src/HOL/Analysis/Cartesian_Euclidean_Space.thy
author paulson <lp15@cam.ac.uk>
Tue May 01 23:25:00 2018 +0100 (13 months ago)
changeset 68062 ee88c0fccbae
parent 68054 ebd179b82e20
child 68069 36209dfb981e
permissions -rw-r--r--
simplified some messy proofs
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(* Title:      HOL/Analysis/Cartesian_Euclidean_Space.thy
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   Some material by Jose Divasón, Tim Makarios and L C Paulson
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*)
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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 Finite_Cartesian_Product Derivative
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begin
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lemma norm_le_componentwise:
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   "(\<And>b. b \<in> Basis \<Longrightarrow> abs(x \<bullet> b) \<le> abs(y \<bullet> b)) \<Longrightarrow> norm x \<le> norm y"
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  by (auto simp: norm_le euclidean_inner [of x x] euclidean_inner [of y y] abs_le_square_iff power2_eq_square intro!: sum_mono)
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lemma norm_le_componentwise_cart:
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  fixes x :: "real^'n"
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  shows "(\<And>i. abs(x$i) \<le> abs(y$i)) \<Longrightarrow> norm x \<le> norm y"
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  unfolding cart_eq_inner_axis
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  by (rule norm_le_componentwise) (metis axis_index)
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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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subsection\<open>Basic componentwise operations on vectors\<close>
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instantiation vec :: (times, finite) times
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begin
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definition "( * ) \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
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instance ..
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end
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instantiation vec :: (one, finite) one
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begin
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definition "1 \<equiv> (\<chi> i. 1)"
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instance ..
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end
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instantiation vec :: (ord, finite) ord
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begin
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definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"
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definition "x < (y::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
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instance ..
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end
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text\<open>The ordering on one-dimensional vectors is linear.\<close>
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class cart_one =
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  assumes UNIV_one: "card (UNIV :: 'a set) = Suc 0"
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begin
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subclass finite
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proof
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  from UNIV_one show "finite (UNIV :: 'a set)"
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    by (auto intro!: card_ge_0_finite)
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qed
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end
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instance vec:: (order, finite) order
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  by standard (auto simp: less_eq_vec_def less_vec_def vec_eq_iff
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      intro: order.trans order.antisym order.strict_implies_order)
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instance vec :: (linorder, cart_one) linorder
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proof
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  obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"
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  proof -
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    have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)
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    then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)
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    then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto
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    then show thesis by (auto intro: that)
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  qed
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  fix x y :: "'a^'b::cart_one"
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  note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
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  show "x \<le> y \<or> y \<le> x" by auto
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qed
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text\<open>Constant Vectors\<close>
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definition "vec x = (\<chi> i. x)"
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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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text\<open>Also the scalar-vector multiplication.\<close>
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definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
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  where "c *s x = (\<chi> i. c * (x$i))"
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subsection \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space\<close>
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lemma sum_cong_aux:
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  "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> sum f A = sum g A"
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  by (auto intro: sum.cong)
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hide_fact (open) sum_cong_aux
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method_setup vector = \<open>
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let
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  val ss1 =
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    simpset_of (put_simpset HOL_basic_ss @{context}
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      addsimps [@{thm sum.distrib} RS sym,
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      @{thm sum_subtractf} RS sym, @{thm sum_distrib_left},
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      @{thm sum_distrib_right}, @{thm sum_negf} RS sym])
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  val ss2 =
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    simpset_of (@{context} addsimps
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             [@{thm plus_vec_def}, @{thm times_vec_def},
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              @{thm minus_vec_def}, @{thm uminus_vec_def},
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              @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
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              @{thm scaleR_vec_def},
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              @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}])
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  fun vector_arith_tac ctxt ths =
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    simp_tac (put_simpset ss1 ctxt)
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    THEN' (fn i => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.sum_cong_aux} i
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         ORELSE resolve_tac ctxt @{thms sum.neutral} i
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         ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i)
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    (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
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    THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths)
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in
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  Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths))
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end
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\<close> "lift trivial vector statements to real arith statements"
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lemma vec_0[simp]: "vec 0 = 0" by vector
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lemma vec_1[simp]: "vec 1 = 1" by vector
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lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
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lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
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lemma vec_add: "vec(x + y) = vec x + vec y"  by vector
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lemma vec_sub: "vec(x - y) = vec x - vec y" by vector
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lemma vec_cmul: "vec(c * x) = c *s vec x " by vector
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lemma vec_neg: "vec(- x) = - vec x " by vector
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lemma vec_scaleR: "vec(c * x) = c *\<^sub>R vec x"
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  by vector
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lemma vec_sum:
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  assumes "finite S"
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  shows "vec(sum f S) = sum (vec \<circ> f) S"
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  using assms
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proof induct
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  case empty
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  then show ?case by simp
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next
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  case insert
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  then show ?case by (auto simp add: vec_add)
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qed
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text\<open>Obvious "component-pushing".\<close>
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lemma vec_component [simp]: "vec x $ i = x"
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  by vector
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lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
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  by vector
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lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
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  by vector
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lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
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lemmas vector_component =
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  vec_component vector_add_component vector_mult_component
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  vector_smult_component vector_minus_component vector_uminus_component
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  vector_scaleR_component cond_component
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subsection \<open>Some frequently useful arithmetic lemmas over vectors\<close>
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instance vec :: (semigroup_mult, finite) semigroup_mult
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  by standard (vector mult.assoc)
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instance vec :: (monoid_mult, finite) monoid_mult
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  by standard vector+
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instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
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  by standard (vector mult.commute)
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instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
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  by standard vector
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instance vec :: (semiring, finite) semiring
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  by standard (vector field_simps)+
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instance vec :: (semiring_0, finite) semiring_0
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  by standard (vector field_simps)+
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instance vec :: (semiring_1, finite) semiring_1
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  by standard vector
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instance vec :: (comm_semiring, finite) comm_semiring
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  by standard (vector field_simps)+
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instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
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instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
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instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
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instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
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instance vec :: (ring, finite) ring ..
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instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
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instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
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instance vec :: (ring_1, finite) ring_1 ..
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instance vec :: (real_algebra, finite) real_algebra
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  by standard (simp_all add: vec_eq_iff)
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instance vec :: (real_algebra_1, finite) real_algebra_1 ..
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lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
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proof (induct n)
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  case 0
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  then show ?case by vector
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next
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  case Suc
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  then show ?case by vector
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qed
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lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) $ i = 1"
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  by vector
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lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) $ i = - 1"
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  by vector
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instance vec :: (semiring_char_0, finite) semiring_char_0
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proof
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  fix m n :: nat
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  show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
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    by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
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qed
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instance vec :: (numeral, finite) numeral ..
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instance vec :: (semiring_numeral, finite) semiring_numeral ..
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lemma numeral_index [simp]: "numeral w $ i = numeral w"
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  by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
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lemma neg_numeral_index [simp]: "- numeral w $ i = - numeral w"
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  by (simp only: vector_uminus_component numeral_index)
