src/HOL/Analysis/Cartesian_Euclidean_Space.thy
author immler
Wed May 02 13:49:38 2018 +0200 (12 months ago)
changeset 68072 493b818e8e10
parent 67986 b65c4a6a015e
child 68073 fad29d2a17a5
permissions -rw-r--r--
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
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section \<open>Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space\<close>
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theory Cartesian_Euclidean_Space
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imports Cartesian_Space Derivative
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begin
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lemma subspace_special_hyperplane: "subspace {x. x $ k = 0}"
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  by (simp add: subspace_def)
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lemma sum_mult_product:
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  "sum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
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  unfolding sum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]
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proof (rule sum.cong, simp, rule sum.reindex_cong)
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  fix i
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  show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
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  show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
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  proof safe
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    fix j assume "j \<in> {i * B..<i * B + B}"
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    then show "j \<in> (\<lambda>j. j + i * B) ` {..<B}"
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      by (auto intro!: image_eqI[of _ _ "j - i * B"])
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  qed simp
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qed simp
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lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b"
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  by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis)
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lemma differentiable_vec:
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  fixes S :: "'a::euclidean_space set"
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  shows "vec differentiable_on S"
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  by (simp add: linear_linear bounded_linear_imp_differentiable_on)
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lemma continuous_vec [continuous_intros]:
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  fixes x :: "'a::euclidean_space"
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  shows "isCont vec x"
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  apply (clarsimp simp add: continuous_def LIM_def dist_vec_def L2_set_def)
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  apply (rule_tac x="r / sqrt (real CARD('b))" in exI)
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  by (simp add: mult.commute pos_less_divide_eq real_sqrt_mult)
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lemma box_vec_eq_empty [simp]:
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  shows "cbox (vec a) (vec b) = {} \<longleftrightarrow> cbox a b = {}"
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        "box (vec a) (vec b) = {} \<longleftrightarrow> box a b = {}"
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  by (auto simp: Basis_vec_def mem_box box_eq_empty inner_axis)
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subsection\<open>Closures and interiors of halfspaces\<close>
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lemma interior_halfspace_le [simp]:
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  assumes "a \<noteq> 0"
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    shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
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proof -
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  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
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  proof -
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    obtain e where "e>0" and e: "cball x e \<subseteq> S"
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      using \<open>open S\<close> open_contains_cball x by blast
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    then have "x + (e / norm a) *\<^sub>R a \<in> cball x e"
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      by (simp add: dist_norm)
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    then have "x + (e / norm a) *\<^sub>R a \<in> S"
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      using e by blast
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    then have "x + (e / norm a) *\<^sub>R a \<in> {x. a \<bullet> x \<le> b}"
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      using S by blast
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    moreover have "e * (a \<bullet> a) / norm a > 0"
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      by (simp add: \<open>0 < e\<close> assms)
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    ultimately show ?thesis
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      by (simp add: algebra_simps)
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  qed
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  show ?thesis
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    by (rule interior_unique) (auto simp: open_halfspace_lt *)
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qed
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lemma interior_halfspace_ge [simp]:
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   "a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"
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using interior_halfspace_le [of "-a" "-b"] by simp
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lemma interior_halfspace_component_le [simp]:
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     "interior {x. x$k \<le> a} = {x :: (real^'n). x$k < a}" (is "?LE")
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  and interior_halfspace_component_ge [simp]:
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     "interior {x. x$k \<ge> a} = {x :: (real^'n). x$k > a}" (is "?GE")
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proof -
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  have "axis k (1::real) \<noteq> 0"
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    by (simp add: axis_def vec_eq_iff)
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  moreover have "axis k (1::real) \<bullet> x = x$k" for x
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    by (simp add: cart_eq_inner_axis inner_commute)
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  ultimately show ?LE ?GE
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    using interior_halfspace_le [of "axis k (1::real)" a]
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          interior_halfspace_ge [of "axis k (1::real)" a] by auto
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qed
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lemma closure_halfspace_lt [simp]:
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  assumes "a \<noteq> 0"
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    shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
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proof -
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  have [simp]: "-{x. a \<bullet> x < b} = {x. a \<bullet> x \<ge> b}"
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    by (force simp:)
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  then show ?thesis
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    using interior_halfspace_ge [of a b] assms
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    by (force simp: closure_interior)
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qed
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lemma closure_halfspace_gt [simp]:
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   "a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"
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using closure_halfspace_lt [of "-a" "-b"] by simp
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lemma closure_halfspace_component_lt [simp]:
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     "closure {x. x$k < a} = {x :: (real^'n). x$k \<le> a}" (is "?LE")
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  and closure_halfspace_component_gt [simp]:
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     "closure {x. x$k > a} = {x :: (real^'n). x$k \<ge> a}" (is "?GE")
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proof -
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  have "axis k (1::real) \<noteq> 0"
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    by (simp add: axis_def vec_eq_iff)
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  moreover have "axis k (1::real) \<bullet> x = x$k" for x
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    by (simp add: cart_eq_inner_axis inner_commute)
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  ultimately show ?LE ?GE
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    using closure_halfspace_lt [of "axis k (1::real)" a]
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          closure_halfspace_gt [of "axis k (1::real)" a] by auto
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qed
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lemma interior_hyperplane [simp]:
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  assumes "a \<noteq> 0"
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    shows "interior {x. a \<bullet> x = b} = {}"
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proof -
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  have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
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    by (force simp:)
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  then show ?thesis
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    by (auto simp: assms)
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qed
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lemma frontier_halfspace_le:
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  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
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    shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
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proof (cases "a = 0")
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  case True with assms show ?thesis by simp
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next
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  case False then show ?thesis
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    by (force simp: frontier_def closed_halfspace_le)
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qed
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lemma frontier_halfspace_ge:
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  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
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    shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"
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proof (cases "a = 0")
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  case True with assms show ?thesis by simp
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next
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  case False then show ?thesis
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    by (force simp: frontier_def closed_halfspace_ge)
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qed
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lemma frontier_halfspace_lt:
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  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
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    shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"
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proof (cases "a = 0")
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  case True with assms show ?thesis by simp
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next
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  case False then show ?thesis
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    by (force simp: frontier_def interior_open open_halfspace_lt)
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qed
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lemma frontier_halfspace_gt:
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  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
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    shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"
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proof (cases "a = 0")
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  case True with assms show ?thesis by simp
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next
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  case False then show ?thesis
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    by (force simp: frontier_def interior_open open_halfspace_gt)
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qed
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lemma interior_standard_hyperplane:
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   "interior {x :: (real^'n). x$k = a} = {}"
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proof -
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  have "axis k (1::real) \<noteq> 0"
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    by (simp add: axis_def vec_eq_iff)
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  moreover have "axis k (1::real) \<bullet> x = x$k" for x
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    by (simp add: cart_eq_inner_axis inner_commute)
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  ultimately show ?thesis
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    using interior_hyperplane [of "axis k (1::real)" a]
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    by force
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qed
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lemma matrix_mult_transpose_dot_column:
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  fixes A :: "real^'n^'n"
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  shows "transpose A ** A = (\<chi> i j. inner (column i A) (column j A))"
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  by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)
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lemma matrix_mult_transpose_dot_row:
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  fixes A :: "real^'n^'n"
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  shows "A ** transpose A = (\<chi> i j. inner (row i A) (row j A))"
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  by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)
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text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
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lemma matrix_mult_dot: "A *v x = (\<chi> i. inner (A$i) x)"
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  by (simp add: matrix_vector_mult_def inner_vec_def)
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lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
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  apply (rule adjoint_unique)
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  apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
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    sum_distrib_right sum_distrib_left)
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  apply (subst sum.swap)
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  apply (simp add:  ac_simps)
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  done
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lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
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  shows "matrix(adjoint f) = transpose(matrix f)"
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proof -
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  have "matrix(adjoint f) = matrix(adjoint (( *v) (matrix f)))"
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    by (simp add: lf)
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  also have "\<dots> = transpose(matrix f)"
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    unfolding adjoint_matrix matrix_of_matrix_vector_mul
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    apply rule
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    done
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  finally show ?thesis .
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qed
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lemma matrix_vector_mul_bounded_linear[intro, simp]: "bounded_linear (( *v) A)" for A :: "real^'n^'m"
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  using matrix_vector_mul_linear[of A]
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  by (simp add: linear_conv_bounded_linear linear_matrix_vector_mul_eq)
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lemma
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  fixes A :: "real^'n^'m"
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  shows matrix_vector_mult_linear_continuous_at [continuous_intros]: "isCont (( *v) A) z"
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    and matrix_vector_mult_linear_continuous_on [continuous_intros]: "continuous_on S (( *v) A)"
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  by (simp_all add: linear_continuous_at linear_continuous_on)
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subsection\<open>Some bounds on components etc. relative to operator norm\<close>
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lemma norm_column_le_onorm:
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  fixes A :: "real^'n^'m"
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  shows "norm(column i A) \<le> onorm(( *v) A)"
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proof -
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  have "norm (\<chi> j. A $ j $ i) \<le> norm (A *v axis i 1)"
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    by (simp add: matrix_mult_dot cart_eq_inner_axis)
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  also have "\<dots> \<le> onorm (( *v) A)"
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    using onorm [OF matrix_vector_mul_bounded_linear, of A "axis i 1"] by auto
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  finally have "norm (\<chi> j. A $ j $ i) \<le> onorm (( *v) A)" .
