src/HOL/Analysis/Cartesian_Euclidean_Space.thy
author nipkow
Wed, 10 Jan 2018 15:25:09 +0100
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child 67673 c8caefb20564
permissions -rw-r--r--
ran isabelle update_op on all sources
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21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
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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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diff changeset
   276
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   277
  by (simp add: vec_eq_iff)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   278
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   279
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   280
lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   281
  by vector
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   282
lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   283
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   284
lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   285
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   286
lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   287
  by (metis vector_mul_lcancel)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   288
lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   289
  by (metis vector_mul_rcancel)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   290
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   291
lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   292
  apply (simp add: norm_vec_def)
67155
9e5b05d54f9d canonical name
nipkow
parents: 66804
diff changeset
   293
  apply (rule member_le_L2_set, simp_all)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   294
  done
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   295
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   296
lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x$i\<bar> <= e"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   297
  by (metis component_le_norm_cart order_trans)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   298
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   299
lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
53595
5078034ade16 prefer theorem name over 'long_thm_list(n)'
huffman
parents: 53593
diff changeset
   300
  by (metis component_le_norm_cart le_less_trans)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   301
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   302
lemma norm_le_l1_cart: "norm x <= sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
67155
9e5b05d54f9d canonical name
nipkow
parents: 66804
diff changeset
   303
  by (simp add: norm_vec_def L2_set_le_sum)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   304
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   305
lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   306
  unfolding scaleR_vec_def vector_scalar_mult_def by simp
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   307
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   308
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   309
  unfolding dist_norm scalar_mult_eq_scaleR
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   310
  unfolding scaleR_right_diff_distrib[symmetric] by simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   311
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   312
lemma sum_component [simp]:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   313
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   314
  shows "(sum f S)$i = sum (\<lambda>x. (f x)$i) S"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   315
proof (cases "finite S")
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   316
  case True
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   317
  then show ?thesis by induct simp_all
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   318
next
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   319
  case False
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   320
  then show ?thesis by simp
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   321
qed
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   322
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   323
lemma sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)$i ) S)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   324
  by (simp add: vec_eq_iff)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   325
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   326
lemma sum_cmul:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   327
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   328
  shows "sum (\<lambda>x. c *s f x) S = c *s sum f S"
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   329
  by (simp add: vec_eq_iff sum_distrib_left)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   330
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   331
lemma sum_norm_allsubsets_bound_cart:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   332
  fixes f:: "'a \<Rightarrow> real ^'n"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   333
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   334
  shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   335
  using sum_norm_allsubsets_bound[OF assms]
57865
dcfb33c26f50 tuned proofs -- fewer warnings;
wenzelm
parents: 57514
diff changeset
   336
  by simp
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   337
62397
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   338
subsection\<open>Closures and interiors of halfspaces\<close>
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   339
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   340
lemma interior_halfspace_le [simp]:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   341
  assumes "a \<noteq> 0"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   342
    shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   343
proof -
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   344
  have *: "a \<bullet> x < b" if x: "x \<in> S" and S: "S \<subseteq> {x. a \<bullet> x \<le> b}" and "open S" for S x
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   345
  proof -
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   346
    obtain e where "e>0" and e: "cball x e \<subseteq> S"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   347
      using \<open>open S\<close> open_contains_cball x by blast
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   348
    then have "x + (e / norm a) *\<^sub>R a \<in> cball x e"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   349
      by (simp add: dist_norm)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   350
    then have "x + (e / norm a) *\<^sub>R a \<in> S"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   351
      using e by blast
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   352
    then have "x + (e / norm a) *\<^sub>R a \<in> {x. a \<bullet> x \<le> b}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   353
      using S by blast
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   354
    moreover have "e * (a \<bullet> a) / norm a > 0"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   355
      by (simp add: \<open>0 < e\<close> assms)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   356
    ultimately show ?thesis
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   357
      by (simp add: algebra_simps)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   358
  qed
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   359
  show ?thesis
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   360
    by (rule interior_unique) (auto simp: open_halfspace_lt *)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   361
qed
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   362
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   363
lemma interior_halfspace_ge [simp]:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   364
   "a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   365
using interior_halfspace_le [of "-a" "-b"] by simp
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   366
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   367
lemma interior_halfspace_component_le [simp]:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   368
     "interior {x. x$k \<le> a} = {x :: (real,'n::finite) vec. x$k < a}" (is "?LE")
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   369
  and interior_halfspace_component_ge [simp]:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   370
     "interior {x. x$k \<ge> a} = {x :: (real,'n::finite) vec. x$k > a}" (is "?GE")
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   371
proof -
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   372