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instance vec :: (comm_ring_1, finite) comm_ring_1 ..
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instance vec :: (ring_char_0, finite) ring_char_0 ..
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lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
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  by (vector mult.assoc)
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lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
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  by (vector field_simps)
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lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
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  by (vector field_simps)
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lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
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lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
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lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
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  by (vector field_simps)
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lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
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lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
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lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
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lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
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lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
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  by (vector field_simps)
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lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
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  by (simp add: vec_eq_iff)
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lemma linear_vec [simp]: "linear vec"
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  by (simp add: linearI vec_add vec_eq_iff)
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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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lemma norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
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  by (simp add: inner_axis' norm_eq_1)
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lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
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  by vector
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lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
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  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
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hoelzl@37489
   312
lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
hoelzl@37489
   313
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
lp15@67683
   314
lp15@67979
   315
lemma component_le_norm_cart: "\<bar>x$i\<bar> \<le> norm x"
huffman@44136
   316
  apply (simp add: norm_vec_def)
nipkow@67155
   317
  apply (rule member_le_L2_set, simp_all)
hoelzl@37489
   318
  done
hoelzl@37489
   319
lp15@67979
   320
lemma norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x$i\<bar> \<le> e"
hoelzl@37489
   321
  by (metis component_le_norm_cart order_trans)
hoelzl@37489
   322
hoelzl@37489
   323
lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
huffman@53595
   324
  by (metis component_le_norm_cart le_less_trans)
hoelzl@37489
   325
lp15@67979
   326
lemma norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
nipkow@67155
   327
  by (simp add: norm_vec_def L2_set_le_sum)
hoelzl@37489
   328
lp15@67969
   329
lemma scalar_mult_eq_scaleR [simp]: "c *s x = c *\<^sub>R x"
huffman@44136
   330
  unfolding scaleR_vec_def vector_scalar_mult_def by simp
hoelzl@37489
   331
hoelzl@37489
   332
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
hoelzl@37489
   333
  unfolding dist_norm scalar_mult_eq_scaleR
hoelzl@37489
   334
  unfolding scaleR_right_diff_distrib[symmetric] by simp
hoelzl@37489
   335
nipkow@64267
   336
lemma sum_component [simp]:
hoelzl@37489
   337
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
nipkow@64267
   338
  shows "(sum f S)$i = sum (\<lambda>x. (f x)$i) S"
wenzelm@49644
   339
proof (cases "finite S")
wenzelm@49644
   340
  case True
wenzelm@49644
   341
  then show ?thesis by induct simp_all
wenzelm@49644
   342
next
wenzelm@49644
   343
  case False
wenzelm@49644
   344
  then show ?thesis by simp
wenzelm@49644
   345
qed
hoelzl@37489
   346
nipkow@64267
   347
lemma sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)$i ) S)"
huffman@44136
   348
  by (simp add: vec_eq_iff)
hoelzl@37489
   349
nipkow@64267
   350
lemma sum_cmul:
hoelzl@37489
   351
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
nipkow@64267
   352
  shows "sum (\<lambda>x. c *s f x) S = c *s sum f S"
nipkow@64267
   353
  by (simp add: vec_eq_iff sum_distrib_left)
hoelzl@37489
   354
nipkow@64267
   355
lemma sum_norm_allsubsets_bound_cart:
hoelzl@37489
   356
  fixes f:: "'a \<Rightarrow> real ^'n"
nipkow@64267
   357
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
nipkow@64267
   358
  shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
nipkow@64267
   359
  using sum_norm_allsubsets_bound[OF assms]
wenzelm@57865
   360
  by simp
hoelzl@37489
   361
lp15@62397
   362
subsection\<open>Closures and interiors of halfspaces\<close>
lp15@62397
   363
lp15@62397
   364
lemma interior_halfspace_le [simp]:
lp15@62397
   365
  assumes "a \<noteq> 0"
lp15@62397
   366
    shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
lp15@62397
   367
proof -
lp15@62397
   368
  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
lp15@62397
   369
  proof -
lp15@62397
   370
    obtain e where "e>0" and e: "cball x e \<subseteq> S"
lp15@62397
   371
      using \<open>open S\<close> open_contains_cball x by blast
lp15@62397
   372
    then have "x + (e / norm a) *\<^sub>R a \<in> cball x e"
lp15@62397
   373
      by (simp add: dist_norm)
lp15@62397
   374
    then have "x + (e / norm a) *\<^sub>R a \<in> S"
lp15@62397
   375
      using e by blast
lp15@62397
   376
    then have "x + (e / norm a) *\<^sub>R a \<in> {x. a \<bullet> x \<le> b}"
lp15@62397
   377
      using S by blast
lp15@62397
   378
    moreover have "e * (a \<bullet> a) / norm a > 0"
lp15@62397
   379
      by (simp add: \<open>0 < e\<close> assms)
lp15@62397
   380
    ultimately show ?thesis
lp15@62397
   381
      by (simp add: algebra_simps)
lp15@62397
   382
  qed
lp15@62397
   383
  show ?thesis
lp15@62397
   384
    by (rule interior_unique) (auto simp: open_halfspace_lt *)
lp15@62397
   385
qed
lp15@62397
   386
lp15@62397
   387
lemma interior_halfspace_ge [simp]:
lp15@62397
   388
   "a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"
lp15@62397
   389
using interior_halfspace_le [of "-a" "-b"] by simp
lp15@62397
   390
lp15@62397
   391
lemma interior_halfspace_component_le [simp]:
wenzelm@67731
   392
     "interior {x. x$k \<le> a} = {x :: (real^'n). x$k < a}" (is "?LE")
lp15@62397
   393
  and interior_halfspace_component_ge [simp]:
wenzelm@67731
   394
     "interior {x. x$k \<ge> a} = {x :: (real^'n). x$k > a}" (is "?GE")
lp15@62397
   395
proof -
lp15@62397
   396
  have "axis k (1::real) \<noteq> 0"
lp15@62397
   397
    by (simp add: axis_def vec_eq_iff)
lp15@62397
   398
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
lp15@62397
   399
    by (simp add: cart_eq_inner_axis inner_commute)
lp15@62397
   400
  ultimately show ?LE ?GE
lp15@62397
   401
    using interior_halfspace_le [of "axis k (1::real)" a]
lp15@62397
   402
          interior_halfspace_ge [of "axis k (1::real)" a] by auto
lp15@62397
   403
qed
lp15@62397
   404
lp15@62397
   405
lemma closure_halfspace_lt [simp]:
lp15@62397
   406
  assumes "a \<noteq> 0"
lp15@62397
   407
    shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
lp15@62397
   408
proof -
lp15@62397
   409
  have [simp]: "-{x. a \<bullet> x < b} = {x. a \<bullet> x \<ge> b}"
lp15@62397
   410
    by (force simp:)
lp15@62397
   411
  then show ?thesis
lp15@62397
   412
    using interior_halfspace_ge [of a b] assms
lp15@62397
   413
    by (force simp: closure_interior)
lp15@62397
   414
qed
lp15@62397
   415
lp15@62397
   416
lemma closure_halfspace_gt [simp]:
lp15@62397
   417
   "a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"
lp15@62397
   418
using closure_halfspace_lt [of "-a" "-b"] by simp
lp15@62397
   419
lp15@62397
   420
lemma closure_halfspace_component_lt [simp]:
wenzelm@67731
   421
     "closure {x. x$k < a} = {x :: (real^'n). x$k \<le> a}" (is "?LE")
lp15@62397
   422
  and closure_halfspace_component_gt [simp]:
wenzelm@67731
   423
     "closure {x. x$k > a} = {x :: (real^'n). x$k \<ge> a}" (is "?GE")
lp15@62397
   424
proof -
lp15@62397
   425
  have "axis k (1::real) \<noteq> 0"
lp15@62397
   426
    by (simp add: axis_def vec_eq_iff)
lp15@62397
   427
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
lp15@62397
   428
    by (simp add: cart_eq_inner_axis inner_commute)
lp15@62397
   429
  ultimately show ?LE ?GE
lp15@62397
   430
    using closure_halfspace_lt [of "axis k (1::real)" a]
lp15@62397
   431
          closure_halfspace_gt [of "axis k (1::real)" a] by auto
lp15@62397
   432
qed
lp15@62397
   433
lp15@62397
   434
lemma interior_hyperplane [simp]:
lp15@62397
   435
  assumes "a \<noteq> 0"
lp15@62397
   436
    shows "interior {x. a \<bullet> x = b} = {}"
lp15@62397
   437
proof -
lp15@62397
   438
  have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
lp15@62397
   439
    by (force simp:)
lp15@62397
   440
  then show ?thesis
lp15@62397
   441
    by (auto simp: assms)
lp15@62397
   442
qed
lp15@62397
   443
lp15@62397
   444
lemma frontier_halfspace_le:
lp15@62397
   445
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   446
    shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
lp15@62397
   447
proof (cases "a = 0")
lp15@62397
   448
  case True with assms show ?thesis by simp
lp15@62397
   449
next
lp15@62397
   450
  case False then show ?thesis
lp15@62397
   451
    by (force simp: frontier_def closed_halfspace_le)
lp15@62397
   452
qed
lp15@62397
   453
lp15@62397
   454
lemma frontier_halfspace_ge:
lp15@62397
   455
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   456
    shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"
lp15@62397
   457
proof (cases "a = 0")
lp15@62397
   458
  case True with assms show ?thesis by simp
lp15@62397
   459
next
lp15@62397
   460
  case False then show ?thesis
lp15@62397
   461
    by (force simp: frontier_def closed_halfspace_ge)
lp15@62397
   462
qed
lp15@62397
   463
lp15@62397
   464
lemma frontier_halfspace_lt:
lp15@62397
   465
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   466
    shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"
lp15@62397
   467
proof (cases "a = 0")
lp15@62397
   468
  case True with assms show ?thesis by simp
lp15@62397
   469
next
lp15@62397
   470
  case False then show ?thesis
lp15@62397
   471
    by (force simp: frontier_def interior_open open_halfspace_lt)
lp15@62397
   472
qed
lp15@62397
   473
lp15@62397
   474
lemma frontier_halfspace_gt:
lp15@62397
   475
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   476
    shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"
lp15@62397
   477
proof (cases "a = 0")
lp15@62397
   478
  case True with assms show ?thesis by simp
lp15@62397
   479
next
lp15@62397
   480
  case False then show ?thesis
lp15@62397
   481
    by (force simp: frontier_def interior_open open_halfspace_gt)
lp15@62397
   482
qed
lp15@62397
   483
lp15@62397
   484
lemma interior_standard_hyperplane:
wenzelm@67731
   485
   "interior {x :: (real^'n). x$k = a} = {}"
lp15@62397
   486
proof -
lp15@62397
   487
  have "axis k (1::real) \<noteq> 0"
lp15@62397
   488
    by (simp add: axis_def vec_eq_iff)
lp15@62397
   489
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
lp15@62397
   490
    by (simp add: cart_eq_inner_axis inner_commute)
lp15@62397
   491
  ultimately show ?thesis
lp15@62397
   492
    using interior_hyperplane [of "axis k (1::real)" a]
lp15@62397
   493
    by force
lp15@62397
   494
qed
lp15@62397
   495
wenzelm@60420
   496
subsection \<open>Matrix operations\<close>
hoelzl@37489
   497
wenzelm@60420
   498
text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close>
hoelzl@37489
   499
immler@67962
   500
definition map_matrix::"('a \<Rightarrow> 'b) \<Rightarrow> (('a, 'i::finite)vec, 'j::finite) vec \<Rightarrow> (('b, 'i)vec, 'j) vec" where
immler@67962
   501
  "map_matrix f x = (\<chi> i j. f (x $ i $ j))"
immler@67962
   502
immler@67962
   503
lemma nth_map_matrix[simp]: "map_matrix f x $ i $ j = f (x $ i $ j)"
immler@67962
   504
  by (simp add: map_matrix_def)
immler@67962
   505
wenzelm@49644
   506
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
wenzelm@49644
   507
    (infixl "**" 70)
nipkow@64267
   508
  where "m ** m' == (\<chi> i j. sum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
hoelzl@37489
   509
wenzelm@49644
   510
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
wenzelm@49644
   511
    (infixl "*v" 70)
nipkow@64267
   512
  where "m *v x \<equiv> (\<chi> i. sum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
hoelzl@37489
   513
wenzelm@49644
   514
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
wenzelm@49644
   515
    (infixl "v*" 70)
nipkow@64267
   516
  where "v v* m == (\<chi> j. sum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
hoelzl@37489
   517
hoelzl@37489
   518
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
hoelzl@63332
   519
definition transpose where
hoelzl@37489
   520
  "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
hoelzl@37489
   521
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
hoelzl@37489
   522
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
hoelzl@37489
   523
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
hoelzl@37489
   524
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
lp15@68038
   525
   
lp15@68038
   526
lemma times0_left [simp]: "(0::'a::semiring_1^'n^'m) ** (A::'a ^'p^'n) = 0" 
lp15@68038
   527
  by (simp add: matrix_matrix_mult_def zero_vec_def)
lp15@68038
   528
lp15@68038
   529
lemma times0_right [simp]: "(A::'a::semiring_1^'n^'m) ** (0::'a ^'p^'n) = 0" 
lp15@68038
   530
  by (simp add: matrix_matrix_mult_def zero_vec_def)
hoelzl@37489
   531
hoelzl@37489
   532
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
hoelzl@37489
   533
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
nipkow@64267
   534
  by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
hoelzl@37489
   535
lp15@67673
   536
lemma matrix_mul_lid [simp]:
hoelzl@37489
   537
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   538
  shows "mat 1 ** A = A"
hoelzl@37489
   539
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   540
  apply vector
nipkow@64267
   541
  apply (auto simp only: if_distrib cond_application_beta sum.delta'[OF finite]
wenzelm@49644
   542
    mult_1_left mult_zero_left if_True UNIV_I)
wenzelm@49644
   543
  done
hoelzl@37489
   544
lp15@67673
   545
lemma matrix_mul_rid [simp]:
hoelzl@37489
   546
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   547
  shows "A ** mat 1 = A"
hoelzl@37489
   548
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   549
  apply vector
nipkow@64267
   550
  apply (auto simp only: if_distrib cond_application_beta sum.delta[OF finite]
wenzelm@49644
   551
    mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
wenzelm@49644
   552
  done
hoelzl@37489
   553
hoelzl@37489
   554
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
nipkow@64267
   555
  apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc)
haftmann@66804
   556
  apply (subst sum.swap)
hoelzl@37489
   557
  apply simp
hoelzl@37489
   558
  done
hoelzl@37489
   559
hoelzl@37489
   560
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
wenzelm@49644
   561
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def
nipkow@64267
   562
    sum_distrib_left sum_distrib_right mult.assoc)
haftmann@66804
   563
  apply (subst sum.swap)
hoelzl@37489
   564
  apply simp
hoelzl@37489
   565
  done
hoelzl@37489
   566
lp15@68038
   567
lemma scalar_matrix_assoc:
lp15@68045
   568
  fixes A :: "('a::real_algebra_1)^'m^'n"
lp15@68038
   569
  shows "k *\<^sub>R (A ** B) = (k *\<^sub>R A) ** B"
lp15@68045
   570
  by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff scaleR_sum_right)
lp15@68038
   571
lp15@68038
   572
lemma matrix_scalar_ac:
lp15@68045
   573
  fixes A :: "('a::real_algebra_1)^'m^'n"
