   373   fixes A :: nat assumes "x < A" "y < A"
   374   shows "x + i * A = y + j * A \<longleftrightarrow> x = y \<and> i = j"
   375 proof
   376   assume *: "x + i * A = y + j * A"
   377   { fix x y i j assume "i < j" "x < A" and *: "x + i * A = y + j * A"
   378     hence "x + i * A < Suc i * A" using `x < A` by simp
   379     also have "\<dots> \<le> j * A" using `i < j` unfolding mult_le_cancel2 by simp
   380     also have "\<dots> \<le> y + j * A" by simp
   381     finally have "i = j" using * by simp }
   382   note eq = this
   383 
   384   have "i = j"
   385   proof (cases rule: linorder_cases)
   386     assume "i < j" from eq[OF this `x < A` *] show "i = j" by simp
   387   next
   388     assume "j < i" from eq[OF this `y < A` *[symmetric]] show "i = j" by simp
   389   qed simp
   390   thus "x = y \<and> i = j" using * by simp
   391 qed simp
   392 
   393 instance vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
   394 proof
   395   fix x y::"'a^'b"
   396   show "(x \<le> y) = (\<forall>i<DIM(('a, 'b) vec). x $$ i \<le> y $$ i)"
   397     unfolding less_eq_vec_def apply(subst eucl_le) by (simp add: cart_simps)
   398   show"(x < y) = (\<forall>i<DIM(('a, 'b) vec). x $$ i < y $$ i)"
   399     unfolding less_vec_def apply(subst eucl_less) by (simp add: cart_simps)
   400 qed
   401 
   402 subsection{* Basis vectors in coordinate directions. *}
   403 
   404 definition "cart_basis k = (\<chi> i. if i = k then 1 else 0)"
   405 
   406 lemma basis_component [simp]: "cart_basis k $ i = (if k=i then 1 else 0)"
   407   unfolding cart_basis_def by simp
   408 
   409 lemma norm_basis[simp]:
   410   shows "norm (cart_basis k :: real ^'n) = 1"
   411   apply (simp add: cart_basis_def norm_eq_sqrt_inner) unfolding inner_vec_def
   412   apply (vector delta_mult_idempotent)
   413   using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"] by auto
   414 
   415 lemma norm_basis_1: "norm(cart_basis 1 :: real ^'n::{finite,one}) = 1"
   416   by (rule norm_basis)
   417 
   418 lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c"
   419   by (rule exI[where x="c *\<^sub>R cart_basis arbitrary"]) simp
   420 
   421 lemma vector_choose_dist: assumes e: "0 <= e"
   422   shows "\<exists>(y::real^'n). dist x y = e"
   423 proof-
   424   from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"
   425     by blast
   426   then have "dist x (x - c) = e" by (simp add: dist_norm)
   427   then show ?thesis by blast
   428 qed
   429 
   430 lemma basis_inj[intro]: "inj (cart_basis :: 'n \<Rightarrow> real ^'n)"
   431   by (simp add: inj_on_def vec_eq_iff)
   432 
   433 lemma basis_expansion:
   434   "setsum (\<lambda>i. (x$i) *s cart_basis i) UNIV = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
   435   by (auto simp add: vec_eq_iff if_distrib setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
   436 
   437 lemma smult_conv_scaleR: "c *s x = scaleR c x"
   438   unfolding vector_scalar_mult_def scaleR_vec_def by simp
   439 
   440 lemma basis_expansion':
   441   "setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) UNIV = x"
   442   by (rule basis_expansion [where 'a=real, unfolded smult_conv_scaleR])
   443 
   444 lemma basis_expansion_unique:
   445   "setsum (\<lambda>i. f i *s cart_basis i) UNIV = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i. f i = x$i)"
   446   by (simp add: vec_eq_iff setsum_delta if_distrib cong del: if_weak_cong)
   447 
   448 lemma dot_basis:
   449   shows "cart_basis i \<bullet> x = x$i" "x \<bullet> (cart_basis i) = (x$i)"
   450   by (auto simp add: inner_vec_def cart_basis_def cond_application_beta if_distrib setsum_delta
   451            cong del: if_weak_cong)
   452 
   453 lemma inner_basis:
   454   fixes x :: "'a::{real_inner, real_algebra_1} ^ 'n"
   455   shows "inner (cart_basis i) x = inner 1 (x $ i)"
   456     and "inner x (cart_basis i) = inner (x $ i) 1"
   457   unfolding inner_vec_def cart_basis_def
   458   by (auto simp add: cond_application_beta if_distrib setsum_delta cong del: if_weak_cong)
   459 
   460 lemma basis_eq_0: "cart_basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
   461   by (auto simp add: vec_eq_iff)
   462 
   463 lemma basis_nonzero:
   464   shows "cart_basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
   465   by (simp add: basis_eq_0)
   466 
   467 text {* some lemmas to map between Eucl and Cart *}
   468 lemma basis_real_n[simp]:"(basis (\<pi>' i)::real^'a) = cart_basis i"
   469   unfolding basis_vec_def using pi'_range[where 'n='a]
   470   by (auto simp: vec_eq_iff)
   471 
   472 subsection {* Orthogonality on cartesian products *}
   473 
   474 lemma orthogonal_basis:
   475   shows "orthogonal (cart_basis i) x \<longleftrightarrow> x$i = (0::real)"
   476   by (auto simp add: orthogonal_def inner_vec_def cart_basis_def if_distrib
   477                      cond_application_beta setsum_delta cong del: if_weak_cong)
   478 
   479 lemma orthogonal_basis_basis:
   480   shows "orthogonal (cart_basis i :: real^'n) (cart_basis j) \<longleftrightarrow> i \<noteq> j"
   481   unfolding orthogonal_basis[of i] basis_component[of j] by simp
   482 
   483 subsection {* Linearity on cartesian products *}
   484 
   485 lemma linear_vmul_component:
   486   assumes lf: "linear f"
   487   shows "linear (\<lambda>x. f x $ k *\<^sub>R v)"
   488   using lf
   489   by (auto simp add: linear_def algebra_simps)
   490 
   491 
   492 subsection{* Adjoints on cartesian products *}
   493 
   494 text {* TODO: The following lemmas about adjoints should hold for any
   495 Hilbert space (i.e. complete inner product space).
   496 (see \url{http://en.wikipedia.org/wiki/Hermitian_adjoint})
   497 *}
   498 
   499 lemma adjoint_works_lemma:
   500   fixes f:: "real ^'n \<Rightarrow> real ^'m"
   501   assumes lf: "linear f"
   502   shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
   503 proof-
   504   let ?N = "UNIV :: 'n set"
   505   let ?M = "UNIV :: 'm set"
   506   have fN: "finite ?N" by simp
   507   have fM: "finite ?M" by simp
   508   {fix y:: "real ^ 'm"
   509     let ?w = "(\<chi> i. (f (cart_basis i) \<bullet> y)) :: real ^ 'n"
   510     {fix x
   511       have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) ?N) \<bullet> y"
   512         by (simp only: basis_expansion')
   513       also have "\<dots> = (setsum (\<lambda>i. (x$i) *\<^sub>R f (cart_basis i)) ?N) \<bullet> y"
   514         unfolding linear_setsum[OF lf fN]
   515         by (simp add: linear_cmul[OF lf])
   516       finally have "f x \<bullet> y = x \<bullet> ?w"
   517         apply (simp only: )
   518         apply (simp add: inner_vec_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps)
   519         done}
   520   }
   521   then show ?thesis unfolding adjoint_def
   522     some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
   523     using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
   524     by metis
   525 qed
   526 
   527 lemma adjoint_works:
   528   fixes f:: "real ^'n \<Rightarrow> real ^'m"
   529   assumes lf: "linear f"
   530   shows "x \<bullet> adjoint f y = f x \<bullet> y"
   531   using adjoint_works_lemma[OF lf] by metis
   532 
   533 lemma adjoint_linear:
   534   fixes f:: "real ^'n \<Rightarrow> real ^'m"
   535   assumes lf: "linear f"
   536   shows "linear (adjoint f)"
   537   unfolding linear_def vector_eq_ldot[where 'a="real^'n", symmetric] apply safe
   538   unfolding inner_simps smult_conv_scaleR adjoint_works[OF lf] by auto
   539 
   540 lemma adjoint_clauses:
   541   fixes f:: "real ^'n \<Rightarrow> real ^'m"
   542   assumes lf: "linear f"
   543   shows "x \<bullet> adjoint f y = f x \<bullet> y"
   544   and "adjoint f y \<bullet> x = y \<bullet> f x"
   545   by (simp_all add: adjoint_works[OF lf] inner_commute)
   546 
   547 lemma adjoint_adjoint:
   548   fixes f:: "real ^'n \<Rightarrow> real ^'m"
   549   assumes lf: "linear f"
   550   shows "adjoint (adjoint f) = f"
   551   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
   552 
   553 
   554 subsection {* Matrix operations *}
   555 
   556 text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
   557 
   558 definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)
   559   where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
   560 
   561 definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)
   562   where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
   563 
   564 definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)
   565   where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
   566 
   567 definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
   568 definition transpose where 
   569   "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
   570 definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
   571 definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
   572 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
   573 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
   574 
   575 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
   576 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
   577   by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps)
   578 
   579 lemma matrix_mul_lid:
   580   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
   581   shows "mat 1 ** A = A"
   582   apply (simp add: matrix_matrix_mult_def mat_def)
   583   apply vector
   584   by (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite]  mult_1_left mult_zero_left if_True UNIV_I)
   585 
   586 
   587 lemma matrix_mul_rid:
   588   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
   589   shows "A ** mat 1 = A"
   590   apply (simp add: matrix_matrix_mult_def mat_def)
   591   apply vector
   592   by (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite]  mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
   593 
   594 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
   595   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
   596   apply (subst setsum_commute)
   597   apply simp
   598   done
   599 
   600 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
   601   apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
   602   apply (subst setsum_commute)
   603   apply simp
   604   done
   605 
   606 lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
   607   apply (vector matrix_vector_mult_def mat_def)
   608   by (simp add: if_distrib cond_application_beta
   609     setsum_delta' cong del: if_weak_cong)
   610 
   611 lemma matrix_transpose_mul: "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
   612   by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute)
   613 
   614 lemma matrix_eq:
   615   fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
   616   shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
   617   apply auto
   618   apply (subst vec_eq_iff)
   619   apply clarify
   620   apply (clarsimp simp add: matrix_vector_mult_def cart_basis_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
   621   apply (erule_tac x="cart_basis ia" in allE)
   622   apply (erule_tac x="i" in allE)
   623   by (auto simp add: cart_basis_def if_distrib cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong)
   624 
   625 lemma matrix_vector_mul_component:
   626   shows "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
   627   by (simp add: matrix_vector_mult_def inner_vec_def)
   628 
   629 lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
   630   apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
   631   apply (subst setsum_commute)
   632   by simp
   633 
   634 lemma transpose_mat: "transpose (mat n) = mat n"
   635   by (vector transpose_def mat_def)
   636 
   637 lemma transpose_transpose: "transpose(transpose A) = A"
   638   by (vector transpose_def)
   639 
   640 lemma row_transpose:
   641   fixes A:: "'a::semiring_1^_^_"
   642   shows "row i (transpose A) = column i A"
   643   by (simp add: row_def column_def transpose_def vec_eq_iff)
   644 
   645 lemma column_transpose:
   646   fixes A:: "'a::semiring_1^_^_"
   647   shows "column i (transpose A) = row i A"
   648   by (simp add: row_def column_def transpose_def vec_eq_iff)
   649 
   650 lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
   651 by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
   652 
   653 lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" by (metis transpose_transpose rows_transpose)
   654 
   655 text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
   656 
   657 lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
   658   by (simp add: matrix_vector_mult_def inner_vec_def)
   659 
   660 lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
   661   by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
   662 
   663 lemma vector_componentwise:
   664   "(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (cart_basis i :: 'a^'n)$j) (UNIV :: 'n set))"
   665   apply (subst basis_expansion[symmetric])
   666   by (vector vec_eq_iff setsum_component)
   667 
   668 lemma linear_componentwise:
   669   fixes f:: "real ^'m \<Rightarrow> real ^ _"
   670   assumes lf: "linear f"
   671   shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (cart_basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
   672 proof-
   673   let ?M = "(UNIV :: 'm set)"
   674   let ?N = "(UNIV :: 'n set)"
   675   have fM: "finite ?M" by simp
   676   have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (cart_basis i) ) ?M)$j"
   677     unfolding vector_smult_component[symmetric] smult_conv_scaleR
   678     unfolding setsum_component[of "(\<lambda>i.(x$i) *\<^sub>R f (cart_basis i :: real^'m))" ?M]
   679     ..
