src/HOL/Analysis/Cartesian_Euclidean_Space.thy
author paulson <lp15@cam.ac.uk>
Wed Feb 21 12:57:49 2018 +0000 (18 months ago)
changeset 67683 817944aeac3f
parent 67673 c8caefb20564
child 67686 2c58505bf151
permissions -rw-r--r--
Lots of new material about matrices, etc.
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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 "( * ) \<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 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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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$i\<bar> < e"
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  by (metis component_le_norm_cart le_less_trans)
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nipkow@64267
   310
lemma norm_le_l1_cart: "norm x <= sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
nipkow@67155
   311
  by (simp add: norm_vec_def L2_set_le_sum)
hoelzl@37489
   312
hoelzl@37489
   313
lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
huffman@44136
   314
  unfolding scaleR_vec_def vector_scalar_mult_def by simp
hoelzl@37489
   315
hoelzl@37489
   316
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
hoelzl@37489
   317
  unfolding dist_norm scalar_mult_eq_scaleR
hoelzl@37489
   318
  unfolding scaleR_right_diff_distrib[symmetric] by simp
hoelzl@37489
   319
nipkow@64267
   320
lemma sum_component [simp]:
hoelzl@37489
   321
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
nipkow@64267
   322
  shows "(sum f S)$i = sum (\<lambda>x. (f x)$i) S"
wenzelm@49644
   323
proof (cases "finite S")
wenzelm@49644
   324
  case True
wenzelm@49644
   325
  then show ?thesis by induct simp_all
wenzelm@49644
   326
next
wenzelm@49644
   327
  case False
wenzelm@49644
   328
  then show ?thesis by simp
wenzelm@49644
   329
qed
hoelzl@37489
   330
nipkow@64267
   331
lemma sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)$i ) S)"
huffman@44136
   332
  by (simp add: vec_eq_iff)
hoelzl@37489
   333
nipkow@64267
   334
lemma sum_cmul:
hoelzl@37489
   335
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
nipkow@64267
   336
  shows "sum (\<lambda>x. c *s f x) S = c *s sum f S"
nipkow@64267
   337
  by (simp add: vec_eq_iff sum_distrib_left)
hoelzl@37489
   338
nipkow@64267
   339
lemma sum_norm_allsubsets_bound_cart:
hoelzl@37489
   340
  fixes f:: "'a \<Rightarrow> real ^'n"
nipkow@64267
   341
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
nipkow@64267
   342
  shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
nipkow@64267
   343
  using sum_norm_allsubsets_bound[OF assms]
wenzelm@57865
   344
  by simp
hoelzl@37489
   345
lp15@62397
   346
subsection\<open>Closures and interiors of halfspaces\<close>
lp15@62397
   347
lp15@62397
   348
lemma interior_halfspace_le [simp]:
lp15@62397
   349
  assumes "a \<noteq> 0"
lp15@62397
   350
    shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
lp15@62397
   351
proof -
lp15@62397
   352
  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
   353
  proof -
lp15@62397
   354
    obtain e where "e>0" and e: "cball x e \<subseteq> S"
lp15@62397
   355
      using \<open>open S\<close> open_contains_cball x by blast
lp15@62397
   356
    then have "x + (e / norm a) *\<^sub>R a \<in> cball x e"
lp15@62397
   357
      by (simp add: dist_norm)
lp15@62397
   358
    then have "x + (e / norm a) *\<^sub>R a \<in> S"
lp15@62397
   359
      using e by blast
lp15@62397
   360
    then have "x + (e / norm a) *\<^sub>R a \<in> {x. a \<bullet> x \<le> b}"
lp15@62397
   361
      using S by blast
lp15@62397
   362
    moreover have "e * (a \<bullet> a) / norm a > 0"
lp15@62397
   363
      by (simp add: \<open>0 < e\<close> assms)
lp15@62397
   364
    ultimately show ?thesis
lp15@62397
   365
      by (simp add: algebra_simps)
lp15@62397
   366
  qed
lp15@62397
   367
  show ?thesis
lp15@62397
   368
    by (rule interior_unique) (auto simp: open_halfspace_lt *)
lp15@62397
   369
qed
lp15@62397
   370
lp15@62397
   371
lemma interior_halfspace_ge [simp]:
lp15@62397
   372
   "a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"
lp15@62397
   373
using interior_halfspace_le [of "-a" "-b"] by simp
lp15@62397
   374
lp15@62397
   375
lemma interior_halfspace_component_le [simp]:
lp15@62397
   376
     "interior {x. x$k \<le> a} = {x :: (real,'n::finite) vec. x$k < a}" (is "?LE")
lp15@62397
   377
  and interior_halfspace_component_ge [simp]:
lp15@62397
   378
     "interior {x. x$k \<ge> a} = {x :: (real,'n::finite) vec. x$k > a}" (is "?GE")
lp15@62397
   379
proof -
lp15@62397
   380
  have "axis k (1::real) \<noteq> 0"
lp15@62397
   381
    by (simp add: axis_def vec_eq_iff)
lp15@62397
   382
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
lp15@62397
   383
    by (simp add: cart_eq_inner_axis inner_commute)
lp15@62397
   384
  ultimately show ?LE ?GE
lp15@62397
   385
    using interior_halfspace_le [of "axis k (1::real)" a]
lp15@62397
   386
          interior_halfspace_ge [of "axis k (1::real)" a] by auto
lp15@62397
   387
qed
lp15@62397
   388
lp15@62397
   389
lemma closure_halfspace_lt [simp]:
lp15@62397
   390
  assumes "a \<noteq> 0"
lp15@62397
   391
    shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
lp15@62397
   392
proof -
lp15@62397
   393
  have [simp]: "-{x. a \<bullet> x < b} = {x. a \<bullet> x \<ge> b}"
lp15@62397
   394
    by (force simp:)
lp15@62397
   395
  then show ?thesis
lp15@62397
   396
    using interior_halfspace_ge [of a b] assms
lp15@62397
   397
    by (force simp: closure_interior)
lp15@62397
   398
qed
lp15@62397
   399
lp15@62397
   400
lemma closure_halfspace_gt [simp]:
lp15@62397
   401
   "a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"
lp15@62397
   402
using closure_halfspace_lt [of "-a" "-b"] by simp
lp15@62397
   403
lp15@62397
   404
lemma closure_halfspace_component_lt [simp]:
lp15@62397
   405
     "closure {x. x$k < a} = {x :: (real,'n::finite) vec. x$k \<le> a}" (is "?LE")
lp15@62397
   406
  and closure_halfspace_component_gt [simp]:
lp15@62397
   407
     "closure {x. x$k > a} = {x :: (real,'n::finite) vec. x$k \<ge> a}" (is "?GE")
lp15@62397
   408
proof -
lp15@62397
   409
  have "axis k (1::real) \<noteq> 0"
lp15@62397
   410
    by (simp add: axis_def vec_eq_iff)
lp15@62397
   411
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
lp15@62397
   412
    by (simp add: cart_eq_inner_axis inner_commute)
lp15@62397
   413
  ultimately show ?LE ?GE
lp15@62397
   414
    using closure_halfspace_lt [of "axis k (1::real)" a]
lp15@62397
   415
          closure_halfspace_gt [of "axis k (1::real)" a] by auto
lp15@62397
   416
qed
lp15@62397
   417
lp15@62397
   418
lemma interior_hyperplane [simp]:
lp15@62397
   419
  assumes "a \<noteq> 0"
lp15@62397
   420
    shows "interior {x. a \<bullet> x = b} = {}"
lp15@62397
   421
proof -
lp15@62397
   422
  have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
lp15@62397
   423
    by (force simp:)
lp15@62397
   424
  then show ?thesis
lp15@62397
   425
    by (auto simp: assms)
lp15@62397
   426
qed
lp15@62397
   427
lp15@62397
   428
lemma frontier_halfspace_le:
lp15@62397
   429
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   430
    shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
lp15@62397
   431
proof (cases "a = 0")
lp15@62397
   432
  case True with assms show ?thesis by simp
lp15@62397
   433
next
lp15@62397
   434
  case False then show ?thesis
lp15@62397
   435
    by (force simp: frontier_def closed_halfspace_le)
lp15@62397
   436
qed
lp15@62397
   437
lp15@62397
   438
lemma frontier_halfspace_ge:
lp15@62397
   439
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   440
    shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"
lp15@62397
   441
proof (cases "a = 0")
lp15@62397
   442
  case True with assms show ?thesis by simp
lp15@62397
   443
next
lp15@62397
   444
  case False then show ?thesis
lp15@62397
   445
