src/HOL/Analysis/Cartesian_Euclidean_Space.thy
author nipkow
Thu Dec 07 15:48:50 2017 +0100 (4 months ago)
changeset 67155 9e5b05d54f9d
parent 66804 3f9bb52082c4
child 67399 eab6ce8368fa
permissions -rw-r--r--
canonical name
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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 subspace_special_hyperplane: "subspace {x. x $ k = 0}"
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  by (simp add: subspace_def)
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lemma delta_mult_idempotent:
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  "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)"
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  by simp
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(*move up?*)
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lemma sum_UNIV_sum:
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  fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
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  shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
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  apply (subst UNIV_Plus_UNIV [symmetric])
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  apply (subst sum.Plus)
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  apply simp_all
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  done
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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 "op * \<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_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 norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
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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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lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
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  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
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lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
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  by (metis vector_mul_lcancel)
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lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
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  by (metis vector_mul_rcancel)
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lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x"
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  apply (simp add: norm_vec_def)
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  apply (rule member_le_L2_set, simp_all)
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  done
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lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x$i\<bar> <= e"
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  by (metis component_le_norm_cart order_trans)
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lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
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  by (metis component_le_norm_cart le_less_trans)
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lemma norm_le_l1_cart: "norm x <= sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
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  by (simp add: norm_vec_def L2_set_le_sum)
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lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
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   306
  unfolding scaleR_vec_def vector_scalar_mult_def by simp
hoelzl@37489
   307
hoelzl@37489
   308
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
hoelzl@37489
   309
  unfolding dist_norm scalar_mult_eq_scaleR
hoelzl@37489
   310
  unfolding scaleR_right_diff_distrib[symmetric] by simp
hoelzl@37489
   311
nipkow@64267
   312
lemma sum_component [simp]:
hoelzl@37489
   313
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
nipkow@64267
   314
  shows "(sum f S)$i = sum (\<lambda>x. (f x)$i) S"
wenzelm@49644
   315
proof (cases "finite S")
wenzelm@49644
   316
  case True
wenzelm@49644
   317
  then show ?thesis by induct simp_all
wenzelm@49644
   318
next
wenzelm@49644
   319
  case False
wenzelm@49644
   320
  then show ?thesis by simp
wenzelm@49644
   321
qed
hoelzl@37489
   322
nipkow@64267
   323
lemma sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)$i ) S)"
huffman@44136
   324
  by (simp add: vec_eq_iff)
hoelzl@37489
   325
nipkow@64267
   326
lemma sum_cmul:
hoelzl@37489
   327
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
nipkow@64267
   328
  shows "sum (\<lambda>x. c *s f x) S = c *s sum f S"
nipkow@64267
   329
  by (simp add: vec_eq_iff sum_distrib_left)
hoelzl@37489
   330
nipkow@64267
   331
lemma sum_norm_allsubsets_bound_cart:
hoelzl@37489
   332
  fixes f:: "'a \<Rightarrow> real ^'n"
nipkow@64267
   333
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
nipkow@64267
   334
  shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
nipkow@64267
   335
  using sum_norm_allsubsets_bound[OF assms]
wenzelm@57865
   336
  by simp
hoelzl@37489
   337
lp15@62397
   338
subsection\<open>Closures and interiors of halfspaces\<close>
lp15@62397
   339
lp15@62397
   340
lemma interior_halfspace_le [simp]:
lp15@62397
   341
  assumes "a \<noteq> 0"
lp15@62397
   342
    shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
lp15@62397
   343
proof -
lp15@62397
   344
  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
   345
  proof -
lp15@62397
   346
    obtain e where "e>0" and e: "cball x e \<subseteq> S"
lp15@62397
   347
      using \<open>open S\<close> open_contains_cball x by blast
lp15@62397
   348
    then have "x + (e / norm a) *\<^sub>R a \<in> cball x e"
lp15@62397
   349
      by (simp add: dist_norm)
lp15@62397
   350
    then have "x + (e / norm a) *\<^sub>R a \<in> S"
lp15@62397
   351
      using e by blast
lp15@62397
   352
    then have "x + (e / norm a) *\<^sub>R a \<in> {x. a \<bullet> x \<le> b}"
lp15@62397
   353
      using S by blast
lp15@62397
   354
    moreover have "e * (a \<bullet> a) / norm a > 0"
lp15@62397
   355
      by (simp add: \<open>0 < e\<close> assms)
lp15@62397
   356
    ultimately show ?thesis
lp15@62397
   357
      by (simp add: algebra_simps)
lp15@62397
   358
  qed
lp15@62397
   359
  show ?thesis
lp15@62397
   360
    by (rule interior_unique) (auto simp: open_halfspace_lt *)
lp15@62397
   361
qed
lp15@62397
   362
lp15@62397
   363
lemma interior_halfspace_ge [simp]:
lp15@62397
   364
   "a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"
lp15@62397
   365
using interior_halfspace_le [of "-a" "-b"] by simp
lp15@62397
   366
lp15@62397
   367
lemma interior_halfspace_component_le [simp]:
lp15@62397
   368
     "interior {x. x$k \<le> a} = {x :: (real,'n::finite) vec. x$k < a}" (is "?LE")
lp15@62397
   369
  and interior_halfspace_component_ge [simp]:
lp15@62397
   370
     "interior {x. x$k \<ge> a} = {x :: (real,'n::finite) vec. x$k > a}" (is "?GE")
lp15@62397
   371
proof -
lp15@62397
   372
  have "axis k (1::real) \<noteq> 0"
lp15@62397
   373
    by (simp add: axis_def vec_eq_iff)
lp15@62397
   374
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
lp15@62397
   375
    by (simp add: cart_eq_inner_axis inner_commute)
lp15@62397
   376
  ultimately show ?LE ?GE
lp15@62397
   377
    using interior_halfspace_le [of "axis k (1::real)" a]
lp15@62397
   378
          interior_halfspace_ge [of "axis k (1::real)" a] by auto
lp15@62397
   379
qed
lp15@62397
   380
lp15@62397
   381
lemma closure_halfspace_lt [simp]:
lp15@62397
   382
  assumes "a \<noteq> 0"
lp15@62397
   383
    shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
lp15@62397
   384
proof -
lp15@62397
   385
  have [simp]: "-{x. a \<bullet> x < b} = {x. a \<bullet> x \<ge> b}"
lp15@62397
   386
    by (force simp:)
lp15@62397
   387
  then show ?thesis
lp15@62397
   388
    using interior_halfspace_ge [of a b] assms
lp15@62397
   389
    by (force simp: closure_interior)
lp15@62397
   390
qed
lp15@62397
   391
lp15@62397
   392
lemma closure_halfspace_gt [simp]:
lp15@62397
   393
   "a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"
lp15@62397
   394
using closure_halfspace_lt [of "-a" "-b"] by simp
lp15@62397
   395
lp15@62397
   396
lemma closure_halfspace_component_lt [simp]:
lp15@62397
   397