lp15@68038
   574
  shows "A ** (k *\<^sub>R B) = k *\<^sub>R A ** B"
lp15@68038
   575
  by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff)
lp15@68038
   576
lp15@67673
   577
lemma matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
hoelzl@37489
   578
  apply (vector matrix_vector_mult_def mat_def)
nipkow@64267
   579
  apply (simp add: if_distrib cond_application_beta sum.delta' cong del: if_weak_cong)
wenzelm@49644
   580
  done
hoelzl@37489
   581
wenzelm@49644
   582
lemma matrix_transpose_mul:
wenzelm@49644
   583
    "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
haftmann@57512
   584
  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
hoelzl@37489
   585
hoelzl@37489
   586
lemma matrix_eq:
hoelzl@37489
   587
  fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
hoelzl@37489
   588
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@37489
   589
  apply auto
huffman@44136
   590
  apply (subst vec_eq_iff)
hoelzl@37489
   591
  apply clarify
hoelzl@50526
   592
  apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
hoelzl@50526
   593
  apply (erule_tac x="axis ia 1" in allE)
hoelzl@37489
   594
  apply (erule_tac x="i" in allE)
hoelzl@50526
   595
  apply (auto simp add: if_distrib cond_application_beta axis_def
nipkow@64267
   596
    sum.delta[OF finite] cong del: if_weak_cong)
wenzelm@49644
   597
  done
hoelzl@37489
   598
lp15@68050
   599
lemma matrix_vector_mul_component: "(A *v x)$k = (A$k) \<bullet> x"
huffman@44136
   600
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   601
hoelzl@37489
   602
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
nipkow@64267
   603
  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps)
haftmann@66804
   604
  apply (subst sum.swap)
wenzelm@49644
   605
  apply simp
wenzelm@49644
   606
  done
hoelzl@37489
   607
lp15@67673
   608
lemma transpose_mat [simp]: "transpose (mat n) = mat n"
hoelzl@37489
   609
  by (vector transpose_def mat_def)
hoelzl@37489
   610
lp15@67683
   611
lemma transpose_transpose [simp]: "transpose(transpose A) = A"
hoelzl@37489
   612
  by (vector transpose_def)
hoelzl@37489
   613
lp15@67673
   614
lemma row_transpose [simp]:
hoelzl@37489
   615
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   616
  shows "row i (transpose A) = column i A"
huffman@44136
   617
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   618
lp15@67673
   619
lemma column_transpose [simp]:
hoelzl@37489
   620
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   621
  shows "column i (transpose A) = row i A"
huffman@44136
   622
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   623
lp15@67683
   624
lemma rows_transpose [simp]: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
wenzelm@49644
   625
  by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
hoelzl@37489
   626
lp15@67683
   627
lemma columns_transpose [simp]: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
wenzelm@49644
   628
  by (metis transpose_transpose rows_transpose)
hoelzl@37489
   629
lp15@68038
   630
lemma transpose_scalar: "transpose (k *\<^sub>R A) = k *\<^sub>R transpose A"
lp15@68038
   631
  unfolding transpose_def
lp15@68038
   632
  by (simp add: vec_eq_iff)
lp15@68038
   633
lp15@68038
   634
lemma transpose_iff [iff]: "transpose A = transpose B \<longleftrightarrow> A = B"
lp15@68038
   635
  by (metis transpose_transpose)
lp15@68038
   636
lp15@67673
   637
lemma matrix_mult_transpose_dot_column:
lp15@67673
   638
  shows "transpose A ** A = (\<chi> i j. (column i A) \<bullet> (column j A))"
lp15@67673
   639
  by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)
lp15@67673
   640
lp15@67673
   641
lemma matrix_mult_transpose_dot_row:
lp15@67673
   642
  shows "A ** transpose A = (\<chi> i j. (row i A) \<bullet> (row j A))"
lp15@67673
   643
  by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)
lp15@67673
   644
wenzelm@60420
   645
text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
hoelzl@37489
   646
hoelzl@37489
   647
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
huffman@44136
   648
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   649
lp15@67673
   650
lemma matrix_mult_sum:
nipkow@64267
   651
  "(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
haftmann@57512
   652
  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
hoelzl@37489
   653
hoelzl@37489
   654
lemma vector_componentwise:
hoelzl@50526
   655
  "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
nipkow@64267
   656
  by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)
hoelzl@50526
   657
nipkow@64267
   658
lemma basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
nipkow@64267
   659
  by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
hoelzl@37489
   660
lp15@63938
   661
lemma linear_componentwise_expansion:
hoelzl@37489
   662
  fixes f:: "real ^'m \<Rightarrow> real ^ _"
hoelzl@37489
   663
  assumes lf: "linear f"
nipkow@64267
   664
  shows "(f x)$j = sum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
wenzelm@49644
   665
proof -
hoelzl@37489
   666
  let ?M = "(UNIV :: 'm set)"
hoelzl@37489
   667
  let ?N = "(UNIV :: 'n set)"
nipkow@64267
   668
  have "?rhs = (sum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
nipkow@64267
   669
    unfolding sum_component by simp
wenzelm@49644
   670
  then show ?thesis
nipkow@64267
   671
    unfolding linear_sum_mul[OF lf, symmetric]
hoelzl@50526
   672
    unfolding scalar_mult_eq_scaleR[symmetric]
hoelzl@50526
   673
    unfolding basis_expansion
hoelzl@50526
   674
    by simp
hoelzl@37489
   675
qed
hoelzl@37489
   676
lp15@67719
   677
subsection\<open>Inverse matrices  (not necessarily square)\<close>
hoelzl@37489
   678
wenzelm@49644
   679
definition
wenzelm@49644
   680
  "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   681
wenzelm@49644
   682
definition
wenzelm@49644
   683
  "matrix_inv(A:: 'a::semiring_1^'n^'m) =
wenzelm@49644
   684
    (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   685
wenzelm@60420
   686
text\<open>Correspondence between matrices and linear operators.\<close>
hoelzl@37489
   687
wenzelm@49644
   688
definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
hoelzl@50526
   689
  where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
hoelzl@37489
   690
lp15@67986
   691
lemma matrix_id_mat_1: "matrix id = mat 1"
lp15@67986
   692
  by (simp add: mat_def matrix_def axis_def)
lp15@67986
   693
lp15@67986
   694
lemma matrix_scaleR: "(matrix (( *\<^sub>R) r)) = mat r"
lp15@67986
   695
  by (simp add: mat_def matrix_def axis_def if_distrib cong: if_cong)
lp15@67986
   696
lp15@68050
   697
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::('a::real_algebra_1) ^ _))"
lp15@68050
   698
  by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff field_simps sum.distrib scaleR_right.sum)
hoelzl@37489
   699
lp15@67683
   700
lemma
lp15@68050
   701
  fixes A :: "'a::{euclidean_space,real_algebra_1}^'n^'m"
lp15@67683
   702
  shows matrix_vector_mult_linear_continuous_at [continuous_intros]: "isCont (( *v) A) z"
lp15@67683
   703
    and matrix_vector_mult_linear_continuous_on [continuous_intros]: "continuous_on S (( *v) A)"
lp15@67683
   704
  by (simp_all add: linear_linear linear_continuous_at linear_continuous_on matrix_vector_mul_linear)
lp15@67683
   705
lp15@68043
   706
lemma vector_matrix_left_distrib [algebra_simps]:
lp15@68043
   707
  shows "(x + y) v* A = x v* A + y v* A"
lp15@68043
   708
  unfolding vector_matrix_mult_def
lp15@68043
   709
  by (simp add: algebra_simps sum.distrib vec_eq_iff)
lp15@68043
   710
lp15@68043
   711
lemma matrix_vector_right_distrib [algebra_simps]:
immler@67728
   712
  "A *v (x + y) = A *v x + A *v y"
immler@67728
   713
  by (vector matrix_vector_mult_def sum.distrib distrib_left)
lp15@67673
   714
lp15@67673
   715
lemma matrix_vector_mult_diff_distrib [algebra_simps]:
immler@67728
   716
  fixes A :: "'a::ring_1^'n^'m"
lp15@67673
   717
  shows "A *v (x - y) = A *v x - A *v y"
immler@67728
   718
  by (vector matrix_vector_mult_def sum_subtractf right_diff_distrib)
lp15@67673
   719
lp15@67673
   720
lemma matrix_vector_mult_scaleR[algebra_simps]:
lp15@67673
   721
  fixes A :: "real^'n^'m"
lp15@67673
   722
  shows "A *v (c *\<^sub>R x) = c *\<^sub>R (A *v x)"
lp15@67673
   723
  using linear_iff matrix_vector_mul_linear by blast
lp15@67673
   724
lp15@67673
   725
lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
lp15@67673
   726
  by (simp add: matrix_vector_mult_def vec_eq_iff)
lp15@67673
   727
lp15@67673
   728
lemma matrix_vector_mult_0 [simp]: "0 *v w = 0"
lp15@67673
   729
  by (simp add: matrix_vector_mult_def vec_eq_iff)
lp15@67673
   730
lp15@67673
   731
lemma matrix_vector_mult_add_rdistrib [algebra_simps]:
immler@67728
   732
  "(A + B) *v x = (A *v x) + (B *v x)"
immler@67728
   733
  by (vector matrix_vector_mult_def sum.distrib distrib_right)
lp15@67673
   734
lp15@67673
   735
lemma matrix_vector_mult_diff_rdistrib [algebra_simps]:
immler@67728
   736
  fixes A :: "'a :: ring_1^'n^'m"
lp15@67673
   737
  shows "(A - B) *v x = (A *v x) - (B *v x)"
immler@67728
   738
  by (vector matrix_vector_mult_def sum_subtractf left_diff_distrib)
lp15@67673
   739
wenzelm@49644
   740
lemma matrix_works:
wenzelm@49644
   741
  assumes lf: "linear f"
wenzelm@49644
   742
  shows "matrix f *v x = f (x::real ^ 'n)"
haftmann@57512
   743
  apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)
lp15@63938
   744
  by (simp add: linear_componentwise_expansion lf)
hoelzl@37489
   745
wenzelm@49644
   746
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
wenzelm@49644
   747
  by (simp add: ext matrix_works)
hoelzl@37489
   748
lp15@67683
   749
declare matrix_vector_mul [symmetric, simp]
lp15@67683
   750
lp15@67673
   751
lemma matrix_of_matrix_vector_mul [simp]: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
hoelzl@37489
   752
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
hoelzl@37489
   753
hoelzl@37489
   754
lemma matrix_compose:
hoelzl@37489
   755
  assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
wenzelm@49644
   756
    and lg: "linear (g::real^'m \<Rightarrow> real^_)"
wenzelm@61736
   757
  shows "matrix (g \<circ> f) = matrix g ** matrix f"
hoelzl@37489
   758
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
wenzelm@49644
   759
  by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
hoelzl@37489
   760
wenzelm@49644
   761
lemma matrix_vector_column:
nipkow@64267
   762
  "(A::'a::comm_semiring_1^'n^_) *v x = sum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
haftmann@57512
   763
  by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
hoelzl@37489
   764
hoelzl@37489
   765
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
hoelzl@37489
   766
  apply (rule adjoint_unique)
wenzelm@49644
   767
  apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
nipkow@64267
   768
    sum_distrib_right sum_distrib_left)
haftmann@66804
   769
  apply (subst sum.swap)
haftmann@57514
   770
  apply (auto simp add: ac_simps)
hoelzl@37489
   771
  done
hoelzl@37489
   772
hoelzl@37489
   773
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
hoelzl@37489
   774
  shows "matrix(adjoint f) = transpose(matrix f)"
hoelzl@37489
   775
  apply (subst matrix_vector_mul[OF lf])
wenzelm@49644
   776
  unfolding adjoint_matrix matrix_of_matrix_vector_mul
wenzelm@49644
   777
  apply rule
wenzelm@49644
   778
  done
wenzelm@49644
   779
lp15@67981
   780
lemma inj_matrix_vector_mult:
lp15@67981
   781
  fixes A::"'a::field^'n^'m"
lp15@67981
   782
  assumes "invertible A"
lp15@67981
   783
  shows "inj (( *v) A)"
lp15@67981
   784
  by (metis assms inj_on_inverseI invertible_def matrix_vector_mul_assoc matrix_vector_mul_lid)
lp15@67981
   785
lp15@68038
   786
lemma scalar_invertible:
lp15@68050
   787
  fixes A :: "('a::real_algebra_1)^'m^'n"
lp15@68038
   788
  assumes "k \<noteq> 0" and "invertible A"
lp15@68038
   789
  shows "invertible (k *\<^sub>R A)"
lp15@68038
   790
proof -
lp15@68038
   791
  obtain A' where "A ** A' = mat 1" and "A' ** A = mat 1"
lp15@68038
   792
    using assms unfolding invertible_def by auto
lp15@68038
   793
  with `k \<noteq> 0`
lp15@68038
   794
  have "(k *\<^sub>R A) ** ((1/k) *\<^sub>R A') = mat 1" "((1/k) *\<^sub>R A') ** (k *\<^sub>R A) = mat 1"
lp15@68038
   795
    by (simp_all add: assms matrix_scalar_ac)
lp15@68038
   796
  thus "invertible (k *\<^sub>R A)"
lp15@68038
   797
    unfolding invertible_def by auto
lp15@68038
   798
qed
lp15@68038
   799
lp15@68038
   800
lemma scalar_invertible_iff:
lp15@68050
   801
  fixes A :: "('a::real_algebra_1)^'m^'n"
lp15@68038
   802
  assumes "k \<noteq> 0" and "invertible A"
lp15@68038
   803
  shows "invertible (k *\<^sub>R A) \<longleftrightarrow> k \<noteq> 0 \<and> invertible A"
lp15@68038
   804
  by (simp add: assms scalar_invertible)
lp15@68038
   805
lp15@68038
   806
lemma vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
lp15@68038
   807
  unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
lp15@68038
   808
  by simp
lp15@68038
   809
lp15@68038
   810
lemma transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
lp15@68038
   811
  unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
lp15@68038
   812
  by simp
lp15@68038
   813
lp15@68043
   814
lemma vector_scalar_commute:
lp15@68043
   815
  fixes A :: "'a::{field}^'m^'n"
lp15@68043
   816
  shows "A *v (c *s x) = c *s (A *v x)"
lp15@68043
   817
  by (simp add: vector_scalar_mult_def matrix_vector_mult_def mult_ac sum_distrib_left)
lp15@68043
   818
lp15@68043
   819
lemma scalar_vector_matrix_assoc:
lp15@68043
   820
  fixes k :: "'a::{field}" and x :: "'a::{field}^'n" and A :: "'a^'m^'n"
lp15@68043
   821
  shows "(k *s x) v* A = k *s (x v* A)"
lp15@68043
   822
  by (metis transpose_matrix_vector vector_scalar_commute)
lp15@68043
   823
 
lp15@68043
   824
lemma vector_matrix_mult_0 [simp]: "0 v* A = 0"
lp15@68043
   825
  unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
lp15@68043
   826
lp15@68043
   827
lemma vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
lp15@68043
   828
  unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
lp15@68043
   829
lp15@68043
   830
lemma vector_matrix_mul_rid [simp]:
lp15@68038
   831
  fixes v :: "('a::semiring_1)^'n"
lp15@68038
   832
  shows "v v* mat 1 = v"
lp15@68038
   833
  by (metis matrix_vector_mul_lid transpose_mat vector_transpose_matrix)
lp15@68038
   834
lp15@68043
   835
lemma scaleR_vector_matrix_assoc:
lp15@68038
   836
  fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
lp15@68038
   837
  shows "(k *\<^sub>R x) v* A = k *\<^sub>R (x v* A)"
lp15@68038
   838
  by (metis matrix_vector_mult_scaleR transpose_matrix_vector)
lp15@68038
   839
lp15@68043
   840
lemma vector_scaleR_matrix_ac:
lp15@68038
   841
  fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
lp15@68038
   842
  shows "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
lp15@68038
   843
proof -
lp15@68038
   844
  have "x v* (k *\<^sub>R A) = (k *\<^sub>R x) v* A"
lp15@68038
   845
    unfolding vector_matrix_mult_def
lp15@68038
   846
    by (simp add: algebra_simps)
lp15@68043
   847
  with scaleR_vector_matrix_assoc
lp15@68038
   848
  show "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
lp15@68038
   849
    by auto
lp15@68038
   850
qed
lp15@68038
   851
hoelzl@37489
   852
nipkow@67968
   853
subsection\<open>Some bounds on components etc. relative to operator norm\<close>
lp15@67719
   854
lp15@67719
   855
lemma norm_column_le_onorm:
lp15@67719
   856
  fixes A :: "real^'n^'m"
lp15@67719
   857
  shows "norm(column i A) \<le> onorm(( *v) A)"
lp15@67719
   858
proof -
lp15@67719
   859
  have bl: "bounded_linear (( *v) A)"
lp15@67719
   860
    by (simp add: linear_linear matrix_vector_mul_linear)
lp15@67719
   861
  have "norm (\<chi> j. A $ j $ i) \<le> norm (A *v axis i 1)"
lp15@67719
   862
    by (simp add: matrix_mult_dot cart_eq_inner_axis)
lp15@67719
   863
  also have "\<dots> \<le> onorm (( *v) A)"
lp15@67982
   864
    using onorm [OF bl, of "axis i 1"] by auto
lp15@67719
   865
  finally have "norm (\<chi> j. A $ j $ i) \<le> onorm (( *v) A)" .
lp15@67719
   866
  then show ?thesis
lp15@67719
   867
    unfolding column_def .
lp15@67719
   868
qed
lp15@67719
   869
lp15@67719
   870
lemma matrix_component_le_onorm:
lp15@67719
   871
  fixes A :: "real^'n^'m"
lp15@67719
   872
  shows "\<bar>A $ i $ j\<bar> \<le> onorm(( *v) A)"
lp15@67719
   873
proof -
lp15@67719
   874
  have "\<bar>A $ i $ j\<bar> \<le> norm (\<chi> n. (A $ n $ j))"
lp15@67719
   875
    by (metis (full_types, lifting) component_le_norm_cart vec_lambda_beta)
lp15@67719
   876
  also have "\<dots> \<le> onorm (( *v) A)"
lp15@67719
   877
    by (metis (no_types) column_def norm_column_le_onorm)
lp15@67719
   878
  finally show ?thesis .
