src/HOL/Analysis/Cartesian_Euclidean_Space.thy
changeset 68833 fde093888c16
parent 68077 ee8c13ae81e9
child 69064 5840724b1d71
     1.1 --- a/src/HOL/Analysis/Cartesian_Euclidean_Space.thy	Mon Aug 27 22:58:36 2018 +0200
     1.2 +++ b/src/HOL/Analysis/Cartesian_Euclidean_Space.thy	Tue Aug 28 13:28:39 2018 +0100
     1.3 @@ -2,19 +2,19 @@
     1.4     Some material by Jose Divasón, Tim Makarios and L C Paulson
     1.5  *)
     1.6  
     1.7 -section \<open>Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space\<close>
     1.8 +section%important \<open>Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space\<close>
     1.9  
    1.10  theory Cartesian_Euclidean_Space
    1.11  imports Cartesian_Space Derivative
    1.12  begin
    1.13  
    1.14 -lemma subspace_special_hyperplane: "subspace {x. x $ k = 0}"
    1.15 +lemma%unimportant subspace_special_hyperplane: "subspace {x. x $ k = 0}"
    1.16    by (simp add: subspace_def)
    1.17  
    1.18 -lemma sum_mult_product:
    1.19 +lemma%important sum_mult_product:
    1.20    "sum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
    1.21    unfolding sum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]
    1.22 -proof (rule sum.cong, simp, rule sum.reindex_cong)
    1.23 +proof%unimportant (rule sum.cong, simp, rule sum.reindex_cong)
    1.24    fix i
    1.25    show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
    1.26    show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
    1.27 @@ -25,32 +25,32 @@
    1.28    qed simp
    1.29  qed simp
    1.30  
    1.31 -lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b"
    1.32 +lemma%unimportant interval_cbox_cart: "{a::real^'n..b} = cbox a b"
    1.33    by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis)
    1.34  
    1.35 -lemma differentiable_vec:
    1.36 +lemma%unimportant differentiable_vec:
    1.37    fixes S :: "'a::euclidean_space set"
    1.38    shows "vec differentiable_on S"
    1.39    by (simp add: linear_linear bounded_linear_imp_differentiable_on)
    1.40  
    1.41 -lemma continuous_vec [continuous_intros]:
    1.42 +lemma%unimportant continuous_vec [continuous_intros]:
    1.43    fixes x :: "'a::euclidean_space"
    1.44    shows "isCont vec x"
    1.45    apply (clarsimp simp add: continuous_def LIM_def dist_vec_def L2_set_def)
    1.46    apply (rule_tac x="r / sqrt (real CARD('b))" in exI)
    1.47    by (simp add: mult.commute pos_less_divide_eq real_sqrt_mult)
    1.48  
    1.49 -lemma box_vec_eq_empty [simp]:
    1.50 +lemma%unimportant box_vec_eq_empty [simp]:
    1.51    shows "cbox (vec a) (vec b) = {} \<longleftrightarrow> cbox a b = {}"
    1.52          "box (vec a) (vec b) = {} \<longleftrightarrow> box a b = {}"
    1.53    by (auto simp: Basis_vec_def mem_box box_eq_empty inner_axis)
    1.54  
    1.55 -subsection\<open>Closures and interiors of halfspaces\<close>
    1.56 +subsection%important\<open>Closures and interiors of halfspaces\<close>
    1.57  
    1.58 -lemma interior_halfspace_le [simp]:
    1.59 +lemma%important interior_halfspace_le [simp]:
    1.60    assumes "a \<noteq> 0"
    1.61      shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
    1.62 -proof -
    1.63 +proof%unimportant -
    1.64    have *: "a \<bullet> x < b" if x: "x \<in> S" and S: "S \<subseteq> {x. a \<bullet> x \<le> b}" and "open S" for S x
    1.65    proof -
    1.66      obtain e where "e>0" and e: "cball x e \<subseteq> S"
    1.67 @@ -70,15 +70,15 @@
    1.68      by (rule interior_unique) (auto simp: open_halfspace_lt *)
    1.69  qed
    1.70  
    1.71 -lemma interior_halfspace_ge [simp]:
    1.72 +lemma%unimportant interior_halfspace_ge [simp]:
    1.73     "a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"
    1.74  using interior_halfspace_le [of "-a" "-b"] by simp
    1.75  
    1.76 -lemma interior_halfspace_component_le [simp]:
    1.77 +lemma%important interior_halfspace_component_le [simp]:
    1.78       "interior {x. x$k \<le> a} = {x :: (real^'n). x$k < a}" (is "?LE")
    1.79    and interior_halfspace_component_ge [simp]:
    1.80       "interior {x. x$k \<ge> a} = {x :: (real^'n). x$k > a}" (is "?GE")
    1.81 -proof -
    1.82 +proof%unimportant -
    1.83    have "axis k (1::real) \<noteq> 0"
    1.84      by (simp add: axis_def vec_eq_iff)
    1.85    moreover have "axis k (1::real) \<bullet> x = x$k" for x
    1.86 @@ -88,7 +88,7 @@
    1.87            interior_halfspace_ge [of "axis k (1::real)" a] by auto
    1.88  qed
    1.89  
    1.90 -lemma closure_halfspace_lt [simp]:
    1.91 +lemma%unimportant closure_halfspace_lt [simp]:
    1.92    assumes "a \<noteq> 0"
    1.93      shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
    1.94  proof -
    1.95 @@ -99,15 +99,15 @@
    1.96      by (force simp: closure_interior)
    1.97  qed
    1.98  
    1.99 -lemma closure_halfspace_gt [simp]:
   1.100 +lemma%unimportant closure_halfspace_gt [simp]:
   1.101     "a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"