  have "axis k (1::real) \<noteq> 0"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   373
    by (simp add: axis_def vec_eq_iff)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   374
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   375
    by (simp add: cart_eq_inner_axis inner_commute)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   376
  ultimately show ?LE ?GE
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   377
    using interior_halfspace_le [of "axis k (1::real)" a]
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   378
          interior_halfspace_ge [of "axis k (1::real)" a] by auto
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   379
qed
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   380
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   381
lemma closure_halfspace_lt [simp]:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   382
  assumes "a \<noteq> 0"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   383
    shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   384
proof -
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   385
  have [simp]: "-{x. a \<bullet> x < b} = {x. a \<bullet> x \<ge> b}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   386
    by (force simp:)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   387
  then show ?thesis
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   388
    using interior_halfspace_ge [of a b] assms
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   389
    by (force simp: closure_interior)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   390
qed
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   391
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   392
lemma closure_halfspace_gt [simp]:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   393
   "a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   394
using closure_halfspace_lt [of "-a" "-b"] by simp
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   395
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   396
lemma closure_halfspace_component_lt [simp]:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   397
     "closure {x. x$k < a} = {x :: (real,'n::finite) vec. x$k \<le> a}" (is "?LE")
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   398
  and closure_halfspace_component_gt [simp]:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   399
     "closure {x. x$k > a} = {x :: (real,'n::finite) vec. x$k \<ge> a}" (is "?GE")
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   400
proof -
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   401
  have "axis k (1::real) \<noteq> 0"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   402
    by (simp add: axis_def vec_eq_iff)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   403
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   404
    by (simp add: cart_eq_inner_axis inner_commute)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   405
  ultimately show ?LE ?GE
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   406
    using closure_halfspace_lt [of "axis k (1::real)" a]
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   407
          closure_halfspace_gt [of "axis k (1::real)" a] by auto
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   408
qed
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   409
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   410
lemma interior_hyperplane [simp]:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   411
  assumes "a \<noteq> 0"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   412
    shows "interior {x. a \<bullet> x = b} = {}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   413
proof -
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   414
  have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   415
    by (force simp:)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   416
  then show ?thesis
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   417
    by (auto simp: assms)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   418
qed
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   419
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   420
lemma frontier_halfspace_le:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   421
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   422
    shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   423
proof (cases "a = 0")
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   424
  case True with assms show ?thesis by simp
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   425
next
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   426
  case False then show ?thesis
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   427
    by (force simp: frontier_def closed_halfspace_le)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   428
qed
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   429
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   430
lemma frontier_halfspace_ge:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   431
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   432
    shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   433
proof (cases "a = 0")
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   434
  case True with assms show ?thesis by simp
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   435
next
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   436
  case False then show ?thesis
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   437
    by (force simp: frontier_def closed_halfspace_ge)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   438
qed
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   439
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   440
lemma frontier_halfspace_lt:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   441
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   442
    shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   443
proof (cases "a = 0")
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   444
  case True with assms show ?thesis by simp
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   445
next
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   446
  case False then show ?thesis
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   447
    by (force simp: frontier_def interior_open open_halfspace_lt)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   448
qed
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   449
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   450
lemma frontier_halfspace_gt:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   451
  assumes "a \<noteq> 0 \<or> b \<noteq> 0"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   452
    shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   453
proof (cases "a = 0")
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   454
  case True with assms show ?thesis by simp
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   455
next
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   456
  case False then show ?thesis
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   457
    by (force simp: frontier_def interior_open open_halfspace_gt)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   458
qed
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   459
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   460
lemma interior_standard_hyperplane:
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   461
   "interior {x :: (real,'n::finite) vec. x$k = a} = {}"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   462
proof -
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   463
  have "axis k (1::real) \<noteq> 0"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   464
    by (simp add: axis_def vec_eq_iff)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   465
  moreover have "axis k (1::real) \<bullet> x = x$k" for x
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   466
    by (simp add: cart_eq_inner_axis inner_commute)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   467
  ultimately show ?thesis
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   468
    using interior_hyperplane [of "axis k (1::real)" a]
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   469
    by force
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   470
qed
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62127
diff changeset
   471
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 58877
diff changeset
   472
subsection \<open>Matrix operations\<close>
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   473
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 58877
diff changeset
   474
text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close>
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   475