   680   then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion' ..
   681 qed
   682 
   683 text{* Inverse matrices  (not necessarily square) *}
   684 
   685 definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
   686 
   687 definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
   688         (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
   689 
   690 text{* Correspondence between matrices and linear operators. *}
   691 
   692 definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
   693 where "matrix f = (\<chi> i j. (f(cart_basis j))$i)"
   694 
   695 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
   696   by (simp add: linear_def matrix_vector_mult_def vec_eq_iff field_simps setsum_right_distrib setsum_addf)
   697 
   698 lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::real ^ 'n)"
   699 apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
   700 apply clarify
   701 apply (rule linear_componentwise[OF lf, symmetric])
   702 done
   703 
   704 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))" by (simp add: ext matrix_works)
   705 
   706 lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
   707   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
   708 
   709 lemma matrix_compose:
   710   assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
   711   and lg: "linear (g::real^'m \<Rightarrow> real^_)"
   712   shows "matrix (g o f) = matrix g ** matrix f"
   713   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
   714   by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
   715 
   716 lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
   717   by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult_commute)
   718 
   719 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
   720   apply (rule adjoint_unique)
   721   apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
   722   apply (subst setsum_commute)
   723   apply (auto simp add: mult_ac)
   724   done
   725 
   726 lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
   727   shows "matrix(adjoint f) = transpose(matrix f)"
   728   apply (subst matrix_vector_mul[OF lf])
   729   unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
   730 
   731 section {* lambda skolemization on cartesian products *}
   732 
   733 (* FIXME: rename do choice_cart *)
   734 
   735 lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
   736    (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
   737 proof-
   738   let ?S = "(UNIV :: 'n set)"
   739   {assume H: "?rhs"
   740     then have ?lhs by auto}
   741   moreover
   742   {assume H: "?lhs"
   743     then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
   744     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
   745     {fix i
   746       from f have "P i (f i)" by metis
   747       then have "P i (?x $ i)" by auto
   748     }
   749     hence "\<forall>i. P i (?x$i)" by metis
   750     hence ?rhs by metis }
   751   ultimately show ?thesis by metis
   752 qed
   753 
   754 subsection {* Standard bases are a spanning set, and obviously finite. *}
   755 
   756 lemma span_stdbasis:"span {cart_basis i :: real^'n | i. i \<in> (UNIV :: 'n set)} = UNIV"
   757 apply (rule set_eqI)
   758 apply auto
   759 apply (subst basis_expansion'[symmetric])
   760 apply (rule span_setsum)
   761 apply simp
   762 apply auto
   763 apply (rule span_mul)
   764 apply (rule span_superset)
   765 apply (auto simp add: Collect_def mem_def)
   766 done
   767 
   768 lemma finite_stdbasis: "finite {cart_basis i ::real^'n |i. i\<in> (UNIV:: 'n set)}" (is "finite ?S")
   769 proof-
   770   have eq: "?S = cart_basis ` UNIV" by blast
   771   show ?thesis unfolding eq by auto
   772 qed
   773 
   774 lemma card_stdbasis: "card {cart_basis i ::real^'n |i. i\<in> (UNIV :: 'n set)} = CARD('n)" (is "card ?S = _")
   775 proof-
   776   have eq: "?S = cart_basis ` UNIV" by blast
   777   show ?thesis unfolding eq using card_image[OF basis_inj] by simp
   778 qed
   779 
   780 
   781 lemma independent_stdbasis_lemma:
   782   assumes x: "(x::real ^ 'n) \<in> span (cart_basis ` S)"
   783   and iS: "i \<notin> S"
   784   shows "(x$i) = 0"
   785 proof-
   786   let ?U = "UNIV :: 'n set"
   787   let ?B = "cart_basis ` S"
   788   let ?P = "\<lambda>(x::real^_). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0"
   789  {fix x::"real^_" assume xS: "x\<in> ?B"
   790    from xS have "?P x" by auto}
   791  moreover
   792  have "subspace ?P"
   793    by (auto simp add: subspace_def Collect_def mem_def)
   794  ultimately show ?thesis
   795    using x span_induct[of ?B ?P x] iS by blast
   796 qed
   797 
   798 lemma independent_stdbasis: "independent {cart_basis i ::real^'n |i. i\<in> (UNIV :: 'n set)}"
   799 proof-
   800   let ?I = "UNIV :: 'n set"
   801   let ?b = "cart_basis :: _ \<Rightarrow> real ^'n"
   802   let ?B = "?b ` ?I"
   803   have eq: "{?b i|i. i \<in> ?I} = ?B"
   804     by auto
   805   {assume d: "dependent ?B"
   806     then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"
   807       unfolding dependent_def by auto
   808     have eq1: "?B - {?b k} = ?B - ?b ` {k}"  by simp
   809     have eq2: "?B - {?b k} = ?b ` (?I - {k})"
   810       unfolding eq1
   811       apply (rule inj_on_image_set_diff[symmetric])
   812       apply (rule basis_inj) using k(1) by auto
   813     from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
   814     from independent_stdbasis_lemma[OF th0, of k, simplified]
   815     have False by simp}
   816   then show ?thesis unfolding eq dependent_def ..
   817 qed
   818 
   819 lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
   820   unfolding inner_simps smult_conv_scaleR by auto
   821 
   822 lemma linear_eq_stdbasis_cart:
   823   assumes lf: "linear (f::real^'m \<Rightarrow> _)" and lg: "linear g"
   824   and fg: "\<forall>i. f (cart_basis i) = g(cart_basis i)"
   825   shows "f = g"
   826 proof-
   827   let ?U = "UNIV :: 'm set"
   828   let ?I = "{cart_basis i:: real^'m|i. i \<in> ?U}"
   829   {fix x assume x: "x \<in> (UNIV :: (real^'m) set)"
   830     from equalityD2[OF span_stdbasis]
   831     have IU: " (UNIV :: (real^'m) set) \<subseteq> span ?I" by blast
   832     from linear_eq[OF lf lg IU] fg x
   833     have "f x = g x" unfolding Collect_def  Ball_def mem_def by metis}
   834   then show ?thesis by (auto intro: ext)
   835 qed
   836 
   837 lemma bilinear_eq_stdbasis_cart:
   838   assumes bf: "bilinear (f:: real^'m \<Rightarrow> real^'n \<Rightarrow> _)"
   839   and bg: "bilinear g"
   840   and fg: "\<forall>i j. f (cart_basis i) (cart_basis j) = g (cart_basis i) (cart_basis j)"
   841   shows "f = g"
   842 proof-
   843   from fg have th: "\<forall>x \<in> {cart_basis i| i. i\<in> (UNIV :: 'm set)}. \<forall>y\<in>  {cart_basis j |j. j \<in> (UNIV :: 'n set)}. f x y = g x y" by blast
   844   from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext)
   845 qed
   846 
   847 lemma left_invertible_transpose:
   848   "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
   849   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
   850 
   851 lemma right_invertible_transpose:
   852   "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
   853   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
   854 
   855 lemma matrix_left_invertible_injective:
   856 "(\<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)"
   857 proof-
   858   {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
   859     from xy have "B*v (A *v x) = B *v (A*v y)" by simp
   860     hence "x = y"
   861       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .}
   862   moreover
   863   {assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
   864     hence i: "inj (op *v A)" unfolding inj_on_def by auto
   865     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
   866     obtain g where g: "linear g" "g o op *v A = id" by blast
   867     have "matrix g ** A = mat 1"
   868       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
   869       using g(2) by (simp add: fun_eq_iff)
   870     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast}
   871   ultimately show ?thesis by blast
   872 qed
   873 
   874 lemma matrix_left_invertible_ker:
   875   "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
   876   unfolding matrix_left_invertible_injective
   877   using linear_injective_0[OF matrix_vector_mul_linear, of A]
   878   by (simp add: inj_on_def)
   879 
   880 lemma matrix_right_invertible_surjective:
   881 "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
   882 proof-
   883   {fix B :: "real ^'m^'n"  assume AB: "A ** B = mat 1"
   884     {fix x :: "real ^ 'm"
   885       have "A *v (B *v x) = x"
   886         by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)}
   887     hence "surj (op *v A)" unfolding surj_def by metis }
   888   moreover
   889   {assume sf: "surj (op *v A)"
   890     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
   891     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
   892       by blast
   893 
   894     have "A ** (matrix g) = mat 1"
   895       unfolding matrix_eq  matrix_vector_mul_lid
   896         matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
   897       using g(2) unfolding o_def fun_eq_iff id_def
   898       .