     "closure {x. x$k < a} = {x :: (real,'n::finite) vec. x$k \<le> a}" (is "?LE")
lp15@62397
   398
  and closure_halfspace_component_gt [simp]:
lp15@62397
   399
     "closure {x. x$k > a} = {x :: (real,'n::finite) vec. x$k \<ge> a}" (is "?GE")
lp15@62397
   400
proof -
lp15@62397
   401
  have "axis k (1::real) \<noteq> 0"
lp15@62397
   402
    by (simp add: axis_def vec_eq_iff)
lp15@62397
   403
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
lp15@62397
   404
    by (simp add: cart_eq_inner_axis inner_commute)
lp15@62397
   405
  ultimately show ?LE ?GE
lp15@62397
   406
    using closure_halfspace_lt [of "axis k (1::real)" a]
lp15@62397
   407
          closure_halfspace_gt [of "axis k (1::real)" a] by auto
lp15@62397
   408
qed
lp15@62397
   409
lp15@62397
   410
lemma interior_hyperplane [simp]:
lp15@62397
   411
  assumes "a \<noteq> 0"
lp15@62397
   412
    shows "interior {x. a \<bullet> x = b} = {}"
lp15@62397
   413
proof -
lp15@62397
   414
  have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
lp15@62397
   415
    by (force simp:)
lp15@62397
   416
  then show ?thesis
lp15@62397
   417
    by (auto simp: assms)
lp15@62397
   418
qed
lp15@62397
   419
lp15@62397
   420
lemma frontier_halfspace_le:
lp15@62397
   421
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   422
    shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
lp15@62397
   423
proof (cases "a = 0")
lp15@62397
   424
  case True with assms show ?thesis by simp
lp15@62397
   425
next
lp15@62397
   426
  case False then show ?thesis
lp15@62397
   427
    by (force simp: frontier_def closed_halfspace_le)
lp15@62397
   428
qed
lp15@62397
   429
lp15@62397
   430
lemma frontier_halfspace_ge:
lp15@62397
   431
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   432
    shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"
lp15@62397
   433
proof (cases "a = 0")
lp15@62397
   434
  case True with assms show ?thesis by simp
lp15@62397
   435
next
lp15@62397
   436
  case False then show ?thesis
lp15@62397
   437
    by (force simp: frontier_def closed_halfspace_ge)
lp15@62397
   438
qed
lp15@62397
   439
lp15@62397
   440
lemma frontier_halfspace_lt:
lp15@62397
   441
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   442
    shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"
lp15@62397
   443
proof (cases "a = 0")
lp15@62397
   444
  case True with assms show ?thesis by simp
lp15@62397
   445
next
lp15@62397
   446
  case False then show ?thesis
lp15@62397
   447
    by (force simp: frontier_def interior_open open_halfspace_lt)
lp15@62397
   448
qed
lp15@62397
   449
lp15@62397
   450
lemma frontier_halfspace_gt:
lp15@62397
   451
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   452
    shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"
lp15@62397
   453
proof (cases "a = 0")
lp15@62397
   454
  case True with assms show ?thesis by simp
lp15@62397
   455
next
lp15@62397
   456
  case False then show ?thesis
lp15@62397
   457
    by (force simp: frontier_def interior_open open_halfspace_gt)
lp15@62397
   458
qed
lp15@62397
   459
lp15@62397
   460
lemma interior_standard_hyperplane:
lp15@62397
   461
   "interior {x :: (real,'n::finite) vec. x$k = a} = {}"
lp15@62397
   462
proof -
lp15@62397
   463
  have "axis k (1::real) \<noteq> 0"
lp15@62397
   464
    by (simp add: axis_def vec_eq_iff)
lp15@62397
   465
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
lp15@62397
   466
    by (simp add: cart_eq_inner_axis inner_commute)
lp15@62397
   467
  ultimately show ?thesis
lp15@62397
   468
    using interior_hyperplane [of "axis k (1::real)" a]
lp15@62397
   469
    by force
lp15@62397
   470
qed
lp15@62397
   471
wenzelm@60420
   472
subsection \<open>Matrix operations\<close>
hoelzl@37489
   473
wenzelm@60420
   474
text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close>
hoelzl@37489
   475
wenzelm@49644
   476
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
wenzelm@49644
   477
    (infixl "**" 70)
nipkow@64267
   478
  where "m ** m' == (\<chi> i j. sum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
hoelzl@37489
   479
wenzelm@49644
   480
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
wenzelm@49644
   481
    (infixl "*v" 70)
nipkow@64267
   482
  where "m *v x \<equiv> (\<chi> i. sum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
hoelzl@37489
   483
wenzelm@49644
   484
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
wenzelm@49644
   485
    (infixl "v*" 70)
nipkow@64267
   486
  where "v v* m == (\<chi> j. sum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
hoelzl@37489
   487
hoelzl@37489
   488
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
hoelzl@63332
   489
definition transpose where
hoelzl@37489
   490
  "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
hoelzl@37489
   491
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
hoelzl@37489
   492
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
hoelzl@37489
   493
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
hoelzl@37489
   494
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
hoelzl@37489
   495
hoelzl@37489
   496
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
hoelzl@37489
   497
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
nipkow@64267
   498
  by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
hoelzl@37489
   499
hoelzl@37489
   500
lemma matrix_mul_lid:
hoelzl@37489
   501
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   502
  shows "mat 1 ** A = A"
hoelzl@37489
   503
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   504
  apply vector
nipkow@64267
   505
  apply (auto simp only: if_distrib cond_application_beta sum.delta'[OF finite]
wenzelm@49644
   506
    mult_1_left mult_zero_left if_True UNIV_I)
wenzelm@49644
   507
  done
hoelzl@37489
   508
hoelzl@37489
   509
hoelzl@37489
   510
lemma matrix_mul_rid:
hoelzl@37489
   511
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   512
  shows "A ** mat 1 = A"
hoelzl@37489
   513
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   514
  apply vector
nipkow@64267
   515
  apply (auto simp only: if_distrib cond_application_beta sum.delta[OF finite]
wenzelm@49644
   516
    mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
wenzelm@49644
   517
  done
hoelzl@37489
   518
hoelzl@37489
   519
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
nipkow@64267
   520
  apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc)
haftmann@66804
   521
  apply (subst sum.swap)
hoelzl@37489
   522
  apply simp
hoelzl@37489
   523
  done
hoelzl@37489
   524
hoelzl@37489
   525
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
wenzelm@49644
   526
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def
nipkow@64267
   527
    sum_distrib_left sum_distrib_right mult.assoc)
haftmann@66804
   528
  apply (subst sum.swap)
hoelzl@37489
   529
  apply simp
hoelzl@37489
   530
  done
hoelzl@37489
   531
hoelzl@37489
   532
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
hoelzl@37489
   533
  apply (vector matrix_vector_mult_def mat_def)
nipkow@64267
   534
  apply (simp add: if_distrib cond_application_beta sum.delta' cong del: if_weak_cong)
wenzelm@49644
   535
  done
hoelzl@37489
   536
wenzelm@49644
   537
lemma matrix_transpose_mul:
wenzelm@49644
   538
    "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
haftmann@57512
   539
  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
hoelzl@37489
   540
hoelzl@37489
   541
lemma matrix_eq:
hoelzl@37489
   542
  fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
hoelzl@37489
   543
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@37489
   544
  apply auto
huffman@44136
   545
  apply (subst vec_eq_iff)
hoelzl@37489
   546
  apply clarify
hoelzl@50526
   547
  apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
hoelzl@50526
   548
  apply (erule_tac x="axis ia 1" in allE)
hoelzl@37489
   549
  apply (erule_tac x="i" in allE)
hoelzl@50526
   550
  apply (auto simp add: if_distrib cond_application_beta axis_def
nipkow@64267
   551
    sum.delta[OF finite] cong del: if_weak_cong)
wenzelm@49644
   552
  done
hoelzl@37489
   553
wenzelm@49644
   554
lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
huffman@44136
   555
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   556
hoelzl@37489
   557
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
nipkow@64267