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   476
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   477
    (infixl "**" 70)
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   478
  where "m ** m' == (\<chi> i j. sum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   479
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   480
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   481
    (infixl "*v" 70)
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   482
  where "m *v x \<equiv> (\<chi> i. sum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   483
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   484
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   485
    (infixl "v*" 70)
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   486
  where "v v* m == (\<chi> j. sum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   487
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   488
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
63332
f164526d8727 move open_Collect_eq/less to HOL
hoelzl
parents: 63075
diff changeset
   489
definition transpose where
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   490
  "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   491
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   492
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   493
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   494
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   495
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   496
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   497
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   498
  by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   499
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   500
lemma matrix_mul_lid:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   501
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   502
  shows "mat 1 ** A = A"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   503
  apply (simp add: matrix_matrix_mult_def mat_def)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   504
  apply vector
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   505
  apply (auto simp only: if_distrib cond_application_beta sum.delta'[OF finite]
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   506
    mult_1_left mult_zero_left if_True UNIV_I)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   507
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   508
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   509
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   510
lemma matrix_mul_rid:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   511
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   512
  shows "A ** mat 1 = A"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   513
  apply (simp add: matrix_matrix_mult_def mat_def)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   514
  apply vector
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   515
  apply (auto simp only: if_distrib cond_application_beta sum.delta[OF finite]
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   516
    mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   517
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   518
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   519
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   520
  apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc)
66804
3f9bb52082c4 avoid name clashes on interpretation of abstract locales
haftmann
parents: 66447
diff changeset
   521
  apply (subst sum.swap)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   522
  apply simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   523
  done
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   524
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   525
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   526
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   527
    sum_distrib_left sum_distrib_right mult.assoc)
66804
3f9bb52082c4 avoid name clashes on interpretation of abstract locales
haftmann
parents: 66447
diff changeset
   528
  apply (subst sum.swap)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   529
  apply simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   530
  done
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   531
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   532
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   533
  apply (vector matrix_vector_mult_def mat_def)
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   534
  apply (simp add: if_distrib cond_application_beta sum.delta' cong del: if_weak_cong)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   535
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   536
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   537
lemma matrix_transpose_mul:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   538
    "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   539
  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   540
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   541
lemma matrix_eq:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   542
  fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   543
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   544
  apply auto
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   545
  apply (subst vec_eq_iff)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   546
  apply clarify
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   547
  apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   548
  apply (erule_tac x="axis ia 1" in allE)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   549
  apply (erule_tac x="i" in allE)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   550
  apply (auto simp add: if_distrib cond_application_beta axis_def
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   551
    sum.delta[OF finite] cong del: if_weak_cong)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   552
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   553
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   554
lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   555
  by (simp add: matrix_vector_mult_def inner_vec_def)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   556
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   557
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   558
  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps)
66804
3f9bb52082c4 avoid name clashes on interpretation of abstract locales
haftmann
parents: 66447
diff changeset
   559
  apply (subst sum.swap)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   560
  apply simp
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   561
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   562
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   563
lemma transpose_mat: "transpose (mat n) = mat n"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   564
  by (vector transpose_def mat_def)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   565
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   566
lemma transpose_transpose: "transpose(transpose A) = A"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   567
  by (vector transpose_def)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   568
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   569
lemma row_transpose:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   570
  fixes A:: "'a::semiring_1^_^_"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   571
  shows "row i (transpose A) = column i A"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   572
  by (simp add: row_def column_def transpose_def vec_eq_iff)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   573
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   574
lemma column_transpose:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   575
  fixes A:: "'a::semiring_1^_^_"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   576
  shows "column i (transpose A) = row i A"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   577
  by (simp add: row_def column_def transpose_def vec_eq_iff)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   578
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   579
lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   580
  by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   581
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   582
lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   583
  by (metis transpose_transpose rows_transpose)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   584
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 58877
diff changeset
   585
text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   586