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  then show ?thesis
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    unfolding column_def .
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qed
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lemma matrix_component_le_onorm:
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  fixes A :: "real^'n^'m"
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  shows "\<bar>A $ i $ j\<bar> \<le> onorm(( *v) A)"
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proof -
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  have "\<bar>A $ i $ j\<bar> \<le> norm (\<chi> n. (A $ n $ j))"
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    by (metis (full_types, lifting) component_le_norm_cart vec_lambda_beta)
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  also have "\<dots> \<le> onorm (( *v) A)"
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    by (metis (no_types) column_def norm_column_le_onorm)
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  finally show ?thesis .
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qed
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lemma component_le_onorm:
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  fixes f :: "real^'m \<Rightarrow> real^'n"
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  shows "linear f \<Longrightarrow> \<bar>matrix f $ i $ j\<bar> \<le> onorm f"
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  by (metis linear_matrix_vector_mul_eq matrix_component_le_onorm matrix_vector_mul)
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lemma onorm_le_matrix_component_sum:
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  fixes A :: "real^'n^'m"
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  shows "onorm(( *v) A) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>)"
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proof (rule onorm_le)
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  fix x
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  have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
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    by (rule norm_le_l1_cart)
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  also have "\<dots> \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)"
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  proof (rule sum_mono)
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    fix i
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    have "\<bar>(A *v x) $ i\<bar> \<le> \<bar>\<Sum>j\<in>UNIV. A $ i $ j * x $ j\<bar>"
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      by (simp add: matrix_vector_mult_def)
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    also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j * x $ j\<bar>)"
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      by (rule sum_abs)
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    also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)"
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      by (rule sum_mono) (simp add: abs_mult component_le_norm_cart mult_left_mono)
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    finally show "\<bar>(A *v x) $ i\<bar> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)" .
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  qed
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  finally show "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>) * norm x"
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    by (simp add: sum_distrib_right)
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qed
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lemma onorm_le_matrix_component:
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  fixes A :: "real^'n^'m"
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  assumes "\<And>i j. abs(A$i$j) \<le> B"
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  shows "onorm(( *v) A) \<le> real (CARD('m)) * real (CARD('n)) * B"
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proof (rule onorm_le)
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  fix x :: "real^'n::_"
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  have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
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    by (rule norm_le_l1_cart)
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  also have "\<dots> \<le> (\<Sum>i::'m \<in>UNIV. real (CARD('n)) * B * norm x)"
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  proof (rule sum_mono)
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    fix i
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    have "\<bar>(A *v x) $ i\<bar> \<le> norm(A $ i) * norm x"
lp15@67719
   289
      by (simp add: matrix_mult_dot Cauchy_Schwarz_ineq2)
lp15@67719
   290
    also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>) * norm x"
lp15@67719
   291
      by (simp add: mult_right_mono norm_le_l1_cart)
lp15@67719
   292
    also have "\<dots> \<le> real (CARD('n)) * B * norm x"
lp15@67719
   293
      by (simp add: assms sum_bounded_above mult_right_mono)
lp15@67719
   294
    finally show "\<bar>(A *v x) $ i\<bar> \<le> real (CARD('n)) * B * norm x" .
lp15@67719
   295
  qed
lp15@67719
   296
  also have "\<dots> \<le> CARD('m) * real (CARD('n)) * B * norm x"
lp15@67719
   297
    by simp
lp15@67719
   298
  finally show "norm (A *v x) \<le> CARD('m) * real (CARD('n)) * B * norm x" .
lp15@67719
   299
qed
lp15@67719
   300
lp15@67719
   301
subsection \<open>lambda skolemization on cartesian products\<close>
hoelzl@37489
   302
hoelzl@37489
   303
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
hoelzl@37494
   304
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   305
proof -
hoelzl@37489
   306
  let ?S = "(UNIV :: 'n set)"
wenzelm@49644
   307
  { assume H: "?rhs"
wenzelm@49644
   308
    then have ?lhs by auto }
hoelzl@37489
   309
  moreover
wenzelm@49644
   310
  { assume H: "?lhs"
hoelzl@37489
   311
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
hoelzl@37489
   312
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
wenzelm@49644
   313
    { fix i
hoelzl@37489
   314
      from f have "P i (f i)" by metis
hoelzl@37494
   315
      then have "P i (?x $ i)" by auto
hoelzl@37489
   316
    }
hoelzl@37489
   317
    hence "\<forall>i. P i (?x$i)" by metis
hoelzl@37489
   318
    hence ?rhs by metis }
hoelzl@37489
   319
  ultimately show ?thesis by metis
hoelzl@37489
   320
qed
hoelzl@37489
   321
lp15@67719
   322
lemma rational_approximation:
lp15@67719
   323
  assumes "e > 0"
lp15@67719
   324
  obtains r::real where "r \<in> \<rat>" "\<bar>r - x\<bar> < e"
lp15@67719
   325
  using Rats_dense_in_real [of "x - e/2" "x + e/2"] assms by auto
lp15@67719
   326
lp15@67719
   327
lemma matrix_rational_approximation:
lp15@67719
   328
  fixes A :: "real^'n^'m"
lp15@67719
   329
  assumes "e > 0"
lp15@67719
   330
  obtains B where "\<And>i j. B$i$j \<in> \<rat>" "onorm(\<lambda>x. (A - B) *v x) < e"
lp15@67719
   331
proof -
lp15@67719
   332
  have "\<forall>i j. \<exists>q \<in> \<rat>. \<bar>q - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))"
lp15@67719
   333
    using assms by (force intro: rational_approximation [of "e / (2 * CARD('m) * CARD('n))"])
lp15@67719
   334
  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))"
lp15@67719
   335
    by (auto simp: lambda_skolem Bex_def)
lp15@67719
   336
  show ?thesis
lp15@67719
   337
  proof
lp15@67719
   338
    have "onorm (( *v) (A - B)) \<le> real CARD('m) * real CARD('n) *
lp15@67719
   339
    (e / (2 * real CARD('m) * real CARD('n)))"
lp15@67719
   340
      apply (rule onorm_le_matrix_component)
lp15@67719
   341
      using Bclo by (simp add: abs_minus_commute less_imp_le)
lp15@67719
   342
    also have "\<dots> < e"
lp15@67719
   343
      using \<open>0 < e\<close> by (simp add: divide_simps)
lp15@67719
   344
    finally show "onorm (( *v) (A - B)) < e" .
lp15@67719
   345
  qed (use B in auto)
lp15@67719
   346
qed
lp15@67719
   347
hoelzl@37489
   348
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
hoelzl@50526
   349
  unfolding inner_simps scalar_mult_eq_scaleR by auto
hoelzl@37489
   350
hoelzl@37489
   351
lemma matrix_left_invertible_injective:
immler@68072
   352
  fixes A :: "'a::field^'n^'m"
lp15@67986
   353
  shows "(\<exists>B. B ** A = mat 1) \<longleftrightarrow> inj (( *v) A)"
lp15@67986
   354
proof safe
lp15@67986
   355
  fix B
lp15@67986
   356
  assume B: "B ** A = mat 1"
lp15@67986
   357
  show "inj (( *v) A)"
lp15@67986
   358
    unfolding inj_on_def
lp15@67986
   359
      by (metis B matrix_vector_mul_assoc matrix_vector_mul_lid)
lp15@67986
   360
next
lp15@67986
   361
  assume "inj (( *v) A)"
immler@68072
   362
  from vec.linear_injective_left_inverse[OF matrix_vector_mul_linear_gen this]
immler@68072
   363
  obtain g where "Vector_Spaces.linear ( *s) ( *s) g" and g: "g \<circ> ( *v) A = id"
lp15@67986
   364
    by blast
lp15@67986
   365
  have "matrix g ** A = mat 1"
immler@68072
   366
    by (metis matrix_vector_mul_linear_gen \<open>Vector_Spaces.linear ( *s) ( *s) g\<close> g matrix_compose_gen
immler@68072
   367
        matrix_eq matrix_id_mat_1 matrix_vector_mul(1))
lp15@67986
   368
  then show "\<exists>B. B ** A = mat 1"
lp15@67986
   369
    by metis
hoelzl@37489
   370
qed
hoelzl@37489
   371
hoelzl@37489
   372
lemma matrix_right_invertible_surjective:
immler@68072
   373
  "(\<exists>B. (A::'a::field^'n^'m) ** (B::'a::field^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
wenzelm@49644
   374
proof -
immler@68072
   375
  { fix B :: "'a ^'m^'n"
wenzelm@49644
   376
    assume AB: "A ** B = mat 1"
immler@68072
   377
    { fix x :: "'a ^ 'm"
hoelzl@37489
   378
      have "A *v (B *v x) = x"
immler@68072
   379
        by (simp add: matrix_vector_mul_assoc AB) }
nipkow@67399
   380
    hence "surj (( *v) A)" unfolding surj_def by metis }
hoelzl@37489
   381
  moreover
nipkow@67399
   382
  { assume sf: "surj (( *v) A)"
immler@68072
   383
    from vec.linear_surjective_right_inverse[OF _ this]
immler@68072
   384
    obtain g:: "'a ^'m \<Rightarrow> 'a ^'n" where g: "Vector_Spaces.linear ( *s) ( *s) g" "( *v) A \<circ> g = id"
hoelzl@37489
   385
      by blast
hoelzl@37489
   386
hoelzl@37489
   387
    have "A ** (matrix g) = mat 1"
hoelzl@37489
   388
      unfolding matrix_eq  matrix_vector_mul_lid
hoelzl@37489
   389
        matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   390
      using g(2) unfolding o_def fun_eq_iff id_def
hoelzl@37489
   391
      .