    by (force simp: frontier_def closed_halfspace_ge)
lp15@62397
   446
qed
lp15@62397
   447
lp15@62397
   448
lemma frontier_halfspace_lt:
lp15@62397
   449
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   450
    shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"
lp15@62397
   451
proof (cases "a = 0")
lp15@62397
   452
  case True with assms show ?thesis by simp
lp15@62397
   453
next
lp15@62397
   454
  case False then show ?thesis
lp15@62397
   455
    by (force simp: frontier_def interior_open open_halfspace_lt)
lp15@62397
   456
qed
lp15@62397
   457
lp15@62397
   458
lemma frontier_halfspace_gt:
lp15@62397
   459
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
lp15@62397
   460
    shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"
lp15@62397
   461
proof (cases "a = 0")
lp15@62397
   462
  case True with assms show ?thesis by simp
lp15@62397
   463
next
lp15@62397
   464
  case False then show ?thesis
lp15@62397
   465
    by (force simp: frontier_def interior_open open_halfspace_gt)
lp15@62397
   466
qed
lp15@62397
   467
lp15@62397
   468
lemma interior_standard_hyperplane:
lp15@62397
   469
   "interior {x :: (real,'n::finite) vec. x$k = a} = {}"
lp15@62397
   470
proof -
lp15@62397
   471
  have "axis k (1::real) \<noteq> 0"
lp15@62397
   472
    by (simp add: axis_def vec_eq_iff)
lp15@62397
   473
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
lp15@62397
   474
    by (simp add: cart_eq_inner_axis inner_commute)
lp15@62397
   475
  ultimately show ?thesis
lp15@62397
   476
    using interior_hyperplane [of "axis k (1::real)" a]
lp15@62397
   477
    by force
lp15@62397
   478
qed
lp15@62397
   479
wenzelm@60420
   480
subsection \<open>Matrix operations\<close>
hoelzl@37489
   481
wenzelm@60420
   482
text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close>
hoelzl@37489
   483
wenzelm@49644
   484
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
wenzelm@49644
   485
    (infixl "**" 70)
nipkow@64267
   486
  where "m ** m' == (\<chi> i j. sum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
hoelzl@37489
   487
wenzelm@49644
   488
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
wenzelm@49644
   489
    (infixl "*v" 70)
nipkow@64267
   490
  where "m *v x \<equiv> (\<chi> i. sum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
hoelzl@37489
   491
wenzelm@49644
   492
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
wenzelm@49644
   493
    (infixl "v*" 70)
nipkow@64267
   494
  where "v v* m == (\<chi> j. sum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
hoelzl@37489
   495
hoelzl@37489
   496
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
hoelzl@63332
   497
definition transpose where
hoelzl@37489
   498
  "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
hoelzl@37489
   499
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
hoelzl@37489
   500
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
hoelzl@37489
   501
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
hoelzl@37489
   502
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
hoelzl@37489
   503
hoelzl@37489
   504
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
hoelzl@37489
   505
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
nipkow@64267
   506
  by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
hoelzl@37489
   507
lp15@67673
   508
lemma matrix_mul_lid [simp]:
hoelzl@37489
   509
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   510
  shows "mat 1 ** A = A"
hoelzl@37489
   511
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   512
  apply vector
nipkow@64267
   513
  apply (auto simp only: if_distrib cond_application_beta sum.delta'[OF finite]
wenzelm@49644
   514
    mult_1_left mult_zero_left if_True UNIV_I)
wenzelm@49644
   515
  done
hoelzl@37489
   516
hoelzl@37489
   517
lp15@67673
   518
lemma matrix_mul_rid [simp]:
hoelzl@37489
   519
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   520
  shows "A ** mat 1 = A"
hoelzl@37489
   521
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   522
  apply vector
nipkow@64267
   523
  apply (auto simp only: if_distrib cond_application_beta sum.delta[OF finite]
wenzelm@49644
   524
    mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
wenzelm@49644
   525
  done
hoelzl@37489
   526
hoelzl@37489
   527
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
nipkow@64267
   528
  apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc)
haftmann@66804
   529
  apply (subst sum.swap)
hoelzl@37489
   530
  apply simp
hoelzl@37489
   531
  done
hoelzl@37489
   532
hoelzl@37489
   533
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
wenzelm@49644
   534
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def
nipkow@64267
   535
    sum_distrib_left sum_distrib_right mult.assoc)
haftmann@66804
   536
  apply (subst sum.swap)
hoelzl@37489
   537
  apply simp
hoelzl@37489
   538
  done
hoelzl@37489
   539
lp15@67673
   540
lemma matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
hoelzl@37489
   541
  apply (vector matrix_vector_mult_def mat_def)
nipkow@64267
   542
  apply (simp add: if_distrib cond_application_beta sum.delta' cong del: if_weak_cong)
wenzelm@49644
   543
  done
hoelzl@37489
   544
wenzelm@49644
   545
lemma matrix_transpose_mul:
wenzelm@49644
   546
    "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
haftmann@57512
   547
  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
hoelzl@37489
   548
hoelzl@37489
   549
lemma matrix_eq:
hoelzl@37489
   550
  fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
hoelzl@37489
   551
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@37489
   552
  apply auto
huffman@44136
   553
  apply (subst vec_eq_iff)
hoelzl@37489
   554
  apply clarify
hoelzl@50526
   555
  apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
hoelzl@50526
   556
  apply (erule_tac x="axis ia 1" in allE)
hoelzl@37489
   557
  apply (erule_tac x="i" in allE)
hoelzl@50526
   558
  apply (auto simp add: if_distrib cond_application_beta axis_def
nipkow@64267
   559
    sum.delta[OF finite] cong del: if_weak_cong)
wenzelm@49644
   560
  done
hoelzl@37489
   561
wenzelm@49644
   562
lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
huffman@44136
   563
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   564
hoelzl@37489
   565
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
nipkow@64267
   566
  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps)
haftmann@66804
   567
  apply (subst sum.swap)
wenzelm@49644
   568
  apply simp
wenzelm@49644
   569
  done
hoelzl@37489
   570
lp15@67673
   571
lemma transpose_mat [simp]: "transpose (mat n) = mat n"
hoelzl@37489
   572
  by (vector transpose_def mat_def)
hoelzl@37489
   573
lp15@67683
   574
lemma transpose_transpose [simp]: "transpose(transpose A) = A"
hoelzl@37489
   575
  by (vector transpose_def)
hoelzl@37489
   576
lp15@67673
   577
lemma row_transpose [simp]:
hoelzl@37489
   578
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   579
  shows "row i (transpose A) = column i A"
huffman@44136
   580
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   581
lp15@67673
   582
lemma column_transpose [simp]:
hoelzl@37489
   583
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   584
  shows "column i (transpose A) = row i A"
huffman@44136
   585
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   586
lp15@67683
   587
lemma rows_transpose [simp]: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
wenzelm@49644
   588
  by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
hoelzl@37489
   589
lp15@67683
   590
lemma columns_transpose [simp]: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
wenzelm@49644
   591
  by (metis transpose_transpose rows_transpose)
hoelzl@37489
   592
lp15@67673
   593
lemma matrix_mult_transpose_dot_column:
lp15@67673
   594
  fixes A :: "real^'n^'n"
lp15@67673
   595