   1.102  using closure_halfspace_lt [of "-a" "-b"] by simp
   1.103  
   1.104 -lemma closure_halfspace_component_lt [simp]:
   1.105 +lemma%important closure_halfspace_component_lt [simp]:
   1.106       "closure {x. x$k < a} = {x :: (real^'n). x$k \<le> a}" (is "?LE")
   1.107    and closure_halfspace_component_gt [simp]:
   1.108       "closure {x. x$k > a} = {x :: (real^'n). x$k \<ge> a}" (is "?GE")
   1.109 -proof -
   1.110 +proof%unimportant -
   1.111    have "axis k (1::real) \<noteq> 0"
   1.112      by (simp add: axis_def vec_eq_iff)
   1.113    moreover have "axis k (1::real) \<bullet> x = x$k" for x
   1.114 @@ -117,17 +117,17 @@
   1.115            closure_halfspace_gt [of "axis k (1::real)" a] by auto
   1.116  qed
   1.117  
   1.118 -lemma interior_hyperplane [simp]:
   1.119 +lemma%unimportant interior_hyperplane [simp]:
   1.120    assumes "a \<noteq> 0"
   1.121      shows "interior {x. a \<bullet> x = b} = {}"
   1.122 -proof -
   1.123 +proof%unimportant -
   1.124    have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
   1.125      by (force simp:)
   1.126    then show ?thesis
   1.127      by (auto simp: assms)
   1.128  qed
   1.129  
   1.130 -lemma frontier_halfspace_le:
   1.131 +lemma%unimportant frontier_halfspace_le:
   1.132    assumes "a \<noteq> 0 \<or> b \<noteq> 0"
   1.133      shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
   1.134  proof (cases "a = 0")
   1.135 @@ -137,7 +137,7 @@
   1.136      by (force simp: frontier_def closed_halfspace_le)
   1.137  qed
   1.138  
   1.139 -lemma frontier_halfspace_ge:
   1.140 +lemma%unimportant frontier_halfspace_ge:
   1.141    assumes "a \<noteq> 0 \<or> b \<noteq> 0"
   1.142      shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"
   1.143  proof (cases "a = 0")
   1.144 @@ -147,7 +147,7 @@
   1.145      by (force simp: frontier_def closed_halfspace_ge)
   1.146  qed
   1.147  
   1.148 -lemma frontier_halfspace_lt:
   1.149 +lemma%unimportant frontier_halfspace_lt:
   1.150    assumes "a \<noteq> 0 \<or> b \<noteq> 0"
   1.151      shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"
   1.152  proof (cases "a = 0")
   1.153 @@ -157,19 +157,19 @@
   1.154      by (force simp: frontier_def interior_open open_halfspace_lt)
   1.155  qed
   1.156  
   1.157 -lemma frontier_halfspace_gt:
   1.158 +lemma%important frontier_halfspace_gt:
   1.159    assumes "a \<noteq> 0 \<or> b \<noteq> 0"
   1.160      shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"
   1.161 -proof (cases "a = 0")
   1.162 +proof%unimportant (cases "a = 0")
   1.163    case True with assms show ?thesis by simp
   1.164  next
   1.165    case False then show ?thesis
   1.166      by (force simp: frontier_def interior_open open_halfspace_gt)
   1.167  qed
   1.168  
   1.169 -lemma interior_standard_hyperplane:
   1.170 +lemma%important interior_standard_hyperplane:
   1.171     "interior {x :: (real^'n). x$k = a} = {}"
   1.172 -proof -
   1.173 +proof%unimportant -
   1.174    have "axis k (1::real) \<noteq> 0"
   1.175      by (simp add: axis_def vec_eq_iff)
   1.176    moreover have "axis k (1::real) \<bullet> x = x$k" for x
   1.177 @@ -179,20 +179,20 @@
   1.178      by force
   1.179  qed
   1.180  
   1.181 -lemma matrix_mult_transpose_dot_column:
   1.182 +lemma%unimportant matrix_mult_transpose_dot_column:
   1.183    shows "transpose A ** A = (\<chi> i j. inner (column i A) (column j A))"
   1.184    by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)
   1.185  
   1.186 -lemma matrix_mult_transpose_dot_row:
   1.187 +lemma%unimportant matrix_mult_transpose_dot_row:
   1.188    shows "A ** transpose A = (\<chi> i j. inner (row i A) (row j A))"
   1.189    by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)
   1.190  
   1.191  text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
   1.192  
   1.193 -lemma matrix_mult_dot: "A *v x = (\<chi> i. inner (A$i) x)"
   1.194 +lemma%important matrix_mult_dot: "A *v x = (\<chi> i. inner (A$i) x)"
   1.195    by (simp add: matrix_vector_mult_def inner_vec_def)
   1.196  
   1.197 -lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
   1.198 +lemma%unimportant adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
   1.199    apply (rule adjoint_unique)
   1.200    apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
   1.201      sum_distrib_right sum_distrib_left)
   1.202 @@ -200,9 +200,9 @@
   1.203    apply (simp add:  ac_simps)
   1.204    done
   1.205  
   1.206 -lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
   1.207 +lemma%important matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
   1.208    shows "matrix(adjoint f) = transpose(matrix f)"
   1.209 -proof -
   1.210 +proof%unimportant -
   1.211    have "matrix(adjoint f) = matrix(adjoint (( *v) (matrix f)))"
   1.212      by (simp add: lf)
   1.213    also have "\<dots> = transpose(matrix f)"
   1.214 @@ -212,17 +212,17 @@
   1.215    finally show ?thesis .