   899     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
   900   }
   901   ultimately show ?thesis unfolding surj_def by blast
   902 qed
   903 
   904 lemma matrix_left_invertible_independent_columns:
   905   fixes A :: "real^'n^'m"
   906   shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
   907    (is "?lhs \<longleftrightarrow> ?rhs")
   908 proof-
   909   let ?U = "UNIV :: 'n set"
   910   {assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
   911     {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0"
   912       and i: "i \<in> ?U"
   913       let ?x = "\<chi> i. c i"
   914       have th0:"A *v ?x = 0"
   915         using c
   916         unfolding matrix_mult_vsum vec_eq_iff
   917         by auto
   918       from k[rule_format, OF th0] i
   919       have "c i = 0" by (vector vec_eq_iff)}
   920     hence ?rhs by blast}
   921   moreover
   922   {assume H: ?rhs
   923     {fix x assume x: "A *v x = 0"
   924       let ?c = "\<lambda>i. ((x$i ):: real)"
   925       from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
   926       have "x = 0" by vector}}
   927   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
   928 qed
   929 
   930 lemma matrix_right_invertible_independent_rows:
   931   fixes A :: "real^'n^'m"
   932   shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
   933   unfolding left_invertible_transpose[symmetric]
   934     matrix_left_invertible_independent_columns
   935   by (simp add: column_transpose)
   936 
   937 lemma matrix_right_invertible_span_columns:
   938   "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")
   939 proof-
   940   let ?U = "UNIV :: 'm set"
   941   have fU: "finite ?U" by simp
   942   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
   943     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
   944     apply (subst eq_commute) ..
   945   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
   946   {assume h: ?lhs
   947     {fix x:: "real ^'n"
   948         from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m"
   949           where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
   950         have "x \<in> span (columns A)"
   951           unfolding y[symmetric]
   952           apply (rule span_setsum[OF fU])
   953           apply clarify
   954           unfolding smult_conv_scaleR
   955           apply (rule span_mul)
   956           apply (rule span_superset)
   957           unfolding columns_def
   958           by blast}
   959     then have ?rhs unfolding rhseq by blast}
   960   moreover
   961   {assume h:?rhs
   962     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
   963     {fix y have "?P y"
   964       proof(rule span_induct_alt[of ?P "columns A", folded smult_conv_scaleR])
   965         show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
   966           by (rule exI[where x=0], simp)
   967       next
   968         fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
   969         from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
   970           unfolding columns_def by blast
   971         from y2 obtain x:: "real ^'m" where
   972           x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
   973         let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
   974         show "?P (c*s y1 + y2)"
   975           proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib right_distrib cond_application_beta cong del: if_weak_cong)
   976             fix j
   977             have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
   978            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)
   979               by (simp add: field_simps)
   980             have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
   981            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"
   982               apply (rule setsum_cong[OF refl])
   983               using th by blast
   984             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"
   985               by (simp add: setsum_addf)
   986             also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
   987               unfolding setsum_delta[OF fU]
   988               using i(1) by simp
   989             finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
   990            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
   991           qed
   992         next
   993           show "y \<in> span (columns A)" unfolding h by blast
   994         qed}
   995     then have ?lhs unfolding lhseq ..}
   996   ultimately show ?thesis by blast
   997 qed
   998 
   999 lemma matrix_left_invertible_span_rows:
  1000   "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
  1001   unfolding right_invertible_transpose[symmetric]
  1002   unfolding columns_transpose[symmetric]
  1003   unfolding matrix_right_invertible_span_columns
  1004  ..
  1005 
  1006 text {* The same result in terms of square matrices. *}
  1007 
  1008 lemma matrix_left_right_inverse:
  1009   fixes A A' :: "real ^'n^'n"
  1010   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
  1011 proof-
  1012   {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"
  1013     have sA: "surj (op *v A)"
  1014       unfolding surj_def
  1015       apply clarify
  1016       apply (rule_tac x="(A' *v y)" in exI)
  1017       by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
  1018     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
  1019     obtain f' :: "real ^'n \<Rightarrow> real ^'n"
  1020       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
  1021     have th: "matrix f' ** A = mat 1"
  1022       by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
  1023     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
  1024     hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
  1025     hence "matrix f' ** A = A' ** A" by simp
  1026     hence "A' ** A = mat 1" by (simp add: th)}
  1027   then show ?thesis by blast
  1028 qed
  1029 
  1030 text {* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *}
  1031 
  1032 definition "rowvector v = (\<chi> i j. (v$j))"
  1033 
  1034 definition "columnvector v = (\<chi> i j. (v$i))"
  1035 
  1036 lemma transpose_columnvector:
  1037  "transpose(columnvector v) = rowvector v"
  1038   by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
  1039 
  1040 lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
  1041   by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
  1042 
  1043 lemma dot_rowvector_columnvector:
  1044   "columnvector (A *v v) = A ** columnvector v"
  1045   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
  1046 
  1047 lemma dot_matrix_product: "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
  1048   by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
  1049 
  1050 lemma dot_matrix_vector_mul:
  1051   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
  1052   shows "(A *v x) \<bullet> (B *v y) =
  1053       (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
  1054 unfolding dot_matrix_product transpose_columnvector[symmetric]
  1055   dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
  1056 
  1057 
  1058 lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in> (UNIV :: 'n set)}"
  1059   unfolding infnorm_def apply(rule arg_cong[where f=Sup]) apply safe
  1060   apply(rule_tac x="\<pi> i" in exI) defer
  1061   apply(rule_tac x="\<pi>' i" in exI) unfolding nth_conv_component using pi'_range by auto
  1062 
  1063 lemma infnorm_set_image_cart:
  1064   "{abs(x$i) |i. i\<in> (UNIV :: _ set)} =
  1065   (\<lambda>i. abs(x$i)) ` (UNIV)" by blast
  1066 
  1067 lemma infnorm_set_lemma_cart:
  1068   shows "finite {abs((x::'a::abs ^'n)$i) |i. i\<in> (UNIV :: 'n set)}"
  1069   and "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"
  1070   unfolding  infnorm_set_image_cart
  1071   by auto
  1072 
  1073 lemma component_le_infnorm_cart:
  1074   shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
  1075   unfolding nth_conv_component
  1076   using component_le_infnorm[of x] .
  1077 
  1078 instance vec :: (perfect_space, finite) perfect_space
  1079 proof
  1080   fix x :: "'a ^ 'b"
  1081   show "x islimpt UNIV"
  1082     apply (rule islimptI)
  1083     apply (unfold open_vec_def)
  1084     apply (drule (1) bspec)
  1085     apply clarsimp
  1086     apply (subgoal_tac "\<forall>i\<in>UNIV. \<exists>y. y \<in> A i \<and> y \<noteq> x $ i")
  1087      apply (drule finite_choice [OF finite_UNIV], erule exE)
  1088      apply (rule_tac x="vec_lambda f" in exI)
  1089      apply (simp add: vec_eq_iff)
  1090     apply (rule ballI, drule_tac x=i in spec, clarify)
  1091     apply (cut_tac x="x $ i" in islimpt_UNIV)
  1092     apply (simp add: islimpt_def)
  1093     done
  1094 qed
  1095 
  1096 lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
  1097 proof-
  1098   let ?U = "UNIV :: 'n set"
  1099   let ?O = "{x::real^'n. \<forall>i. x$i\<ge>0}"
  1100   {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"
  1101     and xi: "x$i < 0"
  1102     from xi have th0: "-x$i > 0" by arith
  1103     from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x $ i" by blast
  1104       have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith
  1105       have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith
  1106       have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using x'(1) xi
  1107         apply (simp only: vector_component)
  1108         by (rule th') auto
  1109       have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using  component_le_norm_cart[of "x'-x" i]
  1110         apply (simp add: dist_norm) by norm
  1111       from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) }
  1112   then show ?thesis unfolding closed_limpt islimpt_approachable
  1113     unfolding not_le[symmetric] by blast
  1114 qed
  1115 lemma Lim_component_cart:
  1116   fixes f :: "'a \<Rightarrow> 'b::metric_space ^ 'n"
  1117   shows "(f ---> l) net \<Longrightarrow> ((\<lambda>a. f a $i) ---> l$i) net"
  1118   unfolding tendsto_iff
  1119   apply (clarify)
  1120   apply (drule spec, drule (1) mp)
  1121   apply (erule eventually_elim1)
  1122   apply (erule le_less_trans [OF dist_vec_nth_le])
  1123   done
  1124 
  1125 lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
  1126 unfolding bounded_def
  1127 apply clarify
  1128 apply (rule_tac x="x $ i" in exI)
  1129 apply (rule_tac x="e" in exI)
  1130 apply clarify
  1131 apply (rule order_trans [OF dist_vec_nth_le], simp)
  1132 done
  1133 
  1134 lemma compact_lemma_cart:
  1135   fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
  1136   assumes "bounded s" and "\<forall>n. f n \<in> s"
  1137   shows "\<forall>d.