immler@68072
   392
    hence "\<exists>B. A ** (B::'a^'m^'n) = mat 1" by blast
hoelzl@37489
   393
  }
hoelzl@37489
   394
  ultimately show ?thesis unfolding surj_def by blast
hoelzl@37489
   395
qed
hoelzl@37489
   396
hoelzl@37489
   397
lemma matrix_right_invertible_span_columns:
immler@68072
   398
  "(\<exists>(B::'a::field ^'n^'m). (A::'a ^'m^'n) ** B = mat 1) \<longleftrightarrow>
immler@68072
   399
    vec.span (columns A) = UNIV" (is "?lhs = ?rhs")
wenzelm@49644
   400
proof -
hoelzl@37489
   401
  let ?U = "UNIV :: 'm set"
hoelzl@37489
   402
  have fU: "finite ?U" by simp
immler@68072
   403
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::'a^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y)"
lp15@67673
   404
    unfolding matrix_right_invertible_surjective matrix_mult_sum surj_def
wenzelm@49644
   405
    apply (subst eq_commute)
wenzelm@49644
   406
    apply rule
wenzelm@49644
   407
    done
immler@68072
   408
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> vec.span (columns A))" by blast
wenzelm@49644
   409
  { assume h: ?lhs
immler@68072
   410
    { fix x:: "'a ^'n"
immler@68072
   411
      from h[unfolded lhseq, rule_format, of x] obtain y :: "'a ^'m"
nipkow@64267
   412
        where y: "sum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
immler@68072
   413
      have "x \<in> vec.span (columns A)"
wenzelm@49644
   414
        unfolding y[symmetric]
immler@68072
   415
        apply (rule vec.span_sum)
immler@68072
   416
        apply (rule vec.span_scale)
immler@68072
   417
        apply (rule vec.span_base)
wenzelm@49644
   418
        unfolding columns_def
wenzelm@49644
   419
        apply blast
wenzelm@49644
   420
        done
wenzelm@49644
   421
    }
wenzelm@49644
   422
    then have ?rhs unfolding rhseq by blast }
hoelzl@37489
   423
  moreover
wenzelm@49644
   424
  { assume h:?rhs
immler@68072
   425
    let ?P = "\<lambda>(y::'a ^'n). \<exists>(x::'a^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y"
wenzelm@49644
   426
    { fix y
immler@68072
   427
      have "y \<in> vec.span (columns A)"
immler@68072
   428
        unfolding h by blast
immler@68072
   429
      then have "?P y"
immler@68072
   430
      proof (induction rule: vec.span_induct_alt)
immler@68072
   431
        show "\<exists>x::'a ^ 'm. sum (\<lambda>i. (x$i) *s column i A) ?U = 0"
hoelzl@37489
   432
          by (rule exI[where x=0], simp)
hoelzl@37489
   433
      next
wenzelm@49644
   434
        fix c y1 y2
wenzelm@49644
   435
        assume y1: "y1 \<in> columns A" and y2: "?P y2"
hoelzl@37489
   436
        from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
hoelzl@37489
   437
          unfolding columns_def by blast
immler@68072
   438
        from y2 obtain x:: "'a ^'m" where
nipkow@64267
   439
          x: "sum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
immler@68072
   440
        let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::'a^'m"
hoelzl@37489
   441
        show "?P (c*s y1 + y2)"
immler@68072
   442
        proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left if_distribR cong del: if_weak_cong)
wenzelm@49644
   443
          fix j
wenzelm@49644
   444
          have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
   445
              else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
wenzelm@49644
   446
            using i(1) by (simp add: field_simps)
nipkow@64267
   447
          have "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
nipkow@64267
   448
              else (x$xa) * ((column xa A$j))) ?U = sum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"
nipkow@64267
   449
            apply (rule sum.cong[OF refl])
wenzelm@49644
   450
            using th apply blast
wenzelm@49644
   451
            done
nipkow@64267
   452
          also have "\<dots> = sum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
nipkow@64267
   453
            by (simp add: sum.distrib)
nipkow@64267
   454
          also have "\<dots> = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
nipkow@64267
   455
            unfolding sum.delta[OF fU]
wenzelm@49644
   456
            using i(1) by simp
nipkow@64267
   457
          finally show "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
nipkow@64267
   458
            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
wenzelm@49644
   459
        qed
wenzelm@49644
   460
      qed
wenzelm@49644
   461
    }
wenzelm@49644
   462
    then have ?lhs unfolding lhseq ..
wenzelm@49644
   463
  }
hoelzl@37489
   464
  ultimately show ?thesis by blast
hoelzl@37489
   465
qed
hoelzl@37489
   466
immler@68072
   467
lemma matrix_left_invertible_span_rows_gen:
immler@68072
   468
  "(\<exists>(B::'a^'m^'n). B ** (A::'a::field^'n^'m) = mat 1) \<longleftrightarrow> vec.span (rows A) = UNIV"
hoelzl@37489
   469
  unfolding right_invertible_transpose[symmetric]
hoelzl@37489
   470
  unfolding columns_transpose[symmetric]
hoelzl@37489
   471
  unfolding matrix_right_invertible_span_columns
wenzelm@49644
   472
  ..
hoelzl@37489
   473
immler@68072
   474
lemma matrix_left_invertible_span_rows:
immler@68072
   475
  "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
immler@68072
   476
  using matrix_left_invertible_span_rows_gen[of A] by (simp add: span_vec_eq)
immler@68072
   477
immler@68072
   478
wenzelm@60420
   479
text \<open>The same result in terms of square matrices.\<close>
hoelzl@37489
   480
hoelzl@37489
   481
wenzelm@60420
   482
text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
hoelzl@37489
   483
hoelzl@37489
   484
definition "rowvector v = (\<chi> i j. (v$j))"
hoelzl@37489
   485
hoelzl@37489
   486
definition "columnvector v = (\<chi> i j. (v$i))"
hoelzl@37489
   487
wenzelm@49644
   488
lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
huffman@44136
   489
  by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
hoelzl@37489
   490
hoelzl@37489
   491
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
huffman@44136
   492
  by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
hoelzl@37489
   493
wenzelm@49644
   494
lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
hoelzl@37489
   495
  by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
hoelzl@37489
   496
wenzelm@49644
   497
lemma dot_matrix_product:
wenzelm@49644
   498
  "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
huffman@44136
   499
  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
hoelzl@37489
   500
hoelzl@37489
   501
lemma dot_matrix_vector_mul:
hoelzl@37489
   502
  fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
hoelzl@37489
   503
  shows "(A *v x) \<bullet> (B *v y) =
hoelzl@37489
   504
      (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
wenzelm@49644
   505
  unfolding dot_matrix_product transpose_columnvector[symmetric]
wenzelm@49644
   506
    dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
hoelzl@37489
   507
wenzelm@61945
   508
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}"
hoelzl@50526
   509
  by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
hoelzl@37489
   510
wenzelm@49644
   511
lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
hoelzl@50526
   512
  using Basis_le_infnorm[of "axis i 1" x]
hoelzl@50526
   513
  by (simp add: Basis_vec_def axis_eq_axis inner_axis)
hoelzl@37489
   514
hoelzl@63334
   515
lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
huffman@44647
   516
  unfolding continuous_def by (rule tendsto_vec_nth)
huffman@44213
   517
hoelzl@63334
   518
lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
huffman@44647
   519
  unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
huffman@44213
   520
hoelzl@63334
   521
lemma continuous_on_vec_lambda[continuous_intros]:
hoelzl@63334
   522
  "(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)"
hoelzl@63334
   523
  unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)
hoelzl@63334
   524
hoelzl@37489
   525
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
hoelzl@63332
   526
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
huffman@44213
   527
hoelzl@37489
   528
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
wenzelm@49644
   529
  unfolding bounded_def
wenzelm@49644
   530
  apply clarify
wenzelm@49644
   531
  apply (rule_tac x="x $ i" in exI)
wenzelm@49644
   532
  apply (rule_tac x="e" in exI)
wenzelm@49644
   533
  apply clarify
wenzelm@49644
   534
  apply (rule order_trans [OF dist_vec_nth_le], simp)
wenzelm@49644
   535
  done
hoelzl@37489
   536
hoelzl@37489
   537
lemma compact_lemma_cart:
hoelzl@37489
   538
  fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
hoelzl@50998
   539
  assumes f: "bounded (range f)"
eberlm@66447
   540
  shows "\<exists>l r. strict_mono r \<and>
hoelzl@37489
   541
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
immler@62127
   542
    (is "?th d")
immler@62127
   543
proof -
immler@62127
   544
  have "\<forall>d' \<subseteq> d. ?th d'"
immler@62127
   545
    by (rule compact_lemma_general[where unproj=vec_lambda])
immler@62127
   546
      (auto intro!: f bounded_component_cart simp: vec_lambda_eta)
immler@62127
   547
  then show "?th d" by simp
hoelzl@37489
   548
qed
hoelzl@37489
   549
huffman@44136
   550
instance vec :: (heine_borel, finite) heine_borel
hoelzl@37489
   551
proof
hoelzl@50998
   552
  fix f :: "nat \<Rightarrow> 'a ^ 'b"
hoelzl@50998
   553
  assume f: "bounded (range f)"
eberlm@66447
   554
  then obtain l r where r: "strict_mono r"
wenzelm@49644
   555
      and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@50998
   556
    using compact_lemma_cart [OF f] by blast
hoelzl@37489
   557
  let ?d = "UNIV::'b set"
hoelzl@37489
   558
  { fix e::real assume "e>0"
hoelzl@37489
   559
    hence "0 < e / (real_of_nat (card ?d))"
wenzelm@49644
   560
      using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
hoelzl@37489
   561
    with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
hoelzl@37489
   562
      by simp
hoelzl@37489
   563
    moreover
wenzelm@49644
   564
    { fix n
wenzelm@49644
   565
      assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
hoelzl@37489
   566
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
nipkow@67155
   567
        unfolding dist_vec_def using zero_le_dist by (rule L2_set_le_sum)
hoelzl@37489
   568
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
nipkow@64267
   569
        by (rule sum_strict_mono) (simp_all add: n)
hoelzl@37489
   570
      finally have "dist (f (r n)) l < e" by simp
hoelzl@37489
   571
    }
hoelzl@37489
   572
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
lp15@61810
   573
      by (rule eventually_mono)
hoelzl@37489
   574
  }
wenzelm@61973
   575
  hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp
eberlm@66447
   576
  with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
hoelzl@37489
   577
qed
hoelzl@37489
   578
wenzelm@49644
   579
lemma interval_cart:
immler@54775
   580
  fixes a :: "real^'n"
immler@54775
   581
  shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
immler@56188
   582
    and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
immler@56188
   583
  by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
hoelzl@37489
   584
lp15@67673
   585
lemma mem_box_cart:
immler@54775
   586
  fixes a :: "real^'n"
immler@54775
   587
  shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
immler@56188
   588
    and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
wenzelm@49644
   589
  using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
hoelzl@37489
   590
wenzelm@49644
   591
lemma interval_eq_empty_cart:
wenzelm@49644
   592
  fixes a :: "real^'n"
immler@54775
   593
  shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
immler@56188
   594
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
wenzelm@49644
   595
proof -
immler@54775
   596
  { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>box a b"
lp15@67673
   597
    hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_box_cart by auto
hoelzl@37489
   598
    hence "a$i < b$i" by auto
wenzelm@49644
   599
    hence False using as by auto }
hoelzl@37489
   600
  moreover
hoelzl@37489
   601
  { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
hoelzl@37489
   602
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
   603
    { fix i
hoelzl@37489
   604
      have "a$i < b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
   605
      hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
hoelzl@37489
   606
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
   607
        by auto }
lp15@67673
   608
    hence "box a b \<noteq> {}" using mem_box_cart(1)[of "?x" a b] by auto }
hoelzl@37489
   609
  ultimately show ?th1 by blast
hoelzl@37489
   610
immler@56188
   611
  { fix i x assume as:"b$i < a$i" and x:"x\<in>cbox a b"
lp15@67673
   612
    hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_box_cart by auto
hoelzl@37489
   613
    hence "a$i \<le> b$i" by auto
wenzelm@49644
   614
    hence False using as by auto }
hoelzl@37489
   615
  moreover
hoelzl@37489
   616
  { assume as:"\<forall>i. \<not> (b$i < a$i)"
hoelzl@37489
   617
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
   618
    { fix i
hoelzl@37489
   619
      have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
   620
      hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
hoelzl@37489
   621
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
   622
        by auto }
lp15@67673
   623
    hence "cbox a b \<noteq> {}" using mem_box_cart(2)[of "?x" a b] by auto  }
hoelzl@37489
   624
  ultimately show ?th2 by blast
hoelzl@37489
   625
qed
hoelzl@37489
   626
wenzelm@49644
   627
lemma interval_ne_empty_cart:
wenzelm@49644
   628
  fixes a :: "real^'n"
immler@56188
   629
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
immler@54775
   630
    and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
hoelzl@37489
   631
  unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
hoelzl@37489
   632
    (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
   633
wenzelm@49644
   634
lemma subset_interval_imp_cart:
wenzelm@49644
   635
  fixes a :: "real^'n"
immler@56188
   636
  shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
immler@56188
   637
    and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
immler@56188
   638
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
immler@54775
   639
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b"
lp15@67673
   640
  unfolding subset_eq[unfolded Ball_def] unfolding mem_box_cart
hoelzl@37489
   641
  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
   642
wenzelm@49644
   643
lemma interval_sing:
wenzelm@49644
   644
  fixes a :: "'a::linorder^'n"
wenzelm@49644
   645
  shows "{a .. a} = {a} \<and> {a<..<a} = {}"
wenzelm@49644
   646
  apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
wenzelm@49644
   647
  done
hoelzl@37489
   648
wenzelm@49644
   649
lemma subset_interval_cart:
wenzelm@49644
   650
  fixes a :: "real^'n"
immler@56188
   651
  shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
immler@56188
   652
    and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
immler@56188
   653
    and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
immler@54775
   654
    and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
immler@56188
   655
  using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
   656
wenzelm@49644
   657
lemma disjoint_interval_cart:
wenzelm@49644
   658
  fixes a::"real^'n"
immler@56188
   659
  shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
immler@56188
   660
    and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
immler@56188
   661
    and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
immler@54775
   662
    and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
hoelzl@50526
   663
  using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
   664
lp15@67719
   665
lemma Int_interval_cart:
immler@54775
   666
  fixes a :: "real^'n"
immler@56188
   667
  shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
lp15@63945
   668
  unfolding Int_interval
immler@56188
   669
  by (auto simp: mem_box less_eq_vec_def)
immler@56188
   670
    (auto simp: Basis_vec_def inner_axis)
hoelzl@37489
   671
wenzelm@49644
   672
lemma closed_interval_left_cart:
wenzelm@49644
   673
  fixes b :: "real^'n"
hoelzl@37489
   674
  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
hoelzl@63332
   675
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
   676
wenzelm@49644
   677
lemma closed_interval_right_cart:
wenzelm@49644
   678
  fixes a::"real^'n"
hoelzl@37489
   679
  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
hoelzl@63332
   680
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
   681
wenzelm@49644
   682
lemma is_interval_cart:
wenzelm@49644
   683
  "is_interval (s::(real^'n) set) \<longleftrightarrow>
wenzelm@49644
   684
    (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
hoelzl@50526
   685
  by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
hoelzl@37489
   686
wenzelm@49644
   687
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
hoelzl@63332
   688
  by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
   689
wenzelm@49644
   690
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
hoelzl@63332
   691
  by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
   692
wenzelm@49644
   693
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
hoelzl@63332
   694
  by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
wenzelm@49644
   695
wenzelm@49644
   696
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
hoelzl@63332
   697
  by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
   698
wenzelm@49644
   699
lemma Lim_component_le_cart:
wenzelm@49644
   700
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
   701
  assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"
hoelzl@37489
   702
  shows "l$i \<le> b"
hoelzl@50526
   703
  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
hoelzl@37489
   704
wenzelm@49644
   705
lemma Lim_component_ge_cart:
wenzelm@49644
   706
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
   707
  assumes "(f \<longlongrightarrow> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
hoelzl@37489
   708
  shows "b \<le> l$i"
hoelzl@50526
   709
  by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
hoelzl@37489
   710
wenzelm@49644
   711
lemma Lim_component_eq_cart:
wenzelm@49644
   712
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
   713
  assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
hoelzl@37489
   714
  shows "l$i = b"
wenzelm@49644
   715
  using ev[unfolded order_eq_iff eventually_conj_iff] and
wenzelm@49644
   716
    Lim_component_ge_cart[OF net, of b i] and
hoelzl@37489
   717
    Lim_component_le_cart[OF net, of i b] by auto
hoelzl@37489
   718
wenzelm@49644
   719
lemma connected_ivt_component_cart:
wenzelm@49644
   720
  fixes x :: "real^'n"
wenzelm@49644
   721
  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$k = a)"
hoelzl@50526
   722
  using connected_ivt_hyperplane[of s x y "axis k 1" a]
hoelzl@50526
   723
  by (auto simp add: inner_axis inner_commute)
hoelzl@37489
   724
immler@68072
   725
lemma subspace_substandard_cart: "vec.subspace {x. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
immler@68072
   726
  unfolding vec.subspace_def by auto
hoelzl@37489
   727
hoelzl@37489
   728
lemma closed_substandard_cart:
huffman@44213
   729
  "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
   730
proof -
huffman@44213
   731
  { fix i::'n
huffman@44213
   732
    have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
hoelzl@63332
   733
      by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) }
huffman@44213
   734
  thus ?thesis
huffman@44213
   735
    unfolding Collect_all_eq by (simp add: closed_INT)
hoelzl@37489
   736
qed
hoelzl@37489
   737
immler@68072
   738
lemma dim_substandard_cart: "vec.dim {x::'a::field^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
immler@68072
   739
  (is "vec.dim ?A = _")
immler@68072
   740
proof (rule vec.dim_unique)
immler@68072
   741
  let ?B = "((\<lambda>x. axis x 1) ` d)"
immler@68072
   742
  have subset_basis: "?B \<subseteq> cart_basis"
immler@68072
   743
    by (auto simp: cart_basis_def)
immler@68072
   744
  show "?B \<subseteq> ?A"
immler@68072
   745
    by (auto simp: axis_def)
immler@68072
   746
  show "vec.independent ((\<lambda>x. axis x 1) ` d)"
immler@68072
   747
    using subset_basis
immler@68072
   748
    by (rule vec.independent_mono[OF vec.independent_Basis])
immler@68072
   749
  have "x \<in> vec.span ?B" if "\<forall>i. i \<notin> d \<longrightarrow> x $ i = 0" for x::"'a^'n"
immler@68072
   750
  proof -
immler@68072
   751
    have "finite ?B"
immler@68072
   752
      using subset_basis finite_cart_basis
immler@68072
   753
      by (rule finite_subset)
immler@68072
   754
    have "x = (\<Sum>i\<in>UNIV. x $ i *s axis i 1)"
immler@68072
   755
      by (rule basis_expansion[symmetric])
immler@68072
   756
    also have "\<dots> = (\<Sum>i\<in>d. (x $ i) *s axis i 1)"
immler@68072
   757
      by (rule sum.mono_neutral_cong_right) (auto simp: that)
immler@68072
   758
    also have "\<dots> \<in> vec.span ?B"
immler@68072
   759
      by (simp add: vec.span_sum vec.span_clauses)
immler@68072
   760
    finally show "x \<in> vec.span ?B" .
immler@68072
   761
  qed
immler@68072
   762
  then show "?A \<subseteq> vec.span ?B" by auto
immler@68072
   763
qed (simp add: card_image inj_on_def axis_eq_axis)
immler@68072
   764
immler@68072
   765
lemma dim_subset_UNIV_cart_gen:
immler@68072
   766
  fixes S :: "('a::field^'n) set"
immler@68072
   767
  shows "vec.dim S \<le> CARD('n)"
immler@68072
   768
  by (metis vec.dim_eq_full vec.dim_subset_UNIV vec.span_UNIV vec_dim_card)
hoelzl@37489
   769
lp15@67719
   770
lemma dim_subset_UNIV_cart:
lp15@67719
   771
  fixes S :: "(real^'n) set"
lp15@67719
   772
  shows "dim S \<le> CARD('n)"
immler@68072
   773
  using dim_subset_UNIV_cart_gen[of S] by (simp add: dim_vec_eq)
lp15@67719
   774
hoelzl@37489
   775
lemma affinity_inverses:
hoelzl@37489
   776
  assumes m0: "m \<noteq> (0::'a::field)"
wenzelm@61736
   777
  shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
wenzelm@61736
   778
  "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
hoelzl@37489
   779
  using m0
immler@68072
   780
  by (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
hoelzl@37489
   781
hoelzl@37489
   782
lemma vector_affinity_eq:
hoelzl@37489
   783
  assumes m0: "(m::'a::field) \<noteq> 0"
hoelzl@37489
   784
  shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
hoelzl@37489
   785
proof
hoelzl@37489
   786
  assume h: "m *s x + c = y"
hoelzl@37489
   787
  hence "m *s x = y - c" by (simp add: field_simps)
hoelzl@37489
   788
  hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
hoelzl@37489
   789
  then show "x = inverse m *s y + - (inverse m *s c)"
hoelzl@37489
   790
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
   791
next
hoelzl@37489
   792
  assume h: "x = inverse m *s y + - (inverse m *s c)"
haftmann@54230
   793
  show "m *s x + c = y" unfolding h
hoelzl@37489
   794
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
   795
qed
hoelzl@37489
   796
hoelzl@37489
   797
lemma vector_eq_affinity:
wenzelm@49644
   798
    "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
hoelzl@37489
   799
  using vector_affinity_eq[where m=m and x=x and y=y and c=c]
hoelzl@37489
   800
  by metis
hoelzl@37489
   801
hoelzl@50526
   802
lemma vector_cart:
hoelzl@50526
   803
  fixes f :: "real^'n \<Rightarrow> real"
hoelzl@50526
   804
  shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
hoelzl@50526
   805
  unfolding euclidean_eq_iff[where 'a="real^'n"]
hoelzl@50526
   806
  by simp (simp add: Basis_vec_def inner_axis)
hoelzl@63332
   807
hoelzl@50526
   808
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
hoelzl@50526
   809
  by (rule vector_cart)
wenzelm@49644
   810
huffman@44360
   811
subsection "Convex Euclidean Space"
hoelzl@37489
   812
hoelzl@50526
   813
lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
hoelzl@50526
   814
  using const_vector_cart[of 1] by (simp add: one_vec_def)
hoelzl@37489
   815
hoelzl@37489
   816
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
hoelzl@37489
   817
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
hoelzl@37489
   818
hoelzl@50526
   819
lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
hoelzl@37489
   820
hoelzl@37489
   821
lemma convex_box_cart:
hoelzl@37489
   822
  assumes "\<And>i. convex {x. P i x}"
hoelzl@37489
   823
  shows "convex {x. \<forall>i. P i (x$i)}"
hoelzl@37489
   824
  using assms unfolding convex_def by auto
hoelzl@37489
   825
hoelzl@37489
   826
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
hoelzl@63334
   827
  by (rule convex_box_cart) (simp add: atLeast_def[symmetric])
hoelzl@37489
   828
hoelzl@37489
   829
lemma unit_interval_convex_hull_cart:
immler@56188
   830
  "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}"
immler@56188
   831
  unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
hoelzl@50526
   832
  by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
hoelzl@37489
   833
hoelzl@37489
   834
lemma cube_convex_hull_cart:
wenzelm@49644
   835
  assumes "0 < d"
wenzelm@49644
   836
  obtains s::"(real^'n) set"
immler@56188
   837
    where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
wenzelm@49644
   838
proof -
wenzelm@55522
   839
  from assms obtain s where "finite s"
nipkow@67399
   840
    and "cbox (x - sum (( *\<^sub>R) d) Basis) (x + sum (( *\<^sub>R) d) Basis) = convex hull s"
wenzelm@55522
   841
    by (rule cube_convex_hull)
wenzelm@55522
   842
  with that[of s] show thesis
wenzelm@55522
   843
    by (simp add: const_vector_cart)
hoelzl@37489
   844
qed
hoelzl@37489
   845
hoelzl@37489
   846
hoelzl@37489
   847
subsection "Derivative"
hoelzl@37489
   848
hoelzl@37489
   849
definition "jacobian f net = matrix(frechet_derivative f net)"
hoelzl@37489
   850
wenzelm@49644
   851
lemma jacobian_works:
wenzelm@49644
   852
  "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
lp15@67986
   853
    (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net" (is "?lhs = ?rhs")
lp15@67986
   854
proof
lp15@67986
   855
  assume ?lhs then show ?rhs
lp15@67986
   856
    by (simp add: frechet_derivative_works has_derivative_linear jacobian_def)
lp15@67986
   857
next
lp15@67986
   858
  assume ?rhs then show ?lhs
lp15@67986
   859
    by (rule differentiableI)
lp15@67986
   860
qed
hoelzl@37489
   861
hoelzl@37489
   862
wenzelm@60420
   863
subsection \<open>Component of the differential must be zero if it exists at a local
nipkow@67968
   864
  maximum or minimum for that corresponding component\<close>
hoelzl@37489
   865
hoelzl@50526
   866
lemma differential_zero_maxmin_cart:
wenzelm@49644
   867
  fixes f::"real^'a \<Rightarrow> real^'b"
wenzelm@49644
   868
  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))"
hoelzl@50526
   869
    "f differentiable (at x)"
hoelzl@50526
   870
  shows "jacobian f (at x) $ k = 0"
hoelzl@50526
   871
  using differential_zero_maxmin_component[of "axis k 1" e x f] assms
hoelzl@50526
   872
    vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
hoelzl@50526
   873
  by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
wenzelm@49644
   874
wenzelm@60420
   875
subsection \<open>Lemmas for working on @{typ "real^1"}\<close>
hoelzl@37489
   876
hoelzl@37489
   877
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
wenzelm@49644
   878
  by (metis (full_types) num1_eq_iff)
hoelzl@37489
   879
hoelzl@37489
   880
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
wenzelm@49644
   881
  by auto (metis (full_types) num1_eq_iff)
hoelzl@37489
   882
hoelzl@37489
   883
lemma exhaust_2:
wenzelm@49644
   884
  fixes x :: 2
wenzelm@49644
   885
  shows "x = 1 \<or> x = 2"
hoelzl@37489
   886
proof (induct x)
hoelzl@37489
   887
  case (of_int z)
lp15@67979
   888
  then have "0 \<le> z" and "z < 2" by simp_all
hoelzl@37489
   889
  then have "z = 0 | z = 1" by arith
hoelzl@37489
   890
  then show ?case by auto
hoelzl@37489
   891
qed
hoelzl@37489
   892
hoelzl@37489
   893
lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
hoelzl@37489
   894
  by (metis exhaust_2)
hoelzl@37489
   895
hoelzl@37489
   896
lemma exhaust_3:
wenzelm@49644
   897
  fixes x :: 3
wenzelm@49644
   898
  shows "x = 1 \<or> x = 2 \<or> x = 3"
hoelzl@37489
   899
proof (induct x)
hoelzl@37489
   900
  case (of_int z)
lp15@67979
   901
  then have "0 \<le> z" and "z < 3" by simp_all
hoelzl@37489
   902
  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
hoelzl@37489
   903
  then show ?case by auto
hoelzl@37489
   904
qed
hoelzl@37489
   905
hoelzl@37489
   906
lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
hoelzl@37489
   907
  by (metis exhaust_3)
hoelzl@37489
   908
hoelzl@37489
   909
lemma UNIV_1 [simp]: "UNIV = {1::1}"
hoelzl@37489
   910
  by (auto simp add: num1_eq_iff)
hoelzl@37489
   911
hoelzl@37489
   912
lemma UNIV_2: "UNIV = {1::2, 2::2}"
hoelzl@37489
   913
  using exhaust_2 by auto
hoelzl@37489
   914
hoelzl@37489
   915
lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
hoelzl@37489
   916
  using exhaust_3 by auto
hoelzl@37489
   917
nipkow@64267
   918
lemma sum_1: "sum f (UNIV::1 set) = f 1"
hoelzl@37489
   919
  unfolding UNIV_1 by simp
hoelzl@37489
   920
nipkow@64267
   921
lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2"
hoelzl@37489
   922
  unfolding UNIV_2 by simp
hoelzl@37489
   923
nipkow@64267
   924
lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3"
haftmann@57514
   925
  unfolding UNIV_3 by (simp add: ac_simps)
hoelzl@37489
   926
lp15@67979
   927
lemma num1_eqI:
lp15@67979
   928
  fixes a::num1 shows "a = b"
lp15@67979
   929
  by (metis (full_types) UNIV_1 UNIV_I empty_iff insert_iff)
lp15@67979
   930
lp15@67979
   931
lemma num1_eq1 [simp]:
lp15@67979
   932
  fixes a::num1 shows "a = 1"
lp15@67979
   933
  by (rule num1_eqI)
lp15@67979
   934
wenzelm@49644
   935
instantiation num1 :: cart_one
wenzelm@49644
   936
begin
wenzelm@49644
   937
wenzelm@49644
   938
instance
wenzelm@49644
   939
proof
hoelzl@37489
   940
  show "CARD(1) = Suc 0" by auto
wenzelm@49644
   941
qed
wenzelm@49644
   942
wenzelm@49644
   943
end
hoelzl@37489
   944
lp15@67979
   945
instantiation num1 :: linorder begin
lp15@67979
   946
definition "a < b \<longleftrightarrow> Rep_num1 a < Rep_num1 b"
lp15@67979
   947
definition "a \<le> b \<longleftrightarrow> Rep_num1 a \<le> Rep_num1 b"
lp15@67979
   948
instance
lp15@67979
   949
  by intro_classes (auto simp: less_eq_num1_def less_num1_def intro: num1_eqI)
lp15@67979
   950
end
lp15@67979
   951
lp15@67979
   952
instance num1 :: wellorder
lp15@67979
   953
  by intro_classes (auto simp: less_eq_num1_def less_num1_def)
lp15@67979
   954
nipkow@67968
   955
subsection\<open>The collapse of the general concepts to dimension one\<close>
hoelzl@37489
   956
hoelzl@37489
   957
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
huffman@44136
   958
  by (simp add: vec_eq_iff)
hoelzl@37489
   959
hoelzl@37489
   960
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
hoelzl@37489
   961
  apply auto
hoelzl@37489
   962
  apply (erule_tac x= "x$1" in allE)
hoelzl@37489
   963
  apply (simp only: vector_one[symmetric])
hoelzl@37489
   964
  done
hoelzl@37489
   965
hoelzl@37489
   966
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
huffman@44136
   967
  by (simp add: norm_vec_def)
hoelzl@37489
   968
lp15@67979
   969
lemma dist_vector_1:
lp15@67979
   970
  fixes x :: "'a::real_normed_vector^1"
lp15@67979
   971
  shows "dist x y = dist (x$1) (y$1)"
lp15@67979
   972
  by (simp add: dist_norm norm_vector_1)
lp15@67979
   973
wenzelm@61945
   974
lemma norm_real: "norm(x::real ^ 1) = \<bar>x$1\<bar>"
hoelzl@37489
   975
  by (simp add: norm_vector_1)
hoelzl@37489
   976
wenzelm@61945
   977
lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x$1) - (y$1)\<bar>"
hoelzl@37489
   978
  by (auto simp add: norm_real dist_norm)
hoelzl@37489
   979
lp15@67986
   980
subsection\<open> Rank of a matrix\<close>
lp15@67986
   981
lp15@67986
   982
text\<open>Equivalence of row and column rank is taken from George Mackiw's paper, Mathematics Magazine 1995, p. 285.\<close>
lp15@67986
   983
immler@68072
   984
lemma matrix_vector_mult_in_columnspace_gen:
immler@68072
   985
  fixes A :: "'a::field^'n^'m"
immler@68072
   986
  shows "(A *v x) \<in> vec.span(columns A)"
immler@68072
   987
  apply (simp add: matrix_vector_column columns_def transpose_def column_def)
immler@68072
   988
  apply (intro vec.span_sum vec.span_scale)
immler@68072
   989
  apply (force intro: vec.span_base)
immler@68072
   990
  done
immler@68072
   991
lp15@67986
   992
lemma matrix_vector_mult_in_columnspace:
lp15@67986
   993
  fixes A :: "real^'n^'m"
lp15@67986
   994
  shows "(A *v x) \<in> span(columns A)"
immler@68072
   995
  using matrix_vector_mult_in_columnspace_gen[of A x] by (simp add: span_vec_eq)
lp15@67986
   996
lp15@67986
   997
lemma orthogonal_nullspace_rowspace:
lp15@67986
   998
  fixes A :: "real^'n^'m"
lp15@67986
   999
  assumes 0: "A *v x = 0" and y: "y \<in> span(rows A)"
lp15@67986
  1000
  shows "orthogonal x y"
lp15@67986
  1001
proof (rule span_induct [OF y])
lp15@67986
  1002
  show "subspace {a. orthogonal x a}"
lp15@67986
  1003
    by (simp add: subspace_orthogonal_to_vector)
lp15@67986
  1004
next
lp15@67986
  1005
  fix v
lp15@67986
  1006
  assume "v \<in> rows A"
lp15@67986
  1007
  then obtain i where "v = row i A"
lp15@67986
  1008
    by (auto simp: rows_def)
lp15@67986
  1009
  with 0 show "orthogonal x v"
lp15@67986
  1010
    unfolding orthogonal_def inner_vec_def matrix_vector_mult_def row_def
lp15@67986
  1011
    by (simp add: mult.commute) (metis (no_types) vec_lambda_beta zero_index)
lp15@67986
  1012
qed
lp15@67986
  1013
lp15@67986
  1014
lemma nullspace_inter_rowspace:
lp15@67986
  1015
  fixes A :: "real^'n^'m"
lp15@67986
  1016
  shows "A *v x = 0 \<and> x \<in> span(rows A) \<longleftrightarrow> x = 0"
immler@68072
  1017
  using orthogonal_nullspace_rowspace orthogonal_self span_zero matrix_vector_mult_0_right
immler@68072
  1018
  by blast
lp15@67986
  1019
lp15@67986
  1020
lemma matrix_vector_mul_injective_on_rowspace:
lp15@67986
  1021
  fixes A :: "real^'n^'m"
lp15@67986
  1022
  shows "\<lbrakk>A *v x = A *v y; x \<in> span(rows A); y \<in> span(rows A)\<rbrakk> \<Longrightarrow> x = y"
lp15@67986
  1023
  using nullspace_inter_rowspace [of A "x-y"]
immler@68072
  1024
  by (metis diff_eq_diff_eq diff_self matrix_vector_mult_diff_distrib span_diff)
lp15@67986
  1025
immler@68072
  1026
definition rank :: "'a::field^'n^'m=>nat"
immler@68072
  1027
  where row_rank_def_gen: "rank A \<equiv> vec.dim(rows A)"
immler@68072
  1028
immler@68072
  1029
lemma row_rank_def: "rank A = dim (rows A)" for A::"real^'n^'m"
immler@68072
  1030
  by (auto simp: row_rank_def_gen dim_vec_eq)
lp15@67986
  1031
lp15@67986
  1032
lemma dim_rows_le_dim_columns:
lp15@67986
  1033
  fixes A :: "real^'n^'m"
lp15@67986
  1034
  shows "dim(rows A) \<le> dim(columns A)"
lp15@67986
  1035
proof -
lp15@67986
  1036
  have "dim (span (rows A)) \<le> dim (span (columns A))"
lp15@67986
  1037
  proof -
lp15@67986
  1038
    obtain B where "independent B" "span(rows A) \<subseteq> span B"
lp15@67986
  1039
              and B: "B \<subseteq> span(rows A)""card B = dim (span(rows A))"
lp15@67986
  1040
      using basis_exists [of "span(rows A)"] by blast
immler@68072
  1041
    then have eq: "span B = span(rows A)"
immler@68072
  1042
      using span_subspace subspace_span by blast
lp15@67986
  1043
    then have inj: "inj_on (( *v) A) (span B)"
immler@68072
  1044
      by (simp add: inj_on_def matrix_vector_mul_injective_on_rowspace)
lp15@67986
  1045
    then have ind: "independent (( *v) A ` B)"
immler@68072
  1046
      by (rule linear_independent_injective_image [OF Finite_Cartesian_Product.matrix_vector_mul_linear \<open>independent B\<close>])
immler@68072
  1047
    have "dim (span (rows A)) \<le> card (( *v) A ` B)"
immler@68072
  1048
      unfolding B(2)[symmetric]
immler@68072
  1049
      using inj
immler@68072
  1050
      by (auto simp: card_image inj_on_subset span_superset)
immler@68072
  1051
    also have "\<dots> \<le> dim (span (columns A))"
immler@68072
  1052
      using _ ind
immler@68072
  1053
      by (rule independent_card_le_dim) (auto intro!: matrix_vector_mult_in_columnspace)
immler@68072
  1054
    finally show ?thesis .
lp15@67986
  1055
  qed
lp15@67986
  1056
  then show ?thesis
immler@68072
  1057
    by (simp add: dim_span)
lp15@67986
  1058
qed
lp15@67986
  1059
immler@68072
  1060
lemma column_rank_def:
lp15@67986
  1061
  fixes A :: "real^'n^'m"
immler@68072
  1062
  shows "rank A = dim(columns A)"
immler@68072
  1063
  unfolding row_rank_def
immler@68072
  1064
  by (metis columns_transpose dim_rows_le_dim_columns le_antisym rows_transpose)
lp15@67986
  1065
lp15@67986
  1066
lemma rank_transpose:
lp15@67986
  1067
  fixes A :: "real^'n^'m"
lp15@67986
  1068
  shows "rank(transpose A) = rank A"
immler@68072
  1069
  by (metis column_rank_def row_rank_def rows_transpose)
lp15@67986
  1070
lp15@67986
  1071
lemma matrix_vector_mult_basis:
lp15@67986
  1072
  fixes A :: "real^'n^'m"
lp15@67986
  1073
  shows "A *v (axis k 1) = column k A"
lp15@67986
  1074
  by (simp add: cart_eq_inner_axis column_def matrix_mult_dot)
lp15@67986
  1075
lp15@67986
  1076
lemma columns_image_basis:
lp15@67986
  1077
  fixes A :: "real^'n^'m"
lp15@67986
  1078
  shows "columns A = ( *v) A ` (range (\<lambda>i. axis i 1))"
lp15@67986
  1079
  by (force simp: columns_def matrix_vector_mult_basis [symmetric])
lp15@67986
  1080
lp15@67986
  1081
lemma rank_dim_range:
lp15@67986
  1082
  fixes A :: "real^'n^'m"
lp15@67986
  1083
  shows "rank A = dim(range (\<lambda>x. A *v x))"
immler@68072
  1084
  unfolding column_rank_def
lp15@67986
  1085
proof (rule span_eq_dim)
immler@68072
  1086
  have "span (columns A) \<subseteq> span (range (( *v) A))" (is "?l \<subseteq> ?r")
immler@68072
  1087
    by (simp add: columns_image_basis image_subsetI span_mono)
immler@68072
  1088
  then show "?l = ?r"
immler@68072
  1089
    by (metis (no_types, lifting) image_subset_iff matrix_vector_mult_in_columnspace
immler@68072
  1090
        span_eq span_span)
lp15@67986
  1091
qed
lp15@67986
  1092
lp15@67986
  1093
lemma rank_bound:
lp15@67986
  1094
  fixes A :: "real^'n^'m"
lp15@67986
  1095
  shows "rank A \<le> min CARD('m) (CARD('n))"
immler@68072
  1096
  by (metis (mono_tags, lifting) dim_subset_UNIV_cart min.bounded_iff
immler@68072
  1097
      column_rank_def row_rank_def)
lp15@67986
  1098
lp15@67986
  1099
lemma full_rank_injective:
lp15@67986
  1100
  fixes A :: "real^'n^'m"
lp15@67986
  1101
  shows "rank A = CARD('n) \<longleftrightarrow> inj (( *v) A)"
immler@68072
  1102
  by (simp add: matrix_left_invertible_injective [symmetric] matrix_left_invertible_span_rows row_rank_def
immler@68072
  1103
      dim_eq_full [symmetric] card_cart_basis vec.dimension_def)
lp15@67986
  1104
lp15@67986
  1105
lemma full_rank_surjective:
lp15@67986
  1106
  fixes A :: "real^'n^'m"
lp15@67986
  1107
  shows "rank A = CARD('m) \<longleftrightarrow> surj (( *v) A)"
lp15@67986
  1108
  by (simp add: matrix_right_invertible_surjective [symmetric] left_invertible_transpose [symmetric]
lp15@67986
  1109
                matrix_left_invertible_injective full_rank_injective [symmetric] rank_transpose)
lp15@67986
  1110
lp15@67986
  1111
lemma rank_I: "rank(mat 1::real^'n^'n) = CARD('n)"
lp15@67986
  1112
  by (simp add: full_rank_injective inj_on_def)
lp15@67986
  1113
lp15@67986
  1114
lemma less_rank_noninjective:
lp15@67986
  1115
  fixes A :: "real^'n^'m"
lp15@67986
  1116
  shows "rank A < CARD('n) \<longleftrightarrow> \<not> inj (( *v) A)"
lp15@67986
  1117
using less_le rank_bound by (auto simp: full_rank_injective [symmetric])
lp15@67986
  1118
lp15@67986
  1119
lemma matrix_nonfull_linear_equations_eq:
lp15@67986
  1120
  fixes A :: "real^'n^'m"
lp15@67986
  1121
  shows "(\<exists>x. (x \<noteq> 0) \<and> A *v x = 0) \<longleftrightarrow> ~(rank A = CARD('n))"
lp15@67986
  1122
  by (meson matrix_left_invertible_injective full_rank_injective matrix_left_invertible_ker)
lp15@67986
  1123
immler@68072
  1124
lemma rank_eq_0: "rank A = 0 \<longleftrightarrow> A = 0" and rank_0 [simp]: "rank (0::real^'n^'m) = 0"
immler@68072
  1125
  for A :: "real^'n^'m"
lp15@67986
  1126
  by (auto simp: rank_dim_range matrix_eq)
lp15@67986
  1127
lp15@67986
  1128
lemma rank_mul_le_right:
lp15@67986
  1129
  fixes A :: "real^'n^'m" and B :: "real^'p^'n"
lp15@67986
  1130
  shows "rank(A ** B) \<le> rank B"
lp15@67986
  1131
proof -
lp15@67986
  1132
  have "rank(A ** B) \<le> dim (( *v) A ` range (( *v) B))"
lp15@67986
  1133
    by (auto simp: rank_dim_range image_comp o_def matrix_vector_mul_assoc)
lp15@67986
  1134
  also have "\<dots> \<le> rank B"
immler@68072
  1135
    by (simp add: rank_dim_range dim_image_le)
lp15@67986
  1136
  finally show ?thesis .
lp15@67986
  1137
qed
lp15@67986
  1138
lp15@67986
  1139
lemma rank_mul_le_left:
lp15@67986
  1140
  fixes A :: "real^'n^'m" and B :: "real^'p^'n"
lp15@67986
  1141
  shows "rank(A ** B) \<le> rank A"
lp15@67986
  1142
  by (metis matrix_transpose_mul rank_mul_le_right rank_transpose)
lp15@67986
  1143
lp15@67981
  1144
subsection\<open>Routine results connecting the types @{typ "real^1"} and @{typ real}\<close>
lp15@67981
  1145
lp15@67981
  1146
lemma vector_one_nth [simp]:
lp15@67981
  1147
  fixes x :: "'a^1" shows "vec (x $ 1) = x"
lp15@67981
  1148
  by (metis vec_def vector_one)
lp15@67981
  1149
lp15@67981
  1150
lemma vec_cbox_1_eq [simp]:
lp15@67981
  1151
  shows "vec ` cbox u v = cbox (vec u) (vec v ::real^1)"
lp15@67981
  1152
  by (force simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box)
lp15@67981
  1153
lp15@67981
  1154
lemma vec_nth_cbox_1_eq [simp]:
lp15@67981
  1155
  fixes u v :: "'a::euclidean_space^1"
lp15@67981
  1156
  shows "(\<lambda>x. x $ 1) ` cbox u v = cbox (u$1) (v$1)"
lp15@67981
  1157
    by (auto simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box image_iff Bex_def inner_axis) (metis vec_component)
lp15@67981
  1158
lp15@67981
  1159
lemma vec_nth_1_iff_cbox [simp]:
lp15@67981
  1160
  fixes a b :: "'a::euclidean_space"
lp15@67981
  1161
  shows "(\<lambda>x::'a^1. x $ 1) ` S = cbox a b \<longleftrightarrow> S = cbox (vec a) (vec b)"
lp15@67981
  1162
    (is "?lhs = ?rhs")
lp15@67981
  1163
proof
lp15@67981
  1164
  assume L: ?lhs show ?rhs
lp15@67981
  1165
  proof (intro equalityI subsetI)
lp15@67981
  1166
    fix x 
lp15@67981
  1167
    assume "x \<in> S"
lp15@67981
  1168
    then have "x $ 1 \<in> (\<lambda>v. v $ (1::1)) ` cbox (vec a) (vec b)"
lp15@67981
  1169
      using L by auto
lp15@67981
  1170
    then show "x \<in> cbox (vec a) (vec b)"
lp15@67981
  1171
      by (metis (no_types, lifting) imageE vector_one_nth)
lp15@67981
  1172
  next
lp15@67981
  1173
    fix x :: "'a^1"
lp15@67981
  1174
    assume "x \<in> cbox (vec a) (vec b)"
lp15@67981
  1175
    then show "x \<in> S"
lp15@67981
  1176
      by (metis (no_types, lifting) L imageE imageI vec_component vec_nth_cbox_1_eq vector_one_nth)
lp15@67981
  1177
  qed
lp15@67981
  1178
qed simp
wenzelm@49644
  1179
lp15@67979
  1180
lemma tendsto_at_within_vector_1:
lp15@67979
  1181
  fixes S :: "'a :: metric_space set"
lp15@67979
  1182
  assumes "(f \<longlongrightarrow> fx) (at x within S)"
lp15@67979
  1183
  shows "((\<lambda>y::'a^1. \<chi> i. f (y $ 1)) \<longlongrightarrow> (vec fx::'a^1)) (at (vec x) within vec ` S)"
lp15@67979
  1184
proof (rule topological_tendstoI)
lp15@67979
  1185
  fix T :: "('a^1) set"
lp15@67979
  1186
  assume "open T" "vec fx \<in> T"
lp15@67979
  1187
  have "\<forall>\<^sub>F x in at x within S. f x \<in> (\<lambda>x. x $ 1) ` T"
lp15@67979
  1188
    using \<open>open T\<close> \<open>vec fx \<in> T\<close> assms open_image_vec_nth tendsto_def by fastforce
lp15@67979
  1189
  then show "\<forall>\<^sub>F x::'a^1 in at (vec x) within vec ` S. (\<chi> i. f (x $ 1)) \<in> T"
lp15@67979
  1190
    unfolding eventually_at dist_norm [symmetric]
lp15@67979
  1191
    by (rule ex_forward)
lp15@67979
  1192
       (use \<open>open T\<close> in 
lp15@67979
  1193
         \<open>fastforce simp: dist_norm dist_vec_def L2_set_def image_iff vector_one open_vec_def\<close>)
lp15@67979
  1194
qed
lp15@67979
  1195
lp15@67979
  1196
lemma has_derivative_vector_1:
lp15@67979
  1197
  assumes der_g: "(g has_derivative (\<lambda>x. x * g' a)) (at a within S)"
lp15@67979
  1198
  shows "((\<lambda>x. vec (g (x $ 1))) has_derivative ( *\<^sub>R) (g' a))
lp15@67979
  1199
         (at ((vec a)::real^1) within vec ` S)"
lp15@67979
  1200
    using der_g
lp15@67979
  1201
    apply (auto simp: Deriv.has_derivative_within bounded_linear_scaleR_right norm_vector_1)
lp15@67979
  1202
    apply (drule tendsto_at_within_vector_1, vector)
lp15@67979
  1203
    apply (auto simp: algebra_simps eventually_at tendsto_def)
lp15@67979
  1204
    done
lp15@67979
  1205
lp15@67979
  1206
nipkow@67968
  1207
subsection\<open>Explicit vector construction from lists\<close>
hoelzl@37489
  1208
hoelzl@43995
  1209
definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
hoelzl@37489
  1210
hoelzl@37489
  1211
lemma vector_1: "(vector[x]) $1 = x"
hoelzl@37489
  1212
  unfolding vector_def by simp
hoelzl@37489
  1213
hoelzl@37489
  1214
lemma vector_2:
hoelzl@37489
  1215
 "(vector[x,y]) $1 = x"
hoelzl@37489
  1216
 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
hoelzl@37489
  1217
  unfolding vector_def by simp_all
hoelzl@37489
  1218
hoelzl@37489
  1219
lemma vector_3:
hoelzl@37489
  1220
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
hoelzl@37489
  1221
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
hoelzl@37489
  1222
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
hoelzl@37489
  1223
  unfolding vector_def by simp_all
hoelzl@37489
  1224
hoelzl@37489
  1225
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
lp15@67719
  1226
  by (metis vector_1 vector_one)
hoelzl@37489
  1227
hoelzl@37489
  1228
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
hoelzl@37489
  1229
  apply auto
hoelzl@37489
  1230
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1231
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1232
  apply (subgoal_tac "vector [v$1, v$2] = v")
hoelzl@37489
  1233
  apply simp
hoelzl@37489
  1234
  apply (vector vector_def)
hoelzl@37489
  1235
  apply (simp add: forall_2)
hoelzl@37489
  1236
  done
hoelzl@37489
  1237
hoelzl@37489
  1238
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
hoelzl@37489
  1239
  apply auto
hoelzl@37489
  1240
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1241
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1242
  apply (erule_tac x="v$3" in allE)
hoelzl@37489
  1243
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
hoelzl@37489
  1244
  apply simp
hoelzl@37489
  1245
  apply (vector vector_def)
hoelzl@37489
  1246
  apply (simp add: forall_3)
hoelzl@37489
  1247
  done
hoelzl@37489
  1248
hoelzl@37489
  1249
lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
wenzelm@49644
  1250
  apply (rule bounded_linearI[where K=1])
hoelzl@37489
  1251
  using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
hoelzl@37489
  1252
hoelzl@37489
  1253
lemma interval_split_cart:
hoelzl@37489
  1254
  "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
immler@56188
  1255
  "cbox a b \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
wenzelm@49644
  1256
  apply (rule_tac[!] set_eqI)
lp15@67673
  1257
  unfolding Int_iff mem_box_cart mem_Collect_eq interval_cbox_cart
wenzelm@49644
  1258
  unfolding vec_lambda_beta
wenzelm@49644
  1259
  by auto
hoelzl@37489
  1260
immler@67685
  1261
lemmas cartesian_euclidean_space_uniform_limit_intros[uniform_limit_intros] =
immler@67685
  1262
  bounded_linear.uniform_limit[OF blinfun.bounded_linear_right]
immler@67685
  1263
  bounded_linear.uniform_limit[OF bounded_linear_vec_nth]
immler@67685
  1264
  bounded_linear.uniform_limit[OF bounded_linear_component_cart]
immler@67685
  1265
immler@68072
  1266
end