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   587
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   588
  by (simp add: matrix_vector_mult_def inner_vec_def)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   589
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   590
lemma matrix_mult_vsum:
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   591
  "(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   592
  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   593
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   594
lemma vector_componentwise:
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   595
  "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   596
  by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   597
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   598
lemma basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   599
  by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   600
63938
f6ce08859d4c More mainly topological results
paulson <lp15@cam.ac.uk>
parents: 63918
diff changeset
   601
lemma linear_componentwise_expansion:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   602
  fixes f:: "real ^'m \<Rightarrow> real ^ _"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   603
  assumes lf: "linear f"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   604
  shows "(f x)$j = sum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   605
proof -
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   606
  let ?M = "(UNIV :: 'm set)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   607
  let ?N = "(UNIV :: 'n set)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   608
  have "?rhs = (sum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   609
    unfolding sum_component by simp
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   610
  then show ?thesis
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   611
    unfolding linear_sum_mul[OF lf, symmetric]
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   612
    unfolding scalar_mult_eq_scaleR[symmetric]
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   613
    unfolding basis_expansion
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   614
    by simp
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   615
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   616
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 58877
diff changeset
   617
text\<open>Inverse matrices  (not necessarily square)\<close>
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   618
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   619
definition
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   620
  "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   621
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   622
definition
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   623
  "matrix_inv(A:: 'a::semiring_1^'n^'m) =
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   624
    (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   625
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 58877
diff changeset
   626
text\<open>Correspondence between matrices and linear operators.\<close>
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   627
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   628
definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   629
  where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   630
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   631
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53595
diff changeset
   632
  by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   633
      field_simps sum_distrib_left sum.distrib)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   634
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   635
lemma matrix_works:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   636
  assumes lf: "linear f"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   637
  shows "matrix f *v x = f (x::real ^ 'n)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   638
  apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)
63938
f6ce08859d4c More mainly topological results
paulson <lp15@cam.ac.uk>
parents: 63918
diff changeset
   639
  by (simp add: linear_componentwise_expansion lf)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   640
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   641
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   642
  by (simp add: ext matrix_works)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   643
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   644
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   645
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   646
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   647
lemma matrix_compose:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   648
  assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   649
    and lg: "linear (g::real^'m \<Rightarrow> real^_)"
61736
d6b2d638af23 more symbols;
wenzelm
parents: 61711
diff changeset
   650
  shows "matrix (g \<circ> f) = matrix g ** matrix f"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   651
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   652
  by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   653
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   654
lemma matrix_vector_column:
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   655
  "(A::'a::comm_semiring_1^'n^_) *v x = sum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   656
  by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   657
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   658
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   659
  apply (rule adjoint_unique)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   660
  apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   661
    sum_distrib_right sum_distrib_left)
66804
3f9bb52082c4 avoid name clashes on interpretation of abstract locales
haftmann
parents: 66447
diff changeset
   662
  apply (subst sum.swap)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   663
  apply (auto simp add: ac_simps)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   664
  done
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   665
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   666
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   667
  shows "matrix(adjoint f) = transpose(matrix f)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   668
  apply (subst matrix_vector_mul[OF lf])
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   669
  unfolding adjoint_matrix matrix_of_matrix_vector_mul
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   670
  apply rule
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   671
  done
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   672
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   673
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 58877
diff changeset
   674
subsection \<open>lambda skolemization on cartesian products\<close>
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   675
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   676
(* FIXME: rename do choice_cart *)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   677
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   678
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
37494
6e9f48cf6adf Make latex happy
hoelzl
parents: 37489
diff changeset
   679
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   680
proof -
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   681
  let ?S = "(UNIV :: 'n set)"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   682
  { assume H: "?rhs"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   683
    then have ?lhs by auto }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   684
  moreover
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   685
  { assume H: "?lhs"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   686
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   687
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   688
    { fix i
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   689
      from f have "P i (f i)" by metis
37494
6e9f48cf6adf Make latex happy
hoelzl
parents: 37489
diff changeset
   690
      then have "P i (?x $ i)" by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   691
    }
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   692
    hence "\<forall>i. P i (?x$i)" by metis
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   693
    hence ?rhs by metis }
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   694
  ultimately show ?thesis by metis
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   695
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   696