  1138         \<exists>l r. subseq r \<and>
  1139         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
  1140 proof
  1141   fix d::"'n set" have "finite d" by simp
  1142   thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
  1143       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
  1144   proof(induct d) case empty thus ?case unfolding subseq_def by auto
  1145   next case (insert k d)
  1146     have s': "bounded ((\<lambda>x. x $ k) ` s)" using `bounded s` by (rule bounded_component_cart)
  1147     obtain l1::"'a^'n" and r1 where r1:"subseq r1" and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
  1148       using insert(3) by auto
  1149     have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` s" using `\<forall>n. f n \<in> s` by simp
  1150     obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
  1151       using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
  1152     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
  1153       using r1 and r2 unfolding r_def o_def subseq_def by auto
  1154     moreover
  1155     def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
  1156     { fix e::real assume "e>0"
  1157       from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially" by blast
  1158       from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially" by (rule tendstoD)
  1159       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
  1160         by (rule eventually_subseq)
  1161       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
  1162         using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
  1163     }
  1164     ultimately show ?case by auto
  1165   qed
  1166 qed
  1167 
  1168 instance vec :: (heine_borel, finite) heine_borel
  1169 proof
  1170   fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"
  1171   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
  1172   then obtain l r where r: "subseq r"
  1173     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
  1174     using compact_lemma_cart [OF s f] by blast
  1175   let ?d = "UNIV::'b set"
  1176   { fix e::real assume "e>0"
  1177     hence "0 < e / (real_of_nat (card ?d))"
  1178       using zero_less_card_finite using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
  1179     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
  1180       by simp
  1181     moreover
  1182     { fix n assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
  1183       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
  1184         unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
  1185       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
  1186         by (rule setsum_strict_mono) (simp_all add: n)
  1187       finally have "dist (f (r n)) l < e" by simp
  1188     }
  1189     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
  1190       by (rule eventually_elim1)
  1191   }
  1192   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
  1193   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
  1194 qed
  1195 
  1196 lemma continuous_at_component: "continuous (at a) (\<lambda>x. x $ i)"
  1197 unfolding continuous_at by (intro tendsto_intros)
  1198 
  1199 lemma continuous_on_component: "continuous_on s (\<lambda>x. x $ i)"
  1200 unfolding continuous_on_def by (intro ballI tendsto_intros)
  1201 
  1202 lemma interval_cart: fixes a :: "'a::ord^'n" shows
  1203   "{a <..< b} = {x::'a^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" and
  1204   "{a .. b} = {x::'a^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
  1205   by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
  1206 
  1207 lemma mem_interval_cart: fixes a :: "'a::ord^'n" shows
  1208   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
  1209   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
  1210   using interval_cart[of a b] by(auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
  1211 
  1212 lemma interval_eq_empty_cart: fixes a :: "real^'n" shows
  1213  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) and
  1214  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
  1215 proof-
  1216   { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>{a <..< b}"
  1217     hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto
  1218     hence "a$i < b$i" by auto
  1219     hence False using as by auto  }
  1220   moreover
  1221   { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
  1222     let ?x = "(1/2) *\<^sub>R (a + b)"
  1223     { fix i
  1224       have "a$i < b$i" using as[THEN spec[where x=i]] by auto
  1225       hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
  1226         unfolding vector_smult_component and vector_add_component
  1227         by auto  }
  1228     hence "{a <..< b} \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto  }
  1229   ultimately show ?th1 by blast
  1230 
  1231   { fix i x assume as:"b$i < a$i" and x:"x\<in>{a .. b}"
  1232     hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto
  1233     hence "a$i \<le> b$i" by auto
  1234     hence False using as by auto  }
  1235   moreover
  1236   { assume as:"\<forall>i. \<not> (b$i < a$i)"
  1237     let ?x = "(1/2) *\<^sub>R (a + b)"
  1238     { fix i
  1239       have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
  1240       hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
  1241         unfolding vector_smult_component and vector_add_component
  1242         by auto  }
  1243     hence "{a .. b} \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
  1244   ultimately show ?th2 by blast
  1245 qed
  1246 
  1247 lemma interval_ne_empty_cart: fixes a :: "real^'n" shows
  1248   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)" and
  1249   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
  1250   unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
  1251     (* BH: Why doesn't just "auto" work here? *)
  1252 
  1253 lemma subset_interval_imp_cart: fixes a :: "real^'n" shows
  1254  "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
  1255  "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
  1256  "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
  1257  "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
  1258   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
  1259   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
  1260 
  1261 lemma interval_sing: fixes a :: "'a::linorder^'n" shows
  1262  "{a .. a} = {a} \<and> {a<..<a} = {}"
  1263 apply(auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
  1264 apply (simp add: order_eq_iff)
  1265 apply (auto simp add: not_less less_imp_le)
  1266 done
  1267 
  1268 lemma interval_open_subset_closed_cart:  fixes a :: "'a::preorder^'n" shows
  1269  "{a<..<b} \<subseteq> {a .. b}"
  1270 proof(simp add: subset_eq, rule)
  1271   fix x
  1272   assume x:"x \<in>{a<..<b}"
  1273   { fix i
  1274     have "a $ i \<le> x $ i"
  1275       using x order_less_imp_le[of "a$i" "x$i"]
  1276       by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
  1277   }
  1278   moreover
  1279   { fix i
  1280     have "x $ i \<le> b $ i"
  1281       using x order_less_imp_le[of "x$i" "b$i"]
  1282       by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
  1283   }
  1284   ultimately
  1285   show "a \<le> x \<and> x \<le> b"
  1286     by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
  1287 qed
  1288 
  1289 lemma subset_interval_cart: fixes a :: "real^'n" shows
  1290  "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1) and
  1291  "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2) and
  1292  "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3) and
  1293  "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
  1294   using subset_interval[of c d a b] by (simp_all add: cart_simps real_euclidean_nth)
  1295 
  1296 lemma disjoint_interval_cart: fixes a::"real^'n" shows
  1297   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1) and
  1298   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2) and
  1299   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3) and
  1300   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
  1301   using disjoint_interval[of a b c d] by (simp_all add: cart_simps real_euclidean_nth)
  1302 
  1303 lemma inter_interval_cart: fixes a :: "'a::linorder^'n" shows
  1304  "{a .. b} \<inter> {c .. d} =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
  1305   unfolding set_eq_iff and Int_iff and mem_interval_cart
  1306   by auto
  1307 
  1308 lemma closed_interval_left_cart: fixes b::"real^'n"
  1309   shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
  1310 proof-
  1311   { fix i
  1312     fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i. x $ i \<le> b $ i}. x' \<noteq> x \<and> dist x' x < e"
  1313     { assume "x$i > b$i"
  1314       then obtain y where "y $ i \<le> b $ i"  "y \<noteq> x"  "dist y x < x$i - b$i" using x[THEN spec[where x="x$i - b$i"]] by auto
  1315       hence False using component_le_norm_cart[of "y - x" i] unfolding dist_norm and vector_minus_component by auto   }
  1316     hence "x$i \<le> b$i" by(rule ccontr)auto  }
  1317   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
  1318 qed
  1319 
  1320 lemma closed_interval_right_cart: fixes a::"real^'n"
  1321   shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
  1322 proof-
  1323   { fix i
  1324     fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i. a $ i \<le> x $ i}. x' \<noteq> x \<and> dist x' x < e"
  1325     { assume "a$i > x$i"
  1326       then obtain y where "a $ i \<le> y $ i"  "y \<noteq> x"  "dist y x < a$i - x$i" using x[THEN spec[where x="a$i - x$i"]] by auto
  1327       hence False using component_le_norm_cart[of "y - x" i] unfolding dist_norm and vector_minus_component by auto   }
  1328     hence "a$i \<le> x$i" by(rule ccontr)auto  }
  1329   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
  1330 qed
  1331 
  1332 lemma is_interval_cart:"is_interval (s::(real^'n) set) \<longleftrightarrow>
  1333   (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