   558
  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps)
haftmann@66804
   559
  apply (subst sum.swap)
wenzelm@49644
   560
  apply simp
wenzelm@49644
   561
  done
hoelzl@37489
   562
hoelzl@37489
   563
lemma transpose_mat: "transpose (mat n) = mat n"
hoelzl@37489
   564
  by (vector transpose_def mat_def)
hoelzl@37489
   565
hoelzl@37489
   566
lemma transpose_transpose: "transpose(transpose A) = A"
hoelzl@37489
   567
  by (vector transpose_def)
hoelzl@37489
   568
hoelzl@37489
   569
lemma row_transpose:
hoelzl@37489
   570
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   571
  shows "row i (transpose A) = column i A"
huffman@44136
   572
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   573
hoelzl@37489
   574
lemma column_transpose:
hoelzl@37489
   575
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   576
  shows "column i (transpose A) = row i A"
huffman@44136
   577
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   578
hoelzl@37489
   579
lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
wenzelm@49644
   580
  by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
hoelzl@37489
   581
wenzelm@49644
   582
lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
wenzelm@49644
   583
  by (metis transpose_transpose rows_transpose)
hoelzl@37489
   584
wenzelm@60420
   585
text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
hoelzl@37489
   586
hoelzl@37489
   587
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
huffman@44136
   588
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   589
wenzelm@49644
   590
lemma matrix_mult_vsum:
nipkow@64267
   591
  "(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
haftmann@57512
   592
  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
hoelzl@37489
   593
hoelzl@37489
   594
lemma vector_componentwise:
hoelzl@50526
   595
  "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
nipkow@64267
   596
  by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)
hoelzl@50526
   597
nipkow@64267
   598
lemma basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
nipkow@64267
   599
  by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
hoelzl@37489
   600
lp15@63938
   601
lemma linear_componentwise_expansion:
hoelzl@37489
   602
  fixes f:: "real ^'m \<Rightarrow> real ^ _"
hoelzl@37489
   603
  assumes lf: "linear f"
nipkow@64267
   604
  shows "(f x)$j = sum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
wenzelm@49644
   605
proof -
hoelzl@37489
   606
  let ?M = "(UNIV :: 'm set)"
hoelzl@37489
   607
  let ?N = "(UNIV :: 'n set)"
nipkow@64267
   608
  have "?rhs = (sum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
nipkow@64267
   609
    unfolding sum_component by simp
wenzelm@49644
   610
  then show ?thesis
nipkow@64267
   611
    unfolding linear_sum_mul[OF lf, symmetric]
hoelzl@50526
   612
    unfolding scalar_mult_eq_scaleR[symmetric]
hoelzl@50526
   613
    unfolding basis_expansion
hoelzl@50526
   614
    by simp
hoelzl@37489
   615
qed
hoelzl@37489
   616
wenzelm@60420
   617
text\<open>Inverse matrices  (not necessarily square)\<close>
hoelzl@37489
   618
wenzelm@49644
   619
definition
wenzelm@49644
   620
  "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   621
wenzelm@49644
   622
definition
wenzelm@49644
   623
  "matrix_inv(A:: 'a::semiring_1^'n^'m) =
wenzelm@49644
   624
    (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   625
wenzelm@60420
   626
text\<open>Correspondence between matrices and linear operators.\<close>
hoelzl@37489
   627
wenzelm@49644
   628
definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
hoelzl@50526
   629
  where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
hoelzl@37489
   630
hoelzl@37489
   631
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
huffman@53600
   632
  by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
nipkow@64267
   633
      field_simps sum_distrib_left sum.distrib)
hoelzl@37489
   634
wenzelm@49644
   635
lemma matrix_works:
wenzelm@49644
   636
  assumes lf: "linear f"
wenzelm@49644
   637
  shows "matrix f *v x = f (x::real ^ 'n)"
haftmann@57512
   638
  apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)
lp15@63938
   639
  by (simp add: linear_componentwise_expansion lf)
hoelzl@37489
   640
wenzelm@49644
   641
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
wenzelm@49644
   642
  by (simp add: ext matrix_works)
hoelzl@37489
   643
hoelzl@37489
   644
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
hoelzl@37489
   645
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
hoelzl@37489
   646
hoelzl@37489
   647
lemma matrix_compose:
hoelzl@37489
   648
  assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
wenzelm@49644
   649
    and lg: "linear (g::real^'m \<Rightarrow> real^_)"
wenzelm@61736
   650
  shows "matrix (g \<circ> f) = matrix g ** matrix f"
hoelzl@37489
   651
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
wenzelm@49644
   652
  by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
hoelzl@37489
   653
wenzelm@49644
   654
lemma matrix_vector_column:
nipkow@64267
   655
  "(A::'a::comm_semiring_1^'n^_) *v x = sum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
haftmann@57512
   656
  by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
hoelzl@37489
   657
hoelzl@37489
   658
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
hoelzl@37489
   659
  apply (rule adjoint_unique)
wenzelm@49644
   660
  apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
nipkow@64267
   661
    sum_distrib_right sum_distrib_left)
haftmann@66804
   662
  apply (subst sum.swap)
haftmann@57514
   663
  apply (auto simp add: ac_simps)
hoelzl@37489
   664
  done
hoelzl@37489
   665
hoelzl@37489
   666
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
hoelzl@37489
   667
  shows "matrix(adjoint f) = transpose(matrix f)"
hoelzl@37489
   668
  apply (subst matrix_vector_mul[OF lf])
wenzelm@49644
   669
  unfolding adjoint_matrix matrix_of_matrix_vector_mul
wenzelm@49644
   670
  apply rule
wenzelm@49644
   671
  done
wenzelm@49644
   672
hoelzl@37489
   673
wenzelm@60420
   674
subsection \<open>lambda skolemization on cartesian products\<close>
hoelzl@37489
   675
hoelzl@37489
   676
(* FIXME: rename do choice_cart *)
hoelzl@37489
   677
hoelzl@37489
   678
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
hoelzl@37494
   679
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   680
proof -
hoelzl@37489
   681
  let ?S = "(UNIV :: 'n set)"
wenzelm@49644
   682
  { assume H: "?rhs"
wenzelm@49644
   683
    then have ?lhs by auto }
hoelzl@37489
   684
  moreover
wenzelm@49644
   685
  { assume H: "?lhs"
hoelzl@37489
   686
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
hoelzl@37489
   687
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
wenzelm@49644
   688
    { fix i
hoelzl@37489
   689
      from f have "P i (f i)" by metis
hoelzl@37494
   690
      then have "P i (?x $ i)" by auto
hoelzl@37489
   691
    }
hoelzl@37489
   692
    hence "\<forall>i. P i (?x$i)" by metis
hoelzl@37489
   693
    hence ?rhs by metis }
hoelzl@37489
   694
  ultimately show ?thesis by metis
hoelzl@37489
   695
qed
hoelzl@37489
   696
hoelzl@37489
   697
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
hoelzl@50526
   698
  unfolding inner_simps scalar_mult_eq_scaleR by auto
hoelzl@37489
   699
hoelzl@37489
   700
lemma left_invertible_transpose:
hoelzl@37489
   701
  "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
hoelzl@37489
   702
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   703
hoelzl@37489
   704
lemma right_invertible_transpose:
hoelzl@37489
   705
  "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
hoelzl@37489
   706
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   707
hoelzl@37489
   708
lemma matrix_left_invertible_injective:
wenzelm@49644
   709
  "(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
wenzelm@49644
   710
proof -
wenzelm@49644
   711
  { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
hoelzl@37489
   712
    from xy have "B*v (A *v x) = B *v (A*v y)" by simp
hoelzl@37489
   713
    hence "x = y"
wenzelm@49644
   714
      unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
hoelzl@37489
   715
  moreover
wenzelm@49644
   716
  { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
hoelzl@37489
   717
    hence i: "inj (op *v A)" unfolding inj_on_def by auto