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   697
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   698
  unfolding inner_simps scalar_mult_eq_scaleR by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   699
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   700
lemma left_invertible_transpose:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   701
  "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   702
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   703
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   704
lemma right_invertible_transpose:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   705
  "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   706
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   707
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   708
lemma matrix_left_invertible_injective:
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   709
  "(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   710
proof -
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   711
  { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   712
    from xy have "B*v (A *v x) = B *v (A*v y)" by simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   713
    hence "x = y"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   714
      unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   715
  moreover
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   716
  { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67155
diff changeset
   717
    hence i: "inj (( *v) A)" unfolding inj_on_def by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   718
    from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67155
diff changeset
   719
    obtain g where g: "linear g" "g \<circ> ( *v) A = id" by blast
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   720
    have "matrix g ** A = mat 1"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   721
      unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
44165
d26a45f3c835 remove lemma stupid_ext
huffman
parents: 44140
diff changeset
   722
      using g(2) by (simp add: fun_eq_iff)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   723
    then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   724
  ultimately show ?thesis by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   725
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   726
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   727
lemma matrix_left_invertible_ker:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   728
  "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   729
  unfolding matrix_left_invertible_injective
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   730
  using linear_injective_0[OF matrix_vector_mul_linear, of A]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   731
  by (simp add: inj_on_def)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   732
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   733
lemma matrix_right_invertible_surjective:
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   734
  "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   735
proof -
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   736
  { fix B :: "real ^'m^'n"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   737
    assume AB: "A ** B = mat 1"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   738
    { fix x :: "real ^ 'm"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   739
      have "A *v (B *v x) = x"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   740
        by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67155
diff changeset
   741
    hence "surj (( *v) A)" unfolding surj_def by metis }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   742
  moreover
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67155
diff changeset
   743
  { assume sf: "surj (( *v) A)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   744
    from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67155
diff changeset
   745
    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "( *v) A \<circ> g = id"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   746
      by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   747
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   748
    have "A ** (matrix g) = mat 1"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   749
      unfolding matrix_eq  matrix_vector_mul_lid
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   750
        matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
44165
d26a45f3c835 remove lemma stupid_ext
huffman
parents: 44140
diff changeset
   751
      using g(2) unfolding o_def fun_eq_iff id_def
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   752
      .
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   753
    hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   754
  }
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   755
  ultimately show ?thesis unfolding surj_def by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   756
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   757
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   758
lemma matrix_left_invertible_independent_columns:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   759
  fixes A :: "real^'n^'m"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   760
  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   761
      (\<forall>c. sum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   762
    (is "?lhs \<longleftrightarrow> ?rhs")
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   763
proof -
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   764
  let ?U = "UNIV :: 'n set"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   765
  { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   766
    { fix c i
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   767
      assume c: "sum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   768
      let ?x = "\<chi> i. c i"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   769
      have th0:"A *v ?x = 0"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   770
        using c
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   771
        unfolding matrix_mult_vsum vec_eq_iff
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   772
        by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   773
      from k[rule_format, OF th0] i
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   774
      have "c i = 0" by (vector vec_eq_iff)}
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   775
    hence ?rhs by blast }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   776
  moreover
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   777
  { assume H: ?rhs
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   778
    { fix x assume x: "A *v x = 0"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   779
      let ?c = "\<lambda>i. ((x$i ):: real)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   780
      from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   781
      have "x = 0" by vector }
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   782
  }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   783
  ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   784
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   785
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   786
lemma matrix_right_invertible_independent_rows:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   787
  fixes A :: "real^'n^'m"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   788
  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   789
    (\<forall>c. sum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   790
  unfolding left_invertible_transpose[symmetric]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   791
    matrix_left_invertible_independent_columns
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   792
  by (simp add: column_transpose)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   793
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   794
lemma matrix_right_invertible_span_columns:
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   795
  "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   796
    span (columns A) = UNIV" (is "?lhs = ?rhs")
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   797
proof -
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   798
  let ?U = "UNIV :: 'm set"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   799
  have fU: "finite ?U" by simp
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   800
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   801
    unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   802
    apply (subst eq_commute)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   803