  shows "transpose A ** A = (\<chi> i j. (column i A) \<bullet> (column j A))"
lp15@67673
   596
  by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)
lp15@67673
   597
lp15@67673
   598
lemma matrix_mult_transpose_dot_row:
lp15@67673
   599
  fixes A :: "real^'n^'n"
lp15@67673
   600
  shows "A ** transpose A = (\<chi> i j. (row i A) \<bullet> (row j A))"
lp15@67673
   601
  by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)
lp15@67673
   602
wenzelm@60420
   603
text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
hoelzl@37489
   604
hoelzl@37489
   605
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
huffman@44136
   606
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   607
lp15@67673
   608
lemma matrix_mult_sum:
nipkow@64267
   609
  "(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
haftmann@57512
   610
  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
hoelzl@37489
   611
hoelzl@37489
   612
lemma vector_componentwise:
hoelzl@50526
   613
  "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
nipkow@64267
   614
  by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)
hoelzl@50526
   615
nipkow@64267
   616
lemma basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
nipkow@64267
   617
  by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
hoelzl@37489
   618
lp15@63938
   619
lemma linear_componentwise_expansion:
hoelzl@37489
   620
  fixes f:: "real ^'m \<Rightarrow> real ^ _"
hoelzl@37489
   621
  assumes lf: "linear f"
nipkow@64267
   622
  shows "(f x)$j = sum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
wenzelm@49644
   623
proof -
hoelzl@37489
   624
  let ?M = "(UNIV :: 'm set)"
hoelzl@37489
   625
  let ?N = "(UNIV :: 'n set)"
nipkow@64267
   626
  have "?rhs = (sum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
nipkow@64267
   627
    unfolding sum_component by simp
wenzelm@49644
   628
  then show ?thesis
nipkow@64267
   629
    unfolding linear_sum_mul[OF lf, symmetric]
hoelzl@50526
   630
    unfolding scalar_mult_eq_scaleR[symmetric]
hoelzl@50526
   631
    unfolding basis_expansion
hoelzl@50526
   632
    by simp
hoelzl@37489
   633
qed
hoelzl@37489
   634
wenzelm@60420
   635
text\<open>Inverse matrices  (not necessarily square)\<close>
hoelzl@37489
   636
wenzelm@49644
   637
definition
wenzelm@49644
   638
  "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   639
wenzelm@49644
   640
definition
wenzelm@49644
   641
  "matrix_inv(A:: 'a::semiring_1^'n^'m) =
wenzelm@49644
   642
    (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   643
wenzelm@60420
   644
text\<open>Correspondence between matrices and linear operators.\<close>
hoelzl@37489
   645
wenzelm@49644
   646
definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
hoelzl@50526
   647
  where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
hoelzl@37489
   648
hoelzl@37489
   649
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
huffman@53600
   650
  by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
nipkow@64267
   651
      field_simps sum_distrib_left sum.distrib)
hoelzl@37489
   652
lp15@67683
   653
lemma
lp15@67683
   654
  fixes A :: "real^'n^'m"
lp15@67683
   655
  shows matrix_vector_mult_linear_continuous_at [continuous_intros]: "isCont (( *v) A) z"
lp15@67683
   656
    and matrix_vector_mult_linear_continuous_on [continuous_intros]: "continuous_on S (( *v) A)"
lp15@67683
   657
  by (simp_all add: linear_linear linear_continuous_at linear_continuous_on matrix_vector_mul_linear)
lp15@67683
   658
lp15@67673
   659
lemma matrix_vector_mult_add_distrib [algebra_simps]:
lp15@67673
   660
  fixes A :: "real^'n^'m"
lp15@67673
   661
  shows "A *v (x + y) = A *v x + A *v y"
lp15@67673
   662
  using matrix_vector_mul_linear [of A]  by (simp add: linear_add)
lp15@67673
   663
lp15@67673
   664
lemma matrix_vector_mult_diff_distrib [algebra_simps]:
lp15@67673
   665
  fixes A :: "real^'n^'m"
lp15@67673
   666
  shows "A *v (x - y) = A *v x - A *v y"
lp15@67673
   667
  using matrix_vector_mul_linear [of A]  by (simp add: linear_diff)
lp15@67673
   668
lp15@67673
   669
lemma matrix_vector_mult_scaleR[algebra_simps]:
lp15@67673
   670
  fixes A :: "real^'n^'m"
lp15@67673
   671
  shows "A *v (c *\<^sub>R x) = c *\<^sub>R (A *v x)"
lp15@67673
   672
  using linear_iff matrix_vector_mul_linear by blast
lp15@67673
   673
lp15@67673
   674
lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
lp15@67673
   675
  by (simp add: matrix_vector_mult_def vec_eq_iff)
lp15@67673
   676
lp15@67673
   677
lemma matrix_vector_mult_0 [simp]: "0 *v w = 0"
lp15@67673
   678
  by (simp add: matrix_vector_mult_def vec_eq_iff)
lp15@67673
   679
lp15@67673
   680
lemma matrix_vector_mult_add_rdistrib [algebra_simps]:
lp15@67673
   681
  fixes A :: "real^'n^'m"
lp15@67673
   682
  shows "(A + B) *v x = (A *v x) + (B *v x)"
lp15@67673
   683
  by (simp add: vec_eq_iff inner_add_left matrix_vector_mul_component)
lp15@67673
   684
lp15@67673
   685
lemma matrix_vector_mult_diff_rdistrib [algebra_simps]:
lp15@67673
   686
  fixes A :: "real^'n^'m"
lp15@67673
   687
  shows "(A - B) *v x = (A *v x) - (B *v x)"
lp15@67673
   688
  by (simp add: vec_eq_iff inner_diff_left matrix_vector_mul_component)
lp15@67673
   689
wenzelm@49644
   690
lemma matrix_works:
wenzelm@49644
   691
  assumes lf: "linear f"
wenzelm@49644
   692
  shows "matrix f *v x = f (x::real ^ 'n)"
haftmann@57512
   693
  apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)
lp15@63938
   694
  by (simp add: linear_componentwise_expansion lf)
hoelzl@37489
   695
wenzelm@49644
   696
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
wenzelm@49644
   697
  by (simp add: ext matrix_works)
hoelzl@37489
   698
lp15@67683
   699
declare matrix_vector_mul [symmetric, simp]
lp15@67683
   700
lp15@67673
   701
lemma matrix_of_matrix_vector_mul [simp]: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
hoelzl@37489
   702
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
hoelzl@37489
   703
hoelzl@37489
   704
lemma matrix_compose:
hoelzl@37489
   705
  assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
wenzelm@49644
   706
    and lg: "linear (g::real^'m \<Rightarrow> real^_)"
wenzelm@61736
   707
  shows "matrix (g \<circ> f) = matrix g ** matrix f"
hoelzl@37489
   708
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
wenzelm@49644
   709
  by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
hoelzl@37489
   710
wenzelm@49644
   711
lemma matrix_vector_column:
nipkow@64267
   712
  "(A::'a::comm_semiring_1^'n^_) *v x = sum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
haftmann@57512
   713
  by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
hoelzl@37489
   714
hoelzl@37489
   715
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
hoelzl@37489
   716
  apply (rule adjoint_unique)
wenzelm@49644
   717
  apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
nipkow@64267
   718
    sum_distrib_right sum_distrib_left)
haftmann@66804
   719
  apply (subst sum.swap)
haftmann@57514
   720
  apply (auto simp add: ac_simps)
hoelzl@37489
   721
  done
hoelzl@37489
   722
hoelzl@37489
   723
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
hoelzl@37489
   724
  shows "matrix(adjoint f) = transpose(matrix f)"
hoelzl@37489
   725
  apply (subst matrix_vector_mul[OF lf])
wenzelm@49644
   726
  unfolding adjoint_matrix matrix_of_matrix_vector_mul
wenzelm@49644
   727
  apply rule
wenzelm@49644
   728
  done
wenzelm@49644
   729
hoelzl@37489
   730
wenzelm@60420
   731
subsection \<open>lambda skolemization on cartesian products\<close>
hoelzl@37489
   732
hoelzl@37489
   733
(* FIXME: rename do choice_cart *)
hoelzl@37489
   734
hoelzl@37489
   735
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
hoelzl@37494
   736
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   737
proof -
hoelzl@37489
   738
  let ?S = "(UNIV :: 'n set)"
wenzelm@49644
   739
  { assume H: "?rhs"
wenzelm@49644