lp15@67719
   879
qed
lp15@67719
   880
lp15@67719
   881
lemma component_le_onorm:
lp15@67719
   882
  fixes f :: "real^'m \<Rightarrow> real^'n"
lp15@67719
   883
  shows "linear f \<Longrightarrow> \<bar>matrix f $ i $ j\<bar> \<le> onorm f"
lp15@67719
   884
  by (metis matrix_component_le_onorm matrix_vector_mul)
hoelzl@37489
   885
lp15@67719
   886
lemma onorm_le_matrix_component_sum:
lp15@67719
   887
  fixes A :: "real^'n^'m"
lp15@67719
   888
  shows "onorm(( *v) A) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>)"
lp15@67719
   889
proof (rule onorm_le)
lp15@67719
   890
  fix x
lp15@67719
   891
  have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
lp15@67719
   892
    by (rule norm_le_l1_cart)
lp15@67719
   893
  also have "\<dots> \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)"
lp15@67719
   894
  proof (rule sum_mono)
lp15@67719
   895
    fix i
lp15@67719
   896
    have "\<bar>(A *v x) $ i\<bar> \<le> \<bar>\<Sum>j\<in>UNIV. A $ i $ j * x $ j\<bar>"
lp15@67719
   897
      by (simp add: matrix_vector_mult_def)
lp15@67719
   898
    also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j * x $ j\<bar>)"
lp15@67719
   899
      by (rule sum_abs)
lp15@67719
   900
    also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)"
lp15@67719
   901
      by (rule sum_mono) (simp add: abs_mult component_le_norm_cart mult_left_mono)
lp15@67719
   902
    finally show "\<bar>(A *v x) $ i\<bar> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)" .
lp15@67719
   903
  qed
lp15@67719
   904
  finally show "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>) * norm x"
lp15@67719
   905
    by (simp add: sum_distrib_right)
lp15@67719
   906
qed
lp15@67719
   907
lp15@67719
   908
lemma onorm_le_matrix_component:
lp15@67719
   909
  fixes A :: "real^'n^'m"
lp15@67719
   910
  assumes "\<And>i j. abs(A$i$j) \<le> B"
lp15@67719
   911
  shows "onorm(( *v) A) \<le> real (CARD('m)) * real (CARD('n)) * B"
lp15@67719
   912
proof (rule onorm_le)
wenzelm@67731
   913
  fix x :: "real^'n::_"
lp15@67719
   914
  have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
lp15@67719
   915
    by (rule norm_le_l1_cart)
lp15@67719
   916
  also have "\<dots> \<le> (\<Sum>i::'m \<in>UNIV. real (CARD('n)) * B * norm x)"
lp15@67719
   917
  proof (rule sum_mono)
lp15@67719
   918
    fix i
lp15@67719
   919
    have "\<bar>(A *v x) $ i\<bar> \<le> norm(A $ i) * norm x"
lp15@67719
   920
      by (simp add: matrix_mult_dot Cauchy_Schwarz_ineq2)
lp15@67719
   921
    also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>) * norm x"
lp15@67719
   922
      by (simp add: mult_right_mono norm_le_l1_cart)
lp15@67719
   923
    also have "\<dots> \<le> real (CARD('n)) * B * norm x"
lp15@67719
   924
      by (simp add: assms sum_bounded_above mult_right_mono)
lp15@67719
   925
    finally show "\<bar>(A *v x) $ i\<bar> \<le> real (CARD('n)) * B * norm x" .
lp15@67719
   926
  qed
lp15@67719
   927
  also have "\<dots> \<le> CARD('m) * real (CARD('n)) * B * norm x"
lp15@67719
   928
    by simp
lp15@67719
   929
  finally show "norm (A *v x) \<le> CARD('m) * real (CARD('n)) * B * norm x" .
lp15@67719
   930
qed
lp15@67719
   931
lp15@67719
   932
subsection \<open>lambda skolemization on cartesian products\<close>
hoelzl@37489
   933
hoelzl@37489
   934
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
hoelzl@37494
   935
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   936
proof -
hoelzl@37489
   937
  let ?S = "(UNIV :: 'n set)"
wenzelm@49644
   938
  { assume H: "?rhs"
wenzelm@49644
   939
    then have ?lhs by auto }
hoelzl@37489
   940
  moreover
wenzelm@49644
   941
  { assume H: "?lhs"
hoelzl@37489
   942
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
hoelzl@37489
   943
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
wenzelm@49644
   944
    { fix i
hoelzl@37489
   945
      from f have "P i (f i)" by metis
hoelzl@37494
   946
      then have "P i (?x $ i)" by auto
hoelzl@37489
   947
    }
hoelzl@37489
   948
    hence "\<forall>i. P i (?x$i)" by metis
hoelzl@37489
   949
    hence ?rhs by metis }
hoelzl@37489
   950
  ultimately show ?thesis by metis
hoelzl@37489
   951
qed
hoelzl@37489
   952
lp15@67719
   953
lemma rational_approximation:
lp15@67719
   954
  assumes "e > 0"
lp15@67719
   955
  obtains r::real where "r \<in> \<rat>" "\<bar>r - x\<bar> < e"
lp15@67719
   956
  using Rats_dense_in_real [of "x - e/2" "x + e/2"] assms by auto
lp15@67719
   957
lp15@67719
   958
lemma matrix_rational_approximation:
lp15@67719
   959
  fixes A :: "real^'n^'m"
lp15@67719
   960
  assumes "e > 0"
lp15@67719
   961
  obtains B where "\<And>i j. B$i$j \<in> \<rat>" "onorm(\<lambda>x. (A - B) *v x) < e"
lp15@67719
   962
proof -
lp15@67719
   963
  have "\<forall>i j. \<exists>q \<in> \<rat>. \<bar>q - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))"
lp15@67719
   964
    using assms by (force intro: rational_approximation [of "e / (2 * CARD('m) * CARD('n))"])
lp15@67719
   965
  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
   966
    by (auto simp: lambda_skolem Bex_def)
lp15@67719
   967
  show ?thesis
lp15@67719
   968
  proof
lp15@67719
   969
    have "onorm (( *v) (A - B)) \<le> real CARD('m) * real CARD('n) *
lp15@67719
   970
    (e / (2 * real CARD('m) * real CARD('n)))"
lp15@67719
   971
      apply (rule onorm_le_matrix_component)
lp15@67719
   972
      using Bclo by (simp add: abs_minus_commute less_imp_le)
lp15@67719
   973
    also have "\<dots> < e"
lp15@67719
   974
      using \<open>0 < e\<close> by (simp add: divide_simps)
lp15@67719
   975
    finally show "onorm (( *v) (A - B)) < e" .
lp15@67719
   976
  qed (use B in auto)
lp15@67719
   977
qed
lp15@67719
   978
hoelzl@37489
   979
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
hoelzl@50526
   980
  unfolding inner_simps scalar_mult_eq_scaleR by auto
hoelzl@37489
   981
hoelzl@37489
   982
lemma left_invertible_transpose:
hoelzl@37489
   983
  "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
hoelzl@37489
   984
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   985
hoelzl@37489
   986
lemma right_invertible_transpose:
hoelzl@37489
   987
  "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
hoelzl@37489
   988
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   989
hoelzl@37489
   990
lemma matrix_left_invertible_injective:
lp15@67986
   991
  fixes A :: "real^'n^'m"
lp15@67986
   992
  shows "(\<exists>B. B ** A = mat 1) \<longleftrightarrow> inj (( *v) A)"
lp15@67986
   993
proof safe
lp15@67986
   994
  fix B
lp15@67986
   995
  assume B: "B ** A = mat 1"
lp15@67986
   996
  show "inj (( *v) A)"
lp15@67986
   997
    unfolding inj_on_def
lp15@67986
   998
      by (metis B matrix_vector_mul_assoc matrix_vector_mul_lid)
lp15@67986
   999
next
lp15@67986
  1000
  assume "inj (( *v) A)"
lp15@67986
  1001
  with linear_injective_left_inverse[OF matrix_vector_mul_linear]
lp15@67986
  1002
  obtain g where "linear g" and g: "g \<circ> ( *v) A = id"
lp15@67986
  1003
    by blast
lp15@67986
  1004
  have "matrix g ** A = mat 1"
lp15@67986
  1005
    by (metis \<open>linear g\<close> g matrix_compose matrix_id_mat_1 matrix_of_matrix_vector_mul matrix_vector_mul_linear)
lp15@67986
  1006
  then show "\<exists>B. B ** A = mat 1"
lp15@67986
  1007
    by metis
hoelzl@37489
  1008
qed
hoelzl@37489
  1009
hoelzl@37489
  1010
lemma matrix_left_invertible_ker:
hoelzl@37489
  1011
  "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
hoelzl@37489
  1012
  unfolding matrix_left_invertible_injective
hoelzl@37489
  1013
  using linear_injective_0[OF matrix_vector_mul_linear, of A]
hoelzl@37489
  1014
  by (simp add: inj_on_def)
hoelzl@37489
  1015
hoelzl@37489
  1016
lemma matrix_right_invertible_surjective:
wenzelm@49644
  1017
  "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
wenzelm@49644
  1018
proof -
wenzelm@49644
  1019
  { fix B :: "real ^'m^'n"
wenzelm@49644
  1020
    assume AB: "A ** B = mat 1"
wenzelm@49644
  1021
    { fix x :: "real ^ 'm"
hoelzl@37489
  1022
      have "A *v (B *v x) = x"
wenzelm@49644
  1023
        by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
nipkow@67399
  1024
    hence "surj (( *v) A)" unfolding surj_def by metis }
hoelzl@37489
  1025
  moreover
nipkow@67399
  1026
  { assume sf: "surj (( *v) A)"
hoelzl@37489
  1027
    from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
nipkow@67399
  1028
    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "( *v) A \<circ> g = id"
hoelzl@37489
  1029
      by blast
hoelzl@37489
  1030
hoelzl@37489
  1031
    have "A ** (matrix g) = mat 1"
hoelzl@37489
  1032
      unfolding matrix_eq  matrix_vector_mul_lid
hoelzl@37489
  1033
        matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
  1034
      using g(2) unfolding o_def fun_eq_iff id_def
hoelzl@37489
  1035
      .
hoelzl@37489
  1036
    hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
hoelzl@37489
  1037
  }
hoelzl@37489
  1038
  ultimately show ?thesis unfolding surj_def by blast
hoelzl@37489
  1039
qed
hoelzl@37489
  1040
hoelzl@37489
  1041
lemma matrix_left_invertible_independent_columns:
hoelzl@37489
  1042
  fixes A :: "real^'n^'m"
wenzelm@49644
  1043
  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
nipkow@64267
  1044
      (\<forall>c. sum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
wenzelm@49644
  1045
    (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
  1046
proof -
hoelzl@37489
  1047
  let ?U = "UNIV :: 'n set"
wenzelm@49644
  1048
  { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
wenzelm@49644
  1049
    { fix c i
nipkow@64267
  1050
      assume c: "sum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
hoelzl@37489
  1051
      let ?x = "\<chi> i. c i"
hoelzl@37489
  1052
      have th0:"A *v ?x = 0"
hoelzl@37489
  1053
        using c
lp15@67673
  1054
        unfolding matrix_mult_sum vec_eq_iff
hoelzl@37489
  1055
        by auto
hoelzl@37489
  1056
      from k[rule_format, OF th0] i
huffman@44136
  1057
      have "c i = 0" by (vector vec_eq_iff)}
wenzelm@49644
  1058
    hence ?rhs by blast }
hoelzl@37489
  1059
  moreover
wenzelm@49644
  1060
  { assume H: ?rhs
wenzelm@49644
  1061
    { fix x assume x: "A *v x = 0"
hoelzl@37489
  1062
      let ?c = "\<lambda>i. ((x$i ):: real)"
lp15@67673
  1063
      from H[rule_format, of ?c, unfolded matrix_mult_sum[symmetric], OF x]
wenzelm@49644
  1064
      have "x = 0" by vector }
wenzelm@49644
  1065
  }
hoelzl@37489
  1066
  ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
hoelzl@37489
  1067
qed
hoelzl@37489
  1068
hoelzl@37489
  1069
lemma matrix_right_invertible_independent_rows:
hoelzl@37489
  1070
  fixes A :: "real^'n^'m"
wenzelm@49644
  1071
  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
nipkow@64267
  1072
    (\<forall>c. sum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
hoelzl@37489
  1073
  unfolding left_invertible_transpose[symmetric]
hoelzl@37489
  1074
    matrix_left_invertible_independent_columns
hoelzl@37489
  1075
  by (simp add: column_transpose)
hoelzl@37489
  1076
hoelzl@37489
  1077
lemma matrix_right_invertible_span_columns:
wenzelm@49644
  1078
  "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
wenzelm@49644
  1079
    span (columns A) = UNIV" (is "?lhs = ?rhs")
wenzelm@49644
  1080
proof -
hoelzl@37489
  1081
  let ?U = "UNIV :: 'm set"
hoelzl@37489
  1082
  have fU: "finite ?U" by simp
nipkow@64267
  1083
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y)"
lp15@67673
  1084
    unfolding matrix_right_invertible_surjective matrix_mult_sum surj_def
lp15@68041
  1085
    by (simp add: eq_commute)
hoelzl@37489
  1086
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
wenzelm@49644
  1087
  { assume h: ?lhs
wenzelm@49644
  1088
    { fix x:: "real ^'n"
wenzelm@49644
  1089
      from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
nipkow@64267
  1090
        where y: "sum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
wenzelm@49644
  1091
      have "x \<in> span (columns A)"
lp15@68041
  1092
        unfolding y[symmetric] scalar_mult_eq_scaleR
lp15@68041
  1093
      proof (rule span_sum [OF span_mul])
lp15@68041
  1094
        show "column i A \<in> span (columns A)" for i
lp15@68041
  1095
          using columns_def span_inc by auto
lp15@68041
  1096
      qed
wenzelm@49644
  1097
    }
wenzelm@49644
  1098
    then have ?rhs unfolding rhseq by blast }
hoelzl@37489
  1099
  moreover
wenzelm@49644
  1100
  { assume h:?rhs
nipkow@64267
  1101
    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y"
wenzelm@49644
  1102
    { fix y
wenzelm@49644
  1103
      have "?P y"
hoelzl@50526
  1104
      proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
nipkow@64267
  1105
        show "\<exists>x::real ^ 'm. sum (\<lambda>i. (x$i) *s column i A) ?U = 0"
hoelzl@37489
  1106
          by (rule exI[where x=0], simp)
hoelzl@37489
  1107
      next
wenzelm@49644
  1108
        fix c y1 y2
wenzelm@49644
  1109
        assume y1: "y1 \<in> columns A" and y2: "?P y2"
hoelzl@37489
  1110
        from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
hoelzl@37489
  1111
          unfolding columns_def by blast
hoelzl@37489
  1112
        from y2 obtain x:: "real ^'m" where
nipkow@64267
  1113
          x: "sum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
hoelzl@37489
  1114
        let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
hoelzl@37489
  1115
        show "?P (c*s y1 + y2)"
webertj@49962
  1116
        proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)
wenzelm@49644
  1117
          fix j
wenzelm@49644
  1118
          have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
  1119
              else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
wenzelm@49644
  1120
            using i(1) by (simp add: field_simps)
nipkow@64267
  1121
          have "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
nipkow@64267
  1122
              else (x$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"
lp15@68041
  1123
            by (rule sum.cong[OF refl]) (use th in blast)
nipkow@64267
  1124
          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)$j))) ?U"
nipkow@64267
  1125
            by (simp add: sum.distrib)
nipkow@64267
  1126
          also have "\<dots> = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
nipkow@64267
  1127
            unfolding sum.delta[OF fU]
wenzelm@49644
  1128
            using i(1) by simp
nipkow@64267
  1129
          finally show "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
nipkow@64267
  1130
            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
  1131
        qed
wenzelm@49644
  1132
      next
wenzelm@49644
  1133
        show "y \<in> span (columns A)"
wenzelm@49644
  1134
          unfolding h by blast
wenzelm@49644
  1135
      qed
wenzelm@49644
  1136
    }
wenzelm@49644
  1137
    then have ?lhs unfolding lhseq ..
wenzelm@49644
  1138
  }
hoelzl@37489
  1139
  ultimately show ?thesis by blast
hoelzl@37489
  1140
qed
hoelzl@37489
  1141
hoelzl@37489
  1142
lemma matrix_left_invertible_span_rows:
hoelzl@37489
  1143
  "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
hoelzl@37489
  1144
  unfolding right_invertible_transpose[symmetric]
hoelzl@37489
  1145
  unfolding columns_transpose[symmetric]
hoelzl@37489
  1146
  unfolding matrix_right_invertible_span_columns
wenzelm@49644
  1147
  ..
hoelzl@37489
  1148
wenzelm@60420
  1149
text \<open>The same result in terms of square matrices.\<close>
hoelzl@37489
  1150
hoelzl@37489
  1151
lemma matrix_left_right_inverse:
hoelzl@37489
  1152
  fixes A A' :: "real ^'n^'n"
hoelzl@37489
  1153
  shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
wenzelm@49644
  1154
proof -
wenzelm@49644
  1155
  { fix A A' :: "real ^'n^'n"
wenzelm@49644
  1156
    assume AA': "A ** A' = mat 1"
nipkow@67399
  1157
    have sA: "surj (( *v) A)"
lp15@68038
  1158
      using AA' matrix_right_invertible_surjective by auto
hoelzl@37489
  1159
    from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
hoelzl@37489
  1160
    obtain f' :: "real ^'n \<Rightarrow> real ^'n"
hoelzl@37489
  1161
      where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
hoelzl@37489
  1162
    have th: "matrix f' ** A = mat 1"
wenzelm@49644
  1163
      by (simp add: matrix_eq matrix_works[OF f'(1)]
lp15@68041
  1164
          matrix_vector_mul_assoc[symmetric] f'(2)[rule_format])
hoelzl@37489
  1165
    hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
wenzelm@49644
  1166
    hence "matrix f' = A'"
lp15@68041
  1167
      by (simp add: matrix_mul_assoc[symmetric] AA')
hoelzl@37489
  1168
    hence "matrix f' ** A = A' ** A" by simp
wenzelm@49644
  1169
    hence "A' ** A = mat 1" by (simp add: th)
wenzelm@49644
  1170
  }
hoelzl@37489
  1171
  then show ?thesis by blast
hoelzl@37489
  1172
qed
hoelzl@37489
  1173
lp15@68038
  1174
lemma invertible_mult:
lp15@68045
  1175
  assumes inv_A: "invertible A"
lp15@68045
  1176
  and inv_B: "invertible B"
lp15@68045
  1177
  shows "invertible (A**B)"
lp15@68045
  1178
proof -
lp15@68045
  1179
  obtain A' where AA': "A ** A' = mat 1" and A'A: "A' ** A = mat 1" 
lp15@68045
  1180
    using inv_A unfolding invertible_def by blast
lp15@68045
  1181
  obtain B' where BB': "B ** B' = mat 1" and B'B: "B' ** B = mat 1" 
lp15@68045
  1182
    using inv_B unfolding invertible_def by blast
lp15@68045
  1183
  show ?thesis
lp15@68045
  1184
  proof (unfold invertible_def, rule exI[of _ "B'**A'"], rule conjI)
lp15@68045
  1185
    have "A ** B ** (B' ** A') = A ** (B ** (B' ** A'))" 
lp15@68045
  1186
      using matrix_mul_assoc[of A B "(B' ** A')", symmetric] .
lp15@68045
  1187
    also have "... = A ** (B ** B' ** A')" unfolding matrix_mul_assoc[of B "B'" "A'"] ..
lp15@68045
  1188
    also have "... = A ** (mat 1 ** A')" unfolding BB' ..
lp15@68045
  1189
    also have "... = A ** A'" unfolding matrix_mul_lid ..
lp15@68045
  1190
    also have "... = mat 1" unfolding AA' ..
lp15@68045
  1191
    finally show "A ** B ** (B' ** A') = mat (1::'a)" .    
lp15@68045
  1192
    have "B' ** A' ** (A ** B) = B' ** (A' ** (A ** B))" using matrix_mul_assoc[of B' A' "(A ** B)", symmetric] .
lp15@68045
  1193
    also have "... =  B' ** (A' ** A ** B)" unfolding matrix_mul_assoc[of A' A B] ..
lp15@68045
  1194
    also have "... =  B' ** (mat 1 ** B)" unfolding A'A ..
lp15@68045
  1195
    also have "... = B' ** B"  unfolding matrix_mul_lid ..
lp15@68045
  1196
    also have "... = mat 1" unfolding B'B ..
lp15@68045
  1197
    finally show "B' ** A' ** (A ** B) = mat 1" .
lp15@68045
  1198
  qed
lp15@68045
  1199
qed
lp15@68038
  1200
lp15@68038
  1201
lemma transpose_invertible:
lp15@68038
  1202
  fixes A :: "real^'n^'n"
lp15@68038
  1203
  assumes "invertible A"
lp15@68038
  1204
  shows "invertible (transpose A)"
lp15@68038
  1205
  by (meson assms invertible_def matrix_left_right_inverse right_invertible_transpose)
lp15@68038
  1206
lp15@68041
  1207
lemma vector_matrix_mul_assoc:
lp15@68041
  1208
  fixes v :: "('a::comm_semiring_1)^'n"
lp15@68041
  1209
  shows "(v v* M) v* N = v v* (M ** N)"
lp15@68041
  1210
proof -
lp15@68041
  1211
  from matrix_vector_mul_assoc
lp15@68041
  1212
  have "transpose N *v (transpose M *v v) = (transpose N ** transpose M) *v v" by fast
lp15@68041
  1213
  thus "(v v* M) v* N = v v* (M ** N)"
lp15@68041
  1214
    by (simp add: matrix_transpose_mul [symmetric])
lp15@68041
  1215
qed
lp15@68041
  1216
lp15@68043
  1217
lemma matrix_scaleR_vector_ac:
lp15@68041
  1218
  fixes A :: "real^('m::finite)^'n"
lp15@68041
  1219
  shows "A *v (k *\<^sub>R v) = k *\<^sub>R A *v v"
lp15@68043
  1220
  by (metis matrix_vector_mult_scaleR transpose_scalar vector_scaleR_matrix_ac vector_transpose_matrix)
lp15@68041
  1221
lp15@68043
  1222
lemma scaleR_matrix_vector_assoc:
lp15@68041
  1223
  fixes A :: "real^('m::finite)^'n"
lp15@68041
  1224
  shows "k *\<^sub>R (A *v v) = k *\<^sub>R A *v v"
lp15@68043
  1225
  by (metis matrix_scaleR_vector_ac matrix_vector_mult_scaleR)
lp15@68041
  1226
wenzelm@60420
  1227
text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
hoelzl@37489
  1228
hoelzl@37489
  1229
definition "rowvector v = (\<chi> i j. (v$j))"
hoelzl@37489
  1230
hoelzl@37489
  1231
definition "columnvector v = (\<chi> i j. (v$i))"
hoelzl@37489
  1232
wenzelm@49644
  1233
lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
huffman@44136
  1234
  by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
hoelzl@37489
  1235
hoelzl@37489
  1236
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
huffman@44136
  1237
  by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
hoelzl@37489
  1238
wenzelm@49644
  1239
lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
hoelzl@37489
  1240
  by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
hoelzl@37489
  1241
wenzelm@49644
  1242
lemma dot_matrix_product:
wenzelm@49644
  1243
  "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
huffman@44136
  1244
  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
hoelzl@37489
  1245
hoelzl@37489
  1246
lemma dot_matrix_vector_mul:
hoelzl@37489
  1247
  fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
hoelzl@37489
  1248
  shows "(A *v x) \<bullet> (B *v y) =
hoelzl@37489
  1249
      (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
wenzelm@49644
  1250
  unfolding dot_matrix_product transpose_columnvector[symmetric]
wenzelm@49644
  1251
    dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
hoelzl@37489
  1252
wenzelm@61945
  1253
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}"
hoelzl@50526
  1254
  by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
hoelzl@37489
  1255
wenzelm@49644
  1256
lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
hoelzl@50526
  1257
  using Basis_le_infnorm[of "axis i 1" x]
hoelzl@50526
  1258
  by (simp add: Basis_vec_def axis_eq_axis inner_axis)
hoelzl@37489
  1259
hoelzl@63334
  1260
lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
huffman@44647
  1261
  unfolding continuous_def by (rule tendsto_vec_nth)
huffman@44213
  1262
hoelzl@63334
  1263
lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
huffman@44647
  1264
  unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
huffman@44213
  1265
hoelzl@63334
  1266
lemma continuous_on_vec_lambda[continuous_intros]:
hoelzl@63334
  1267
  "(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)"
hoelzl@63334
  1268
  unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)
hoelzl@63334
  1269
hoelzl@37489
  1270
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
hoelzl@63332
  1271
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
huffman@44213
  1272
hoelzl@37489
  1273
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
wenzelm@49644
  1274
  unfolding bounded_def
wenzelm@49644
  1275
  apply clarify
wenzelm@49644
  1276
  apply (rule_tac x="x $ i" in exI)
wenzelm@49644
  1277
  apply (rule_tac x="e" in exI)
wenzelm@49644
  1278
  apply clarify
wenzelm@49644
  1279
  apply (rule order_trans [OF dist_vec_nth_le], simp)
wenzelm@49644
  1280
  done
hoelzl@37489
  1281
hoelzl@37489
  1282
lemma compact_lemma_cart:
hoelzl@37489
  1283
  fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
hoelzl@50998
  1284
  assumes f: "bounded (range f)"
eberlm@66447
  1285
  shows "\<exists>l r. strict_mono r \<and>
hoelzl@37489
  1286
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
immler@62127
  1287
    (is "?th d")
immler@62127
  1288
proof -
immler@62127
  1289
  have "\<forall>d' \<subseteq> d. ?th d'"
immler@62127
  1290
    by (rule compact_lemma_general[where unproj=vec_lambda])
immler@62127
  1291
      (auto intro!: f bounded_component_cart simp: vec_lambda_eta)
immler@62127
  1292
  then show "?th d" by simp
hoelzl@37489
  1293
qed
hoelzl@37489
  1294
huffman@44136
  1295
instance vec :: (heine_borel, finite) heine_borel
hoelzl@37489
  1296
proof
hoelzl@50998
  1297
  fix f :: "nat \<Rightarrow> 'a ^ 'b"
hoelzl@50998
  1298
  assume f: "bounded (range f)"
eberlm@66447
  1299
  then obtain l r where r: "strict_mono r"
wenzelm@49644
  1300
      and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@50998
  1301
    using compact_lemma_cart [OF f] by blast
hoelzl@37489
  1302
  let ?d = "UNIV::'b set"
hoelzl@37489
  1303
  { fix e::real assume "e>0"
hoelzl@37489
  1304
    hence "0 < e / (real_of_nat (card ?d))"
wenzelm@49644
  1305
      using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
hoelzl@37489
  1306
    with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
hoelzl@37489
  1307
      by simp
hoelzl@37489
  1308
    moreover
wenzelm@49644
  1309
    { fix n
wenzelm@49644
  1310
      assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
hoelzl@37489
  1311
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
nipkow@67155
  1312
        unfolding dist_vec_def using zero_le_dist by (rule L2_set_le_sum)
hoelzl@37489
  1313
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
nipkow@64267
  1314
        by (rule sum_strict_mono) (simp_all add: n)
hoelzl@37489
  1315
      finally have "dist (f (r n)) l < e" by simp
hoelzl@37489
  1316
    }
hoelzl@37489
  1317
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
lp15@61810
  1318
      by (rule eventually_mono)
hoelzl@37489
  1319
  }
wenzelm@61973
  1320
  hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp
eberlm@66447
  1321
  with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
hoelzl@37489
  1322
qed
hoelzl@37489
  1323
wenzelm@49644
  1324
lemma interval_cart:
immler@54775
  1325
  fixes a :: "real^'n"
immler@54775
  1326
  shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
immler@56188
  1327
    and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
immler@56188
  1328
  by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
hoelzl@37489
  1329
lp15@67673
  1330
lemma mem_box_cart:
immler@54775
  1331
  fixes a :: "real^'n"
immler@54775
  1332
  shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
immler@56188
  1333
    and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
wenzelm@49644
  1334
  using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
hoelzl@37489
  1335
wenzelm@49644
  1336
lemma interval_eq_empty_cart:
wenzelm@49644
  1337
  fixes a :: "real^'n"
immler@54775
  1338
  shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
immler@56188
  1339
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
wenzelm@49644
  1340
proof -
immler@54775
  1341
  { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>box a b"
lp15@67673
  1342
    hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_box_cart by auto
hoelzl@37489
  1343
    hence "a$i < b$i" by auto
wenzelm@49644
  1344
    hence False using as by auto }
hoelzl@37489
  1345
  moreover
hoelzl@37489
  1346
  { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
hoelzl@37489
  1347
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
  1348
    { fix i
hoelzl@37489
  1349
      have "a$i < b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
  1350
      hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
hoelzl@37489
  1351
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
  1352
        by auto }
lp15@67673
  1353
    hence "box a b \<noteq> {}" using mem_box_cart(1)[of "?x" a b] by auto }
hoelzl@37489
  1354
  ultimately show ?th1 by blast
hoelzl@37489
  1355
immler@56188
  1356
  { fix i x assume as:"b$i < a$i" and x:"x\<in>cbox a b"
lp15@67673
  1357
    hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_box_cart by auto
hoelzl@37489
  1358
    hence "a$i \<le> b$i" by auto
wenzelm@49644
  1359
    hence False using as by auto }
hoelzl@37489
  1360
  moreover
hoelzl@37489
  1361
  { assume as:"\<forall>i. \<not> (b$i < a$i)"
hoelzl@37489
  1362
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
  1363
    { fix i
hoelzl@37489
  1364
      have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
  1365
      hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
hoelzl@37489
  1366
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
  1367
        by auto }
lp15@67673
  1368
    hence "cbox a b \<noteq> {}" using mem_box_cart(2)[of "?x" a b] by auto  }
hoelzl@37489
  1369
  ultimately show ?th2 by blast
hoelzl@37489
  1370
qed
hoelzl@37489
  1371
wenzelm@49644
  1372
lemma interval_ne_empty_cart:
wenzelm@49644
  1373
  fixes a :: "real^'n"
immler@56188
  1374
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
immler@54775
  1375
    and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
hoelzl@37489
  1376
  unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
hoelzl@37489
  1377
    (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
  1378
wenzelm@49644
  1379
lemma subset_interval_imp_cart:
wenzelm@49644
  1380
  fixes a :: "real^'n"
immler@56188
  1381
  shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
immler@56188
  1382
    and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
immler@56188
  1383
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
immler@54775
  1384
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b"
lp15@67673
  1385
  unfolding subset_eq[unfolded Ball_def] unfolding mem_box_cart
hoelzl@37489
  1386
  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
  1387
wenzelm@49644
  1388
lemma interval_sing:
wenzelm@49644
  1389
  fixes a :: "'a::linorder^'n"
wenzelm@49644
  1390
  shows "{a .. a} = {a} \<and> {a<..<a} = {}"
wenzelm@49644
  1391
  apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
wenzelm@49644
  1392
  done
hoelzl@37489
  1393
wenzelm@49644
  1394
lemma subset_interval_cart:
wenzelm@49644
  1395
  fixes a :: "real^'n"
immler@56188
  1396
  shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
immler@56188
  1397
    and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
immler@56188
  1398
    and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
immler@54775
  1399
    and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
immler@56188
  1400
  using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
  1401
wenzelm@49644
  1402
lemma disjoint_interval_cart:
wenzelm@49644
  1403
  fixes a::"real^'n"
immler@56188
  1404
  shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
immler@56188
  1405
    and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
immler@56188
  1406
    and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
immler@54775
  1407
    and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
hoelzl@50526
  1408
  using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
  1409
lp15@67719
  1410
lemma Int_interval_cart:
immler@54775
  1411
  fixes a :: "real^'n"
immler@56188
  1412
  shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
lp15@63945
  1413
  unfolding Int_interval
immler@56188
  1414
  by (auto simp: mem_box less_eq_vec_def)
immler@56188
  1415
    (auto simp: Basis_vec_def inner_axis)
hoelzl@37489
  1416
wenzelm@49644
  1417
lemma closed_interval_left_cart:
wenzelm@49644
  1418
  fixes b :: "real^'n"
hoelzl@37489
  1419
  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
hoelzl@63332
  1420
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1421
wenzelm@49644
  1422
lemma closed_interval_right_cart:
wenzelm@49644
  1423
  fixes a::"real^'n"
hoelzl@37489
  1424
  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
hoelzl@63332
  1425
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1426
wenzelm@49644
  1427
lemma is_interval_cart:
wenzelm@49644
  1428
  "is_interval (s::(real^'n) set) \<longleftrightarrow>
wenzelm@49644
  1429
    (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
hoelzl@50526
  1430
  by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
hoelzl@37489
  1431
wenzelm@49644
  1432
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
hoelzl@63332
  1433
  by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1434
wenzelm@49644
  1435
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
hoelzl@63332
  1436
  by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1437
wenzelm@49644
  1438
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
hoelzl@63332
  1439
  by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
wenzelm@49644
  1440
wenzelm@49644
  1441
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
hoelzl@63332
  1442
  by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1443
wenzelm@49644
  1444
lemma Lim_component_le_cart:
wenzelm@49644
  1445
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
  1446
  assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"
hoelzl@37489
  1447
  shows "l$i \<le> b"
hoelzl@50526
  1448
  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
hoelzl@37489
  1449
wenzelm@49644
  1450
lemma Lim_component_ge_cart:
wenzelm@49644
  1451
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
  1452
  assumes "(f \<longlongrightarrow> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
hoelzl@37489
  1453
  shows "b \<le> l$i"
hoelzl@50526
  1454
  by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
hoelzl@37489
  1455
wenzelm@49644
  1456
lemma Lim_component_eq_cart:
wenzelm@49644
  1457
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
  1458
  assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
hoelzl@37489
  1459
  shows "l$i = b"
wenzelm@49644
  1460
  using ev[unfolded order_eq_iff eventually_conj_iff] and
wenzelm@49644
  1461
    Lim_component_ge_cart[OF net, of b i] and
hoelzl@37489
  1462
    Lim_component_le_cart[OF net, of i b] by auto
hoelzl@37489
  1463
wenzelm@49644
  1464
lemma connected_ivt_component_cart:
wenzelm@49644
  1465
  fixes x :: "real^'n"
wenzelm@49644
  1466
  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$k = a)"
hoelzl@50526
  1467
  using connected_ivt_hyperplane[of s x y "axis k 1" a]
hoelzl@50526
  1468
  by (auto simp add: inner_axis inner_commute)
hoelzl@37489
  1469
wenzelm@49644
  1470
lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
hoelzl@37489
  1471
  unfolding subspace_def by auto
hoelzl@37489
  1472
hoelzl@37489
  1473
lemma closed_substandard_cart:
huffman@44213
  1474
  "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
  1475
proof -
huffman@44213
  1476
  { fix i::'n
huffman@44213
  1477
    have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
hoelzl@63332
  1478
      by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) }
huffman@44213
  1479
  thus ?thesis
huffman@44213
  1480
    unfolding Collect_all_eq by (simp add: closed_INT)
hoelzl@37489
  1481
qed
hoelzl@37489
  1482
wenzelm@49644
  1483
lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
wenzelm@49644
  1484
  (is "dim ?A = _")
wenzelm@49644
  1485
proof -
hoelzl@50526
  1486
  let ?a = "\<lambda>x. axis x 1 :: real^'n"
hoelzl@50526
  1487
  have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
hoelzl@50526
  1488
    by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
hoelzl@50526
  1489
  have "?a ` d \<subseteq> Basis"
hoelzl@50526
  1490
    by (auto simp: Basis_vec_def)
wenzelm@49644
  1491
  thus ?thesis
hoelzl@50526
  1492
    using dim_substandard[of "?a ` d"] card_image[of ?a d]
hoelzl@50526
  1493
    by (auto simp: axis_eq_axis inj_on_def *)
hoelzl@37489
  1494
qed
hoelzl@37489
  1495
lp15@67719
  1496
lemma dim_subset_UNIV_cart:
lp15@67719
  1497
  fixes S :: "(real^'n) set"
lp15@67719
  1498
  shows "dim S \<le> CARD('n)"
lp15@67719
  1499
  by (metis dim_subset_UNIV DIM_cart DIM_real mult.right_neutral)
lp15@67719
  1500
hoelzl@37489
  1501
lemma affinity_inverses:
hoelzl@37489
  1502
  assumes m0: "m \<noteq> (0::'a::field)"
wenzelm@61736
  1503
  shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
wenzelm@61736
  1504
  "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
hoelzl@37489
  1505
  using m0
haftmann@54230
  1506
  apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
haftmann@54230
  1507
  apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
wenzelm@49644
  1508
  done
hoelzl@37489
  1509
hoelzl@37489
  1510
lemma vector_affinity_eq:
hoelzl@37489
  1511
  assumes m0: "(m::'a::field) \<noteq> 0"
hoelzl@37489
  1512
  shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
hoelzl@37489
  1513
proof
hoelzl@37489
  1514
  assume h: "m *s x + c = y"
hoelzl@37489
  1515
  hence "m *s x = y - c" by (simp add: field_simps)
hoelzl@37489
  1516
  hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
hoelzl@37489
  1517
  then show "x = inverse m *s y + - (inverse m *s c)"
hoelzl@37489
  1518
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1519
next
hoelzl@37489
  1520
  assume h: "x = inverse m *s y + - (inverse m *s c)"
haftmann@54230
  1521
  show "m *s x + c = y" unfolding h
hoelzl@37489
  1522
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1523
qed
hoelzl@37489
  1524
hoelzl@37489
  1525
lemma vector_eq_affinity:
wenzelm@49644
  1526
    "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
hoelzl@37489
  1527
  using vector_affinity_eq[where m=m and x=x and y=y and c=c]
hoelzl@37489
  1528
  by metis
hoelzl@37489
  1529
hoelzl@50526
  1530
lemma vector_cart:
hoelzl@50526
  1531
  fixes f :: "real^'n \<Rightarrow> real"
hoelzl@50526
  1532
  shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
hoelzl@50526
  1533
  unfolding euclidean_eq_iff[where 'a="real^'n"]
hoelzl@50526
  1534
  by simp (simp add: Basis_vec_def inner_axis)
hoelzl@63332
  1535
hoelzl@50526
  1536
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
hoelzl@50526
  1537
  by (rule vector_cart)
wenzelm@49644
  1538
huffman@44360
  1539
subsection "Convex Euclidean Space"
hoelzl@37489
  1540
hoelzl@50526
  1541
lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
hoelzl@50526
  1542
  using const_vector_cart[of 1] by (simp add: one_vec_def)
hoelzl@37489
  1543
hoelzl@37489
  1544
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
hoelzl@37489
  1545
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
hoelzl@37489
  1546
hoelzl@50526
  1547
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
  1548
hoelzl@37489
  1549
lemma convex_box_cart:
hoelzl@37489
  1550
  assumes "\<And>i. convex {x. P i x}"
hoelzl@37489
  1551
  shows "convex {x. \<forall>i. P i (x$i)}"
hoelzl@37489
  1552
  using assms unfolding convex_def by auto
hoelzl@37489
  1553
hoelzl@37489
  1554
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
hoelzl@63334
  1555
  by (rule convex_box_cart) (simp add: atLeast_def[symmetric])
hoelzl@37489
  1556
hoelzl@37489
  1557
lemma unit_interval_convex_hull_cart:
immler@56188
  1558
  "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}"
immler@56188
  1559
  unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
hoelzl@50526
  1560
  by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
hoelzl@37489
  1561
hoelzl@37489
  1562
lemma cube_convex_hull_cart:
wenzelm@49644
  1563
  assumes "0 < d"
wenzelm@49644
  1564
  obtains s::"(real^'n) set"
immler@56188
  1565
    where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
wenzelm@49644
  1566
proof -
wenzelm@55522
  1567
  from assms obtain s where "finite s"
nipkow@67399
  1568
    and "cbox (x - sum (( *\<^sub>R) d) Basis) (x + sum (( *\<^sub>R) d) Basis) = convex hull s"
wenzelm@55522
  1569
    by (rule cube_convex_hull)
wenzelm@55522
  1570
  with that[of s] show thesis
wenzelm@55522
  1571
    by (simp add: const_vector_cart)
hoelzl@37489
  1572
qed
hoelzl@37489
  1573
hoelzl@37489
  1574
hoelzl@37489
  1575
subsection "Derivative"
hoelzl@37489
  1576
hoelzl@37489
  1577
definition "jacobian f net = matrix(frechet_derivative f net)"
hoelzl@37489
  1578
wenzelm@49644
  1579
lemma jacobian_works:
wenzelm@49644
  1580
  "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
lp15@67986
  1581
    (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net" (is "?lhs = ?rhs")
lp15@67986
  1582
proof
lp15@67986
  1583
  assume ?lhs then show ?rhs
lp15@67986
  1584
    by (simp add: frechet_derivative_works has_derivative_linear jacobian_def)
lp15@67986
  1585
next
lp15@67986
  1586
  assume ?rhs then show ?lhs
lp15@67986
  1587
    by (rule differentiableI)
lp15@67986
  1588
qed
hoelzl@37489
  1589
hoelzl@37489
  1590
wenzelm@60420
  1591
subsection \<open>Component of the differential must be zero if it exists at a local
nipkow@67968
  1592
  maximum or minimum for that corresponding component\<close>
hoelzl@37489
  1593
hoelzl@50526
  1594
lemma differential_zero_maxmin_cart:
wenzelm@49644
  1595
  fixes f::"real^'a \<Rightarrow> real^'b"
wenzelm@49644
  1596
  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
  1597
    "f differentiable (at x)"
hoelzl@50526
  1598
  shows "jacobian f (at x) $ k = 0"
hoelzl@50526
  1599
  using differential_zero_maxmin_component[of "axis k 1" e x f] assms
hoelzl@50526
  1600
    vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
hoelzl@50526
  1601
  by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
wenzelm@49644
  1602
wenzelm@60420
  1603
subsection \<open>Lemmas for working on @{typ "real^1"}\<close>
hoelzl@37489
  1604
hoelzl@37489
  1605
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
wenzelm@49644
  1606
  by (metis (full_types) num1_eq_iff)
hoelzl@37489
  1607
hoelzl@37489
  1608
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
wenzelm@49644
  1609
  by auto (metis (full_types) num1_eq_iff)
hoelzl@37489
  1610
hoelzl@37489
  1611
lemma exhaust_2:
wenzelm@49644
  1612
  fixes x :: 2
wenzelm@49644
  1613
  shows "x = 1 \<or> x = 2"
hoelzl@37489
  1614
proof (induct x)
hoelzl@37489
  1615
  case (of_int z)
lp15@67979
  1616
  then have "0 \<le> z" and "z < 2" by simp_all
hoelzl@37489
  1617
  then have "z = 0 | z = 1" by arith
hoelzl@37489
  1618
  then show ?case by auto
hoelzl@37489
  1619
qed
hoelzl@37489
  1620
hoelzl@37489
  1621
lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
hoelzl@37489
  1622
  by (metis exhaust_2)
hoelzl@37489
  1623
hoelzl@37489
  1624
lemma exhaust_3:
wenzelm@49644
  1625
  fixes x :: 3
wenzelm@49644
  1626
  shows "x = 1 \<or> x = 2 \<or> x = 3"
hoelzl@37489
  1627
proof (induct x)
hoelzl@37489
  1628
  case (of_int z)
lp15@67979
  1629
  then have "0 \<le> z" and "z < 3" by simp_all
hoelzl@37489
  1630
  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
hoelzl@37489
  1631
  then show ?case by auto
hoelzl@37489
  1632
qed
hoelzl@37489
  1633
hoelzl@37489
  1634
lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
hoelzl@37489
  1635
  by (metis exhaust_3)
hoelzl@37489
  1636
hoelzl@37489
  1637
lemma UNIV_1 [simp]: "UNIV = {1::1}"
hoelzl@37489
  1638
  by (auto simp add: num1_eq_iff)
hoelzl@37489
  1639
hoelzl@37489
  1640
lemma UNIV_2: "UNIV = {1::2, 2::2}"
hoelzl@37489
  1641
  using exhaust_2 by auto
hoelzl@37489
  1642
hoelzl@37489
  1643
lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
hoelzl@37489
  1644
  using exhaust_3 by auto
hoelzl@37489
  1645
nipkow@64267
  1646
lemma sum_1: "sum f (UNIV::1 set) = f 1"
hoelzl@37489
  1647
  unfolding UNIV_1 by simp
hoelzl@37489
  1648
nipkow@64267
  1649
lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2"
hoelzl@37489
  1650
  unfolding UNIV_2 by simp
hoelzl@37489
  1651
nipkow@64267
  1652
lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3"
haftmann@57514
  1653
  unfolding UNIV_3 by (simp add: ac_simps)
hoelzl@37489
  1654
lp15@67979
  1655
lemma num1_eqI:
lp15@67979
  1656
  fixes a::num1 shows "a = b"
lp15@67979
  1657
  by (metis (full_types) UNIV_1 UNIV_I empty_iff insert_iff)
lp15@67979
  1658
lp15@67979
  1659
lemma num1_eq1 [simp]:
lp15@67979
  1660
  fixes a::num1 shows "a = 1"
lp15@67979
  1661
  by (rule num1_eqI)
lp15@67979
  1662
wenzelm@49644
  1663
instantiation num1 :: cart_one
wenzelm@49644
  1664
begin
wenzelm@49644
  1665
wenzelm@49644
  1666
instance
wenzelm@49644
  1667
proof
hoelzl@37489
  1668
  show "CARD(1) = Suc 0" by auto
wenzelm@49644
  1669
qed
wenzelm@49644
  1670
wenzelm@49644
  1671
end
hoelzl@37489
  1672
lp15@67979
  1673
instantiation num1 :: linorder begin
lp15@67979
  1674
definition "a < b \<longleftrightarrow> Rep_num1 a < Rep_num1 b"
lp15@67979
  1675
definition "a \<le> b \<longleftrightarrow> Rep_num1 a \<le> Rep_num1 b"
lp15@67979
  1676
instance
lp15@67979
  1677
  by intro_classes (auto simp: less_eq_num1_def less_num1_def intro: num1_eqI)
lp15@67979
  1678
end
lp15@67979
  1679
lp15@67979
  1680
instance num1 :: wellorder
lp15@67979
  1681
  by intro_classes (auto simp: less_eq_num1_def less_num1_def)
lp15@67979
  1682
nipkow@67968
  1683
subsection\<open>The collapse of the general concepts to dimension one\<close>
hoelzl@37489
  1684
hoelzl@37489
  1685
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
huffman@44136
  1686
  by (simp add: vec_eq_iff)
hoelzl@37489
  1687
hoelzl@37489
  1688
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
hoelzl@37489
  1689
  apply auto
hoelzl@37489
  1690
  apply (erule_tac x= "x$1" in allE)
hoelzl@37489
  1691
  apply (simp only: vector_one[symmetric])
hoelzl@37489
  1692
  done
hoelzl@37489
  1693
hoelzl@37489
  1694
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
huffman@44136
  1695
  by (simp add: norm_vec_def)
hoelzl@37489
  1696
lp15@67979
  1697
lemma dist_vector_1:
lp15@67979
  1698
  fixes x :: "'a::real_normed_vector^1"
lp15@67979
  1699
  shows "dist x y = dist (x$1) (y$1)"
lp15@67979
  1700
  by (simp add: dist_norm norm_vector_1)
lp15@67979
  1701
wenzelm@61945
  1702
lemma norm_real: "norm(x::real ^ 1) = \<bar>x$1\<bar>"
hoelzl@37489
  1703
  by (simp add: norm_vector_1)
hoelzl@37489
  1704
wenzelm@61945
  1705
lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x$1) - (y$1)\<bar>"
hoelzl@37489
  1706
  by (auto simp add: norm_real dist_norm)
hoelzl@37489
  1707
lp15@67986
  1708
subsection\<open> Rank of a matrix\<close>
lp15@67986
  1709
lp15@67986
  1710
text\<open>Equivalence of row and column rank is taken from George Mackiw's paper, Mathematics Magazine 1995, p. 285.\<close>
lp15@67986
  1711
lp15@67986
  1712
lemma matrix_vector_mult_in_columnspace:
lp15@67986
  1713
  fixes A :: "real^'n^'m"
lp15@67986
  1714
  shows "(A *v x) \<in> span(columns A)"
lp15@67986
  1715
  apply (simp add: matrix_vector_column columns_def transpose_def column_def)
lp15@67986
  1716
  apply (intro span_sum span_mul)
lp15@67986
  1717
  apply (force intro: span_superset)
lp15@67986
  1718
  done
lp15@67986
  1719
lp15@67986
  1720
lemma orthogonal_nullspace_rowspace:
lp15@67986
  1721
  fixes A :: "real^'n^'m"
lp15@67986
  1722
  assumes 0: "A *v x = 0" and y: "y \<in> span(rows A)"
lp15@67986
  1723
  shows "orthogonal x y"
lp15@67986
  1724
proof (rule span_induct [OF y])
lp15@67986
  1725
  show "subspace {a. orthogonal x a}"
lp15@67986
  1726
    by (simp add: subspace_orthogonal_to_vector)
lp15@67986
  1727
next
lp15@67986
  1728
  fix v
lp15@67986
  1729
  assume "v \<in> rows A"
lp15@67986
  1730
  then obtain i where "v = row i A"
lp15@67986
  1731
    by (auto simp: rows_def)
lp15@67986
  1732
  with 0 show "orthogonal x v"
lp15@67986
  1733
    unfolding orthogonal_def inner_vec_def matrix_vector_mult_def row_def
lp15@67986
  1734
    by (simp add: mult.commute) (metis (no_types) vec_lambda_beta zero_index)
lp15@67986
  1735
qed
lp15@67986
  1736
lp15@67986
  1737
lemma nullspace_inter_rowspace:
lp15@67986
  1738
  fixes A :: "real^'n^'m"
lp15@67986
  1739
  shows "A *v x = 0 \<and> x \<in> span(rows A) \<longleftrightarrow> x = 0"
lp15@67986
  1740
  using orthogonal_nullspace_rowspace orthogonal_self by auto
lp15@67986
  1741
lp15@67986
  1742
lemma matrix_vector_mul_injective_on_rowspace:
lp15@67986
  1743
  fixes A :: "real^'n^'m"
lp15@67986
  1744
  shows "\<lbrakk>A *v x = A *v y; x \<in> span(rows A); y \<in> span(rows A)\<rbrakk> \<Longrightarrow> x = y"
lp15@67986
  1745
  using nullspace_inter_rowspace [of A "x-y"]
lp15@67986
  1746
  by (metis eq_iff_diff_eq_0 matrix_vector_mult_diff_distrib span_diff)
lp15@67986
  1747
lp15@67986
  1748
definition rank :: "real^'n^'m=>nat"
lp15@67986
  1749
  where "rank A \<equiv> dim(columns A)"
lp15@67986
  1750
lp15@67986
  1751
lemma dim_rows_le_dim_columns:
lp15@67986
  1752
  fixes A :: "real^'n^'m"
lp15@67986
  1753
  shows "dim(rows A) \<le> dim(columns A)"
lp15@67986
  1754
proof -
lp15@67986
  1755
  have "dim (span (rows A)) \<le> dim (span (columns A))"
lp15@67986
  1756
  proof -
lp15@67986
  1757
    obtain B where "independent B" "span(rows A) \<subseteq> span B"
lp15@67986
  1758
              and B: "B \<subseteq> span(rows A)""card B = dim (span(rows A))"
lp15@67986
  1759
      using basis_exists [of "span(rows A)"] by blast
lp15@67986
  1760
    with span_subspace have eq: "span B = span(rows A)"
lp15@67986
  1761
      by auto
lp15@67986
  1762
    then have inj: "inj_on (( *v) A) (span B)"
lp15@67986
  1763
      using inj_on_def matrix_vector_mul_injective_on_rowspace by blast
lp15@67986
  1764
    then have ind: "independent (( *v) A ` B)"
lp15@67986
  1765
      by (rule independent_inj_on_image [OF \<open>independent B\<close> matrix_vector_mul_linear])
lp15@67986
  1766
    then have "finite (( *v) A ` B) \<and> card (( *v) A ` B) \<le> dim (( *v) A ` B)"
lp15@67986
  1767
      by (rule independent_bound_general)
lp15@67986
  1768
    then show ?thesis
lp15@67986
  1769
      by (metis (no_types, lifting) B ind inj eq card_image image_subset_iff independent_card_le_dim inj_on_subset matrix_vector_mult_in_columnspace)
lp15@67986
  1770
  qed
lp15@67986
  1771
  then show ?thesis
lp15@67986
  1772
    by simp
lp15@67986
  1773
qed
lp15@67986
  1774
lp15@67986
  1775
lemma rank_row:
lp15@67986
  1776
  fixes A :: "real^'n^'m"
lp15@67986
  1777
  shows "rank A = dim(rows A)"
lp15@67986
  1778
  unfolding rank_def
lp15@67986
  1779
  by (metis dim_rows_le_dim_columns columns_transpose dual_order.antisym rows_transpose)
lp15@67986
  1780
lp15@67986
  1781
lemma rank_transpose:
lp15@67986
  1782
  fixes A :: "real^'n^'m"
lp15@67986
  1783
  shows "rank(transpose A) = rank A"
lp15@67986
  1784
  by (metis rank_def rank_row rows_transpose)
lp15@67986
  1785
lp15@67986
  1786
lemma matrix_vector_mult_basis:
lp15@67986
  1787
  fixes A :: "real^'n^'m"
lp15@67986
  1788
  shows "A *v (axis k 1) = column k A"
lp15@67986
  1789
  by (simp add: cart_eq_inner_axis column_def matrix_mult_dot)
lp15@67986
  1790
lp15@67986
  1791
lemma columns_image_basis:
lp15@67986
  1792
  fixes A :: "real^'n^'m"
lp15@67986
  1793
  shows "columns A = ( *v) A ` (range (\<lambda>i. axis i 1))"
lp15@67986
  1794
  by (force simp: columns_def matrix_vector_mult_basis [symmetric])
lp15@67986
  1795
lp15@67986
  1796
lemma rank_dim_range:
lp15@67986
  1797
  fixes A :: "real^'n^'m"
lp15@67986
  1798
  shows "rank A = dim(range (\<lambda>x. A *v x))"
lp15@67986
  1799
  unfolding rank_def
lp15@67986
  1800
proof (rule span_eq_dim)
lp15@67986
  1801
  show "span (columns A) = span (range (( *v) A))"
lp15@67986
  1802
    apply (auto simp: columns_image_basis span_linear_image matrix_vector_mul_linear)
lp15@67986
  1803
    by (metis columns_image_basis matrix_vector_mul_linear matrix_vector_mult_in_columnspace span_linear_image)
lp15@67986
  1804
qed
lp15@67986
  1805
lp15@67986
  1806
lemma rank_bound:
lp15@67986
  1807
  fixes A :: "real^'n^'m"
lp15@67986
  1808
  shows "rank A \<le> min CARD('m) (CARD('n))"
lp15@67986
  1809
  by (metis (mono_tags, hide_lams) min.bounded_iff DIM_cart DIM_real dim_subset_UNIV mult.right_neutral rank_def rank_transpose)
lp15@67986
  1810
lp15@67986
  1811
lemma full_rank_injective:
lp15@67986
  1812
  fixes A :: "real^'n^'m"
lp15@67986
  1813
  shows "rank A = CARD('n) \<longleftrightarrow> inj (( *v) A)"
lp15@67986
  1814
  by (simp add: matrix_left_invertible_injective [symmetric] matrix_left_invertible_span_rows rank_row dim_eq_full [symmetric])
lp15@67986
  1815
lp15@67986
  1816
lemma full_rank_surjective:
lp15@67986
  1817
  fixes A :: "real^'n^'m"
lp15@67986
  1818
  shows "rank A = CARD('m) \<longleftrightarrow> surj (( *v) A)"
lp15@67986
  1819
  by (simp add: matrix_right_invertible_surjective [symmetric] left_invertible_transpose [symmetric]
lp15@67986
  1820
                matrix_left_invertible_injective full_rank_injective [symmetric] rank_transpose)
lp15@67986
  1821
lp15@67986
  1822
lemma rank_I: "rank(mat 1::real^'n^'n) = CARD('n)"
lp15@67986
  1823
  by (simp add: full_rank_injective inj_on_def)
lp15@67986
  1824
lp15@67986
  1825
lemma less_rank_noninjective:
lp15@67986
  1826
  fixes A :: "real^'n^'m"
lp15@67986
  1827
  shows "rank A < CARD('n) \<longleftrightarrow> \<not> inj (( *v) A)"
lp15@67986
  1828
using less_le rank_bound by (auto simp: full_rank_injective [symmetric])
lp15@67986
  1829
lp15@67986
  1830
lemma matrix_nonfull_linear_equations_eq:
lp15@67986
  1831
  fixes A :: "real^'n^'m"
lp15@67986
  1832
  shows "(\<exists>x. (x \<noteq> 0) \<and> A *v x = 0) \<longleftrightarrow> ~(rank A = CARD('n))"
lp15@67986
  1833
  by (meson matrix_left_invertible_injective full_rank_injective matrix_left_invertible_ker)
lp15@67986
  1834
lp15@67986
  1835
lemma rank_eq_0: "rank A = 0 \<longleftrightarrow> A = 0" and rank_0 [simp]: "rank 0 = 0"
lp15@67986
  1836
  by (auto simp: rank_dim_range matrix_eq)
lp15@67986
  1837
lp15@67986
  1838
lp15@67986
  1839
lemma rank_mul_le_right:
lp15@67986
  1840
  fixes A :: "real^'n^'m" and B :: "real^'p^'n"
lp15@67986
  1841
  shows "rank(A ** B) \<le> rank B"
lp15@67986
  1842
proof -
lp15@67986
  1843
  have "rank(A ** B) \<le> dim (( *v) A ` range (( *v) B))"
lp15@67986
  1844
    by (auto simp: rank_dim_range image_comp o_def matrix_vector_mul_assoc)
lp15@67986
  1845
  also have "\<dots> \<le> rank B"
lp15@67986
  1846
    by (simp add: rank_dim_range matrix_vector_mul_linear dim_image_le)
lp15@67986
  1847
  finally show ?thesis .
lp15@67986
  1848
qed
lp15@67986
  1849
lp15@67986
  1850
lemma rank_mul_le_left:
lp15@67986
  1851
  fixes A :: "real^'n^'m" and B :: "real^'p^'n"
lp15@67986
  1852
  shows "rank(A ** B) \<le> rank A"
lp15@67986
  1853
  by (metis matrix_transpose_mul rank_mul_le_right rank_transpose)
lp15@67986
  1854
lp15@67981
  1855
subsection\<open>Routine results connecting the types @{typ "real^1"} and @{typ real}\<close>
lp15@67981
  1856
lp15@67981
  1857
lemma vector_one_nth [simp]:
lp15@67981
  1858
  fixes x :: "'a^1" shows "vec (x $ 1) = x"
lp15@67981
  1859
  by (metis vec_def vector_one)
lp15@67981
  1860
lp15@67981
  1861
lemma vec_cbox_1_eq [simp]:
lp15@67981
  1862
  shows "vec ` cbox u v = cbox (vec u) (vec v ::real^1)"
lp15@67981
  1863
  by (force simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box)
lp15@67981
  1864
lp15@67981
  1865
lemma vec_nth_cbox_1_eq [simp]:
lp15@67981
  1866
  fixes u v :: "'a::euclidean_space^1"
lp15@67981
  1867
  shows "(\<lambda>x. x $ 1) ` cbox u v = cbox (u$1) (v$1)"
lp15@67981
  1868
    by (auto simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box image_iff Bex_def inner_axis) (metis vec_component)
lp15@67981
  1869
lp15@67981
  1870
lemma vec_nth_1_iff_cbox [simp]:
lp15@67981
  1871
  fixes a b :: "'a::euclidean_space"
lp15@67981
  1872
  shows "(\<lambda>x::'a^1. x $ 1) ` S = cbox a b \<longleftrightarrow> S = cbox (vec a) (vec b)"
lp15@67981
  1873
    (is "?lhs = ?rhs")
lp15@67981
  1874
proof
lp15@67981
  1875
  assume L: ?lhs show ?rhs
lp15@67981
  1876
  proof (intro equalityI subsetI)
lp15@67981
  1877
    fix x 
lp15@67981
  1878
    assume "x \<in> S"
lp15@67981
  1879
    then have "x $ 1 \<in> (\<lambda>v. v $ (1::1)) ` cbox (vec a) (vec b)"
lp15@67981
  1880
      using L by auto
lp15@67981
  1881
    then show "x \<in> cbox (vec a) (vec b)"
lp15@67981
  1882
      by (metis (no_types, lifting) imageE vector_one_nth)
lp15@67981
  1883
  next
lp15@67981
  1884
    fix x :: "'a^1"
lp15@67981
  1885
    assume "x \<in> cbox (vec a) (vec b)"
lp15@67981
  1886
    then show "x \<in> S"
lp15@67981
  1887
      by (metis (no_types, lifting) L imageE imageI vec_component vec_nth_cbox_1_eq vector_one_nth)
lp15@67981
  1888
  qed
lp15@67981
  1889
qed simp
wenzelm@49644
  1890
lp15@67979
  1891
lemma tendsto_at_within_vector_1:
lp15@67979
  1892
  fixes S :: "'a :: metric_space set"
lp15@67979
  1893
  assumes "(f \<longlongrightarrow> fx) (at x within S)"
lp15@67979
  1894
  shows "((\<lambda>y::'a^1. \<chi> i. f (y $ 1)) \<longlongrightarrow> (vec fx::'a^1)) (at (vec x) within vec ` S)"
lp15@67979
  1895
proof (rule topological_tendstoI)
lp15@67979
  1896
  fix T :: "('a^1) set"
lp15@67979
  1897
  assume "open T" "vec fx \<in> T"
lp15@67979
  1898
  have "\<forall>\<^sub>F x in at x within S. f x \<in> (\<lambda>x. x $ 1) ` T"
lp15@67979
  1899
    using \<open>open T\<close> \<open>vec fx \<in> T\<close> assms open_image_vec_nth tendsto_def by fastforce
lp15@67979
  1900
  then show "\<forall>\<^sub>F x::'a^1 in at (vec x) within vec ` S. (\<chi> i. f (x $ 1)) \<in> T"
lp15@67979
  1901
    unfolding eventually_at dist_norm [symmetric]
lp15@67979
  1902
    by (rule ex_forward)
lp15@67979
  1903
       (use \<open>open T\<close> in 
lp15@67979
  1904
         \<open>fastforce simp: dist_norm dist_vec_def L2_set_def image_iff vector_one open_vec_def\<close>)
lp15@67979
  1905
qed
lp15@67979
  1906
lp15@67979
  1907
lemma has_derivative_vector_1:
lp15@67979
  1908
  assumes der_g: "(g has_derivative (\<lambda>x. x * g' a)) (at a within S)"
lp15@67979
  1909
  shows "((\<lambda>x. vec (g (x $ 1))) has_derivative ( *\<^sub>R) (g' a))
lp15@67979
  1910
         (at ((vec a)::real^1) within vec ` S)"
lp15@67979
  1911
    using der_g
lp15@67979
  1912
    apply (auto simp: Deriv.has_derivative_within bounded_linear_scaleR_right norm_vector_1)
lp15@67979
  1913
    apply (drule tendsto_at_within_vector_1, vector)
lp15@67979
  1914
    apply (auto simp: algebra_simps eventually_at tendsto_def)
lp15@67979
  1915
    done
lp15@67979
  1916
lp15@67979
  1917
nipkow@67968
  1918
subsection\<open>Explicit vector construction from lists\<close>
hoelzl@37489
  1919
hoelzl@43995
  1920
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
  1921
lp15@68054
  1922
lemma vector_1 [simp]: "(vector[x]) $1 = x"
hoelzl@37489
  1923
  unfolding vector_def by simp
hoelzl@37489
  1924
lp15@68054
  1925
lemma vector_2 [simp]: "(vector[x,y]) $1 = x" "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
hoelzl@37489
  1926
  unfolding vector_def by simp_all
hoelzl@37489
  1927
lp15@68054
  1928
lemma vector_3 [simp]:
hoelzl@37489
  1929
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
hoelzl@37489
  1930
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
hoelzl@37489
  1931
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
hoelzl@37489
  1932
  unfolding vector_def by simp_all
hoelzl@37489
  1933
hoelzl@37489
  1934
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
lp15@67719
  1935
  by (metis vector_1 vector_one)
hoelzl@37489
  1936
hoelzl@37489
  1937
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
hoelzl@37489
  1938
  apply auto
hoelzl@37489
  1939
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1940
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1941
  apply (subgoal_tac "vector [v$1, v$2] = v")
hoelzl@37489
  1942
  apply simp
hoelzl@37489
  1943
  apply (vector vector_def)
hoelzl@37489
  1944
  apply (simp add: forall_2)
hoelzl@37489
  1945
  done
hoelzl@37489
  1946
hoelzl@37489
  1947
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
hoelzl@37489
  1948
  apply auto
hoelzl@37489
  1949
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1950
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1951
  apply (erule_tac x="v$3" in allE)
hoelzl@37489
  1952
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
hoelzl@37489
  1953
  apply simp
hoelzl@37489
  1954
  apply (vector vector_def)
hoelzl@37489
  1955
  apply (simp add: forall_3)
hoelzl@37489
  1956
  done
hoelzl@37489
  1957
hoelzl@37489
  1958
lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
lp15@68062
  1959
  apply (rule bounded_linear_intro[where K=1])
hoelzl@37489
  1960
  using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
hoelzl@37489
  1961
hoelzl@37489
  1962
lemma interval_split_cart:
hoelzl@37489
  1963
  "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
immler@56188
  1964
  "cbox a b \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
wenzelm@49644
  1965
  apply (rule_tac[!] set_eqI)
lp15@67673
  1966
  unfolding Int_iff mem_box_cart mem_Collect_eq interval_cbox_cart
wenzelm@49644
  1967
  unfolding vec_lambda_beta
wenzelm@49644
  1968
  by auto
hoelzl@37489
  1969
immler@67685
  1970
lemmas cartesian_euclidean_space_uniform_limit_intros[uniform_limit_intros] =
immler@67685
  1971
  bounded_linear.uniform_limit[OF blinfun.bounded_linear_right]
immler@67685
  1972
  bounded_linear.uniform_limit[OF bounded_linear_vec_nth]
immler@67685
  1973
  bounded_linear.uniform_limit[OF bounded_linear_component_cart]
immler@67685
  1974
hoelzl@37489
  1975
end