    apply rule
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   804
    done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   805
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   806
  { assume h: ?lhs
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   807
    { fix x:: "real ^'n"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   808
      from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   809
        where y: "sum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   810
      have "x \<in> span (columns A)"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   811
        unfolding y[symmetric]
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   812
        apply (rule span_sum)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   813
        unfolding scalar_mult_eq_scaleR
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   814
        apply (rule span_mul)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   815
        apply (rule span_superset)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   816
        unfolding columns_def
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   817
        apply blast
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   818
        done
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   819
    }
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   820
    then have ?rhs unfolding rhseq by blast }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   821
  moreover
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   822
  { assume h:?rhs
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   823
    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   824
    { fix y
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   825
      have "?P y"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   826
      proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   827
        show "\<exists>x::real ^ 'm. sum (\<lambda>i. (x$i) *s column i A) ?U = 0"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   828
          by (rule exI[where x=0], simp)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   829
      next
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   830
        fix c y1 y2
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   831
        assume y1: "y1 \<in> columns A" and y2: "?P y2"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   832
        from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   833
          unfolding columns_def by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   834
        from y2 obtain x:: "real ^'m" where
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   835
          x: "sum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   836
        let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   837
        show "?P (c*s y1 + y2)"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 49644
diff changeset
   838
        proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   839
          fix j
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   840
          have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   841
              else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   842
            using i(1) by (simp add: field_simps)
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   843
          have "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   844
              else (x$xa) * ((column xa A$j))) ?U = sum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   845
            apply (rule sum.cong[OF refl])
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   846
            using th apply blast
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   847
            done
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   848
          also have "\<dots> = sum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   849
            by (simp add: sum.distrib)
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   850
          also have "\<dots> = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   851
            unfolding sum.delta[OF fU]
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   852
            using i(1) by simp
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   853
          finally show "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   854
            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   855
        qed
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   856
      next
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   857
        show "y \<in> span (columns A)"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   858
          unfolding h by blast
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   859
      qed
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   860
    }
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   861
    then have ?lhs unfolding lhseq ..
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   862
  }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   863
  ultimately show ?thesis by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   864
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   865
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   866
lemma matrix_left_invertible_span_rows:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   867
  "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   868
  unfolding right_invertible_transpose[symmetric]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   869
  unfolding columns_transpose[symmetric]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   870
  unfolding matrix_right_invertible_span_columns
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   871
  ..
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   872
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 58877
diff changeset
   873
text \<open>The same result in terms of square matrices.\<close>
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   874
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   875
lemma matrix_left_right_inverse:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   876
  fixes A A' :: "real ^'n^'n"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   877
  shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   878
proof -
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   879
  { fix A A' :: "real ^'n^'n"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   880
    assume AA': "A ** A' = mat 1"
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67155
diff changeset
   881
    have sA: "surj (( *v) A)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   882
      unfolding surj_def
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   883
      apply clarify
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   884
      apply (rule_tac x="(A' *v y)" in exI)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   885
      apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   886
      done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   887
    from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   888
    obtain f' :: "real ^'n \<Rightarrow> real ^'n"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   889
      where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   890
    have th: "matrix f' ** A = mat 1"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   891
      by (simp add: matrix_eq matrix_works[OF f'(1)]
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   892
          matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   893
    hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   894
    hence "matrix f' = A'"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   895
      by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   896
    hence "matrix f' ** A = A' ** A" by simp
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   897
    hence "A' ** A = mat 1" by (simp add: th)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   898
  }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   899
  then show ?thesis by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   900
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   901
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 58877
diff changeset
   902
text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   903
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   904
definition "rowvector v = (\<chi> i j. (v$j))"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   905
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   906
definition "columnvector v = (\<chi> i j. (v$i))"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   907
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   908
lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   909