   740
    then have ?lhs by auto }
hoelzl@37489
   741
  moreover
wenzelm@49644
   742
  { assume H: "?lhs"
hoelzl@37489
   743
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
hoelzl@37489
   744
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
wenzelm@49644
   745
    { fix i
hoelzl@37489
   746
      from f have "P i (f i)" by metis
hoelzl@37494
   747
      then have "P i (?x $ i)" by auto
hoelzl@37489
   748
    }
hoelzl@37489
   749
    hence "\<forall>i. P i (?x$i)" by metis
hoelzl@37489
   750
    hence ?rhs by metis }
hoelzl@37489
   751
  ultimately show ?thesis by metis
hoelzl@37489
   752
qed
hoelzl@37489
   753
hoelzl@37489
   754
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
hoelzl@50526
   755
  unfolding inner_simps scalar_mult_eq_scaleR by auto
hoelzl@37489
   756
hoelzl@37489
   757
lemma left_invertible_transpose:
hoelzl@37489
   758
  "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
hoelzl@37489
   759
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   760
hoelzl@37489
   761
lemma right_invertible_transpose:
hoelzl@37489
   762
  "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
hoelzl@37489
   763
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   764
hoelzl@37489
   765
lemma matrix_left_invertible_injective:
wenzelm@49644
   766
  "(\<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
   767
proof -
wenzelm@49644
   768
  { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
hoelzl@37489
   769
    from xy have "B*v (A *v x) = B *v (A*v y)" by simp
hoelzl@37489
   770
    hence "x = y"
wenzelm@49644
   771
      unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
hoelzl@37489
   772
  moreover
wenzelm@49644
   773
  { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
nipkow@67399
   774
    hence i: "inj (( *v) A)" unfolding inj_on_def by auto
hoelzl@37489
   775
    from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
nipkow@67399
   776
    obtain g where g: "linear g" "g \<circ> ( *v) A = id" by blast
hoelzl@37489
   777
    have "matrix g ** A = mat 1"
hoelzl@37489
   778
      unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   779
      using g(2) by (simp add: fun_eq_iff)
wenzelm@49644
   780
    then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
hoelzl@37489
   781
  ultimately show ?thesis by blast
hoelzl@37489
   782
qed
hoelzl@37489
   783
hoelzl@37489
   784
lemma matrix_left_invertible_ker:
hoelzl@37489
   785
  "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
hoelzl@37489
   786
  unfolding matrix_left_invertible_injective
hoelzl@37489
   787
  using linear_injective_0[OF matrix_vector_mul_linear, of A]
hoelzl@37489
   788
  by (simp add: inj_on_def)
hoelzl@37489
   789
hoelzl@37489
   790
lemma matrix_right_invertible_surjective:
wenzelm@49644
   791
  "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
wenzelm@49644
   792
proof -
wenzelm@49644
   793
  { fix B :: "real ^'m^'n"
wenzelm@49644
   794
    assume AB: "A ** B = mat 1"
wenzelm@49644
   795
    { fix x :: "real ^ 'm"
hoelzl@37489
   796
      have "A *v (B *v x) = x"
wenzelm@49644
   797
        by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
nipkow@67399
   798
    hence "surj (( *v) A)" unfolding surj_def by metis }
hoelzl@37489
   799
  moreover
nipkow@67399
   800
  { assume sf: "surj (( *v) A)"
hoelzl@37489
   801
    from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
nipkow@67399
   802
    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "( *v) A \<circ> g = id"
hoelzl@37489
   803
      by blast
hoelzl@37489
   804
hoelzl@37489
   805
    have "A ** (matrix g) = mat 1"
hoelzl@37489
   806
      unfolding matrix_eq  matrix_vector_mul_lid
hoelzl@37489
   807
        matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   808
      using g(2) unfolding o_def fun_eq_iff id_def
hoelzl@37489
   809
      .
hoelzl@37489
   810
    hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
hoelzl@37489
   811
  }
hoelzl@37489
   812
  ultimately show ?thesis unfolding surj_def by blast
hoelzl@37489
   813
qed
hoelzl@37489
   814
hoelzl@37489
   815
lemma matrix_left_invertible_independent_columns:
hoelzl@37489
   816
  fixes A :: "real^'n^'m"
wenzelm@49644
   817
  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
nipkow@64267
   818
      (\<forall>c. sum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
wenzelm@49644
   819
    (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   820
proof -
hoelzl@37489
   821
  let ?U = "UNIV :: 'n set"
wenzelm@49644
   822
  { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
wenzelm@49644
   823
    { fix c i
nipkow@64267
   824
      assume c: "sum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
hoelzl@37489
   825
      let ?x = "\<chi> i. c i"
hoelzl@37489
   826
      have th0:"A *v ?x = 0"
hoelzl@37489
   827
        using c
lp15@67673
   828
        unfolding matrix_mult_sum vec_eq_iff
hoelzl@37489
   829
        by auto
hoelzl@37489
   830
      from k[rule_format, OF th0] i
huffman@44136
   831
      have "c i = 0" by (vector vec_eq_iff)}
wenzelm@49644
   832
    hence ?rhs by blast }
hoelzl@37489
   833
  moreover
wenzelm@49644
   834
  { assume H: ?rhs
wenzelm@49644
   835
    { fix x assume x: "A *v x = 0"
hoelzl@37489
   836
      let ?c = "\<lambda>i. ((x$i ):: real)"
lp15@67673
   837
      from H[rule_format, of ?c, unfolded matrix_mult_sum[symmetric], OF x]
wenzelm@49644
   838
      have "x = 0" by vector }
wenzelm@49644
   839
  }
hoelzl@37489
   840
  ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
hoelzl@37489
   841
qed
hoelzl@37489
   842
hoelzl@37489
   843
lemma matrix_right_invertible_independent_rows:
hoelzl@37489
   844
  fixes A :: "real^'n^'m"
wenzelm@49644
   845
  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
nipkow@64267
   846
    (\<forall>c. sum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
hoelzl@37489
   847
  unfolding left_invertible_transpose[symmetric]
hoelzl@37489
   848
    matrix_left_invertible_independent_columns
hoelzl@37489
   849
  by (simp add: column_transpose)
hoelzl@37489
   850
hoelzl@37489
   851
lemma matrix_right_invertible_span_columns:
wenzelm@49644
   852
  "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
wenzelm@49644
   853
    span (columns A) = UNIV" (is "?lhs = ?rhs")
wenzelm@49644
   854
proof -
hoelzl@37489
   855
  let ?U = "UNIV :: 'm set"
hoelzl@37489
   856
  have fU: "finite ?U" by simp
nipkow@64267
   857
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y)"
lp15@67673
   858
    unfolding matrix_right_invertible_surjective matrix_mult_sum surj_def
wenzelm@49644
   859
    apply (subst eq_commute)
wenzelm@49644
   860
    apply rule
wenzelm@49644
   861
    done
hoelzl@37489
   862
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
wenzelm@49644
   863
  { assume h: ?lhs
wenzelm@49644
   864
    { fix x:: "real ^'n"
wenzelm@49644
   865
      from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
nipkow@64267
   866
        where y: "sum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
wenzelm@49644
   867
      have "x \<in> span (columns A)"
wenzelm@49644
   868
        unfolding y[symmetric]
nipkow@64267
   869
        apply (rule span_sum)
hoelzl@50526
   870
        unfolding scalar_mult_eq_scaleR
wenzelm@49644
   871
        apply (rule span_mul)
wenzelm@49644
   872
        apply (rule span_superset)
wenzelm@49644
   873
        unfolding columns_def
wenzelm@49644
   874
        apply blast
wenzelm@49644
   875
        done
wenzelm@49644
   876
    }
wenzelm@49644
   877
    then have ?rhs unfolding rhseq by blast }
hoelzl@37489
   878
  moreover
wenzelm@49644
   879
  { assume h:?rhs
nipkow@64267
   880
    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y"
wenzelm@49644
   881
    { fix y
wenzelm@49644
   882
      have "?P y"
hoelzl@50526
   883
      proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
nipkow@64267
   884
        show "\<exists>x::real ^ 'm. sum (\<lambda>i. (x$i) *s column i A) ?U = 0"
hoelzl@37489
   885
          by (rule exI[where x=0], simp)
hoelzl@37489
   886
      next
wenzelm@49644
   887
        fix c y1 y2
wenzelm@49644
   888
        assume y1: "y1 \<in> columns A" and y2: "?P y2"
hoelzl@37489
   889
        from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
hoelzl@37489
   890
          unfolding columns_def by blast
hoelzl@37489
   891
        from y2 obtain x:: "real ^'m" where
nipkow@64267
   892
          x: "sum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
hoelzl@37489
   893
        let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
hoelzl@37489
   894
        show "?P (c*s y1 + y2)"
webertj@49962
   895
        proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)
wenzelm@49644
   896
          fix j
wenzelm@49644
   897
          have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
   898
              else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
wenzelm@49644
   899
            using i(1) by (simp add: field_simps)
nipkow@64267
   900
          have "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
nipkow@64267
   901
              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)$j))) ?U"
nipkow@64267
   902
            apply (rule sum.cong[OF refl])
wenzelm@49644
   903
            using th apply blast
wenzelm@49644
   904
            done
nipkow@64267
   905
          also have "\<dots> = sum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
nipkow@64267
   906
            by (simp add: sum.distrib)
nipkow@64267
   907
          also have "\<dots> = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
nipkow@64267
   908
            unfolding sum.delta[OF fU]
wenzelm@49644
   909
            using i(1) by simp
nipkow@64267
   910
          finally show "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
nipkow@64267
   911
            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
wenzelm@49644
   912
        qed
wenzelm@49644
   913
      next
wenzelm@49644
   914
        show "y \<in> span (columns A)"
wenzelm@49644
   915
          unfolding h by blast
wenzelm@49644
   916
      qed
wenzelm@49644
   917
    }
wenzelm@49644
   918
    then have ?lhs unfolding lhseq ..
wenzelm@49644
   919
  }
hoelzl@37489
   920
  ultimately show ?thesis by blast
hoelzl@37489
   921
qed
hoelzl@37489
   922
hoelzl@37489
   923
lemma matrix_left_invertible_span_rows:
hoelzl@37489
   924
  "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
hoelzl@37489
   925
  unfolding right_invertible_transpose[symmetric]
hoelzl@37489
   926
  unfolding columns_transpose[symmetric]
hoelzl@37489
   927
  unfolding matrix_right_invertible_span_columns
wenzelm@49644
   928
  ..
hoelzl@37489
   929
wenzelm@60420
   930
text \<open>The same result in terms of square matrices.\<close>
hoelzl@37489
   931
hoelzl@37489
   932
lemma matrix_left_right_inverse:
hoelzl@37489
   933
  fixes A A' :: "real ^'n^'n"
hoelzl@37489
   934
  shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
wenzelm@49644
   935
proof -
wenzelm@49644
   936
  { fix A A' :: "real ^'n^'n"
wenzelm@49644
   937
    assume AA': "A ** A' = mat 1"
nipkow@67399
   938
    have sA: "surj (( *v) A)"
hoelzl@37489
   939
      unfolding surj_def
hoelzl@37489
   940
      apply clarify
hoelzl@37489
   941
      apply (rule_tac x="(A' *v y)" in exI)
wenzelm@49644
   942
      apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
wenzelm@49644
   943
      done
hoelzl@37489
   944
    from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
hoelzl@37489
   945
    obtain f' :: "real ^'n \<Rightarrow> real ^'n"
hoelzl@37489
   946
      where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
hoelzl@37489
   947
    have th: "matrix f' ** A = mat 1"
wenzelm@49644
   948
      by (simp add: matrix_eq matrix_works[OF f'(1)]
wenzelm@49644
   949
          matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
hoelzl@37489
   950
    hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
wenzelm@49644
   951
    hence "matrix f' = A'"
wenzelm@49644
   952
      by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
hoelzl@37489
   953
    hence "matrix f' ** A = A' ** A" by simp
wenzelm@49644
   954
    hence "A' ** A = mat 1" by (simp add: th)
wenzelm@49644
   955
  }
hoelzl@37489
   956
  then show ?thesis by blast
hoelzl@37489
   957
qed
hoelzl@37489
   958
wenzelm@60420
   959
text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
hoelzl@37489
   960
hoelzl@37489
   961
definition "rowvector v = (\<chi> i j. (v$j))"
hoelzl@37489
   962
hoelzl@37489
   963
definition "columnvector v = (\<chi> i j. (v$i))"
hoelzl@37489
   964
wenzelm@49644
   965
lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
huffman@44136
   966
  by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
hoelzl@37489
   967
hoelzl@37489
   968
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
huffman@44136
   969
  by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
hoelzl@37489
   970
wenzelm@49644
   971
lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
hoelzl@37489
   972
  by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
hoelzl@37489
   973
wenzelm@49644
   974
lemma dot_matrix_product:
wenzelm@49644
   975
  "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
huffman@44136
   976
  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
hoelzl@37489
   977
hoelzl@37489
   978
lemma dot_matrix_vector_mul:
hoelzl@37489
   979
  fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
hoelzl@37489
   980
  shows "(A *v x) \<bullet> (B *v y) =
hoelzl@37489
   981
      (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
wenzelm@49644
   982
  unfolding dot_matrix_product transpose_columnvector[symmetric]
wenzelm@49644
   983
    dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
hoelzl@37489
   984
wenzelm@61945
   985
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}"
hoelzl@50526
   986
  by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
hoelzl@37489
   987
wenzelm@49644
   988
lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
hoelzl@50526
   989
  using Basis_le_infnorm[of "axis i 1" x]
hoelzl@50526
   990
  by (simp add: Basis_vec_def axis_eq_axis inner_axis)
hoelzl@37489
   991
hoelzl@63334
   992
lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
huffman@44647
   993
  unfolding continuous_def by (rule tendsto_vec_nth)
huffman@44213
   994
hoelzl@63334
   995
lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
huffman@44647
   996
  unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
huffman@44213
   997
hoelzl@63334
   998
lemma continuous_on_vec_lambda[continuous_intros]:
hoelzl@63334
   999
  "(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)"
hoelzl@63334
  1000
  unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)
hoelzl@63334
  1001
hoelzl@37489
  1002
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
hoelzl@63332
  1003
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
huffman@44213
  1004
hoelzl@37489
  1005
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
wenzelm@49644
  1006
  unfolding bounded_def
wenzelm@49644
  1007
  apply clarify
wenzelm@49644
  1008
  apply (rule_tac x="x $ i" in exI)
wenzelm@49644
  1009
  apply (rule_tac x="e" in exI)
wenzelm@49644
  1010
  apply clarify
wenzelm@49644
  1011
  apply (rule order_trans [OF dist_vec_nth_le], simp)
wenzelm@49644
  1012
  done
hoelzl@37489
  1013
hoelzl@37489
  1014
lemma compact_lemma_cart:
hoelzl@37489
  1015
  fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
hoelzl@50998
  1016
  assumes f: "bounded (range f)"
eberlm@66447
  1017
  shows "\<exists>l r. strict_mono r \<and>
hoelzl@37489
  1018
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
immler@62127
  1019
    (is "?th d")
immler@62127
  1020
proof -
immler@62127
  1021
  have "\<forall>d' \<subseteq> d. ?th d'"
immler@62127
  1022
    by (rule compact_lemma_general[where unproj=vec_lambda])
immler@62127
  1023
      (auto intro!: f bounded_component_cart simp: vec_lambda_eta)
immler@62127
  1024
  then show "?th d" by simp
hoelzl@37489
  1025
qed
hoelzl@37489
  1026
huffman@44136
  1027
instance vec :: (heine_borel, finite) heine_borel
hoelzl@37489
  1028
proof
hoelzl@50998
  1029
  fix f :: "nat \<Rightarrow> 'a ^ 'b"
hoelzl@50998
  1030
  assume f: "bounded (range f)"
eberlm@66447
  1031
  then obtain l r where r: "strict_mono r"
wenzelm@49644
  1032
      and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@50998
  1033
    using compact_lemma_cart [OF f] by blast
hoelzl@37489
  1034
  let ?d = "UNIV::'b set"
hoelzl@37489
  1035
  { fix e::real assume "e>0"
hoelzl@37489
  1036
    hence "0 < e / (real_of_nat (card ?d))"
wenzelm@49644
  1037
      using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
hoelzl@37489
  1038
    with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
hoelzl@37489
  1039
      by simp
hoelzl@37489
  1040
    moreover
wenzelm@49644
  1041
    { fix n
wenzelm@49644
  1042
      assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
hoelzl@37489
  1043
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
nipkow@67155
  1044
        unfolding dist_vec_def using zero_le_dist by (rule L2_set_le_sum)
hoelzl@37489
  1045
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
nipkow@64267
  1046
        by (rule sum_strict_mono) (simp_all add: n)
hoelzl@37489
  1047
      finally have "dist (f (r n)) l < e" by simp
hoelzl@37489
  1048
    }
hoelzl@37489
  1049
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
lp15@61810
  1050
      by (rule eventually_mono)
hoelzl@37489
  1051
  }
wenzelm@61973
  1052
  hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp
eberlm@66447
  1053
  with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
hoelzl@37489
  1054
qed
hoelzl@37489
  1055
wenzelm@49644
  1056
lemma interval_cart:
immler@54775
  1057
  fixes a :: "real^'n"
immler@54775
  1058
  shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
immler@56188
  1059
    and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
immler@56188
  1060
  by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
hoelzl@37489
  1061
lp15@67673
  1062
lemma mem_box_cart:
immler@54775
  1063
  fixes a :: "real^'n"
immler@54775
  1064
  shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
immler@56188
  1065
    and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
wenzelm@49644
  1066
  using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
hoelzl@37489
  1067
wenzelm@49644
  1068
lemma interval_eq_empty_cart:
wenzelm@49644
  1069
  fixes a :: "real^'n"
immler@54775
  1070
  shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
immler@56188
  1071
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
wenzelm@49644
  1072
proof -
immler@54775
  1073
  { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>box a b"
lp15@67673
  1074
    hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_box_cart by auto
hoelzl@37489
  1075
    hence "a$i < b$i" by auto
wenzelm@49644
  1076
    hence False using as by auto }
hoelzl@37489
  1077
  moreover
hoelzl@37489
  1078
  { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
hoelzl@37489
  1079
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
  1080
    { fix i
hoelzl@37489
  1081
      have "a$i < b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
  1082
      hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
hoelzl@37489
  1083
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
  1084
        by auto }
lp15@67673
  1085
    hence "box a b \<noteq> {}" using mem_box_cart(1)[of "?x" a b] by auto }
hoelzl@37489
  1086
  ultimately show ?th1 by blast
hoelzl@37489
  1087
immler@56188
  1088
  { fix i x assume as:"b$i < a$i" and x:"x\<in>cbox a b"
lp15@67673
  1089
    hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_box_cart by auto
hoelzl@37489
  1090
    hence "a$i \<le> b$i" by auto
wenzelm@49644
  1091
    hence False using as by auto }
hoelzl@37489
  1092
  moreover
hoelzl@37489
  1093
  { assume as:"\<forall>i. \<not> (b$i < a$i)"
hoelzl@37489
  1094
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
  1095
    { fix i
hoelzl@37489
  1096
      have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
  1097
      hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
hoelzl@37489
  1098
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
  1099
        by auto }
lp15@67673
  1100
    hence "cbox a b \<noteq> {}" using mem_box_cart(2)[of "?x" a b] by auto  }
hoelzl@37489
  1101
  ultimately show ?th2 by blast
hoelzl@37489
  1102
qed
hoelzl@37489
  1103
wenzelm@49644
  1104
lemma interval_ne_empty_cart:
wenzelm@49644
  1105
  fixes a :: "real^'n"
immler@56188
  1106
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
immler@54775
  1107
    and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
hoelzl@37489
  1108
  unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
hoelzl@37489
  1109
    (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
  1110
wenzelm@49644
  1111
lemma subset_interval_imp_cart:
wenzelm@49644
  1112
  fixes a :: "real^'n"
immler@56188
  1113
  shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
immler@56188
  1114
    and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
immler@56188
  1115
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
immler@54775
  1116
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b"
lp15@67673
  1117
  unfolding subset_eq[unfolded Ball_def] unfolding mem_box_cart
hoelzl@37489
  1118
  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
  1119
wenzelm@49644
  1120
lemma interval_sing:
wenzelm@49644
  1121
  fixes a :: "'a::linorder^'n"
wenzelm@49644
  1122
  shows "{a .. a} = {a} \<and> {a<..<a} = {}"
wenzelm@49644
  1123
  apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
wenzelm@49644
  1124
  done
hoelzl@37489
  1125
wenzelm@49644
  1126
lemma subset_interval_cart:
wenzelm@49644
  1127
  fixes a :: "real^'n"
immler@56188
  1128
  shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
immler@56188
  1129
    and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
immler@56188
  1130
    and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
immler@54775
  1131
    and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
immler@56188
  1132
  using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
  1133
wenzelm@49644
  1134
lemma disjoint_interval_cart:
wenzelm@49644
  1135
  fixes a::"real^'n"
immler@56188
  1136
  shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
immler@56188
  1137
    and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
immler@56188
  1138
    and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
immler@54775
  1139
    and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
hoelzl@50526
  1140
  using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
  1141
wenzelm@49644
  1142
lemma inter_interval_cart:
immler@54775
  1143
  fixes a :: "real^'n"
immler@56188
  1144
  shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
lp15@63945
  1145
  unfolding Int_interval
immler@56188
  1146
  by (auto simp: mem_box less_eq_vec_def)
immler@56188
  1147
    (auto simp: Basis_vec_def inner_axis)
hoelzl@37489
  1148
wenzelm@49644
  1149
lemma closed_interval_left_cart:
wenzelm@49644
  1150
  fixes b :: "real^'n"
hoelzl@37489
  1151
  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
hoelzl@63332
  1152
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1153
wenzelm@49644
  1154
lemma closed_interval_right_cart:
wenzelm@49644
  1155
  fixes a::"real^'n"
hoelzl@37489
  1156
  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
hoelzl@63332
  1157
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1158
wenzelm@49644
  1159
lemma is_interval_cart:
wenzelm@49644
  1160
  "is_interval (s::(real^'n) set) \<longleftrightarrow>
wenzelm@49644
  1161
    (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
hoelzl@50526
  1162
  by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
hoelzl@37489
  1163
wenzelm@49644
  1164
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
hoelzl@63332
  1165
  by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1166
wenzelm@49644
  1167
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
hoelzl@63332
  1168
  by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1169
wenzelm@49644
  1170
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
hoelzl@63332
  1171
  by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
wenzelm@49644
  1172
wenzelm@49644
  1173
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
hoelzl@63332
  1174
  by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
hoelzl@37489
  1175
wenzelm@49644
  1176
lemma Lim_component_le_cart:
wenzelm@49644
  1177
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
  1178
  assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"
hoelzl@37489
  1179
  shows "l$i \<le> b"
hoelzl@50526
  1180
  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
hoelzl@37489
  1181
wenzelm@49644
  1182
lemma Lim_component_ge_cart:
wenzelm@49644
  1183
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
  1184
  assumes "(f \<longlongrightarrow> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
hoelzl@37489
  1185
  shows "b \<le> l$i"
hoelzl@50526
  1186
  by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
hoelzl@37489
  1187
wenzelm@49644
  1188
lemma Lim_component_eq_cart:
wenzelm@49644
  1189
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@61973
  1190
  assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
hoelzl@37489
  1191
  shows "l$i = b"
wenzelm@49644
  1192
  using ev[unfolded order_eq_iff eventually_conj_iff] and
wenzelm@49644
  1193
    Lim_component_ge_cart[OF net, of b i] and
hoelzl@37489
  1194
    Lim_component_le_cart[OF net, of i b] by auto
hoelzl@37489
  1195
wenzelm@49644
  1196
lemma connected_ivt_component_cart:
wenzelm@49644
  1197
  fixes x :: "real^'n"
wenzelm@49644
  1198
  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$k = a)"
hoelzl@50526
  1199
  using connected_ivt_hyperplane[of s x y "axis k 1" a]
hoelzl@50526
  1200
  by (auto simp add: inner_axis inner_commute)
hoelzl@37489
  1201
wenzelm@49644
  1202
lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
hoelzl@37489
  1203
  unfolding subspace_def by auto
hoelzl@37489
  1204
hoelzl@37489
  1205
lemma closed_substandard_cart:
huffman@44213
  1206
  "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
  1207
proof -
huffman@44213
  1208
  { fix i::'n
huffman@44213
  1209
    have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
hoelzl@63332
  1210
      by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) }
huffman@44213
  1211
  thus ?thesis
huffman@44213
  1212
    unfolding Collect_all_eq by (simp add: closed_INT)
hoelzl@37489
  1213
qed
hoelzl@37489
  1214
wenzelm@49644
  1215
lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
wenzelm@49644
  1216
  (is "dim ?A = _")
wenzelm@49644
  1217
proof -
hoelzl@50526
  1218
  let ?a = "\<lambda>x. axis x 1 :: real^'n"
hoelzl@50526
  1219
  have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
hoelzl@50526
  1220
    by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
hoelzl@50526
  1221
  have "?a ` d \<subseteq> Basis"
hoelzl@50526
  1222
    by (auto simp: Basis_vec_def)
wenzelm@49644
  1223
  thus ?thesis
hoelzl@50526
  1224
    using dim_substandard[of "?a ` d"] card_image[of ?a d]
hoelzl@50526
  1225
    by (auto simp: axis_eq_axis inj_on_def *)
hoelzl@37489
  1226
qed
hoelzl@37489
  1227
hoelzl@37489
  1228
lemma affinity_inverses:
hoelzl@37489
  1229
  assumes m0: "m \<noteq> (0::'a::field)"
wenzelm@61736
  1230
  shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
wenzelm@61736
  1231
  "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
hoelzl@37489
  1232
  using m0
haftmann@54230
  1233
  apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
haftmann@54230
  1234
  apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
wenzelm@49644
  1235
  done
hoelzl@37489
  1236
hoelzl@37489
  1237
lemma vector_affinity_eq:
hoelzl@37489
  1238
  assumes m0: "(m::'a::field) \<noteq> 0"
hoelzl@37489
  1239
  shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
hoelzl@37489
  1240
proof
hoelzl@37489
  1241
  assume h: "m *s x + c = y"
hoelzl@37489
  1242
  hence "m *s x = y - c" by (simp add: field_simps)
hoelzl@37489
  1243
  hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
hoelzl@37489
  1244
  then show "x = inverse m *s y + - (inverse m *s c)"
hoelzl@37489
  1245
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1246
next
hoelzl@37489
  1247
  assume h: "x = inverse m *s y + - (inverse m *s c)"
haftmann@54230
  1248
  show "m *s x + c = y" unfolding h
hoelzl@37489
  1249
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1250
qed
hoelzl@37489
  1251
hoelzl@37489
  1252
lemma vector_eq_affinity:
wenzelm@49644
  1253
    "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
hoelzl@37489
  1254
  using vector_affinity_eq[where m=m and x=x and y=y and c=c]
hoelzl@37489
  1255
  by metis
hoelzl@37489
  1256
hoelzl@50526
  1257
lemma vector_cart:
hoelzl@50526
  1258
  fixes f :: "real^'n \<Rightarrow> real"
hoelzl@50526
  1259
  shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
hoelzl@50526
  1260
  unfolding euclidean_eq_iff[where 'a="real^'n"]
hoelzl@50526
  1261
  by simp (simp add: Basis_vec_def inner_axis)
hoelzl@63332
  1262
hoelzl@50526
  1263
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
hoelzl@50526
  1264
  by (rule vector_cart)
wenzelm@49644
  1265
huffman@44360
  1266
subsection "Convex Euclidean Space"
hoelzl@37489
  1267
hoelzl@50526
  1268
lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
hoelzl@50526
  1269
  using const_vector_cart[of 1] by (simp add: one_vec_def)
hoelzl@37489
  1270
hoelzl@37489
  1271
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
hoelzl@37489
  1272
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
hoelzl@37489
  1273
hoelzl@50526
  1274
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
  1275
hoelzl@37489
  1276
lemma convex_box_cart:
hoelzl@37489
  1277
  assumes "\<And>i. convex {x. P i x}"
hoelzl@37489
  1278
  shows "convex {x. \<forall>i. P i (x$i)}"
hoelzl@37489
  1279
  using assms unfolding convex_def by auto
hoelzl@37489
  1280
hoelzl@37489
  1281
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
hoelzl@63334
  1282
  by (rule convex_box_cart) (simp add: atLeast_def[symmetric])
hoelzl@37489
  1283
hoelzl@37489
  1284
lemma unit_interval_convex_hull_cart:
immler@56188
  1285
  "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}"
immler@56188
  1286
  unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
hoelzl@50526
  1287
  by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
hoelzl@37489
  1288
hoelzl@37489
  1289
lemma cube_convex_hull_cart:
wenzelm@49644
  1290
  assumes "0 < d"
wenzelm@49644
  1291
  obtains s::"(real^'n) set"
immler@56188
  1292
    where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
wenzelm@49644
  1293
proof -
wenzelm@55522
  1294
  from assms obtain s where "finite s"
nipkow@67399
  1295
    and "cbox (x - sum (( *\<^sub>R) d) Basis) (x + sum (( *\<^sub>R) d) Basis) = convex hull s"
wenzelm@55522
  1296
    by (rule cube_convex_hull)
wenzelm@55522
  1297
  with that[of s] show thesis
wenzelm@55522
  1298
    by (simp add: const_vector_cart)
hoelzl@37489
  1299
qed
hoelzl@37489
  1300
hoelzl@37489
  1301
hoelzl@37489
  1302
subsection "Derivative"
hoelzl@37489
  1303
hoelzl@37489
  1304
definition "jacobian f net = matrix(frechet_derivative f net)"
hoelzl@37489
  1305
wenzelm@49644
  1306
lemma jacobian_works:
wenzelm@49644
  1307
  "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
wenzelm@49644
  1308
    (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
wenzelm@49644
  1309
  apply rule
wenzelm@49644
  1310
  unfolding jacobian_def
wenzelm@49644
  1311
  apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
wenzelm@49644
  1312
  apply (rule differentiableI)
wenzelm@49644
  1313
  apply assumption
wenzelm@49644
  1314
  unfolding frechet_derivative_works
wenzelm@49644
  1315
  apply assumption
wenzelm@49644
  1316
  done
hoelzl@37489
  1317
hoelzl@37489
  1318
wenzelm@60420
  1319
subsection \<open>Component of the differential must be zero if it exists at a local
wenzelm@60420
  1320
  maximum or minimum for that corresponding component.\<close>
hoelzl@37489
  1321
hoelzl@50526
  1322
lemma differential_zero_maxmin_cart:
wenzelm@49644
  1323
  fixes f::"real^'a \<Rightarrow> real^'b"
wenzelm@49644
  1324
  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
  1325
    "f differentiable (at x)"
hoelzl@50526
  1326
  shows "jacobian f (at x) $ k = 0"
hoelzl@50526
  1327
  using differential_zero_maxmin_component[of "axis k 1" e x f] assms
hoelzl@50526
  1328
    vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
hoelzl@50526
  1329
  by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
wenzelm@49644
  1330
wenzelm@60420
  1331
subsection \<open>Lemmas for working on @{typ "real^1"}\<close>
hoelzl@37489
  1332
hoelzl@37489
  1333
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
wenzelm@49644
  1334
  by (metis (full_types) num1_eq_iff)
hoelzl@37489
  1335
hoelzl@37489
  1336
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
wenzelm@49644
  1337
  by auto (metis (full_types) num1_eq_iff)
hoelzl@37489
  1338
hoelzl@37489
  1339
lemma exhaust_2:
wenzelm@49644
  1340
  fixes x :: 2
wenzelm@49644
  1341
  shows "x = 1 \<or> x = 2"
hoelzl@37489
  1342
proof (induct x)
hoelzl@37489
  1343
  case (of_int z)
hoelzl@37489
  1344
  then have "0 <= z" and "z < 2" by simp_all
hoelzl@37489
  1345
  then have "z = 0 | z = 1" by arith
hoelzl@37489
  1346
  then show ?case by auto
hoelzl@37489
  1347
qed
hoelzl@37489
  1348
hoelzl@37489
  1349
lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
hoelzl@37489
  1350
  by (metis exhaust_2)
hoelzl@37489
  1351
hoelzl@37489
  1352
lemma exhaust_3:
wenzelm@49644
  1353
  fixes x :: 3
wenzelm@49644
  1354
  shows "x = 1 \<or> x = 2 \<or> x = 3"
hoelzl@37489
  1355
proof (induct x)
hoelzl@37489
  1356
  case (of_int z)
hoelzl@37489
  1357
  then have "0 <= z" and "z < 3" by simp_all
hoelzl@37489
  1358
  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
hoelzl@37489
  1359
  then show ?case by auto
hoelzl@37489
  1360
qed
hoelzl@37489
  1361
hoelzl@37489
  1362
lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
hoelzl@37489
  1363
  by (metis exhaust_3)
hoelzl@37489
  1364
hoelzl@37489
  1365
lemma UNIV_1 [simp]: "UNIV = {1::1}"
hoelzl@37489
  1366
  by (auto simp add: num1_eq_iff)
hoelzl@37489
  1367
hoelzl@37489
  1368
lemma UNIV_2: "UNIV = {1::2, 2::2}"
hoelzl@37489
  1369
  using exhaust_2 by auto
hoelzl@37489
  1370
hoelzl@37489
  1371
lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
hoelzl@37489
  1372
  using exhaust_3 by auto
hoelzl@37489
  1373
nipkow@64267
  1374
lemma sum_1: "sum f (UNIV::1 set) = f 1"
hoelzl@37489
  1375
  unfolding UNIV_1 by simp
hoelzl@37489
  1376
nipkow@64267
  1377
lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2"
hoelzl@37489
  1378
  unfolding UNIV_2 by simp
hoelzl@37489
  1379
nipkow@64267
  1380
lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3"
haftmann@57514
  1381
  unfolding UNIV_3 by (simp add: ac_simps)
hoelzl@37489
  1382
wenzelm@49644
  1383
instantiation num1 :: cart_one
wenzelm@49644
  1384
begin
wenzelm@49644
  1385
wenzelm@49644
  1386
instance
wenzelm@49644
  1387
proof
hoelzl@37489
  1388
  show "CARD(1) = Suc 0" by auto
wenzelm@49644
  1389
qed
wenzelm@49644
  1390
wenzelm@49644
  1391
end
hoelzl@37489
  1392
wenzelm@60420
  1393
subsection\<open>The collapse of the general concepts to dimension one.\<close>
hoelzl@37489
  1394
hoelzl@37489
  1395
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
huffman@44136
  1396
  by (simp add: vec_eq_iff)
hoelzl@37489
  1397
hoelzl@37489
  1398
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
hoelzl@37489
  1399
  apply auto
hoelzl@37489
  1400
  apply (erule_tac x= "x$1" in allE)
hoelzl@37489
  1401
  apply (simp only: vector_one[symmetric])
hoelzl@37489
  1402
  done
hoelzl@37489
  1403
hoelzl@37489
  1404
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
huffman@44136
  1405
  by (simp add: norm_vec_def)
hoelzl@37489
  1406
wenzelm@61945
  1407
lemma norm_real: "norm(x::real ^ 1) = \<bar>x$1\<bar>"
hoelzl@37489
  1408
  by (simp add: norm_vector_1)
hoelzl@37489
  1409
wenzelm@61945
  1410
lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x$1) - (y$1)\<bar>"
hoelzl@37489
  1411
  by (auto simp add: norm_real dist_norm)
hoelzl@37489
  1412
wenzelm@49644
  1413
wenzelm@60420
  1414
subsection\<open>Explicit vector construction from lists.\<close>
hoelzl@37489
  1415
hoelzl@43995
  1416
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
  1417
hoelzl@37489
  1418
lemma vector_1: "(vector[x]) $1 = x"
hoelzl@37489
  1419
  unfolding vector_def by simp
hoelzl@37489
  1420
hoelzl@37489
  1421
lemma vector_2:
hoelzl@37489
  1422
 "(vector[x,y]) $1 = x"
hoelzl@37489
  1423
 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
hoelzl@37489
  1424
  unfolding vector_def by simp_all
hoelzl@37489
  1425
hoelzl@37489
  1426
lemma vector_3:
hoelzl@37489
  1427
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
hoelzl@37489
  1428
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
hoelzl@37489
  1429
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
hoelzl@37489
  1430
  unfolding vector_def by simp_all
hoelzl@37489
  1431
hoelzl@37489
  1432
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
hoelzl@37489
  1433
  apply auto
hoelzl@37489
  1434
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1435
  apply (subgoal_tac "vector [v$1] = v")
hoelzl@37489
  1436
  apply simp
hoelzl@37489
  1437
  apply (vector vector_def)
hoelzl@37489
  1438
  apply simp
hoelzl@37489
  1439
  done
hoelzl@37489
  1440
hoelzl@37489
  1441
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
hoelzl@37489
  1442
  apply auto
hoelzl@37489
  1443
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1444
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1445
  apply (subgoal_tac "vector [v$1, v$2] = v")
hoelzl@37489
  1446
  apply simp
hoelzl@37489
  1447
  apply (vector vector_def)
hoelzl@37489
  1448
  apply (simp add: forall_2)
hoelzl@37489
  1449
  done
hoelzl@37489
  1450
hoelzl@37489
  1451
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
hoelzl@37489
  1452
  apply auto
hoelzl@37489
  1453
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1454
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1455
  apply (erule_tac x="v$3" in allE)
hoelzl@37489
  1456
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
hoelzl@37489
  1457
  apply simp
hoelzl@37489
  1458
  apply (vector vector_def)
hoelzl@37489
  1459
  apply (simp add: forall_3)
hoelzl@37489
  1460
  done
hoelzl@37489
  1461
hoelzl@37489
  1462
lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
wenzelm@49644
  1463
  apply (rule bounded_linearI[where K=1])
hoelzl@37489
  1464
  using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
hoelzl@37489
  1465
hoelzl@37489
  1466
lemma interval_split_cart:
hoelzl@37489
  1467
  "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
immler@56188
  1468
  "cbox a b \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
wenzelm@49644
  1469
  apply (rule_tac[!] set_eqI)
lp15@67673
  1470
  unfolding Int_iff mem_box_cart mem_Collect_eq interval_cbox_cart
wenzelm@49644
  1471
  unfolding vec_lambda_beta
wenzelm@49644
  1472
  by auto
hoelzl@37489
  1473
hoelzl@37489
  1474
end