  1334   unfolding is_interval_def Ball_def by (simp add: cart_simps real_euclidean_nth)
  1335 
  1336 lemma closed_halfspace_component_le_cart:
  1337   shows "closed {x::real^'n. x$i \<le> a}"
  1338   using closed_halfspace_le[of "(cart_basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto
  1339 
  1340 lemma closed_halfspace_component_ge_cart:
  1341   shows "closed {x::real^'n. x$i \<ge> a}"
  1342   using closed_halfspace_ge[of a "(cart_basis i)::real^'n"] unfolding inner_basis[OF assms] by auto
  1343 
  1344 lemma open_halfspace_component_lt_cart:
  1345   shows "open {x::real^'n. x$i < a}"
  1346   using open_halfspace_lt[of "(cart_basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto
  1347 
  1348 lemma open_halfspace_component_gt_cart:
  1349   shows "open {x::real^'n. x$i  > a}"
  1350   using open_halfspace_gt[of a "(cart_basis i)::real^'n"] unfolding inner_basis[OF assms] by auto
  1351 
  1352 lemma Lim_component_le_cart: fixes f :: "'a \<Rightarrow> real^'n"
  1353   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)$i \<le> b) net"
  1354   shows "l$i \<le> b"
  1355 proof-
  1356   { fix x have "x \<in> {x::real^'n. inner (cart_basis i) x \<le> b} \<longleftrightarrow> x$i \<le> b" unfolding inner_basis by auto } note * = this
  1357   show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<le> b}" f net l] unfolding *
  1358     using closed_halfspace_le[of "(cart_basis i)::real^'n" b] and assms(1,2,3) by auto
  1359 qed
  1360 
  1361 lemma Lim_component_ge_cart: fixes f :: "'a \<Rightarrow> real^'n"
  1362   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
  1363   shows "b \<le> l$i"
  1364 proof-
  1365   { fix x have "x \<in> {x::real^'n. inner (cart_basis i) x \<ge> b} \<longleftrightarrow> x$i \<ge> b" unfolding inner_basis by auto } note * = this
  1366   show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<ge> b}" f net l] unfolding *
  1367     using closed_halfspace_ge[of b "(cart_basis i)::real^'n"] and assms(1,2,3) by auto
  1368 qed
  1369 
  1370 lemma Lim_component_eq_cart: fixes f :: "'a \<Rightarrow> real^'n"
  1371   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
  1372   shows "l$i = b"
  1373   using ev[unfolded order_eq_iff eventually_and] using Lim_component_ge_cart[OF net, of b i] and
  1374     Lim_component_le_cart[OF net, of i b] by auto
  1375 
  1376 lemma connected_ivt_component_cart: fixes x::"real^'n" shows
  1377  "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s.  z$k = a)"
  1378   using connected_ivt_hyperplane[of s x y "(cart_basis k)::real^'n" a] by (auto simp add: inner_basis)
  1379 
  1380 lemma subspace_substandard_cart:
  1381  "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
  1382   unfolding subspace_def by auto
  1383 
  1384 lemma closed_substandard_cart:
  1385  "closed {x::real^'n. \<forall>i. P i --> x$i = 0}" (is "closed ?A")
  1386 proof-
  1387   let ?D = "{i. P i}"
  1388   let ?Bs = "{{x::real^'n. inner (cart_basis i) x = 0}| i. i \<in> ?D}"
  1389   { fix x
  1390     { assume "x\<in>?A"
  1391       hence x:"\<forall>i\<in>?D. x $ i = 0" by auto
  1392       hence "x\<in> \<Inter> ?Bs" by(auto simp add: inner_basis x) }
  1393     moreover
  1394     { assume x:"x\<in>\<Inter>?Bs"
  1395       { fix i assume i:"i \<in> ?D"
  1396         then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::real^'n. inner (cart_basis i) x = 0}" by auto
  1397         hence "x $ i = 0" unfolding B using x unfolding inner_basis by auto  }
  1398       hence "x\<in>?A" by auto }
  1399     ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" .. }
  1400   hence "?A = \<Inter> ?Bs" by auto
  1401   thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
  1402 qed
  1403 
  1404 lemma dim_substandard_cart:
  1405   shows "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d" (is "dim ?A = _")
  1406 proof- have *:"{x. \<forall>i<DIM((real, 'n) vec). i \<notin> \<pi>' ` d \<longrightarrow> x $$ i = 0} = 
  1407     {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0}"apply safe
  1408     apply(erule_tac x="\<pi>' i" in allE) defer
  1409     apply(erule_tac x="\<pi> i" in allE)
  1410     unfolding image_iff real_euclidean_nth[symmetric] by (auto simp: pi'_inj[THEN inj_eq])
  1411   have " \<pi>' ` d \<subseteq> {..<DIM((real, 'n) vec)}" using pi'_range[where 'n='n] by auto
  1412   thus ?thesis using dim_substandard[of "\<pi>' ` d", where 'a="real^'n"] 
  1413     unfolding * using card_image[of "\<pi>'" d] using pi'_inj unfolding inj_on_def by auto
  1414 qed
  1415 
  1416 lemma affinity_inverses:
  1417   assumes m0: "m \<noteq> (0::'a::field)"
  1418   shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
  1419   "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
  1420   using m0
  1421 apply (auto simp add: fun_eq_iff vector_add_ldistrib)
  1422 by (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1[symmetric])
  1423 
  1424 lemma vector_affinity_eq:
  1425   assumes m0: "(m::'a::field) \<noteq> 0"
  1426   shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
  1427 proof
  1428   assume h: "m *s x + c = y"
  1429   hence "m *s x = y - c" by (simp add: field_simps)
  1430   hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
  1431   then show "x = inverse m *s y + - (inverse m *s c)"
  1432     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
  1433 next
  1434   assume h: "x = inverse m *s y + - (inverse m *s c)"
  1435   show "m *s x + c = y" unfolding h diff_minus[symmetric]
  1436     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
  1437 qed
  1438 
  1439 lemma vector_eq_affinity:
  1440  "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
  1441   using vector_affinity_eq[where m=m and x=x and y=y and c=c]
  1442   by metis
  1443 
  1444 lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<chi>\<chi> i. d)"
  1445   apply(subst euclidean_eq)
  1446 proof safe case goal1
  1447   hence *:"(basis i::real^'n) = cart_basis (\<pi> i)"
  1448     unfolding basis_real_n[THEN sym] by auto
  1449   have "((\<chi> i. d)::real^'n) $$ i = d" unfolding euclidean_component_def *
  1450     unfolding dot_basis by auto
  1451   thus ?case using goal1 by auto
  1452 qed
  1453 
  1454 section "Convex Euclidean Space"
  1455 
  1456 lemma Cart_1:"(1::real^'n) = (\<chi>\<chi> i. 1)"
  1457   apply(subst euclidean_eq)
  1458 proof safe case goal1 thus ?case using nth_conv_component[THEN sym,where i1="\<pi> i" and x1="1::real^'n"] by auto
  1459 qed
  1460 
  1461 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
  1462 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
  1463 
  1464 lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta basis_component vector_uminus_component
  1465 
  1466 lemma convex_box_cart:
  1467   assumes "\<And>i. convex {x. P i x}"
  1468   shows "convex {x. \<forall>i. P i (x$i)}"
  1469   using assms unfolding convex_def by auto
  1470 
  1471 lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
  1472   by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)
  1473 
  1474 lemma unit_interval_convex_hull_cart:
  1475   "{0::real^'n .. 1} = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" (is "?int = convex hull ?points")
  1476   unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"]
  1477   apply(rule arg_cong[where f="\<lambda>x. convex hull x"]) apply(rule set_eqI) unfolding mem_Collect_eq
  1478   apply safe apply(erule_tac x="\<pi>' i" in allE) unfolding nth_conv_component defer
  1479   apply(erule_tac x="\<pi> i" in allE) by auto
  1480 
  1481 lemma cube_convex_hull_cart:
  1482   assumes "0 < d" obtains s::"(real^'n) set" where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s" 
  1483 proof- from cube_convex_hull[OF assms, where 'a="real^'n" and x=x] guess s . note s=this
  1484   show thesis apply(rule that[OF s(1)]) unfolding s(2)[THEN sym] const_vector_cart ..
  1485 qed
  1486 
  1487 lemma std_simplex_cart:
  1488   "(insert (0::real^'n) { cart_basis i | i. i\<in>UNIV}) =
  1489   (insert 0 { basis i | i. i<DIM((real,'n) vec)})"
  1490   apply(rule arg_cong[where f="\<lambda>s. (insert 0 s)"])
  1491   unfolding basis_real_n[THEN sym] apply safe
  1492   apply(rule_tac x="\<pi>' i" in exI) defer
  1493   apply(rule_tac x="\<pi> i" in exI) using pi'_range[where 'n='n] by auto
  1494 
  1495 subsection "Brouwer Fixpoint"
  1496 
  1497 lemma kuhn_labelling_lemma_cart:
  1498   assumes "(\<forall>x::real^_. P x \<longrightarrow> P (f x))"  "\<forall>x. P x \<longrightarrow> (\<forall>i. Q i \<longrightarrow> 0 \<le> x$i \<and> x$i \<le> 1)"
  1499   shows "\<exists>l. (\<forall>x i. l x i \<le> (1::nat)) \<and>
  1500              (\<forall>x i. P x \<and> Q i \<and> (x$i = 0) \<longrightarrow> (l x i = 0)) \<and>
  1501              (\<forall>x i. P x \<and> Q i \<and> (x$i = 1) \<longrightarrow> (l x i = 1)) \<and>
  1502              (\<forall>x i. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x$i \<le> f(x)$i) \<and>
  1503              (\<forall>x i. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)$i \<le> x$i)" proof-
  1504   have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)" by auto
  1505   have *:"\<forall>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow> (x \<noteq> 1 \<and> x \<le> y \<or> x \<noteq> 0 \<and> y \<le> x)" by auto
  1506   show ?thesis unfolding and_forall_thm apply(subst choice_iff[THEN sym])+ proof(rule,rule) case goal1
  1507     let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x $ xa = 0 \<longrightarrow> y = (0::nat)) \<and>
  1508         (P x \<and> Q xa \<and> x $ xa = 1 \<longrightarrow> y = 1) \<and> (P x \<and> Q xa \<and> y = 0 \<longrightarrow> x $ xa \<le> f x $ xa) \<and> (P x \<and> Q xa \<and> y = 1 \<longrightarrow> f x $ xa \<le> x $ xa)"
  1509     { assume "P x" "Q xa" hence "0 \<le> f x $ xa \<and> f x $ xa \<le> 1" using assms(2)[rule_format,of "f x" xa]
  1510         apply(drule_tac assms(1)[rule_format]) by auto }
  1511     hence "?R 0 \<or> ?R 1" by auto thus ?case by auto qed qed 
  1512 
  1513 lemma interval_bij_cart:"interval_bij = (\<lambda> (a,b) (u,v) (x::real^'n).
  1514     (\<chi> i. u$i + (x$i - a$i) / (b$i - a$i) * (v$i - u$i))::real^'n)"
  1515   unfolding interval_bij_def apply(rule ext)+ apply safe
  1516   unfolding vec_eq_iff vec_lambda_beta unfolding nth_conv_component
  1517   apply rule apply(subst euclidean_lambda_beta) using pi'_range by auto
  1518 
  1519 lemma interval_bij_affine_cart:
  1520  "interval_bij (a,b) (u,v) = (\<lambda>x. (\<chi> i. (v$i - u$i) / (b$i - a$i) * x$i) +
  1521             (\<chi> i. u$i - (v$i - u$i) / (b$i - a$i) * a$i)::real^'n)"
  1522   apply rule unfolding vec_eq_iff interval_bij_cart vector_component_simps
  1523   by(auto simp add: field_simps add_divide_distrib[THEN sym]) 
  1524 
  1525 subsection "Derivative"
  1526 
  1527 lemma has_derivative_vmul_component_cart: fixes c::"real^'a \<Rightarrow> real^'b" and v::"real^'c"
  1528   assumes "(c has_derivative c') net"
  1529   shows "((\<lambda>x. c(x)$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$k *\<^sub>R v)) net"
  1530   unfolding nth_conv_component
  1531   by (intro has_derivative_intros assms)
  1532 
  1533 lemma differentiable_at_imp_differentiable_on: "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
  1534   unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI)
  1535 
  1536 definition "jacobian f net = matrix(frechet_derivative f net)"
  1537 
  1538 lemma jacobian_works: "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow> (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
  1539   apply rule unfolding jacobian_def apply(simp only: matrix_works[OF linear_frechet_derivative]) defer
  1540   apply(rule differentiableI) apply assumption unfolding frechet_derivative_works by assumption
  1541 
  1542 subsection {* Component of the differential must be zero if it exists at a local        *)
  1543 (* maximum or minimum for that corresponding component. *}
  1544 
  1545 lemma differential_zero_maxmin_component: fixes f::"real^'a \<Rightarrow> real^'b"
  1546   assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
  1547   "f differentiable (at x)" shows "jacobian f (at x) $ k = 0"
  1548 (* FIXME: reuse proof of generic differential_zero_maxmin_component*)
  1549 
  1550 proof(rule ccontr)
  1551   def D \<equiv> "jacobian f (at x)" assume "jacobian f (at x) $ k \<noteq> 0"
  1552   then obtain j where j:"D$k$j \<noteq> 0" unfolding vec_eq_iff D_def by auto
  1553   hence *:"abs (jacobian f (at x) $ k $ j) / 2 > 0" unfolding D_def by auto
  1554   note as = assms(3)[unfolded jacobian_works has_derivative_at_alt]
  1555   guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this
  1556   guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this
  1557   { fix c assume "abs c \<le> d" 
  1558     hence *:"norm (x + c *\<^sub>R cart_basis j - x) < e'" using norm_basis[of j] d by auto
  1559     have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le> norm (f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j))" 
  1560       by(rule component_le_norm_cart)
  1561     also have "\<dots> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j] unfolding D_def[symmetric] by auto
  1562     finally have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" by simp
  1563     hence "\<bar>f (x + c *\<^sub>R cart_basis j) $ k - f x $ k - c * D $ k $ j\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"
  1564       unfolding vector_component_simps matrix_vector_mul_component unfolding smult_conv_scaleR[symmetric] 
  1565       unfolding inner_simps dot_basis smult_conv_scaleR by simp  } note * = this
  1566   have "x + d *\<^sub>R cart_basis j \<in> ball x e" "x - d *\<^sub>R cart_basis j \<in> ball x e"
  1567     unfolding mem_ball dist_norm using norm_basis[of j] d by auto
  1568   hence **:"((f (x - d *\<^sub>R cart_basis j))$k \<le> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<le> (f x)$k) \<or>
  1569          ((f (x - d *\<^sub>R cart_basis j))$k \<ge> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<ge> (f x)$k)" using assms(2) by auto
  1570   have ***:"\<And>y y1 y2 d dx::real. (y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow> d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
  1571   show False apply(rule ***[OF **, where dx="d * D $ k $ j" and d="\<bar>D $ k $ j\<bar> / 2 * \<bar>d\<bar>"]) 
  1572     using *[of "-d"] and *[of d] and d[THEN conjunct1] and j unfolding mult_minus_left
  1573     unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding algebra_simps by (auto intro: mult_pos_pos)
  1574 qed
  1575 
  1576 subsection {* Lemmas for working on @{typ "real^1"} *}
  1577 
  1578 lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
  1579   by (metis num1_eq_iff)
  1580 
  1581 lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
  1582   by auto (metis num1_eq_iff)
  1583 
  1584 lemma exhaust_2:
  1585   fixes x :: 2 shows "x = 1 \<or> x = 2"
  1586 proof (induct x)
  1587   case (of_int z)
  1588   then have "0 <= z" and "z < 2" by simp_all
  1589   then have "z = 0 | z = 1" by arith
  1590   then show ?case by auto
  1591 qed
  1592 
  1593 lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
  1594   by (metis exhaust_2)
  1595 
  1596 lemma exhaust_3:
  1597   fixes x :: 3 shows "x = 1 \<or> x = 2 \<or> x = 3"
  1598 proof (induct x)
  1599   case (of_int z)
  1600   then have "0 <= z" and "z < 3" by simp_all
  1601   then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
  1602   then show ?case by auto
  1603 qed
  1604 
  1605 lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
  1606   by (metis exhaust_3)
  1607 
  1608 lemma UNIV_1 [simp]: "UNIV = {1::1}"
  1609   by (auto simp add: num1_eq_iff)
  1610 
  1611 lemma UNIV_2: "UNIV = {1::2, 2::2}"
  1612   using exhaust_2 by auto
  1613 
  1614 lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
  1615   using exhaust_3 by auto
  1616 
  1617 lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
  1618   unfolding UNIV_1 by simp
  1619 
  1620 lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
  1621   unfolding UNIV_2 by simp
  1622 
  1623 lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
  1624   unfolding UNIV_3 by (simp add: add_ac)
  1625 
  1626 instantiation num1 :: cart_one begin
  1627 instance proof
  1628   show "CARD(1) = Suc 0" by auto
  1629 qed end
  1630 
  1631 (* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)
  1632 
  1633 abbreviation vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x \<equiv> vec x"
  1634 
  1635 abbreviation dest_vec1:: "'a ^1 \<Rightarrow> 'a"
  1636   where "dest_vec1 x \<equiv> (x$1)"
  1637 
  1638 lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
  1639   by (simp_all add:  vec_eq_iff)
  1640 
  1641 lemma vec1_component[simp]: "(vec1 x)$1 = x"
  1642   by (simp_all add:  vec_eq_iff)
  1643 
  1644 lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))"
  1645   by (metis vec1_dest_vec1(1))
  1646 
  1647 lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))"
  1648   by (metis vec1_dest_vec1(1))
  1649 
  1650 lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y"
  1651   by (metis vec1_dest_vec1(2))
  1652 
  1653 lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y"
  1654   by (metis vec1_dest_vec1(1))
  1655 
  1656 subsection{* The collapse of the general concepts to dimension one. *}
  1657 
  1658 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
  1659   by (simp add: vec_eq_iff)
  1660 
  1661 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
  1662   apply auto
  1663   apply (erule_tac x= "x$1" in allE)
  1664   apply (simp only: vector_one[symmetric])
  1665   done
  1666 
  1667 lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
  1668   by (simp add: norm_vec_def)
  1669 
  1670 lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
  1671   by (simp add: norm_vector_1)
  1672 
  1673 lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
  1674   by (auto simp add: norm_real dist_norm)
  1675 
  1676 subsection{* Explicit vector construction from lists. *}
  1677 
  1678 definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
  1679 
  1680 lemma vector_1: "(vector[x]) $1 = x"
  1681   unfolding vector_def by simp
  1682 
  1683 lemma vector_2:
  1684  "(vector[x,y]) $1 = x"
  1685  "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
  1686   unfolding vector_def by simp_all
  1687 
  1688 lemma vector_3:
  1689  "(vector [x,y,z] ::('a::zero)^3)$1 = x"
  1690  "(vector [x,y,z] ::('a::zero)^3)$2 = y"
  1691  "(vector [x,y,z] ::('a::zero)^3)$3 = z"
  1692   unfolding vector_def by simp_all
  1693 
  1694 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
  1695   apply auto
  1696   apply (erule_tac x="v$1" in allE)
  1697   apply (subgoal_tac "vector [v$1] = v")
  1698   apply simp
  1699   apply (vector vector_def)
  1700   apply simp
  1701   done
  1702 
  1703 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
  1704   apply auto
  1705   apply (erule_tac x="v$1" in allE)
  1706   apply (erule_tac x="v$2" in allE)
  1707   apply (subgoal_tac "vector [v$1, v$2] = v")
  1708   apply simp
  1709   apply (vector vector_def)
  1710   apply (simp add: forall_2)
  1711   done
  1712 
  1713 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
  1714   apply auto
  1715   apply (erule_tac x="v$1" in allE)
  1716   apply (erule_tac x="v$2" in allE)
  1717   apply (erule_tac x="v$3" in allE)
  1718   apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
  1719   apply simp
  1720   apply (vector vector_def)
  1721   apply (simp add: forall_3)
  1722   done
  1723 
  1724 lemma range_vec1[simp]:"range vec1 = UNIV" apply(rule set_eqI,rule) unfolding image_iff defer
  1725   apply(rule_tac x="dest_vec1 x" in bexI) by auto
  1726 
  1727 lemma dest_vec1_lambda: "dest_vec1(\<chi> i. x i) = x 1"
  1728   by (simp)
  1729 
  1730 lemma dest_vec1_vec: "dest_vec1(vec x) = x"
  1731   by (simp)
  1732 
  1733 lemma dest_vec1_sum: assumes fS: "finite S"
  1734   shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"
  1735   apply (induct rule: finite_induct[OF fS])
  1736   apply simp
  1737   apply auto
  1738   done
  1739 
  1740 lemma norm_vec1 [simp]: "norm(vec1 x) = abs(x)"
  1741   by (simp add: vec_def norm_real)
  1742 
  1743 lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)"
  1744   by (simp only: dist_real vec1_component)
  1745 lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
  1746   by (metis vec1_dest_vec1(1) norm_vec1)
  1747 
  1748 lemmas vec1_dest_vec1_simps = forall_vec1 vec_add[THEN sym] dist_vec1 vec_sub[THEN sym] vec1_dest_vec1 norm_vec1 vector_smult_component
  1749    vec1_eq vec_cmul[THEN sym] smult_conv_scaleR[THEN sym] o_def dist_real_def real_norm_def
  1750 
  1751 lemma bounded_linear_vec1:"bounded_linear (vec1::real\<Rightarrow>real^1)"
  1752   unfolding bounded_linear_def additive_def bounded_linear_axioms_def 
  1753   unfolding smult_conv_scaleR[THEN sym] unfolding vec1_dest_vec1_simps
  1754   apply(rule conjI) defer apply(rule conjI) defer apply(rule_tac x=1 in exI) by auto
  1755 
  1756 lemma linear_vmul_dest_vec1:
  1757   fixes f:: "real^_ \<Rightarrow> real^1"
  1758   shows "linear f \<Longrightarrow> linear (\<lambda>x. dest_vec1(f x) *s v)"
  1759   unfolding smult_conv_scaleR
  1760   by (rule linear_vmul_component)
  1761 
  1762 lemma linear_from_scalars:
  1763   assumes lf: "linear (f::real^1 \<Rightarrow> real^_)"
  1764   shows "f = (\<lambda>x. dest_vec1 x *s column 1 (matrix f))"
  1765   unfolding smult_conv_scaleR
  1766   apply (rule ext)
  1767   apply (subst matrix_works[OF lf, symmetric])
  1768   apply (auto simp add: vec_eq_iff matrix_vector_mult_def column_def mult_commute)
  1769   done
  1770 
  1771 lemma linear_to_scalars: assumes lf: "linear (f::real ^'n \<Rightarrow> real^1)"
  1772   shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
  1773   apply (rule ext)
  1774   apply (subst matrix_works[OF lf, symmetric])
  1775   apply (simp add: vec_eq_iff matrix_vector_mult_def row_def inner_vec_def mult_commute)
  1776   done
  1777 
  1778 lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
  1779   by (simp add: dest_vec1_eq[symmetric])
  1780 
  1781 lemma setsum_scalars: assumes fS: "finite S"
  1782   shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)"
  1783   unfolding vec_setsum[OF fS] by simp
  1784 
  1785 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"
  1786   apply (cases "dest_vec1 x \<le> dest_vec1 y")
  1787   apply simp
  1788   apply (subgoal_tac "dest_vec1 y \<le> dest_vec1 x")
  1789   apply (auto)
  1790   done
  1791 
  1792 text{* Lifting and dropping *}
  1793 
  1794 lemma continuous_on_o_dest_vec1: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
  1795   assumes "continuous_on {a..b::real} f" shows "continuous_on {vec1 a..vec1 b} (f o dest_vec1)"
  1796   using assms unfolding continuous_on_iff apply safe
  1797   apply(erule_tac x="x$1" in ballE,erule_tac x=e in allE) apply safe
  1798   apply(rule_tac x=d in exI) apply safe unfolding o_def dist_real_def dist_real 
  1799   apply(erule_tac x="dest_vec1 x'" in ballE) by(auto simp add:less_eq_vec_def)
  1800 
  1801 lemma continuous_on_o_vec1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
  1802   assumes "continuous_on {a..b} f" shows "continuous_on {dest_vec1 a..dest_vec1 b} (f o vec1)"
  1803   using assms unfolding continuous_on_iff apply safe
  1804   apply(erule_tac x="vec x" in ballE,erule_tac x=e in allE) apply safe
  1805   apply(rule_tac x=d in exI) apply safe unfolding o_def dist_real_def dist_real 
  1806   apply(erule_tac x="vec1 x'" in ballE) by(auto simp add:less_eq_vec_def)
  1807 
  1808 lemma continuous_on_vec1:"continuous_on A (vec1::real\<Rightarrow>real^1)"
  1809   by(rule linear_continuous_on[OF bounded_linear_vec1])
  1810 
  1811 lemma mem_interval_1: fixes x :: "real^1" shows
  1812  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
  1813  "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
  1814 by(simp_all add: vec_eq_iff less_vec_def less_eq_vec_def)
  1815 
  1816 lemma vec1_interval:fixes a::"real" shows
  1817   "vec1 ` {a .. b} = {vec1 a .. vec1 b}"
  1818   "vec1 ` {a<..<b} = {vec1 a<..<vec1 b}"
  1819   apply(rule_tac[!] set_eqI) unfolding image_iff less_vec_def unfolding mem_interval_cart
  1820   unfolding forall_1 unfolding vec1_dest_vec1_simps
  1821   apply rule defer apply(rule_tac x="dest_vec1 x" in bexI) prefer 3 apply rule defer
  1822   apply(rule_tac x="dest_vec1 x" in bexI) by auto
  1823 
  1824 (* Some special cases for intervals in R^1.                                  *)
  1825 
  1826 lemma interval_cases_1: fixes x :: "real^1" shows
  1827  "x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"
  1828   unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart by(auto simp del:dest_vec1_eq)
  1829 
  1830 lemma in_interval_1: fixes x :: "real^1" shows
  1831  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b) \<and>
  1832   (x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
  1833   unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart by(auto simp del:dest_vec1_eq)
  1834 
  1835 lemma interval_eq_empty_1: fixes a :: "real^1" shows
  1836   "{a .. b} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a"
  1837   "{a<..<b} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
  1838   unfolding interval_eq_empty_cart and ex_1 by auto
  1839 
  1840 lemma subset_interval_1: fixes a :: "real^1" shows
  1841  "({a .. b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>
  1842                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
  1843  "({a .. b} \<subseteq> {c<..<d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>
  1844                 dest_vec1 c < dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b < dest_vec1 d)"
  1845  "({a<..<b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b \<le> dest_vec1 a \<or>
  1846                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
  1847  "({a<..<b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
  1848                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
  1849   unfolding subset_interval_cart[of a b c d] unfolding forall_1 by auto
  1850 
  1851 lemma eq_interval_1: fixes a :: "real^1" shows
  1852  "{a .. b} = {c .. d} \<longleftrightarrow>
  1853           dest_vec1 b < dest_vec1 a \<and> dest_vec1 d < dest_vec1 c \<or>
  1854           dest_vec1 a = dest_vec1 c \<and> dest_vec1 b = dest_vec1 d"
  1855 unfolding set_eq_subset[of "{a .. b}" "{c .. d}"]
  1856 unfolding subset_interval_1(1)[of a b c d]
  1857 unfolding subset_interval_1(1)[of c d a b]
  1858 by auto
  1859 
  1860 lemma disjoint_interval_1: fixes a :: "real^1" shows
  1861   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or> dest_vec1 d < dest_vec1 c  \<or>  dest_vec1 b < dest_vec1 c \<or> dest_vec1 d < dest_vec1 a"
  1862   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
  1863   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d < dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
  1864   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
  1865   unfolding disjoint_interval_cart and ex_1 by auto
  1866 
  1867 lemma open_closed_interval_1: fixes a :: "real^1" shows
  1868  "{a<..<b} = {a .. b} - {a, b}"
  1869   unfolding set_eq_iff apply simp unfolding less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)
  1870 
  1871 lemma closed_open_interval_1: "dest_vec1 (a::real^1) \<le> dest_vec1 b ==> {a .. b} = {a<..<b} \<union> {a,b}"
  1872   unfolding set_eq_iff apply simp unfolding less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)
  1873 
  1874 lemma Lim_drop_le: fixes f :: "'a \<Rightarrow> real^1" shows
  1875   "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. dest_vec1 (f x) \<le> b) net ==> dest_vec1 l \<le> b"
  1876   using Lim_component_le_cart[of f l net 1 b] by auto
  1877 
  1878 lemma Lim_drop_ge: fixes f :: "'a \<Rightarrow> real^1" shows
  1879  "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. b \<le> dest_vec1 (f x)) net ==> b \<le> dest_vec1 l"
  1880   using Lim_component_ge_cart[of f l net b 1] by auto
  1881 
  1882 text{* Also more convenient formulations of monotone convergence.                *}
  1883 
  1884 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real^1"
  1885   assumes "bounded {s n| n::nat. True}"  "\<forall>n. dest_vec1(s n) \<le> dest_vec1(s(Suc n))"
  1886   shows "\<exists>l. (s ---> l) sequentially"
  1887 proof-
  1888   obtain a where a:"\<forall>n. \<bar>dest_vec1 (s n)\<bar> \<le>  a" using assms(1)[unfolded bounded_iff abs_dest_vec1] by auto
  1889   { fix m::nat
  1890     have "\<And> n. n\<ge>m \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
  1891       apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE) by(auto simp add: not_less_eq_eq)  }
  1892   hence "\<forall>m n. m \<le> n \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)" by auto
  1893   then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>dest_vec1 (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a] unfolding monoseq_def by auto
  1894   thus ?thesis unfolding Lim_sequentially apply(rule_tac x="vec1 l" in exI)
  1895     unfolding dist_norm unfolding abs_dest_vec1  by auto
  1896 qed
  1897 
  1898 lemma dest_vec1_simps[simp]: fixes a::"real^1"
  1899   shows "a$1 = 0 \<longleftrightarrow> a = 0" (*"a \<le> 1 \<longleftrightarrow> dest_vec1 a \<le> 1" "0 \<le> a \<longleftrightarrow> 0 \<le> dest_vec1 a"*)
  1900   "a \<le> b \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 b" "dest_vec1 (1::real^1) = 1"
  1901   by(auto simp add: less_eq_vec_def vec_eq_iff)
  1902 
  1903 lemma dest_vec1_inverval:
  1904   "dest_vec1 ` {a .. b} = {dest_vec1 a .. dest_vec1 b}"
  1905   "dest_vec1 ` {a<.. b} = {dest_vec1 a<.. dest_vec1 b}"
  1906   "dest_vec1 ` {a ..<b} = {dest_vec1 a ..<dest_vec1 b}"
  1907   "dest_vec1 ` {a<..<b} = {dest_vec1 a<..<dest_vec1 b}"
  1908   apply(rule_tac [!] equalityI)
  1909   unfolding subset_eq Ball_def Bex_def mem_interval_1 image_iff
  1910   apply(rule_tac [!] allI)apply(rule_tac [!] impI)
  1911   apply(rule_tac[2] x="vec1 x" in exI)apply(rule_tac[4] x="vec1 x" in exI)
  1912   apply(rule_tac[6] x="vec1 x" in exI)apply(rule_tac[8] x="vec1 x" in exI)
  1913   by (auto simp add: less_vec_def less_eq_vec_def)
  1914 
  1915 lemma dest_vec1_setsum: assumes "finite S"
  1916   shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
  1917   using dest_vec1_sum[OF assms] by auto
  1918 
  1919 lemma open_dest_vec1_vimage: "open S \<Longrightarrow> open (dest_vec1 -` S)"
  1920 unfolding open_vec_def forall_1 by auto
  1921 
  1922 lemma tendsto_dest_vec1 [tendsto_intros]:
  1923   "(f ---> l) net \<Longrightarrow> ((\<lambda>x. dest_vec1 (f x)) ---> dest_vec1 l) net"
  1924 by(rule tendsto_vec_nth)
  1925 
  1926 lemma continuous_dest_vec1: "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
  1927   unfolding continuous_def by (rule tendsto_dest_vec1)
  1928 
  1929 lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))" 
  1930   apply safe defer apply(erule_tac x="vec1 x" in allE) by auto
  1931 
  1932 lemma forall_of_dest_vec1: "(\<forall>v. P (\<lambda>x. dest_vec1 (v x))) \<longleftrightarrow> (\<forall>x. P x)"
  1933   apply rule apply rule apply(erule_tac x="(vec1 \<circ> x)" in allE) unfolding o_def vec1_dest_vec1 by auto 
  1934 
  1935 lemma forall_of_dest_vec1': "(\<forall>v. P (dest_vec1 v)) \<longleftrightarrow> (\<forall>x. P x)"
  1936   apply rule apply rule apply(erule_tac x="(vec1 x)" in allE) defer apply rule 
  1937   apply(erule_tac x="dest_vec1 v" in allE) unfolding o_def vec1_dest_vec1 by auto
  1938 
  1939 lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" unfolding dist_norm by auto
  1940 
  1941 lemma bounded_linear_vec1_dest_vec1: fixes f::"real \<Rightarrow> real"
  1942   shows "linear (vec1 \<circ> f \<circ> dest_vec1) = bounded_linear f" (is "?l = ?r") proof-
  1943   { assume ?l guess K using linear_bounded[OF `?l`] ..
  1944     hence "\<exists>K. \<forall>x. \<bar>f x\<bar> \<le> \<bar>x\<bar> * K" apply(rule_tac x=K in exI)
  1945       unfolding vec1_dest_vec1_simps by (auto simp add:field_simps) }
  1946   thus ?thesis unfolding linear_def bounded_linear_def additive_def bounded_linear_axioms_def o_def
  1947     unfolding vec1_dest_vec1_simps by auto qed
  1948 
  1949 lemma vec1_le[simp]:fixes a::real shows "vec1 a \<le> vec1 b \<longleftrightarrow> a \<le> b"
  1950   unfolding less_eq_vec_def by auto
  1951 lemma vec1_less[simp]:fixes a::real shows "vec1 a < vec1 b \<longleftrightarrow> a < b"
  1952   unfolding less_vec_def by auto
  1953 
  1954 
  1955 subsection {* Derivatives on real = Derivatives on @{typ "real^1"} *}
  1956 
  1957 lemma has_derivative_within_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows
  1958   "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s)
  1959   = (f has_derivative f') (at x within s)"
  1960   unfolding has_derivative_within unfolding bounded_linear_vec1_dest_vec1[unfolded linear_conv_bounded_linear]
  1961   unfolding o_def Lim_within Ball_def unfolding forall_vec1 
  1962   unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto  
  1963 
  1964 lemma has_derivative_at_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows
  1965   "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
  1966   using has_derivative_within_vec1_dest_vec1[where s=UNIV, unfolded range_vec1 within_UNIV] by auto
  1967 
  1968 lemma bounded_linear_vec1': fixes f::"'a::real_normed_vector\<Rightarrow>real"
  1969   shows "bounded_linear f = bounded_linear (vec1 \<circ> f)"
  1970   unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
  1971   unfolding vec1_dest_vec1_simps by auto
  1972 
  1973 lemma bounded_linear_dest_vec1: fixes f::"real\<Rightarrow>'a::real_normed_vector"
  1974   shows "bounded_linear f = bounded_linear (f \<circ> dest_vec1)"
  1975   unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
  1976   unfolding vec1_dest_vec1_simps by auto
  1977 
  1978 lemma has_derivative_at_vec1: fixes f::"'a::real_normed_vector\<Rightarrow>real" shows
  1979   "(f has_derivative f') (at x) = ((vec1 \<circ> f) has_derivative (vec1 \<circ> f')) (at x)"
  1980   unfolding has_derivative_at unfolding bounded_linear_vec1'[unfolded linear_conv_bounded_linear]
  1981   unfolding o_def Lim_at unfolding vec1_dest_vec1_simps dist_vec1_0 by auto
  1982 
  1983 lemma has_derivative_within_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows
  1984   "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s) = (f has_derivative f') (at x within s)"
  1985   unfolding has_derivative_within bounded_linear_dest_vec1 unfolding o_def Lim_within Ball_def
  1986   unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto
  1987 
  1988 lemma has_derivative_at_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows
  1989   "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
  1990   using has_derivative_within_dest_vec1[where s=UNIV] by(auto simp add:within_UNIV)
  1991 
  1992 subsection {* In particular if we have a mapping into @{typ "real^1"}. *}
  1993 
  1994 lemma onorm_vec1: fixes f::"real \<Rightarrow> real"
  1995   shows "onorm (\<lambda>x. vec1 (f (dest_vec1 x))) = onorm f" proof-
  1996   have "\<forall>x::real^1. norm x = 1 \<longleftrightarrow> x\<in>{vec1 -1, vec1 (1::real)}" unfolding forall_vec1 by(auto simp add:vec_eq_iff)
  1997   hence 1:"{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by auto
  1998   have 2:"{norm (vec1 (f (dest_vec1 x))) |x. norm x = 1} = (\<lambda>x. norm (vec1 (f (dest_vec1 x)))) ` {x. norm x=1}" by auto
  1999   have "\<forall>x::real. norm x = 1 \<longleftrightarrow> x\<in>{-1, 1}" by auto hence 3:"{x. norm x = 1} = {-1, (1::real)}" by auto
  2000   have 4:"{norm (f x) |x. norm x = 1} = (\<lambda>x. norm (f x)) ` {x. norm x=1}" by auto
  2001   show ?thesis unfolding onorm_def 1 2 3 4 by(simp add:Sup_finite_Max) qed
  2002 
  2003 lemma convex_vec1:"convex (vec1 ` s) = convex (s::real set)"
  2004   unfolding convex_def Ball_def forall_vec1 unfolding vec1_dest_vec1_simps image_iff by auto
  2005 
  2006 lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
  2007   apply(rule bounded_linearI[where K=1]) 
  2008   using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
  2009 
  2010 lemma bounded_vec1[intro]:  "bounded s \<Longrightarrow> bounded (vec1 ` (s::real set))"
  2011   unfolding bounded_def apply safe apply(rule_tac x="vec1 x" in exI,rule_tac x=e in exI)
  2012   by(auto simp add: dist_real dist_real_def)
  2013 
  2014 (*lemma content_closed_interval_cases_cart:
  2015   "content {a..b::real^'n} =
  2016   (if {a..b} = {} then 0 else setprod (\<lambda>i. b$i - a$i) UNIV)" 
  2017 proof(cases "{a..b} = {}")
  2018   case True thus ?thesis unfolding content_def by auto
  2019 next case Falsethus ?thesis unfolding content_def unfolding if_not_P[OF False]
  2020   proof(cases "\<forall>i. a $ i \<le> b $ i")
  2021     case False thus ?thesis unfolding content_def using t by auto
  2022   next case True note interval_eq_empty
  2023    apply auto 
  2024   
  2025   sorry*)
  2026 
  2027 lemma integral_component_eq_cart[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real^'m"
  2028   assumes "f integrable_on s" shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"
  2029   using integral_linear[OF assms(1) bounded_linear_component_cart,unfolded o_def] .
  2030 
  2031 lemma interval_split_cart:
  2032   "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
  2033   "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
  2034   apply(rule_tac[!] set_eqI) unfolding Int_iff mem_interval_cart mem_Collect_eq
  2035   unfolding vec_lambda_beta by auto
  2036 
  2037 (*lemma content_split_cart:
  2038   "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"
  2039 proof- note simps = interval_split_cart content_closed_interval_cases_cart vec_lambda_beta less_eq_vec_def
  2040   { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }
  2041   have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto
  2042   have *:"\<And>X Y Z. (\<Prod>i\<in>UNIV. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>UNIV-{k}. Z i (Y i))"
  2043     "(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)" 
  2044     apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
  2045   assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c
  2046     \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"
  2047     by  (auto simp add:field_simps)
  2048   moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"
  2049     unfolding not_le using as[unfolded less_eq_vec_def,rule_format,of k] by auto
  2050   ultimately show ?thesis 
  2051     unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)
  2052 qed*)
  2053 
  2054 lemma has_integral_vec1: assumes "(f has_integral k) {a..b}"
  2055   shows "((\<lambda>x. vec1 (f x)) has_integral (vec1 k)) {a..b}"
  2056 proof- have *:"\<And>p. (\<Sum>(x, k)\<in>p. content k *\<^sub>R vec1 (f x)) - vec1 k = vec1 ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k)"
  2057     unfolding vec_sub vec_eq_iff by(auto simp add: split_beta)
  2058   show ?thesis using assms unfolding has_integral apply safe
  2059     apply(erule_tac x=e in allE,safe) apply(rule_tac x=d in exI,safe)
  2060     apply(erule_tac x=p in allE,safe) unfolding * norm_vector_1 by auto qed
  2061 
  2062 end