  by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   910
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   911
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   912
  by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   913
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   914
lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   915
  by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   916
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   917
lemma dot_matrix_product:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   918
  "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   919
  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   920
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   921
lemma dot_matrix_vector_mul:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   922
  fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   923
  shows "(A *v x) \<bullet> (B *v y) =
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   924
      (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   925
  unfolding dot_matrix_product transpose_columnvector[symmetric]
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   926
    dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   927
61945
1135b8de26c3 more symbols;
wenzelm
parents: 61810
diff changeset
   928
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   929
  by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   930
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   931
lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   932
  using Basis_le_infnorm[of "axis i 1" x]
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   933
  by (simp add: Basis_vec_def axis_eq_axis inner_axis)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   934
63334
bd37a72a940a Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents: 63332
diff changeset
   935
lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
44647
e4de7750cdeb modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents: 44571
diff changeset
   936
  unfolding continuous_def by (rule tendsto_vec_nth)
44213
6fb54701a11b add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents: 44211
diff changeset
   937
63334
bd37a72a940a Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents: 63332
diff changeset
   938
lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
44647
e4de7750cdeb modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents: 44571
diff changeset
   939
  unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
44213
6fb54701a11b add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents: 44211
diff changeset
   940
63334
bd37a72a940a Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents: 63332
diff changeset
   941
lemma continuous_on_vec_lambda[continuous_intros]:
bd37a72a940a Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents: 63332
diff changeset
   942
  "(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)"
bd37a72a940a Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents: 63332
diff changeset
   943
  unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)
bd37a72a940a Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents: 63332
diff changeset
   944
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   945
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
63332
f164526d8727 move open_Collect_eq/less to HOL
hoelzl
parents: 63075
diff changeset
   946
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
44213
6fb54701a11b add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents: 44211
diff changeset
   947
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   948
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   949
  unfolding bounded_def
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   950
  apply clarify
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   951
  apply (rule_tac x="x $ i" in exI)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   952
  apply (rule_tac x="e" in exI)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   953
  apply clarify
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   954
  apply (rule order_trans [OF dist_vec_nth_le], simp)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   955
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   956
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   957
lemma compact_lemma_cart:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   958
  fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
50998
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   959
  assumes f: "bounded (range f)"
66447
a1f5c5c26fa6 Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   960
  shows "\<exists>l r. strict_mono r \<and>
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   961
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
62127
d8e7738bd2e9 generalized proofs
immler
parents: 61973
diff changeset
   962
    (is "?th d")
d8e7738bd2e9 generalized proofs
immler
parents: 61973
diff changeset
   963
proof -
d8e7738bd2e9 generalized proofs
immler
parents: 61973
diff changeset
   964
  have "\<forall>d' \<subseteq> d. ?th d'"
d8e7738bd2e9 generalized proofs
immler
parents: 61973
diff changeset
   965
    by (rule compact_lemma_general[where unproj=vec_lambda])
d8e7738bd2e9 generalized proofs
immler
parents: 61973
diff changeset
   966
      (auto intro!: f bounded_component_cart simp: vec_lambda_eta)
d8e7738bd2e9 generalized proofs
immler
parents: 61973
diff changeset
   967
  then show "?th d" by simp
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   968
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   969
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   970
instance vec :: (heine_borel, finite) heine_borel
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   971
proof
50998
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   972
  fix f :: "nat \<Rightarrow> 'a ^ 'b"
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   973
  assume f: "bounded (range f)"
66447
a1f5c5c26fa6 Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   974
  then obtain l r where r: "strict_mono r"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   975
      and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
50998
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   976
    using compact_lemma_cart [OF f] by blast
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   977
  let ?d = "UNIV::'b set"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   978
  { fix e::real assume "e>0"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   979
    hence "0 < e / (real_of_nat (card ?d))"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   980
      using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   981
    with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   982
      by simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   983
    moreover
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   984
    { fix n
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   985
      assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   986
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
67155
9e5b05d54f9d canonical name
nipkow
parents: 66804
diff changeset
   987
        unfolding dist_vec_def using zero_le_dist by (rule L2_set_le_sum)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   988
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63945
diff changeset
   989
        by (rule sum_strict_mono) (simp_all add: n)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   990
      finally have "dist (f (r n)) l < e" by simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   991
    }
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   992
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61736
diff changeset
   993
      by (rule eventually_mono)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   994
  }
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61945
diff changeset
   995
  hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp
66447
a1f5c5c26fa6 Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   996
  with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   997
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   998
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   999
lemma interval_cart:
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
  1000
  fixes a :: "real^'n"
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
  1001
  shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1002
    and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1003
  by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1004
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1005
lemma mem_interval_cart:
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
  1006
  fixes a :: "real^'n"
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