   1.216  qed
   1.217  
   1.218 -lemma matrix_vector_mul_bounded_linear[intro, simp]: "bounded_linear (( *v) A)" for A :: "'a::{euclidean_space,real_algebra_1}^'n^'m"
   1.219 +lemma%unimportant matrix_vector_mul_bounded_linear[intro, simp]: "bounded_linear (( *v) A)" for A :: "'a::{euclidean_space,real_algebra_1}^'n^'m"
   1.220    using matrix_vector_mul_linear[of A]
   1.221    by (simp add: linear_conv_bounded_linear linear_matrix_vector_mul_eq)
   1.222  
   1.223 -lemma
   1.224 +lemma%unimportant (* FIX ME needs name*)
   1.225    fixes A :: "'a::{euclidean_space,real_algebra_1}^'n^'m"
   1.226    shows matrix_vector_mult_linear_continuous_at [continuous_intros]: "isCont (( *v) A) z"
   1.227      and matrix_vector_mult_linear_continuous_on [continuous_intros]: "continuous_on S (( *v) A)"
   1.228    by (simp_all add: linear_continuous_at linear_continuous_on)
   1.229  
   1.230 -lemma scalar_invertible:
   1.231 +lemma%unimportant scalar_invertible:
   1.232    fixes A :: "('a::real_algebra_1)^'m^'n"
   1.233    assumes "k \<noteq> 0" and "invertible A"
   1.234    shows "invertible (k *\<^sub>R A)"
   1.235 @@ -236,50 +236,50 @@
   1.236      unfolding invertible_def by auto
   1.237  qed
   1.238  
   1.239 -lemma scalar_invertible_iff:
   1.240 +lemma%unimportant scalar_invertible_iff:
   1.241    fixes A :: "('a::real_algebra_1)^'m^'n"
   1.242    assumes "k \<noteq> 0" and "invertible A"
   1.243    shows "invertible (k *\<^sub>R A) \<longleftrightarrow> k \<noteq> 0 \<and> invertible A"
   1.244    by (simp add: assms scalar_invertible)
   1.245  
   1.246 -lemma vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
   1.247 +lemma%unimportant vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
   1.248    unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
   1.249    by simp
   1.250  
   1.251 -lemma transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
   1.252 +lemma%unimportant transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
   1.253    unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
   1.254    by simp
   1.255  
   1.256 -lemma vector_scalar_commute:
   1.257 +lemma%unimportant vector_scalar_commute:
   1.258    fixes A :: "'a::{field}^'m^'n"
   1.259    shows "A *v (c *s x) = c *s (A *v x)"
   1.260    by (simp add: vector_scalar_mult_def matrix_vector_mult_def mult_ac sum_distrib_left)
   1.261  
   1.262 -lemma scalar_vector_matrix_assoc:
   1.263 +lemma%unimportant scalar_vector_matrix_assoc:
   1.264    fixes k :: "'a::{field}" and x :: "'a::{field}^'n" and A :: "'a^'m^'n"
   1.265    shows "(k *s x) v* A = k *s (x v* A)"
   1.266    by (metis transpose_matrix_vector vector_scalar_commute)
   1.267   
   1.268 -lemma vector_matrix_mult_0 [simp]: "0 v* A = 0"
   1.269 +lemma%unimportant vector_matrix_mult_0 [simp]: "0 v* A = 0"
   1.270    unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
   1.271  
   1.272 -lemma vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
   1.273 +lemma%unimportant vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
   1.274    unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
   1.275  
   1.276 -lemma vector_matrix_mul_rid [simp]:
   1.277 +lemma%unimportant vector_matrix_mul_rid [simp]:
   1.278    fixes v :: "('a::semiring_1)^'n"
   1.279    shows "v v* mat 1 = v"
   1.280    by (metis matrix_vector_mul_lid transpose_mat vector_transpose_matrix)
   1.281  
   1.282 -lemma scaleR_vector_matrix_assoc:
   1.283 +lemma%unimportant scaleR_vector_matrix_assoc:
   1.284    fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
   1.285    shows "(k *\<^sub>R x) v* A = k *\<^sub>R (x v* A)"
   1.286    by (metis matrix_vector_mult_scaleR transpose_matrix_vector)
   1.287  
   1.288 -lemma vector_scaleR_matrix_ac:
   1.289 +lemma%important vector_scaleR_matrix_ac:
   1.290    fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
   1.291    shows "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
   1.292 -proof -
   1.293 +proof%unimportant -
   1.294    have "x v* (k *\<^sub>R A) = (k *\<^sub>R x) v* A"
   1.295      unfolding vector_matrix_mult_def
   1.296      by (simp add: algebra_simps)
   1.297 @@ -289,12 +289,12 @@
   1.298  qed
   1.299  
   1.300  
   1.301 -subsection\<open>Some bounds on components etc. relative to operator norm\<close>
   1.302 +subsection%important\<open>Some bounds on components etc. relative to operator norm\<close>
   1.303  
   1.304 -lemma norm_column_le_onorm:
   1.305 +lemma%important norm_column_le_onorm:
   1.306    fixes A :: "real^'n^'m"
   1.307    shows "norm(column i A) \<le> onorm(( *v) A)"
   1.308 -proof -
   1.309 +proof%unimportant -
   1.310    have "norm (\<chi> j. A $ j $ i) \<le> norm (A *v axis i 1)"
   1.311      by (simp add: matrix_mult_dot cart_eq_inner_axis)
   1.312    also have "\<dots> \<le> onorm (( *v) A)"
   1.313 @@ -304,10 +304,10 @@
   1.314      unfolding column_def .
   1.315  qed
   1.316  
   1.317 -lemma matrix_component_le_onorm:
   1.318 +lemma%important matrix_component_le_onorm:
   1.319    fixes A :: "real^'n^'m"
   1.320    shows "\<bar>A $ i $ j\<bar> \<le> onorm(( *v) A)"
   1.321 -proof -
   1.322 +proof%unimportant -
   1.323    have "\<bar>A $ i $ j\<bar> \<le> norm (\<chi> n. (A $ n $ j))"
   1.324      by (metis (full_types, lifting) component_le_norm_cart vec_lambda_beta)
   1.325    also have "\<dots> \<le> onorm (( *v) A)"
   1.326 @@ -315,15 +315,15 @@
   1.327    finally show ?thesis .
   1.328  qed
   1.329  
   1.330 -lemma component_le_onorm:
   1.331 +lemma%unimportant component_le_onorm:
   1.332    fixes f :: "real^'m \<Rightarrow> real^'n"
   1.333    shows "linear f \<Longrightarrow> \<bar>matrix f $ i $ j\<bar> \<le> onorm f"
   1.334    by (metis linear_matrix_vector_mul_eq matrix_component_le_onorm matrix_vector_mul)
   1.335  
   1.336 -lemma onorm_le_matrix_component_sum:
   1.337 +lemma%important onorm_le_matrix_component_sum:
   1.338    fixes A :: "real^'n^'m"
   1.339    shows "onorm(( *v) A) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>)"
   1.340 -proof (rule onorm_le)
   1.341 +proof%unimportant (rule onorm_le)
   1.342    fix x
   1.343    have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
   1.344      by (rule norm_le_l1_cart)
   1.345 @@ -342,11 +342,11 @@
   1.346      by (simp add: sum_distrib_right)
   1.347  qed
   1.348  
   1.349 -lemma onorm_le_matrix_component:
   1.350 +lemma%important onorm_le_matrix_component:
   1.351    fixes A :: "real^'n^'m"
   1.352    assumes "\<And>i j. abs(A$i$j) \<le> B"
   1.353    shows "onorm(( *v) A) \<le> real (CARD('m)) * real (CARD('n)) * B"
   1.354 -proof (rule onorm_le)
   1.355 +proof%unimportant (rule onorm_le)
   1.356    fix x :: "real^'n::_"
   1.357    have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
   1.358      by (rule norm_le_l1_cart)
   1.359 @@ -366,11 +366,11 @@
   1.360    finally show "norm (A *v x) \<le> CARD('m) * real (CARD('n)) * B * norm x" .
   1.361  qed
   1.362  
   1.363 -subsection \<open>lambda skolemization on cartesian products\<close>
   1.364 +subsection%important \<open>lambda skolemization on cartesian products\<close>
   1.365  
   1.366 -lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
   1.367 +lemma%important lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
   1.368     (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
   1.369 -proof -
   1.370 +proof%unimportant -
   1.371    let ?S = "(UNIV :: 'n set)"
   1.372    { assume H: "?rhs"
   1.373      then have ?lhs by auto }
   1.374 @@ -387,16 +387,16 @@
   1.375    ultimately show ?thesis by metis
   1.376  qed
   1.377  
   1.378 -lemma rational_approximation:
   1.379 +lemma%unimportant rational_approximation:
   1.380    assumes "e > 0"
   1.381    obtains r::real where "r \<in> \<rat>" "\<bar>r - x\<bar> < e"
   1.382    using Rats_dense_in_real [of "x - e/2" "x + e/2"] assms by auto
   1.383  
   1.384 -lemma matrix_rational_approximation:
   1.385 +lemma%important matrix_rational_approximation:
   1.386    fixes A :: "real^'n^'m"
   1.387    assumes "e > 0"
   1.388    obtains B where "\<And>i j. B$i$j \<in> \<rat>" "onorm(\<lambda>x. (A - B) *v x) < e"
   1.389 -proof -
   1.390 +proof%unimportant -
   1.391    have "\<forall>i j. \<exists>q \<in> \<rat>. \<bar>q - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))"
   1.392      using assms by (force intro: rational_approximation [of "e / (2 * CARD('m) * CARD('n))"])
   1.393    then obtain B where B: "\<And>i j. B$i$j \<in> \<rat>" and Bclo: "\<And>i j. \<bar>B$i$j - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))"
   1.394 @@ -413,7 +413,7 @@
   1.395    qed (use B in auto)
   1.396  qed
   1.397  
   1.398 -lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
   1.399 +lemma%unimportant vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
   1.400    unfolding inner_simps scalar_mult_eq_scaleR by auto
   1.401  
   1.402  
   1.403 @@ -422,51 +422,51 @@
   1.404  
   1.405  text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
   1.406  
   1.407 -definition "rowvector v = (\<chi> i j. (v$j))"
   1.408 +definition%unimportant "rowvector v = (\<chi> i j. (v$j))"
   1.409  
   1.410 -definition "columnvector v = (\<chi> i j. (v$i))"
   1.411 +definition%unimportant "columnvector v = (\<chi> i j. (v$i))"
   1.412  
   1.413 -lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
   1.414 +lemma%unimportant transpose_columnvector: "transpose(columnvector v) = rowvector v"
   1.415    by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
   1.416  
   1.417 -lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
   1.418 +lemma%unimportant transpose_rowvector: "transpose(rowvector v) = columnvector v"
   1.419    by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
   1.420  
   1.421 -lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
   1.422 +lemma%unimportant dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
   1.423    by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
   1.424  
   1.425 -lemma dot_matrix_product:
   1.426 +lemma%unimportant dot_matrix_product:
   1.427    "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
   1.428    by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
   1.429  
   1.430 -lemma dot_matrix_vector_mul:
   1.431 +lemma%unimportant dot_matrix_vector_mul:
   1.432    fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
   1.433    shows "(A *v x) \<bullet> (B *v y) =
   1.434        (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
   1.435    unfolding dot_matrix_product transpose_columnvector[symmetric]
   1.436      dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
   1.437  
   1.438 -lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}"
   1.439 +lemma%unimportant infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}"
   1.440    by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
   1.441  
   1.442 -lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
   1.443 +lemma%unimportant component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
   1.444    using Basis_le_infnorm[of "axis i 1" x]
   1.445    by (simp add: Basis_vec_def axis_eq_axis inner_axis)
   1.446  
   1.447 -lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
   1.448 +lemma%unimportant continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
   1.449    unfolding continuous_def by (rule tendsto_vec_nth)
   1.450  
   1.451 -lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
   1.452 +lemma%unimportant continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
   1.453    unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
   1.454  
   1.455 -lemma continuous_on_vec_lambda[continuous_intros]:
   1.456 +lemma%unimportant continuous_on_vec_lambda[continuous_intros]:
   1.457    "(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)"
   1.458    unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)
   1.459  
   1.460 -lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
   1.461 +lemma%unimportant closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
   1.462    by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
   1.463  
   1.464 -lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
   1.465 +lemma%unimportant bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
   1.466    unfolding bounded_def
   1.467    apply clarify
   1.468    apply (rule_tac x="x $ i" in exI)
   1.469 @@ -475,13 +475,13 @@
   1.470    apply (rule order_trans [OF dist_vec_nth_le], simp)
   1.471    done
   1.472  
   1.473 -lemma compact_lemma_cart:
   1.474 +lemma%important compact_lemma_cart:
   1.475    fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
   1.476    assumes f: "bounded (range f)"
   1.477    shows "\<exists>l r. strict_mono r \<and>
   1.478          (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
   1.479      (is "?th d")
   1.480 -proof -
   1.481 +proof%unimportant -
   1.482    have "\<forall>d' \<subseteq> d. ?th d'"
   1.483      by (rule compact_lemma_general[where unproj=vec_lambda])
   1.484        (auto intro!: f bounded_component_cart simp: vec_lambda_eta)
   1.485 @@ -517,19 +517,19 @@
   1.486    with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
   1.487  qed
   1.488  
   1.489 -lemma interval_cart:
   1.490 +lemma%unimportant interval_cart:
   1.491    fixes a :: "real^'n"
   1.492    shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
   1.493      and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
   1.494    by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
   1.495  
   1.496 -lemma mem_box_cart:
   1.497 +lemma%unimportant mem_box_cart:
   1.498    fixes a :: "real^'n"
   1.499    shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
   1.500      and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
   1.501    using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
   1.502  
   1.503 -lemma interval_eq_empty_cart:
   1.504 +lemma%unimportant interval_eq_empty_cart:
   1.505    fixes a :: "real^'n"
   1.506    shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
   1.507      and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
   1.508 @@ -565,14 +565,14 @@
   1.509    ultimately show ?th2 by blast
   1.510  qed
   1.511  
   1.512 -lemma interval_ne_empty_cart:
   1.513 +lemma%unimportant interval_ne_empty_cart:
   1.514    fixes a :: "real^'n"
   1.515    shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
   1.516      and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
   1.517    unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
   1.518      (* BH: Why doesn't just "auto" work here? *)
   1.519  
   1.520 -lemma subset_interval_imp_cart:
   1.521 +lemma%unimportant subset_interval_imp_cart:
   1.522    fixes a :: "real^'n"
   1.523    shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
   1.524      and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
   1.525 @@ -581,13 +581,13 @@
   1.526    unfolding subset_eq[unfolded Ball_def] unfolding mem_box_cart
   1.527    by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
   1.528  
   1.529 -lemma interval_sing:
   1.530 +lemma%unimportant interval_sing:
   1.531    fixes a :: "'a::linorder^'n"
   1.532    shows "{a .. a} = {a} \<and> {a<..<a} = {}"
   1.533    apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
   1.534    done
   1.535  
   1.536 -lemma subset_interval_cart:
   1.537 +lemma%unimportant subset_interval_cart:
   1.538    fixes a :: "real^'n"
   1.539    shows "cbox c d \<subseteq> cbox 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)
   1.540      and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
   1.541 @@ -595,7 +595,7 @@
   1.542      and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
   1.543    using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
   1.544  
   1.545 -lemma disjoint_interval_cart:
   1.546 +lemma%unimportant disjoint_interval_cart:
   1.547    fixes a::"real^'n"
   1.548    shows "cbox a b \<inter> cbox 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)
   1.549      and "cbox a b \<inter> box 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)
   1.550 @@ -603,53 +603,53 @@
   1.551      and "box a b \<inter> box 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)
   1.552    using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
   1.553  
   1.554 -lemma Int_interval_cart:
   1.555 +lemma%unimportant Int_interval_cart:
   1.556    fixes a :: "real^'n"
   1.557    shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
   1.558    unfolding Int_interval
   1.559    by (auto simp: mem_box less_eq_vec_def)
   1.560      (auto simp: Basis_vec_def inner_axis)
   1.561  
   1.562 -lemma closed_interval_left_cart:
   1.563 +lemma%unimportant closed_interval_left_cart:
   1.564    fixes b :: "real^'n"
   1.565    shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
   1.566    by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
   1.567  
   1.568 -lemma closed_interval_right_cart:
   1.569 +lemma%unimportant closed_interval_right_cart:
   1.570    fixes a::"real^'n"
   1.571    shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
   1.572    by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
   1.573  
   1.574 -lemma is_interval_cart:
   1.575 +lemma%unimportant is_interval_cart:
   1.576    "is_interval (s::(real^'n) set) \<longleftrightarrow>
   1.577      (\<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)"
   1.578    by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
   1.579  
   1.580 -lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
   1.581 +lemma%unimportant closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
   1.582    by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
   1.583  
   1.584 -lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
   1.585 +lemma%unimportant closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
   1.586    by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
   1.587  
   1.588 -lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
   1.589 +lemma%unimportant open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
   1.590    by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
   1.591  
   1.592 -lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
   1.593 +lemma%unimportant open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
   1.594    by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
   1.595  
   1.596 -lemma Lim_component_le_cart:
   1.597 +lemma%unimportant Lim_component_le_cart:
   1.598    fixes f :: "'a \<Rightarrow> real^'n"
   1.599    assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"
   1.600    shows "l$i \<le> b"
   1.601    by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
   1.602  
   1.603 -lemma Lim_component_ge_cart:
   1.604 +lemma%unimportant Lim_component_ge_cart:
   1.605    fixes f :: "'a \<Rightarrow> real^'n"
   1.606    assumes "(f \<longlongrightarrow> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
   1.607    shows "b \<le> l$i"
   1.608    by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
   1.609  
   1.610 -lemma Lim_component_eq_cart:
   1.611 +lemma%unimportant Lim_component_eq_cart:
   1.612    fixes f :: "'a \<Rightarrow> real^'n"
   1.613    assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
   1.614    shows "l$i = b"
   1.615 @@ -657,18 +657,18 @@
   1.616      Lim_component_ge_cart[OF net, of b i] and
   1.617      Lim_component_le_cart[OF net, of i b] by auto
   1.618  
   1.619 -lemma connected_ivt_component_cart:
   1.620 +lemma%unimportant connected_ivt_component_cart:
   1.621    fixes x :: "real^'n"
   1.622    shows "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$k = a)"
   1.623    using connected_ivt_hyperplane[of s x y "axis k 1" a]
   1.624    by (auto simp add: inner_axis inner_commute)
   1.625  
   1.626 -lemma subspace_substandard_cart: "vec.subspace {x. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
   1.627 +lemma%unimportant subspace_substandard_cart: "vec.subspace {x. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
   1.628    unfolding vec.subspace_def by auto
   1.629  
   1.630 -lemma closed_substandard_cart:
   1.631 +lemma%important closed_substandard_cart:
   1.632    "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
   1.633 -proof -
   1.634 +proof%unimportant -
   1.635    { fix i::'n
   1.636      have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
   1.637        by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) }
   1.638 @@ -676,9 +676,9 @@
   1.639      unfolding Collect_all_eq by (simp add: closed_INT)
   1.640  qed
   1.641  
   1.642 -lemma dim_substandard_cart: "vec.dim {x::'a::field^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
   1.643 +lemma%important dim_substandard_cart: "vec.dim {x::'a::field^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
   1.644    (is "vec.dim ?A = _")
   1.645 -proof (rule vec.dim_unique)
   1.646 +proof%unimportant (rule vec.dim_unique)
   1.647    let ?B = "((\<lambda>x. axis x 1) ` d)"
   1.648    have subset_basis: "?B \<subseteq> cart_basis"
   1.649      by (auto simp: cart_basis_def)
   1.650 @@ -703,27 +703,27 @@
   1.651    then show "?A \<subseteq> vec.span ?B" by auto
   1.652  qed (simp add: card_image inj_on_def axis_eq_axis)
   1.653  
   1.654 -lemma dim_subset_UNIV_cart_gen:
   1.655 +lemma%unimportant dim_subset_UNIV_cart_gen:
   1.656    fixes S :: "('a::field^'n) set"
   1.657    shows "vec.dim S \<le> CARD('n)"
   1.658    by (metis vec.dim_eq_full vec.dim_subset_UNIV vec.span_UNIV vec_dim_card)
   1.659  
   1.660 -lemma dim_subset_UNIV_cart:
   1.661 +lemma%unimportant dim_subset_UNIV_cart:
   1.662    fixes S :: "(real^'n) set"
   1.663    shows "dim S \<le> CARD('n)"
   1.664    using dim_subset_UNIV_cart_gen[of S] by (simp add: dim_vec_eq)
   1.665  
   1.666 -lemma affinity_inverses:
   1.667 +lemma%unimportant affinity_inverses:
   1.668    assumes m0: "m \<noteq> (0::'a::field)"
   1.669    shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
   1.670    "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
   1.671    using m0
   1.672    by (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
   1.673  
   1.674 -lemma vector_affinity_eq:
   1.675 +lemma%important vector_affinity_eq:
   1.676    assumes m0: "(m::'a::field) \<noteq> 0"
   1.677    shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
   1.678 -proof
   1.679 +proof%unimportant
   1.680    assume h: "m *s x + c = y"
   1.681    hence "m *s x = y - c" by (simp add: field_simps)
   1.682    hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
   1.683 @@ -735,48 +735,48 @@
   1.684      using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
   1.685  qed
   1.686  
   1.687 -lemma vector_eq_affinity:
   1.688 +lemma%unimportant vector_eq_affinity:
   1.689      "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
   1.690    using vector_affinity_eq[where m=m and x=x and y=y and c=c]
   1.691    by metis
   1.692  
   1.693 -lemma vector_cart:
   1.694 +lemma%unimportant vector_cart:
   1.695    fixes f :: "real^'n \<Rightarrow> real"
   1.696    shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
   1.697    unfolding euclidean_eq_iff[where 'a="real^'n"]
   1.698    by simp (simp add: Basis_vec_def inner_axis)
   1.699  
   1.700 -lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
   1.701 +lemma%unimportant const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
   1.702    by (rule vector_cart)
   1.703  
   1.704 -subsection "Convex Euclidean Space"
   1.705 +subsection%important "Convex Euclidean Space"
   1.706  
   1.707 -lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
   1.708 +lemma%unimportant Cart_1:"(1::real^'n) = \<Sum>Basis"
   1.709    using const_vector_cart[of 1] by (simp add: one_vec_def)
   1.710  
   1.711  declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
   1.712  declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
   1.713  
   1.714 -lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
   1.715 +lemmas%unimportant vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
   1.716  
   1.717 -lemma convex_box_cart:
   1.718 +lemma%unimportant convex_box_cart:
   1.719    assumes "\<And>i. convex {x. P i x}"
   1.720    shows "convex {x. \<forall>i. P i (x$i)}"
   1.721    using assms unfolding convex_def by auto
   1.722  
   1.723 -lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
   1.724 +lemma%unimportant convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
   1.725    by (rule convex_box_cart) (simp add: atLeast_def[symmetric])
   1.726  
   1.727 -lemma unit_interval_convex_hull_cart:
   1.728 +lemma%unimportant unit_interval_convex_hull_cart:
   1.729    "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}"
   1.730    unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
   1.731    by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
   1.732  
   1.733 -lemma cube_convex_hull_cart:
   1.734 +lemma%important cube_convex_hull_cart:
   1.735    assumes "0 < d"
   1.736    obtains s::"(real^'n) set"
   1.737      where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
   1.738 -proof -
   1.739 +proof%unimportant -
   1.740    from assms obtain s where "finite s"
   1.741      and "cbox (x - sum (( *\<^sub>R) d) Basis) (x + sum (( *\<^sub>R) d) Basis) = convex hull s"
   1.742      by (rule cube_convex_hull)
   1.743 @@ -785,14 +785,14 @@
   1.744  qed
   1.745  
   1.746  
   1.747 -subsection "Derivative"
   1.748 +subsection%important "Derivative"
   1.749  
   1.750 -definition "jacobian f net = matrix(frechet_derivative f net)"
   1.751 +definition%important "jacobian f net = matrix(frechet_derivative f net)"
   1.752  
   1.753 -lemma jacobian_works:
   1.754 +lemma%important jacobian_works:
   1.755    "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
   1.756      (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net" (is "?lhs = ?rhs")
   1.757 -proof
   1.758 +proof%unimportant
   1.759    assume ?lhs then show ?rhs
   1.760      by (simp add: frechet_derivative_works has_derivative_linear jacobian_def)
   1.761  next
   1.762 @@ -801,10 +801,10 @@
   1.763  qed
   1.764  
   1.765  
   1.766 -subsection \<open>Component of the differential must be zero if it exists at a local
   1.767 +subsection%important \<open>Component of the differential must be zero if it exists at a local
   1.768    maximum or minimum for that corresponding component\<close>
   1.769  
   1.770 -lemma differential_zero_maxmin_cart:
   1.771 +lemma%important differential_zero_maxmin_cart:
   1.772    fixes f::"real^'a \<Rightarrow> real^'b"
   1.773    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))"
   1.774      "f differentiable (at x)"
   1.775 @@ -813,7 +813,7 @@
   1.776      vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
   1.777    by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
   1.778  
   1.779 -subsection \<open>Lemmas for working on @{typ "real^1"}\<close>
   1.780 +subsection%unimportant \<open>Lemmas for working on @{typ "real^1"}\<close>
   1.781  
   1.782  lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
   1.783    by (metis (full_types) num1_eq_iff)
   1.784 @@ -893,7 +893,7 @@
   1.785  instance num1 :: wellorder
   1.786    by intro_classes (auto simp: less_eq_num1_def less_num1_def)
   1.787  
   1.788 -subsection\<open>The collapse of the general concepts to dimension one\<close>
   1.789 +subsection%unimportant\<open>The collapse of the general concepts to dimension one\<close>
   1.790  
   1.791  lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
   1.792    by (simp add: vec_eq_iff)
   1.793 @@ -918,11 +918,11 @@
   1.794  lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x$1) - (y$1)\<bar>"
   1.795    by (auto simp add: norm_real dist_norm)
   1.796  
   1.797 -subsection\<open> Rank of a matrix\<close>
   1.798 +subsection%important\<open> Rank of a matrix\<close>
   1.799  
   1.800  text\<open>Equivalence of row and column rank is taken from George Mackiw's paper, Mathematics Magazine 1995, p. 285.\<close>
   1.801  
   1.802 -lemma matrix_vector_mult_in_columnspace_gen:
   1.803 +lemma%unimportant matrix_vector_mult_in_columnspace_gen:
   1.804    fixes A :: "'a::field^'n^'m"
   1.805    shows "(A *v x) \<in> vec.span(columns A)"
   1.806    apply (simp add: matrix_vector_column columns_def transpose_def column_def)
   1.807 @@ -930,17 +930,17 @@
   1.808    apply (force intro: vec.span_base)
   1.809    done
   1.810  
   1.811 -lemma matrix_vector_mult_in_columnspace:
   1.812 +lemma%unimportant matrix_vector_mult_in_columnspace:
   1.813    fixes A :: "real^'n^'m"
   1.814    shows "(A *v x) \<in> span(columns A)"
   1.815    using matrix_vector_mult_in_columnspace_gen[of A x] by (simp add: span_vec_eq)
   1.816  
   1.817 -lemma orthogonal_nullspace_rowspace:
   1.818 +lemma%important orthogonal_nullspace_rowspace:
   1.819    fixes A :: "real^'n^'m"
   1.820    assumes 0: "A *v x = 0" and y: "y \<in> span(rows A)"
   1.821    shows "orthogonal x y"
   1.822    using y
   1.823 -proof (induction rule: span_induct)
   1.824 +proof%unimportant (induction rule: span_induct)
   1.825    case base
   1.826    then show ?case
   1.827      by (simp add: subspace_orthogonal_to_vector)
   1.828 @@ -953,28 +953,28 @@
   1.829      by (simp add: mult.commute) (metis (no_types) vec_lambda_beta zero_index)
   1.830  qed
   1.831  
   1.832 -lemma nullspace_inter_rowspace:
   1.833 +lemma%unimportant nullspace_inter_rowspace:
   1.834    fixes A :: "real^'n^'m"
   1.835    shows "A *v x = 0 \<and> x \<in> span(rows A) \<longleftrightarrow> x = 0"
   1.836    using orthogonal_nullspace_rowspace orthogonal_self span_zero matrix_vector_mult_0_right
   1.837    by blast
   1.838  
   1.839 -lemma matrix_vector_mul_injective_on_rowspace:
   1.840 +lemma%unimportant matrix_vector_mul_injective_on_rowspace:
   1.841    fixes A :: "real^'n^'m"
   1.842    shows "\<lbrakk>A *v x = A *v y; x \<in> span(rows A); y \<in> span(rows A)\<rbrakk> \<Longrightarrow> x = y"
   1.843    using nullspace_inter_rowspace [of A "x-y"]
   1.844    by (metis diff_eq_diff_eq diff_self matrix_vector_mult_diff_distrib span_diff)
   1.845  
   1.846 -definition rank :: "'a::field^'n^'m=>nat"
   1.847 +definition%important rank :: "'a::field^'n^'m=>nat"
   1.848    where row_rank_def_gen: "rank A \<equiv> vec.dim(rows A)"
   1.849  
   1.850 -lemma row_rank_def: "rank A = dim (rows A)" for A::"real^'n^'m"
   1.851 -  by (auto simp: row_rank_def_gen dim_vec_eq)
   1.852 +lemma%important row_rank_def: "rank A = dim (rows A)" for A::"real^'n^'m"
   1.853 +  by%unimportant (auto simp: row_rank_def_gen dim_vec_eq)
   1.854  
   1.855 -lemma dim_rows_le_dim_columns:
   1.856 +lemma%important dim_rows_le_dim_columns:
   1.857    fixes A :: "real^'n^'m"
   1.858    shows "dim(rows A) \<le> dim(columns A)"
   1.859 -proof -
   1.860 +proof%unimportant -
   1.861    have "dim (span (rows A)) \<le> dim (span (columns A))"
   1.862    proof -
   1.863      obtain B where "independent B" "span(rows A) \<subseteq> span B"
   1.864 @@ -999,32 +999,32 @@
   1.865      by (simp add: dim_span)
   1.866  qed
   1.867  
   1.868 -lemma column_rank_def:
   1.869 +lemma%unimportant column_rank_def:
   1.870    fixes A :: "real^'n^'m"
   1.871    shows "rank A = dim(columns A)"
   1.872    unfolding row_rank_def
   1.873    by (metis columns_transpose dim_rows_le_dim_columns le_antisym rows_transpose)
   1.874  
   1.875 -lemma rank_transpose:
   1.876 +lemma%unimportant rank_transpose:
   1.877    fixes A :: "real^'n^'m"
   1.878    shows "rank(transpose A) = rank A"
   1.879    by (metis column_rank_def row_rank_def rows_transpose)
   1.880  
   1.881 -lemma matrix_vector_mult_basis:
   1.882 +lemma%unimportant matrix_vector_mult_basis:
   1.883    fixes A :: "real^'n^'m"
   1.884    shows "A *v (axis k 1) = column k A"
   1.885    by (simp add: cart_eq_inner_axis column_def matrix_mult_dot)
   1.886  
   1.887 -lemma columns_image_basis:
   1.888 +lemma%unimportant columns_image_basis:
   1.889    fixes A :: "real^'n^'m"
   1.890    shows "columns A = ( *v) A ` (range (\<lambda>i. axis i 1))"
   1.891    by (force simp: columns_def matrix_vector_mult_basis [symmetric])
   1.892  
   1.893 -lemma rank_dim_range:
   1.894 +lemma%important rank_dim_range:
   1.895    fixes A :: "real^'n^'m"
   1.896    shows "rank A = dim(range (\<lambda>x. A *v x))"
   1.897    unfolding column_rank_def
   1.898 -proof (rule span_eq_dim)
   1.899 +proof%unimportant (rule span_eq_dim)
   1.900    have "span (columns A) \<subseteq> span (range (( *v) A))" (is "?l \<subseteq> ?r")
   1.901      by (simp add: columns_image_basis image_subsetI span_mono)
   1.902    then show "?l = ?r"
   1.903 @@ -1032,45 +1032,45 @@
   1.904          span_eq span_span)
   1.905  qed
   1.906  
   1.907 -lemma rank_bound:
   1.908 +lemma%unimportant rank_bound:
   1.909    fixes A :: "real^'n^'m"
   1.910    shows "rank A \<le> min CARD('m) (CARD('n))"
   1.911    by (metis (mono_tags, lifting) dim_subset_UNIV_cart min.bounded_iff
   1.912        column_rank_def row_rank_def)
   1.913  
   1.914 -lemma full_rank_injective:
   1.915 +lemma%unimportant full_rank_injective:
   1.916    fixes A :: "real^'n^'m"
   1.917    shows "rank A = CARD('n) \<longleftrightarrow> inj (( *v) A)"
   1.918    by (simp add: matrix_left_invertible_injective [symmetric] matrix_left_invertible_span_rows row_rank_def
   1.919        dim_eq_full [symmetric] card_cart_basis vec.dimension_def)
   1.920  
   1.921 -lemma full_rank_surjective:
   1.922 +lemma%unimportant full_rank_surjective:
   1.923    fixes A :: "real^'n^'m"
   1.924    shows "rank A = CARD('m) \<longleftrightarrow> surj (( *v) A)"
   1.925    by (simp add: matrix_right_invertible_surjective [symmetric] left_invertible_transpose [symmetric]
   1.926                  matrix_left_invertible_injective full_rank_injective [symmetric] rank_transpose)
   1.927  
   1.928 -lemma rank_I: "rank(mat 1::real^'n^'n) = CARD('n)"
   1.929 +lemma%unimportant rank_I: "rank(mat 1::real^'n^'n) = CARD('n)"
   1.930    by (simp add: full_rank_injective inj_on_def)
   1.931  
   1.932 -lemma less_rank_noninjective:
   1.933 +lemma%unimportant less_rank_noninjective:
   1.934    fixes A :: "real^'n^'m"
   1.935    shows "rank A < CARD('n) \<longleftrightarrow> \<not> inj (( *v) A)"
   1.936  using less_le rank_bound by (auto simp: full_rank_injective [symmetric])
   1.937  
   1.938 -lemma matrix_nonfull_linear_equations_eq:
   1.939 +lemma%unimportant matrix_nonfull_linear_equations_eq:
   1.940    fixes A :: "real^'n^'m"
   1.941    shows "(\<exists>x. (x \<noteq> 0) \<and> A *v x = 0) \<longleftrightarrow> ~(rank A = CARD('n))"
   1.942    by (meson matrix_left_invertible_injective full_rank_injective matrix_left_invertible_ker)
   1.943  
   1.944 -lemma rank_eq_0: "rank A = 0 \<longleftrightarrow> A = 0" and rank_0 [simp]: "rank (0::real^'n^'m) = 0"
   1.945 +lemma%unimportant rank_eq_0: "rank A = 0 \<longleftrightarrow> A = 0" and rank_0 [simp]: "rank (0::real^'n^'m) = 0"
   1.946    for A :: "real^'n^'m"
   1.947    by (auto simp: rank_dim_range matrix_eq)
   1.948  
   1.949 -lemma rank_mul_le_right:
   1.950 +lemma%important rank_mul_le_right:
   1.951    fixes A :: "real^'n^'m" and B :: "real^'p^'n"
   1.952    shows "rank(A ** B) \<le> rank B"
   1.953 -proof -
   1.954 +proof%unimportant -
   1.955    have "rank(A ** B) \<le> dim (( *v) A ` range (( *v) B))"
   1.956      by (auto simp: rank_dim_range image_comp o_def matrix_vector_mul_assoc)
   1.957    also have "\<dots> \<le> rank B"
   1.958 @@ -1078,12 +1078,12 @@
   1.959    finally show ?thesis .
   1.960  qed
   1.961  
   1.962 -lemma rank_mul_le_left:
   1.963 +lemma%unimportant rank_mul_le_left:
   1.964    fixes A :: "real^'n^'m" and B :: "real^'p^'n"
   1.965    shows "rank(A ** B) \<le> rank A"
   1.966    by (metis matrix_transpose_mul rank_mul_le_right rank_transpose)
   1.967  
   1.968 -subsection\<open>Routine results connecting the types @{typ "real^1"} and @{typ real}\<close>
   1.969 +subsection%unimportant\<open>Routine results connecting the types @{typ "real^1"} and @{typ real}\<close>
   1.970  
   1.971  lemma vector_one_nth [simp]:
   1.972    fixes x :: "'a^1" shows "vec (x $ 1) = x"
   1.973 @@ -1146,7 +1146,7 @@
   1.974      done
   1.975  
   1.976  
   1.977 -subsection\<open>Explicit vector construction from lists\<close>
   1.978 +subsection%unimportant\<open>Explicit vector construction from lists\<close>
   1.979  
   1.980  definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
   1.981