hoelzl@37489
   718
    from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
wenzelm@61736
   719
    obtain g where g: "linear g" "g \<circ> op *v A = id" by blast
hoelzl@37489
   720
    have "matrix g ** A = mat 1"
hoelzl@37489
   721
      unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   722
      using g(2) by (simp add: fun_eq_iff)
wenzelm@49644
   723
    then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
hoelzl@37489
   724
  ultimately show ?thesis by blast
hoelzl@37489
   725
qed
hoelzl@37489
   726
hoelzl@37489
   727
lemma matrix_left_invertible_ker:
hoelzl@37489
   728
  "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
hoelzl@37489
   729
  unfolding matrix_left_invertible_injective
hoelzl@37489
   730
  using linear_injective_0[OF matrix_vector_mul_linear, of A]
hoelzl@37489
   731
  by (simp add: inj_on_def)
hoelzl@37489
   732
hoelzl@37489
   733
lemma matrix_right_invertible_surjective:
wenzelm@49644
   734
  "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
wenzelm@49644
   735
proof -
wenzelm@49644
   736
  { fix B :: "real ^'m^'n"
wenzelm@49644
   737
    assume AB: "A ** B = mat 1"
wenzelm@49644
   738
    { fix x :: "real ^ 'm"
hoelzl@37489
   739
      have "A *v (B *v x) = x"
wenzelm@49644
   740
        by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
hoelzl@37489
   741
    hence "surj (op *v A)" unfolding surj_def by metis }
hoelzl@37489
   742
  moreover
wenzelm@49644
   743
  { assume sf: "surj (op *v A)"
hoelzl@37489
   744
    from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
wenzelm@61736
   745
    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A \<circ> g = id"
hoelzl@37489
   746
      by blast
hoelzl@37489
   747
hoelzl@37489
   748
    have "A ** (matrix g) = mat 1"
hoelzl@37489
   749
      unfolding matrix_eq  matrix_vector_mul_lid
hoelzl@37489
   750
        matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   751
      using g(2) unfolding o_def fun_eq_iff id_def
hoelzl@37489
   752
      .
hoelzl@37489
   753
    hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
hoelzl@37489
   754
  }
hoelzl@37489
   755
  ultimately show ?thesis unfolding surj_def by blast
hoelzl@37489
   756
qed
hoelzl@37489
   757
hoelzl@37489
   758
lemma matrix_left_invertible_independent_columns:
hoelzl@37489
   759
  fixes A :: "real^'n^'m"
wenzelm@49644
   760
  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
nipkow@64267
   761
      (\<forall>c. sum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
wenzelm@49644
   762
    (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   763
proof -
hoelzl@37489
   764
  let ?U = "UNIV :: 'n set"
wenzelm@49644
   765
  { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
wenzelm@49644
   766
    { fix c i
nipkow@64267
   767
      assume c: "sum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
hoelzl@37489
   768
      let ?x = "\<chi> i. c i"
hoelzl@37489
   769
      have th0:"A *v ?x = 0"
hoelzl@37489
   770
        using c
huffman@44136
   771
        unfolding matrix_mult_vsum vec_eq_iff
hoelzl@37489
   772
        by auto
hoelzl@37489
   773
      from k[rule_format, OF th0] i
huffman@44136
   774
      have "c i = 0" by (vector vec_eq_iff)}
wenzelm@49644
   775
    hence ?rhs by blast }
hoelzl@37489
   776
  moreover
wenzelm@49644
   777
  { assume H: ?rhs
wenzelm@49644
   778
    { fix x assume x: "A *v x = 0"
hoelzl@37489
   779
      let ?c = "\<lambda>i. ((x$i ):: real)"
hoelzl@37489
   780
      from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
wenzelm@49644
   781
      have "x = 0" by vector }
wenzelm@49644
   782
  }
hoelzl@37489
   783
  ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
hoelzl@37489
   784
qed
hoelzl@37489
   785
hoelzl@37489
   786
lemma matrix_right_invertible_independent_rows:
hoelzl@37489
   787
  fixes A :: "real^'n^'m"
wenzelm@49644
   788
  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
nipkow@64267
   789
    (\<forall>c. sum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
hoelzl@37489
   790
  unfolding left_invertible_transpose[symmetric]
hoelzl@37489
   791
    matrix_left_invertible_independent_columns
hoelzl@37489
   792
  by (simp add: column_transpose)
hoelzl@37489
   793
hoelzl@37489
   794
lemma matrix_right_invertible_span_columns:
wenzelm@49644
   795
  "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
wenzelm@49644
   796
    span (columns A) = UNIV" (is "?lhs = ?rhs")
wenzelm@49644
   797
proof -
hoelzl@37489
   798
  let ?U = "UNIV :: 'm set"
hoelzl@37489
   799
  have fU: "finite ?U" by simp
nipkow@64267
   800
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y)"
hoelzl@37489
   801
    unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
wenzelm@49644
   802
    apply (subst eq_commute)
wenzelm@49644
   803
    apply rule
wenzelm@49644
   804
    done
hoelzl@37489
   805
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
wenzelm@49644
   806
  { assume h: ?lhs
wenzelm@49644
   807
    { fix x:: "real ^'n"
wenzelm@49644
   808
      from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
nipkow@64267
   809
        where y: "sum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
wenzelm@49644
   810
      have "x \<in> span (columns A)"
wenzelm@49644
   811
        unfolding y[symmetric]
nipkow@64267
   812
        apply (rule span_sum)
hoelzl@50526
   813
        unfolding scalar_mult_eq_scaleR
wenzelm@49644
   814
        apply (rule span_mul)
wenzelm@49644
   815
        apply (rule span_superset)
wenzelm@49644
   816
        unfolding columns_def
wenzelm@49644
   817
        apply blast
wenzelm@49644
   818
        done
wenzelm@49644
   819
    }
wenzelm@49644
   820
    then have ?rhs unfolding rhseq by blast }
hoelzl@37489
   821
  moreover
wenzelm@49644
   822
  { assume h:?rhs
nipkow@64267
   823
    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y"
wenzelm@49644
   824
    { fix y
wenzelm@49644
   825
      have "?P y"
hoelzl@50526
   826
      proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
nipkow@64267
   827
        show "\<exists>x::real ^ 'm. sum (\<lambda>i. (x$i) *s column i A) ?U = 0"
hoelzl@37489
   828
          by (rule exI[where x=0], simp)
hoelzl@37489
   829
      next
wenzelm@49644
   830
        fix c y1 y2
wenzelm@49644
   831
        assume y1: "y1 \<in> columns A" and y2: "?P y2"
hoelzl@37489
   832
        from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
hoelzl@37489
   833
          unfolding columns_def by blast
hoelzl@37489
   834
        from y2 obtain x:: "real ^'m" where
nipkow@64267
   835
          x: "sum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
hoelzl@37489
   836
        let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
hoelzl@37489
   837
        show "?P (c*s y1 + y2)"
webertj@49962
   838
        proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)
wenzelm@49644
   839
          fix j
wenzelm@49644
   840
          have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
   841
              else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
wenzelm@49644
   842
            using i(1) by (simp add: field_simps)
nipkow@64267
   843
          have "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
nipkow@64267
   844
              else (x$xa) * ((column xa A$j))) ?U = sum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"
nipkow@64267
   845
            apply (rule sum.cong[OF refl])
wenzelm@49644
   846
            using th apply blast
wenzelm@49644
   847
            done
nipkow@64267
   848
          also have "\<dots> = sum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
nipkow@64267
   849
            by (simp add: sum.distrib)
nipkow@64267
   850
          also have "\<dots> = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
nipkow@64267
   851
            unfolding sum.delta[OF fU]
wenzelm@49644
   852
            using i(1) by simp
nipkow@64267
   853
          finally show "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
nipkow@64267
   854
            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
wenzelm@49644
   855
        qed
wenzelm@49644
   856
      next
wenzelm@49644
   857
        show "y \<in> span (columns A)"
wenzelm@49644
   858
          unfolding h by blast
wenzelm@49644
   859
      qed
wenzelm@49644
   860
    }
wenzelm@49644
   861
    then have ?lhs unfolding lhseq ..
wenzelm@49644
   862
  }
hoelzl@37489
   863
  ultimately show ?thesis by blast
hoelzl@37489
   864
qed
hoelzl@37489
   865
hoelzl@37489
   866
lemma matrix_left_invertible_span_rows:
hoelzl@37489
   867
  "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
hoelzl@37489
   868
  unfolding right_invertible_transpose[symmetric]
hoelzl@37489
   869
  unfolding columns_transpose[symmetric]
hoelzl@37489
   870
  unfolding matrix_right_invertible_span_columns
wenzelm@49644
   871
  ..
hoelzl@37489
   872
wenzelm@60420
   873
text \<open>The same result in terms of square matrices.\<close>
hoelzl@37489
   874
hoelzl@37489
   875
lemma matrix_left_right_inverse:
hoelzl@37489
   876
  fixes A A' :: "real ^'n^'n"
hoelzl@37489
   877
  shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
wenzelm@49644
   878
proof -
wenzelm@49644
   879
  { fix A A' :: "real ^'n^'n"
wenzelm@49644
   880
    assume AA': "A ** A' = mat 1"
hoelzl@37489
   881
    have sA: "surj (op *v A)"
hoelzl@37489
   882
      unfolding surj_def
hoelzl@37489
   883
      apply clarify
hoelzl@37489
   884
      apply (rule_tac x="(A' *v y)" in exI)
wenzelm@49644
   885
      apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
wenzelm@49644
   886
      done
hoelzl@37489
   887
    from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
hoelzl@37489
   888
    obtain f' :: "real ^'n \<Rightarrow> real ^'n"
hoelzl@37489
   889
      where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
hoelzl@37489
   890
    have th: "matrix f' ** A = mat 1"
wenzelm@49644
   891
      by (simp add: matrix_eq matrix_works[OF f'(1)]
wenzelm@49644
   892
          matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
hoelzl@37489
   893
    hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
wenzelm@49644
   894
    hence "matrix f' = A'"
wenzelm@49644
   895
      by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
hoelzl@37489
   896
    hence "matrix f' ** A = A' ** A" by simp
wenzelm@49644
   897
    hence "A' ** A = mat 1" by (simp add: th)
wenzelm@49644
   898
  }
hoelzl@37489
   899
  then show ?thesis by blast
hoelzl@37489
   900
qed
hoelzl@37489
   901
wenzelm@60420
   902
text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
hoelzl@37489
   903
hoelzl@37489
   904
definition "rowvector v = (\<chi> i j. (v$j))"
hoelzl@37489
   905
hoelzl@37489
   906
definition "columnvector v = (\<chi> i j. (v$i))"
hoelzl@37489
   907
wenzelm@49644
   908
lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
huffman@44136
   909
  by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
hoelzl@37489
   910
hoelzl@37489
   911
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
huffman@44136
   912
  by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
hoelzl@37489
   913
wenzelm@49644
   914
lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
hoelzl@37489
   915
  by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
hoelzl@37489
   916
wenzelm@49644
   917
lemma dot_matrix_product:
wenzelm@49644
   918
  "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
huffman@44136
   919
  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
hoelzl@37489
   920
hoelzl@37489
   921
lemma dot_matrix_vector_mul:
hoelzl@37489
   922
  fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
hoelzl@37489
   923
  shows "(A *v x) \<bullet> (B *v y) =
hoelzl@37489
   924
      (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
wenzelm@49644
   925
  unfolding dot_matrix_product transpose_columnvector[symmetric]
wenzelm@49644
   926
    dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
hoelzl@37489
   927
wenzelm@61945
   928
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}"
hoelzl@50526
   929
  by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
hoelzl@37489
   930
wenzelm@49644
   931
lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
hoelzl@50526
   932
  using Basis_le_infnorm[of "axis i 1" x]
hoelzl@50526
   933
  by (simp add: Basis_vec_def axis_eq_axis inner_axis)
hoelzl@37489
   934
hoelzl@63334
   935
lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
huffman@44647
   936
  unfolding continuous_def by (rule tendsto_vec_nth)
huffman@44213
   937
hoelzl@63334
   938
lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
huffman@44647
   939
  unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
huffman@44213
   940
hoelzl@63334
   941
lemma continuous_on_vec_lambda[continuous_intros]:
hoelzl@63334
   942
  "(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)"
hoelzl@63334
   943
  unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)
hoelzl@63334
   944
hoelzl@37489
   945
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
hoelzl@63332
   946
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
huffman@44213
   947
hoelzl@37489
   948
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
wenzelm@49644
   949
  unfolding bounded_def
wenzelm@49644
   950
  apply clarify
wenzelm@49644
   951
  apply (rule_tac x="x $ i" in exI)
wenzelm@49644
   952
  apply (rule_tac x="e" in exI)
wenzelm@49644
   953
  apply clarify
wenzelm@49644
   954
  apply (rule order_trans [OF dist_vec_nth_le], simp)
wenzelm@49644
   955
  done
hoelzl@37489
   956
hoelzl@37489
   957
lemma compact_lemma_cart:
hoelzl@37489
   958
  fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
hoelzl@50998
   959
  assumes f: "bounded (range f)"
eberlm@66447
   960
  shows "\<exists>l r. strict_mono r \<and>
hoelzl@37489
   961
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
immler@62127
   962
    (is "?th d")
immler@62127
   963
proof -
immler@62127
   964
  have "\<forall>d' \<subseteq> d. ?th d'"
immler@62127
   965
    by (rule compact_lemma_general[where unproj=vec_lambda])
immler@62127
   966
      (auto intro!: f bounded_component_cart simp: vec_lambda_eta)
immler@62127
   967
  then show "?th d" by simp
hoelzl@37489
   968
qed
hoelzl@37489
   969
huffman@44136
   970
instance vec :: (heine_borel, finite) heine_borel
hoelzl@37489
   971
proof
hoelzl@50998
   972
  fix f :: "nat \<Rightarrow> 'a ^ 'b"
hoelzl@50998
   973
  assume f: "bounded (range f)"
eberlm@66447
   974
  then obtain l r where r: "strict_mono r"
wenzelm@49644
   975
      and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@50998
   976
    using compact_lemma_cart [OF f] by blast
hoelzl@37489
   977
  let ?d = "UNIV::'b set"
hoelzl@37489
   978
  { fix e::real assume "e>0"
hoelzl@37489
   979
    hence "0 < e / (real_of_nat (card ?d))"
wenzelm@49644
   980
      using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
hoelzl@37489
   981
    with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
hoelzl@37489
   982
      by simp
hoelzl@37489
   983
    moreover
wenzelm@49644
   984
    { fix n
wenzelm@49644
   985
      assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
hoelzl@37489
   986
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
nipkow@67155
   987
        unfolding dist_vec_def using zero_le_dist by (rule L2_set_le_sum)
hoelzl@37489
   988
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
nipkow@64267
   989
        by (rule sum_strict_mono) (simp_all add: n)
hoelzl@37489
   990
      finally have "dist (f (r n)) l < e" by simp
hoelzl@37489
   991
    }
hoelzl@37489
   992
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
lp15@61810
   993
      by (rule eventually_mono)
hoelzl@37489
   994
  }
wenzelm@61973
   995
  hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp
eberlm@66447
   996
  with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
hoelzl@37489
   997
qed
hoelzl@37489
   998
wenzelm@49644
   999
lemma interval_cart:
immler@54775
  1000
  fixes a :: "real^'n"
immler@54775
  1001
  shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
immler@56188
  1002
    and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
immler@56188
  1003
  by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
hoelzl@37489
  1004
wenzelm@49644
  1005
lemma mem_interval_cart:
immler@54775
  1006
  fixes a :: "real^'n"
immler@54775
  1007
  shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
immler@56188
  1008
    and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
wenzelm@49644
  1009
  using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
hoelzl@37489
  1010
wenzelm@49644
  1011
lemma interval_eq_empty_cart:
wenzelm@49644
  1012
  fixes a :: "real^'n"
immler@54775
  1013
  shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
immler@56188
  1014
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
wenzelm@49644
  1015
proof -
immler@54775
  1016
  { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>box a b"
hoelzl@37489
  1017
    hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto
hoelzl@37489
  1018
    hence "a$i < b$i" by auto
wenzelm@49644
  1019
    hence False using as by auto }
hoelzl@37489
  1020
  moreover
hoelzl@37489
  1021
  { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
hoelzl@37489
  1022
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
  1023
    { fix i
hoelzl@37489
  1024
      have "a$i < b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
  1025
      hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
hoelzl@37489
  1026
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
  1027
        by auto }
immler@54775
  1028
    hence "box a b \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
hoelzl@37489
  1029
  ultimately show ?th1 by blast
hoelzl@37489
  1030
immler@56188
  1031
  { fix i x assume as:"b$i < a$i" and x:"x\<in>cbox a b"
hoelzl@37489
  1032
    hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto
hoelzl@37489
  1033
    hence "a$i \<le> b$i" by auto
wenzelm@49644
  1034
    hence False using as by auto }
hoelzl@37489
  1035
  moreover
hoelzl@37489
  1036
  { assume as:"\<forall>i. \<not> (b$i < a$i)"
hoelzl@37489
  1037
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
  1038
    { fix i
hoelzl@37489
  1039
      have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
  1040
      hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
hoelzl@37489
  1041
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
  1042
        by auto }
immler@56188
  1043
    hence "cbox a b \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
hoelzl@37489
  1044
  ultimately show ?th2 by blast
hoelzl@37489
  1045
qed
hoelzl@37489
  1046
wenzelm@49644
  1047
lemma interval_ne_empty_cart:
wenzelm@49644
  1048
  fixes a :: "real^'n"
immler@56188
  1049
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
immler@54775
  1050
    and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
hoelzl@37489
  1051
  unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
hoelzl@37489
  1052
    (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
  1053
wenzelm@49644
  1054
lemma subset_interval_imp_cart:
wenzelm@49644
  1055
  fixes a :: "real^'n"
immler@56188
  1056
  shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
immler@56188
  1057
    and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
immler@56188
  1058
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
immler@54775
  1059
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b"
hoelzl@37489
  1060
  unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
hoelzl@37489
  1061
  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
  1062
wenzelm@49644
  1063
lemma interval_sing:
wenzelm@49644
  1064
  fixes a :: "'a::linorder^'n"
wenzelm@49644
  1065
  shows "{a .. a} = {a} \<and> {a<..<a} = {}"
wenzelm@49644
  1066
  apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
wenzelm@49644
  1067
  done
hoelzl@37489
  1068
wenzelm@49644
  1069
lemma subset_interval_cart:
wenzelm@49644
  1070
  fixes a :: "real^'n"
immler@56188
  1071
  shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
immler@56188
  1072
    and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
immler@56188
  1073
    and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
immler@54775
  1074
    and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
immler@56188
  1075
  using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
  1076
wenzelm@49644
  1077
lemma disjoint_interval_cart:
wenzelm@49644
  1078
  fixes a::"real^'n"
immler@56188
  1079
  shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
immler@56188
  1080
    and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
immler@56188
  1081
    and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
immler@54775
  1082
    and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
hoelzl@50526
  1083
  using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
  1084
wenzelm@49644
  1085
lemma inter_interval_cart:
immler@54775
  1086
  fixes a :: "real^'n"
immler@56188
  1087
  shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
lp15@63945
  1088
  unfolding Int_interval
immler@56188
  1089
  by (auto simp: mem_box less_eq_vec_def)
immler@56188
  1090
    (auto simp: Basis_vec_def inner_axis)
hoelzl@37489
  1091
wenzelm@49644
  1092
lemma closed_interval_left_cart:
wenzelm@49644
  1093
  fixes b :: "real^'n"
hoelzl@37489
  1094
  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
hoelzl@63332
  1095
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1096
wenzelm@49644
  1097
lemma closed_interval_right_cart:
wenzelm@49644
  1098
  fixes a::"real^'n"
hoelzl@37489
  1099
  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
hoelzl@63332
  1100
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1101
wenzelm@49644
  1102
lemma is_interval_cart:
wenzelm@49644
  1103
  "is_interval (s::(real^'n) set) \<longleftrightarrow>
wenzelm@49644
  1104
    (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
hoelzl@50526
  1105
  by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
hoelzl@37489
  1106
wenzelm@49644
  1107
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
hoelzl@63332
  1108
  by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1109
wenzelm@49644
  1110
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
hoelzl@63332
  1111
  by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1112
wenzelm@49644
  1113
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
hoelzl@63332
  1114
  by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1115
wenzelm@49644
  1116
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
hoelzl@63332
  1117
  by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1118
wenzelm@49644
  1119
lemma Lim_component_le_cart:
wenzelm@49644
  1120
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
  1121
  assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"
hoelzl@37489
  1122
  shows "l$i \<le> b"
hoelzl@50526
  1123
  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
hoelzl@37489
  1124
wenzelm@49644
  1125
lemma Lim_component_ge_cart:
wenzelm@49644
  1126
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
  1127
  assumes "(f \<longlongrightarrow> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
hoelzl@37489
  1128
  shows "b \<le> l$i"
hoelzl@50526
  1129
  by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
hoelzl@37489
  1130
wenzelm@49644
  1131
lemma Lim_component_eq_cart:
wenzelm@49644
  1132
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
  1133
  assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
hoelzl@37489
  1134
  shows "l$i = b"
wenzelm@49644
  1135
  using ev[unfolded order_eq_iff eventually_conj_iff] and
wenzelm@49644
  1136
    Lim_component_ge_cart[OF net, of b i] and
hoelzl@37489
  1137
    Lim_component_le_cart[OF net, of i b] by auto
hoelzl@37489
  1138
wenzelm@49644
  1139
lemma connected_ivt_component_cart:
wenzelm@49644
  1140
  fixes x :: "real^'n"
wenzelm@49644
  1141
  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$k = a)"
hoelzl@50526
  1142
  using connected_ivt_hyperplane[of s x y "axis k 1" a]
hoelzl@50526
  1143
  by (auto simp add: inner_axis inner_commute)
hoelzl@37489
  1144
wenzelm@49644
  1145
lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
hoelzl@37489
  1146
  unfolding subspace_def by auto
hoelzl@37489
  1147
hoelzl@37489
  1148
lemma closed_substandard_cart:
huffman@44213
  1149
  "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
  1150
proof -
huffman@44213
  1151
  { fix i::'n
huffman@44213
  1152
    have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
hoelzl@63332
  1153
      by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) }
huffman@44213
  1154
  thus ?thesis
huffman@44213
  1155
    unfolding Collect_all_eq by (simp add: closed_INT)
hoelzl@37489
  1156
qed
hoelzl@37489
  1157
wenzelm@49644
  1158
lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
wenzelm@49644
  1159
  (is "dim ?A = _")
wenzelm@49644
  1160
proof -
hoelzl@50526
  1161
  let ?a = "\<lambda>x. axis x 1 :: real^'n"
hoelzl@50526
  1162
  have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
hoelzl@50526
  1163
    by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
hoelzl@50526
  1164
  have "?a ` d \<subseteq> Basis"
hoelzl@50526
  1165
    by (auto simp: Basis_vec_def)
wenzelm@49644
  1166
  thus ?thesis
hoelzl@50526
  1167
    using dim_substandard[of "?a ` d"] card_image[of ?a d]
hoelzl@50526
  1168
    by (auto simp: axis_eq_axis inj_on_def *)
hoelzl@37489
  1169
qed
hoelzl@37489
  1170
hoelzl@37489
  1171
lemma affinity_inverses:
hoelzl@37489
  1172
  assumes m0: "m \<noteq> (0::'a::field)"
wenzelm@61736
  1173
  shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
wenzelm@61736
  1174
  "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
hoelzl@37489
  1175
  using m0
haftmann@54230
  1176
  apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
haftmann@54230
  1177
  apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
wenzelm@49644
  1178
  done
hoelzl@37489
  1179
hoelzl@37489
  1180
lemma vector_affinity_eq:
hoelzl@37489
  1181
  assumes m0: "(m::'a::field) \<noteq> 0"
hoelzl@37489
  1182
  shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
hoelzl@37489
  1183
proof
hoelzl@37489
  1184
  assume h: "m *s x + c = y"
hoelzl@37489
  1185
  hence "m *s x = y - c" by (simp add: field_simps)
hoelzl@37489
  1186
  hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
hoelzl@37489
  1187
  then show "x = inverse m *s y + - (inverse m *s c)"
hoelzl@37489
  1188
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1189
next
hoelzl@37489
  1190
  assume h: "x = inverse m *s y + - (inverse m *s c)"
haftmann@54230
  1191
  show "m *s x + c = y" unfolding h
hoelzl@37489
  1192
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1193
qed
hoelzl@37489
  1194
hoelzl@37489
  1195
lemma vector_eq_affinity:
wenzelm@49644
  1196
    "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
hoelzl@37489
  1197
  using vector_affinity_eq[where m=m and x=x and y=y and c=c]
hoelzl@37489
  1198
  by metis
hoelzl@37489
  1199
hoelzl@50526
  1200
lemma vector_cart:
hoelzl@50526
  1201
  fixes f :: "real^'n \<Rightarrow> real"
hoelzl@50526
  1202
  shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
hoelzl@50526
  1203
  unfolding euclidean_eq_iff[where 'a="real^'n"]
hoelzl@50526
  1204
  by simp (simp add: Basis_vec_def inner_axis)
hoelzl@63332
  1205
hoelzl@50526
  1206
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
hoelzl@50526
  1207
  by (rule vector_cart)
wenzelm@49644
  1208
huffman@44360
  1209
subsection "Convex Euclidean Space"
hoelzl@37489
  1210
hoelzl@50526
  1211
lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
hoelzl@50526
  1212
  using const_vector_cart[of 1] by (simp add: one_vec_def)
hoelzl@37489
  1213
hoelzl@37489
  1214
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
hoelzl@37489
  1215
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
hoelzl@37489
  1216
hoelzl@50526
  1217
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
  1218
hoelzl@37489
  1219
lemma convex_box_cart:
hoelzl@37489
  1220
  assumes "\<And>i. convex {x. P i x}"
hoelzl@37489
  1221
  shows "convex {x. \<forall>i. P i (x$i)}"
hoelzl@37489
  1222
  using assms unfolding convex_def by auto
hoelzl@37489
  1223
hoelzl@37489
  1224
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
hoelzl@63334
  1225
  by (rule convex_box_cart) (simp add: atLeast_def[symmetric])
hoelzl@37489
  1226
hoelzl@37489
  1227
lemma unit_interval_convex_hull_cart:
immler@56188
  1228
  "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}"
immler@56188
  1229
  unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
hoelzl@50526
  1230
  by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
hoelzl@37489
  1231
hoelzl@37489
  1232
lemma cube_convex_hull_cart:
wenzelm@49644
  1233
  assumes "0 < d"
wenzelm@49644
  1234
  obtains s::"(real^'n) set"
immler@56188
  1235
    where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
wenzelm@49644
  1236
proof -
wenzelm@55522
  1237
  from assms obtain s where "finite s"
nipkow@64267
  1238
    and "cbox (x - sum (op *\<^sub>R d) Basis) (x + sum (op *\<^sub>R d) Basis) = convex hull s"
wenzelm@55522
  1239
    by (rule cube_convex_hull)
wenzelm@55522
  1240
  with that[of s] show thesis
wenzelm@55522
  1241
    by (simp add: const_vector_cart)
hoelzl@37489
  1242
qed
hoelzl@37489
  1243
hoelzl@37489
  1244
hoelzl@37489
  1245
subsection "Derivative"
hoelzl@37489
  1246
hoelzl@37489
  1247
definition "jacobian f net = matrix(frechet_derivative f net)"
hoelzl@37489
  1248
wenzelm@49644
  1249
lemma jacobian_works:
wenzelm@49644
  1250
  "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
wenzelm@49644
  1251
    (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
wenzelm@49644
  1252
  apply rule
wenzelm@49644
  1253
  unfolding jacobian_def
wenzelm@49644
  1254
  apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
wenzelm@49644
  1255
  apply (rule differentiableI)
wenzelm@49644
  1256
  apply assumption
wenzelm@49644
  1257
  unfolding frechet_derivative_works
wenzelm@49644
  1258
  apply assumption
wenzelm@49644
  1259
  done
hoelzl@37489
  1260
hoelzl@37489
  1261
wenzelm@60420
  1262
subsection \<open>Component of the differential must be zero if it exists at a local
wenzelm@60420
  1263
  maximum or minimum for that corresponding component.\<close>
wenzelm@49644
  1264
hoelzl@50526
  1265
lemma differential_zero_maxmin_cart:
wenzelm@49644
  1266
  fixes f::"real^'a \<Rightarrow> real^'b"
hoelzl@37489
  1267
  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
  1268
    "f differentiable (at x)"
hoelzl@50526
  1269
  shows "jacobian f (at x) $ k = 0"
hoelzl@50526
  1270
  using differential_zero_maxmin_component[of "axis k 1" e x f] assms
hoelzl@50526
  1271
    vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
hoelzl@50526
  1272
  by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
wenzelm@49644
  1273
wenzelm@60420
  1274
subsection \<open>Lemmas for working on @{typ "real^1"}\<close>
hoelzl@37489
  1275
hoelzl@37489
  1276
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
wenzelm@49644
  1277
  by (metis (full_types) num1_eq_iff)
hoelzl@37489
  1278
hoelzl@37489
  1279
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
wenzelm@49644
  1280
  by auto (metis (full_types) num1_eq_iff)
hoelzl@37489
  1281
hoelzl@37489
  1282
lemma exhaust_2:
wenzelm@49644
  1283
  fixes x :: 2
wenzelm@49644
  1284
  shows "x = 1 \<or> x = 2"
hoelzl@37489
  1285
proof (induct x)
hoelzl@37489
  1286
  case (of_int z)
hoelzl@37489
  1287
  then have "0 <= z" and "z < 2" by simp_all
hoelzl@37489
  1288
  then have "z = 0 | z = 1" by arith
hoelzl@37489
  1289
  then show ?case by auto
hoelzl@37489
  1290
qed
hoelzl@37489
  1291
hoelzl@37489
  1292
lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
hoelzl@37489
  1293
  by (metis exhaust_2)
hoelzl@37489
  1294
hoelzl@37489
  1295
lemma exhaust_3:
wenzelm@49644
  1296
  fixes x :: 3
wenzelm@49644
  1297
  shows "x = 1 \<or> x = 2 \<or> x = 3"
hoelzl@37489
  1298
proof (induct x)
hoelzl@37489
  1299
  case (of_int z)
hoelzl@37489
  1300
  then have "0 <= z" and "z < 3" by simp_all
hoelzl@37489
  1301
  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
hoelzl@37489
  1302
  then show ?case by auto
hoelzl@37489
  1303
qed
hoelzl@37489
  1304
hoelzl@37489
  1305
lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
hoelzl@37489
  1306
  by (metis exhaust_3)
hoelzl@37489
  1307
hoelzl@37489
  1308
lemma UNIV_1 [simp]: "UNIV = {1::1}"
hoelzl@37489
  1309
  by (auto simp add: num1_eq_iff)
hoelzl@37489
  1310
hoelzl@37489
  1311
lemma UNIV_2: "UNIV = {1::2, 2::2}"
hoelzl@37489
  1312
  using exhaust_2 by auto
hoelzl@37489
  1313
hoelzl@37489
  1314
lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
hoelzl@37489
  1315
  using exhaust_3 by auto
hoelzl@37489
  1316
nipkow@64267
  1317
lemma sum_1: "sum f (UNIV::1 set) = f 1"
hoelzl@37489
  1318
  unfolding UNIV_1 by simp
hoelzl@37489
  1319
nipkow@64267
  1320
lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2"
hoelzl@37489
  1321
  unfolding UNIV_2 by simp
hoelzl@37489
  1322
nipkow@64267
  1323
lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3"
haftmann@57514
  1324
  unfolding UNIV_3 by (simp add: ac_simps)
hoelzl@37489
  1325
wenzelm@49644
  1326
instantiation num1 :: cart_one
wenzelm@49644
  1327
begin
wenzelm@49644
  1328
wenzelm@49644
  1329
instance
wenzelm@49644
  1330
proof
hoelzl@37489
  1331
  show "CARD(1) = Suc 0" by auto
wenzelm@49644
  1332
qed
wenzelm@49644
  1333
wenzelm@49644
  1334
end
hoelzl@37489
  1335
wenzelm@60420
  1336
subsection\<open>The collapse of the general concepts to dimension one.\<close>
hoelzl@37489
  1337
hoelzl@37489
  1338
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
huffman@44136
  1339
  by (simp add: vec_eq_iff)
hoelzl@37489
  1340
hoelzl@37489
  1341
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
hoelzl@37489
  1342
  apply auto
hoelzl@37489
  1343
  apply (erule_tac x= "x$1" in allE)
hoelzl@37489
  1344
  apply (simp only: vector_one[symmetric])
hoelzl@37489
  1345
  done
hoelzl@37489
  1346
hoelzl@37489
  1347
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
huffman@44136
  1348
  by (simp add: norm_vec_def)
hoelzl@37489
  1349
wenzelm@61945
  1350
lemma norm_real: "norm(x::real ^ 1) = \<bar>x$1\<bar>"
hoelzl@37489
  1351
  by (simp add: norm_vector_1)
hoelzl@37489
  1352
wenzelm@61945
  1353
lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x$1) - (y$1)\<bar>"
hoelzl@37489
  1354
  by (auto simp add: norm_real dist_norm)
hoelzl@37489
  1355
wenzelm@49644
  1356
wenzelm@60420
  1357
subsection\<open>Explicit vector construction from lists.\<close>
hoelzl@37489
  1358
hoelzl@43995
  1359
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
  1360
hoelzl@37489
  1361
lemma vector_1: "(vector[x]) $1 = x"
hoelzl@37489
  1362
  unfolding vector_def by simp
hoelzl@37489
  1363
hoelzl@37489
  1364
lemma vector_2:
hoelzl@37489
  1365
 "(vector[x,y]) $1 = x"
hoelzl@37489
  1366
 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
hoelzl@37489
  1367
  unfolding vector_def by simp_all
hoelzl@37489
  1368
hoelzl@37489
  1369
lemma vector_3:
hoelzl@37489
  1370
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
hoelzl@37489
  1371
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
hoelzl@37489
  1372
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
hoelzl@37489
  1373
  unfolding vector_def by simp_all
hoelzl@37489
  1374
hoelzl@37489
  1375
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
hoelzl@37489
  1376
  apply auto
hoelzl@37489
  1377
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1378
  apply (subgoal_tac "vector [v$1] = v")
hoelzl@37489
  1379
  apply simp
hoelzl@37489
  1380
  apply (vector vector_def)
hoelzl@37489
  1381
  apply simp
hoelzl@37489
  1382
  done
hoelzl@37489
  1383
hoelzl@37489
  1384
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
hoelzl@37489
  1385
  apply auto
hoelzl@37489
  1386
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1387
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1388
  apply (subgoal_tac "vector [v$1, v$2] = v")
hoelzl@37489
  1389
  apply simp
hoelzl@37489
  1390
  apply (vector vector_def)
hoelzl@37489
  1391
  apply (simp add: forall_2)
hoelzl@37489
  1392
  done
hoelzl@37489
  1393
hoelzl@37489
  1394
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
hoelzl@37489
  1395
  apply auto
hoelzl@37489
  1396
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1397
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1398
  apply (erule_tac x="v$3" in allE)
hoelzl@37489
  1399
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
hoelzl@37489
  1400
  apply simp
hoelzl@37489
  1401
  apply (vector vector_def)
hoelzl@37489
  1402
  apply (simp add: forall_3)
hoelzl@37489
  1403
  done
hoelzl@37489
  1404
hoelzl@37489
  1405
lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
wenzelm@49644
  1406
  apply (rule bounded_linearI[where K=1])
hoelzl@37489
  1407
  using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
hoelzl@37489
  1408
hoelzl@37489
  1409
lemma interval_split_cart:
hoelzl@37489
  1410
  "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
immler@56188
  1411
  "cbox a b \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
wenzelm@49644
  1412
  apply (rule_tac[!] set_eqI)
immler@56188
  1413
  unfolding Int_iff mem_interval_cart mem_Collect_eq interval_cbox_cart
wenzelm@49644
  1414
  unfolding vec_lambda_beta
wenzelm@49644
  1415
  by auto
hoelzl@37489
  1416
hoelzl@37489
  1417
end