  1007
  shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1008
    and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1009
  using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1010
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1011
lemma interval_eq_empty_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1012
  fixes a :: "real^'n"
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
  1013
  shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1014
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1015
proof -
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
  1016
  { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>box a b"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1017
    hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1018
    hence "a$i < b$i" by auto
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1019
    hence False using as by auto }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1020
  moreover
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1021
  { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1022
    let ?x = "(1/2) *\<^sub>R (a + b)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1023
    { fix i
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1024
      have "a$i < b$i" using as[THEN spec[where x=i]] by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1025
      hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1026
        unfolding vector_smult_component and vector_add_component
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1027
        by auto }
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
  1028
    hence "box a b \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1029
  ultimately show ?th1 by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1030
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1031
  { fix i x assume as:"b$i < a$i" and x:"x\<in>cbox a b"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1032
    hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1033
    hence "a$i \<le> b$i" by auto
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1034
    hence False using as by auto }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1035
  moreover
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1036
  { assume as:"\<forall>i. \<not> (b$i < a$i)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1037
    let ?x = "(1/2) *\<^sub>R (a + b)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1038
    { fix i
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1039
      have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1040
      hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1041
        unfolding vector_smult_component and vector_add_component
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1042
        by auto }
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1043
    hence "cbox a b \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1044
  ultimately show ?th2 by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1045
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1046
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1047
lemma interval_ne_empty_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1048
  fixes a :: "real^'n"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1049
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
  1050
    and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1051
  unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1052
    (* BH: Why doesn't just "auto" work here? *)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1053
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1054
lemma subset_interval_imp_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1055
  fixes a :: "real^'n"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1056
  shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1057
    and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1058
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
  1059
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1060
  unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1061
  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1062
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1063
lemma interval_sing:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1064
  fixes a :: "'a::linorder^'n"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1065
  shows "{a .. a} = {a} \<and> {a<..<a} = {}"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1066
  apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1067
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1068
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1069
lemma subset_interval_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1070
  fixes a :: "real^'n"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1071
  shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1072
    and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1073
    and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
  1074
    and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1075
  using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1076
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1077
lemma disjoint_interval_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1078
  fixes a::"real^'n"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1079
  shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1080
    and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1081
    and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
  1082
    and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1083
  using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1084
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1085
lemma inter_interval_cart:
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
  1086
  fixes a :: "real^'n"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1087
  shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
63945
444eafb6e864 a few new theorems and a renaming
paulson <lp15@cam.ac.uk>
parents: 63938
diff changeset
  1088
  unfolding Int_interval
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1089
  by (auto simp: mem_box less_eq_vec_def)
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1090
    (auto simp: Basis_vec_def inner_axis)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1091
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1092
lemma closed_interval_left_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1093
  fixes b :: "real^'n"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1094
  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
63332
f164526d8727 move open_Collect_eq/less to HOL
hoelzl
parents: 63075
diff changeset
  1095
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1096
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1097
lemma closed_interval_right_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1098
  fixes a::"real^'n"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1099
  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
63332
f164526d8727 move open_Collect_eq/less to HOL
hoelzl
parents: 63075
diff changeset
  1100
  by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1101
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1102
lemma is_interval_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1103
  "is_interval (s::(real^'n) set) \<longleftrightarrow>
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1104
    (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1105
  by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1106
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1107
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
63332
f164526d8727 move open_Collect_eq/less to HOL
hoelzl
parents: 63075
diff changeset
  1108
  by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1109
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1110
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
63332
f164526d8727 move open_Collect_eq/less to HOL
hoelzl
parents: 63075
diff changeset
  1111
  by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1112
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1113
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
63332
f164526d8727 move open_Collect_eq/less to HOL
hoelzl
parents: 63075
diff changeset
  1114
  by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1115
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1116
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
63332
f164526d8727 move open_Collect_eq/less to HOL
hoelzl
parents: 63075
diff changeset
  1117
  by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1118
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1119
lemma Lim_component_le_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset