src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
author wenzelm
Sun Sep 13 22:56:52 2015 +0200 (2015-09-13)
changeset 61169 4de9ff3ea29a
parent 61104 3c2d4636cebc
child 61711 21d7910d6816
permissions -rw-r--r--
tuned proofs -- less legacy;
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section \<open>Instanciates 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 Integration
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begin
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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 (cases "k=a") auto
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lemma setsum_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 setsum.Plus)
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  apply simp_all
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  done
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lemma setsum_mult_product:
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  "setsum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
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  unfolding setsum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]
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proof (rule setsum.cong, simp, rule setsum.reindex_cong)
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  fix i
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  show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
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  show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
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  proof safe
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    fix j assume "j \<in> {i * B..<i * B + B}"
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    then show "j \<in> (\<lambda>j. j + i * B) ` {..<B}"
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      by (auto intro!: image_eqI[of _ _ "j - i * B"])
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  qed simp
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qed simp
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subsection\<open>Basic componentwise operations on vectors.\<close>
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instantiation vec :: (times, finite) times
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begin
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definition "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
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instance ..
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end
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instantiation vec :: (one, finite) one
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begin
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definition "1 \<equiv> (\<chi> i. 1)"
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instance ..
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end
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instantiation vec :: (ord, finite) ord
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begin
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definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"
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definition "x < (y::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
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instance ..
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end
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text\<open>The ordering on one-dimensional vectors is linear.\<close>
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class cart_one =
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  assumes UNIV_one: "card (UNIV :: 'a set) = Suc 0"
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begin
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subclass finite
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proof
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  from UNIV_one show "finite (UNIV :: 'a set)"
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    by (auto intro!: card_ge_0_finite)
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qed
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end
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instance vec:: (order, finite) order
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  by standard (auto simp: less_eq_vec_def less_vec_def vec_eq_iff
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      intro: order.trans order.antisym order.strict_implies_order)
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instance vec :: (linorder, cart_one) linorder
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proof
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  obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"
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  proof -
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    have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)
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    then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)
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    then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto
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    then show thesis by (auto intro: that)
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  qed
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  fix x y :: "'a^'b::cart_one"
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  note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
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  show "x \<le> y \<or> y \<le> x" by auto
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qed
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text\<open>Constant Vectors\<close>
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definition "vec x = (\<chi> i. x)"
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lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b"
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  by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis)
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text\<open>Also the scalar-vector multiplication.\<close>
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definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
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  where "c *s x = (\<chi> i. c * (x$i))"
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subsection \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space.\<close>
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lemma setsum_cong_aux:
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  "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
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  by (auto intro: setsum.cong)
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hide_fact (open) setsum_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 setsum.distrib} RS sym,
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      @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
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      @{thm setsum_left_distrib}, @{thm setsum_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.setsum_cong_aux} i
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         ORELSE resolve_tac ctxt @{thms setsum.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_setsum:
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  assumes "finite S"
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  shows "vec(setsum f S) = setsum (vec o f) S"
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  using assms
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proof induct
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  case empty
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  then show ?case by simp
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next
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  case insert
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  then show ?case by (auto simp add: vec_add)
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qed
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text\<open>Obvious "component-pushing".\<close>
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lemma vec_component [simp]: "vec x $ i = x"
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  by vector
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lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
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  by vector
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lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
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  by vector
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lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
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lemmas vector_component =
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  vec_component vector_add_component vector_mult_component
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  vector_smult_component vector_minus_component vector_uminus_component
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  vector_scaleR_component cond_component
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subsection \<open>Some frequently useful arithmetic lemmas over vectors.\<close>
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instance vec :: (semigroup_mult, finite) semigroup_mult
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  by standard (vector mult.assoc)
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instance vec :: (monoid_mult, finite) monoid_mult
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  by standard vector+
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instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
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  by standard (vector mult.commute)
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instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
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  by standard vector
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instance vec :: (semiring, finite) semiring
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  by standard (vector field_simps)+
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instance vec :: (semiring_0, finite) semiring_0
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  by standard (vector field_simps)+
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instance vec :: (semiring_1, finite) semiring_1
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  by standard vector
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instance vec :: (comm_semiring, finite) comm_semiring
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  by standard (vector field_simps)+
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instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
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instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
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instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
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instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
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instance vec :: (ring, finite) ring ..
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instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
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instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
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instance vec :: (ring_1, finite) ring_1 ..
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instance vec :: (real_algebra, finite) real_algebra
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  by standard (simp_all add: vec_eq_iff)
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instance vec :: (real_algebra_1, finite) real_algebra_1 ..
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lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
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proof (induct n)
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  case 0
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  then show ?case by vector
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next
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  case Suc
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  then show ?case by vector
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qed
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lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) $ i = 1"
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  by vector
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lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) $ i = - 1"
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  by vector
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instance vec :: (semiring_char_0, finite) semiring_char_0
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proof
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  fix m n :: nat
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  show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
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    by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
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qed
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instance vec :: (numeral, finite) numeral ..
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instance vec :: (semiring_numeral, finite) semiring_numeral ..
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lemma numeral_index [simp]: "numeral w $ i = numeral w"
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  by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
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lemma neg_numeral_index [simp]: "- numeral w $ i = - numeral w"
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  by (simp only: vector_uminus_component numeral_index)
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instance vec :: (comm_ring_1, finite) comm_ring_1 ..
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instance vec :: (ring_char_0, finite) ring_char_0 ..
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lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
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  by (vector mult.assoc)
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lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
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  by (vector field_simps)
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lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
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  by (vector field_simps)
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lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
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lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
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lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
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  by (vector field_simps)
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lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
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lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
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lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
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lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
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lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
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  by (vector field_simps)
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lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
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  by (simp add: vec_eq_iff)
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lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
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lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
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  by vector
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lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
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  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
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lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
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  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
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lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
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  by (metis vector_mul_lcancel)
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lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
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  by (metis vector_mul_rcancel)
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lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x"
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  apply (simp add: norm_vec_def)
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  apply (rule member_le_setL2, simp_all)
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  done
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lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x$i\<bar> <= e"
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  by (metis component_le_norm_cart order_trans)
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lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
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  by (metis component_le_norm_cart le_less_trans)
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lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
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  by (simp add: norm_vec_def setL2_le_setsum)
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lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
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  unfolding scaleR_vec_def vector_scalar_mult_def by simp
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   304
hoelzl@37489
   305
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
hoelzl@37489
   306
  unfolding dist_norm scalar_mult_eq_scaleR
hoelzl@37489
   307
  unfolding scaleR_right_diff_distrib[symmetric] by simp
hoelzl@37489
   308
hoelzl@37489
   309
lemma setsum_component [simp]:
hoelzl@37489
   310
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
hoelzl@37489
   311
  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
wenzelm@49644
   312
proof (cases "finite S")
wenzelm@49644
   313
  case True
wenzelm@49644
   314
  then show ?thesis by induct simp_all
wenzelm@49644
   315
next
wenzelm@49644
   316
  case False
wenzelm@49644
   317
  then show ?thesis by simp
wenzelm@49644
   318
qed
hoelzl@37489
   319
hoelzl@37489
   320
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
huffman@44136
   321
  by (simp add: vec_eq_iff)
hoelzl@37489
   322
hoelzl@37489
   323
lemma setsum_cmul:
hoelzl@37489
   324
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
hoelzl@37489
   325
  shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
huffman@44136
   326
  by (simp add: vec_eq_iff setsum_right_distrib)
hoelzl@37489
   327
hoelzl@37489
   328
lemma setsum_norm_allsubsets_bound_cart:
hoelzl@37489
   329
  fixes f:: "'a \<Rightarrow> real ^'n"
hoelzl@37489
   330
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
hoelzl@37489
   331
  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
hoelzl@50526
   332
  using setsum_norm_allsubsets_bound[OF assms]
wenzelm@57865
   333
  by simp
hoelzl@37489
   334
wenzelm@60420
   335
subsection \<open>Matrix operations\<close>
hoelzl@37489
   336
wenzelm@60420
   337
text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close>
hoelzl@37489
   338
wenzelm@49644
   339
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
wenzelm@49644
   340
    (infixl "**" 70)
hoelzl@37489
   341
  where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
hoelzl@37489
   342
wenzelm@49644
   343
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
wenzelm@49644
   344
    (infixl "*v" 70)
hoelzl@37489
   345
  where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
hoelzl@37489
   346
wenzelm@49644
   347
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
wenzelm@49644
   348
    (infixl "v*" 70)
hoelzl@37489
   349
  where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
hoelzl@37489
   350
hoelzl@37489
   351
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
hoelzl@37489
   352
definition transpose where 
hoelzl@37489
   353
  "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
hoelzl@37489
   354
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
hoelzl@37489
   355
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
hoelzl@37489
   356
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
hoelzl@37489
   357
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
hoelzl@37489
   358
hoelzl@37489
   359
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
hoelzl@37489
   360
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
haftmann@57418
   361
  by (vector matrix_matrix_mult_def setsum.distrib[symmetric] field_simps)
hoelzl@37489
   362
hoelzl@37489
   363
lemma matrix_mul_lid:
hoelzl@37489
   364
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   365
  shows "mat 1 ** A = A"
hoelzl@37489
   366
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   367
  apply vector
haftmann@57418
   368
  apply (auto simp only: if_distrib cond_application_beta setsum.delta'[OF finite]
wenzelm@49644
   369
    mult_1_left mult_zero_left if_True UNIV_I)
wenzelm@49644
   370
  done
hoelzl@37489
   371
hoelzl@37489
   372
hoelzl@37489
   373
lemma matrix_mul_rid:
hoelzl@37489
   374
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   375
  shows "A ** mat 1 = A"
hoelzl@37489
   376
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   377
  apply vector
haftmann@57418
   378
  apply (auto simp only: if_distrib cond_application_beta setsum.delta[OF finite]
wenzelm@49644
   379
    mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
wenzelm@49644
   380
  done
hoelzl@37489
   381
hoelzl@37489
   382
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
haftmann@57512
   383
  apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult.assoc)
haftmann@57418
   384
  apply (subst setsum.commute)
hoelzl@37489
   385
  apply simp
hoelzl@37489
   386
  done
hoelzl@37489
   387
hoelzl@37489
   388
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
wenzelm@49644
   389
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def
haftmann@57512
   390
    setsum_right_distrib setsum_left_distrib mult.assoc)
haftmann@57418
   391
  apply (subst setsum.commute)
hoelzl@37489
   392
  apply simp
hoelzl@37489
   393
  done
hoelzl@37489
   394
hoelzl@37489
   395
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
hoelzl@37489
   396
  apply (vector matrix_vector_mult_def mat_def)
haftmann@57418
   397
  apply (simp add: if_distrib cond_application_beta setsum.delta' cong del: if_weak_cong)
wenzelm@49644
   398
  done
hoelzl@37489
   399
wenzelm@49644
   400
lemma matrix_transpose_mul:
wenzelm@49644
   401
    "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
haftmann@57512
   402
  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
hoelzl@37489
   403
hoelzl@37489
   404
lemma matrix_eq:
hoelzl@37489
   405
  fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
hoelzl@37489
   406
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@37489
   407
  apply auto
huffman@44136
   408
  apply (subst vec_eq_iff)
hoelzl@37489
   409
  apply clarify
hoelzl@50526
   410
  apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
hoelzl@50526
   411
  apply (erule_tac x="axis ia 1" in allE)
hoelzl@37489
   412
  apply (erule_tac x="i" in allE)
hoelzl@50526
   413
  apply (auto simp add: if_distrib cond_application_beta axis_def
haftmann@57418
   414
    setsum.delta[OF finite] cong del: if_weak_cong)
wenzelm@49644
   415
  done
hoelzl@37489
   416
wenzelm@49644
   417
lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
huffman@44136
   418
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   419
hoelzl@37489
   420
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
haftmann@57514
   421
  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib ac_simps)
haftmann@57418
   422
  apply (subst setsum.commute)
wenzelm@49644
   423
  apply simp
wenzelm@49644
   424
  done
hoelzl@37489
   425
hoelzl@37489
   426
lemma transpose_mat: "transpose (mat n) = mat n"
hoelzl@37489
   427
  by (vector transpose_def mat_def)
hoelzl@37489
   428
hoelzl@37489
   429
lemma transpose_transpose: "transpose(transpose A) = A"
hoelzl@37489
   430
  by (vector transpose_def)
hoelzl@37489
   431
hoelzl@37489
   432
lemma row_transpose:
hoelzl@37489
   433
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   434
  shows "row i (transpose A) = column i A"
huffman@44136
   435
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   436
hoelzl@37489
   437
lemma column_transpose:
hoelzl@37489
   438
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   439
  shows "column i (transpose A) = row i A"
huffman@44136
   440
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   441
hoelzl@37489
   442
lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
wenzelm@49644
   443
  by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
hoelzl@37489
   444
wenzelm@49644
   445
lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
wenzelm@49644
   446
  by (metis transpose_transpose rows_transpose)
hoelzl@37489
   447
wenzelm@60420
   448
text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
hoelzl@37489
   449
hoelzl@37489
   450
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
huffman@44136
   451
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   452
wenzelm@49644
   453
lemma matrix_mult_vsum:
wenzelm@49644
   454
  "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
haftmann@57512
   455
  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
hoelzl@37489
   456
hoelzl@37489
   457
lemma vector_componentwise:
hoelzl@50526
   458
  "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
haftmann@57418
   459
  by (simp add: axis_def if_distrib setsum.If_cases vec_eq_iff)
hoelzl@50526
   460
hoelzl@50526
   461
lemma basis_expansion: "setsum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
haftmann@57418
   462
  by (auto simp add: axis_def vec_eq_iff if_distrib setsum.If_cases cong del: if_weak_cong)
hoelzl@37489
   463
hoelzl@37489
   464
lemma linear_componentwise:
hoelzl@37489
   465
  fixes f:: "real ^'m \<Rightarrow> real ^ _"
hoelzl@37489
   466
  assumes lf: "linear f"
hoelzl@50526
   467
  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
wenzelm@49644
   468
proof -
hoelzl@37489
   469
  let ?M = "(UNIV :: 'm set)"
hoelzl@37489
   470
  let ?N = "(UNIV :: 'n set)"
hoelzl@50526
   471
  have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
hoelzl@50526
   472
    unfolding setsum_component by simp
wenzelm@49644
   473
  then show ?thesis
huffman@56196
   474
    unfolding linear_setsum_mul[OF lf, symmetric]
hoelzl@50526
   475
    unfolding scalar_mult_eq_scaleR[symmetric]
hoelzl@50526
   476
    unfolding basis_expansion
hoelzl@50526
   477
    by simp
hoelzl@37489
   478
qed
hoelzl@37489
   479
wenzelm@60420
   480
text\<open>Inverse matrices  (not necessarily square)\<close>
hoelzl@37489
   481
wenzelm@49644
   482
definition
wenzelm@49644
   483
  "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   484
wenzelm@49644
   485
definition
wenzelm@49644
   486
  "matrix_inv(A:: 'a::semiring_1^'n^'m) =
wenzelm@49644
   487
    (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   488
wenzelm@60420
   489
text\<open>Correspondence between matrices and linear operators.\<close>
hoelzl@37489
   490
wenzelm@49644
   491
definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
hoelzl@50526
   492
  where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
hoelzl@37489
   493
hoelzl@37489
   494
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
huffman@53600
   495
  by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
haftmann@57418
   496
      field_simps setsum_right_distrib setsum.distrib)
hoelzl@37489
   497
wenzelm@49644
   498
lemma matrix_works:
wenzelm@49644
   499
  assumes lf: "linear f"
wenzelm@49644
   500
  shows "matrix f *v x = f (x::real ^ 'n)"
haftmann@57512
   501
  apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)
wenzelm@49644
   502
  apply clarify
wenzelm@49644
   503
  apply (rule linear_componentwise[OF lf, symmetric])
wenzelm@49644
   504
  done
hoelzl@37489
   505
wenzelm@49644
   506
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
wenzelm@49644
   507
  by (simp add: ext matrix_works)
hoelzl@37489
   508
hoelzl@37489
   509
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
hoelzl@37489
   510
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
hoelzl@37489
   511
hoelzl@37489
   512
lemma matrix_compose:
hoelzl@37489
   513
  assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
wenzelm@49644
   514
    and lg: "linear (g::real^'m \<Rightarrow> real^_)"
hoelzl@37489
   515
  shows "matrix (g o f) = matrix g ** matrix f"
hoelzl@37489
   516
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
wenzelm@49644
   517
  by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
hoelzl@37489
   518
wenzelm@49644
   519
lemma matrix_vector_column:
wenzelm@49644
   520
  "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
haftmann@57512
   521
  by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
hoelzl@37489
   522
hoelzl@37489
   523
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
hoelzl@37489
   524
  apply (rule adjoint_unique)
wenzelm@49644
   525
  apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
wenzelm@49644
   526
    setsum_left_distrib setsum_right_distrib)
haftmann@57418
   527
  apply (subst setsum.commute)
haftmann@57514
   528
  apply (auto simp add: ac_simps)
hoelzl@37489
   529
  done
hoelzl@37489
   530
hoelzl@37489
   531
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
hoelzl@37489
   532
  shows "matrix(adjoint f) = transpose(matrix f)"
hoelzl@37489
   533
  apply (subst matrix_vector_mul[OF lf])
wenzelm@49644
   534
  unfolding adjoint_matrix matrix_of_matrix_vector_mul
wenzelm@49644
   535
  apply rule
wenzelm@49644
   536
  done
wenzelm@49644
   537
hoelzl@37489
   538
wenzelm@60420
   539
subsection \<open>lambda skolemization on cartesian products\<close>
hoelzl@37489
   540
hoelzl@37489
   541
(* FIXME: rename do choice_cart *)
hoelzl@37489
   542
hoelzl@37489
   543
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
hoelzl@37494
   544
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   545
proof -
hoelzl@37489
   546
  let ?S = "(UNIV :: 'n set)"
wenzelm@49644
   547
  { assume H: "?rhs"
wenzelm@49644
   548
    then have ?lhs by auto }
hoelzl@37489
   549
  moreover
wenzelm@49644
   550
  { assume H: "?lhs"
hoelzl@37489
   551
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
hoelzl@37489
   552
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
wenzelm@49644
   553
    { fix i
hoelzl@37489
   554
      from f have "P i (f i)" by metis
hoelzl@37494
   555
      then have "P i (?x $ i)" by auto
hoelzl@37489
   556
    }
hoelzl@37489
   557
    hence "\<forall>i. P i (?x$i)" by metis
hoelzl@37489
   558
    hence ?rhs by metis }
hoelzl@37489
   559
  ultimately show ?thesis by metis
hoelzl@37489
   560
qed
hoelzl@37489
   561
hoelzl@37489
   562
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
hoelzl@50526
   563
  unfolding inner_simps scalar_mult_eq_scaleR by auto
hoelzl@37489
   564
hoelzl@37489
   565
lemma left_invertible_transpose:
hoelzl@37489
   566
  "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
hoelzl@37489
   567
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   568
hoelzl@37489
   569
lemma right_invertible_transpose:
hoelzl@37489
   570
  "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
hoelzl@37489
   571
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   572
hoelzl@37489
   573
lemma matrix_left_invertible_injective:
wenzelm@49644
   574
  "(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
wenzelm@49644
   575
proof -
wenzelm@49644
   576
  { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
hoelzl@37489
   577
    from xy have "B*v (A *v x) = B *v (A*v y)" by simp
hoelzl@37489
   578
    hence "x = y"
wenzelm@49644
   579
      unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
hoelzl@37489
   580
  moreover
wenzelm@49644
   581
  { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
hoelzl@37489
   582
    hence i: "inj (op *v A)" unfolding inj_on_def by auto
hoelzl@37489
   583
    from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
hoelzl@37489
   584
    obtain g where g: "linear g" "g o op *v A = id" by blast
hoelzl@37489
   585
    have "matrix g ** A = mat 1"
hoelzl@37489
   586
      unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   587
      using g(2) by (simp add: fun_eq_iff)
wenzelm@49644
   588
    then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
hoelzl@37489
   589
  ultimately show ?thesis by blast
hoelzl@37489
   590
qed
hoelzl@37489
   591
hoelzl@37489
   592
lemma matrix_left_invertible_ker:
hoelzl@37489
   593
  "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
hoelzl@37489
   594
  unfolding matrix_left_invertible_injective
hoelzl@37489
   595
  using linear_injective_0[OF matrix_vector_mul_linear, of A]
hoelzl@37489
   596
  by (simp add: inj_on_def)
hoelzl@37489
   597
hoelzl@37489
   598
lemma matrix_right_invertible_surjective:
wenzelm@49644
   599
  "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
wenzelm@49644
   600
proof -
wenzelm@49644
   601
  { fix B :: "real ^'m^'n"
wenzelm@49644
   602
    assume AB: "A ** B = mat 1"
wenzelm@49644
   603
    { fix x :: "real ^ 'm"
hoelzl@37489
   604
      have "A *v (B *v x) = x"
wenzelm@49644
   605
        by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
hoelzl@37489
   606
    hence "surj (op *v A)" unfolding surj_def by metis }
hoelzl@37489
   607
  moreover
wenzelm@49644
   608
  { assume sf: "surj (op *v A)"
hoelzl@37489
   609
    from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
hoelzl@37489
   610
    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
hoelzl@37489
   611
      by blast
hoelzl@37489
   612
hoelzl@37489
   613
    have "A ** (matrix g) = mat 1"
hoelzl@37489
   614
      unfolding matrix_eq  matrix_vector_mul_lid
hoelzl@37489
   615
        matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   616
      using g(2) unfolding o_def fun_eq_iff id_def
hoelzl@37489
   617
      .
hoelzl@37489
   618
    hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
hoelzl@37489
   619
  }
hoelzl@37489
   620
  ultimately show ?thesis unfolding surj_def by blast
hoelzl@37489
   621
qed
hoelzl@37489
   622
hoelzl@37489
   623
lemma matrix_left_invertible_independent_columns:
hoelzl@37489
   624
  fixes A :: "real^'n^'m"
wenzelm@49644
   625
  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
wenzelm@49644
   626
      (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
wenzelm@49644
   627
    (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   628
proof -
hoelzl@37489
   629
  let ?U = "UNIV :: 'n set"
wenzelm@49644
   630
  { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
wenzelm@49644
   631
    { fix c i
wenzelm@49644
   632
      assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
hoelzl@37489
   633
      let ?x = "\<chi> i. c i"
hoelzl@37489
   634
      have th0:"A *v ?x = 0"
hoelzl@37489
   635
        using c
huffman@44136
   636
        unfolding matrix_mult_vsum vec_eq_iff
hoelzl@37489
   637
        by auto
hoelzl@37489
   638
      from k[rule_format, OF th0] i
huffman@44136
   639
      have "c i = 0" by (vector vec_eq_iff)}
wenzelm@49644
   640
    hence ?rhs by blast }
hoelzl@37489
   641
  moreover
wenzelm@49644
   642
  { assume H: ?rhs
wenzelm@49644
   643
    { fix x assume x: "A *v x = 0"
hoelzl@37489
   644
      let ?c = "\<lambda>i. ((x$i ):: real)"
hoelzl@37489
   645
      from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
wenzelm@49644
   646
      have "x = 0" by vector }
wenzelm@49644
   647
  }
hoelzl@37489
   648
  ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
hoelzl@37489
   649
qed
hoelzl@37489
   650
hoelzl@37489
   651
lemma matrix_right_invertible_independent_rows:
hoelzl@37489
   652
  fixes A :: "real^'n^'m"
wenzelm@49644
   653
  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
wenzelm@49644
   654
    (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
hoelzl@37489
   655
  unfolding left_invertible_transpose[symmetric]
hoelzl@37489
   656
    matrix_left_invertible_independent_columns
hoelzl@37489
   657
  by (simp add: column_transpose)
hoelzl@37489
   658
hoelzl@37489
   659
lemma matrix_right_invertible_span_columns:
wenzelm@49644
   660
  "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
wenzelm@49644
   661
    span (columns A) = UNIV" (is "?lhs = ?rhs")
wenzelm@49644
   662
proof -
hoelzl@37489
   663
  let ?U = "UNIV :: 'm set"
hoelzl@37489
   664
  have fU: "finite ?U" by simp
hoelzl@37489
   665
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
hoelzl@37489
   666
    unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
wenzelm@49644
   667
    apply (subst eq_commute)
wenzelm@49644
   668
    apply rule
wenzelm@49644
   669
    done
hoelzl@37489
   670
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
wenzelm@49644
   671
  { assume h: ?lhs
wenzelm@49644
   672
    { fix x:: "real ^'n"
wenzelm@49644
   673
      from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
wenzelm@49644
   674
        where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
wenzelm@49644
   675
      have "x \<in> span (columns A)"
wenzelm@49644
   676
        unfolding y[symmetric]
huffman@56196
   677
        apply (rule span_setsum)
wenzelm@49644
   678
        apply clarify
hoelzl@50526
   679
        unfolding scalar_mult_eq_scaleR
wenzelm@49644
   680
        apply (rule span_mul)
wenzelm@49644
   681
        apply (rule span_superset)
wenzelm@49644
   682
        unfolding columns_def
wenzelm@49644
   683
        apply blast
wenzelm@49644
   684
        done
wenzelm@49644
   685
    }
wenzelm@49644
   686
    then have ?rhs unfolding rhseq by blast }
hoelzl@37489
   687
  moreover
wenzelm@49644
   688
  { assume h:?rhs
hoelzl@37489
   689
    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
wenzelm@49644
   690
    { fix y
wenzelm@49644
   691
      have "?P y"
hoelzl@50526
   692
      proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
wenzelm@61076
   693
        show "\<exists>x::real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
hoelzl@37489
   694
          by (rule exI[where x=0], simp)
hoelzl@37489
   695
      next
wenzelm@49644
   696
        fix c y1 y2
wenzelm@49644
   697
        assume y1: "y1 \<in> columns A" and y2: "?P y2"
hoelzl@37489
   698
        from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
hoelzl@37489
   699
          unfolding columns_def by blast
hoelzl@37489
   700
        from y2 obtain x:: "real ^'m" where
hoelzl@37489
   701
          x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
hoelzl@37489
   702
        let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
hoelzl@37489
   703
        show "?P (c*s y1 + y2)"
webertj@49962
   704
        proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)
wenzelm@49644
   705
          fix j
wenzelm@49644
   706
          have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
   707
              else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
wenzelm@49644
   708
            using i(1) by (simp add: field_simps)
wenzelm@49644
   709
          have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
   710
              else (x$xa) * ((column xa A$j))) ?U = setsum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"
haftmann@57418
   711
            apply (rule setsum.cong[OF refl])
wenzelm@49644
   712
            using th apply blast
wenzelm@49644
   713
            done
wenzelm@49644
   714
          also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
haftmann@57418
   715
            by (simp add: setsum.distrib)
wenzelm@49644
   716
          also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
haftmann@57418
   717
            unfolding setsum.delta[OF fU]
wenzelm@49644
   718
            using i(1) by simp
wenzelm@49644
   719
          finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
   720
            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
wenzelm@49644
   721
        qed
wenzelm@49644
   722
      next
wenzelm@49644
   723
        show "y \<in> span (columns A)"
wenzelm@49644
   724
          unfolding h by blast
wenzelm@49644
   725
      qed
wenzelm@49644
   726
    }
wenzelm@49644
   727
    then have ?lhs unfolding lhseq ..
wenzelm@49644
   728
  }
hoelzl@37489
   729
  ultimately show ?thesis by blast
hoelzl@37489
   730
qed
hoelzl@37489
   731
hoelzl@37489
   732
lemma matrix_left_invertible_span_rows:
hoelzl@37489
   733
  "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
hoelzl@37489
   734
  unfolding right_invertible_transpose[symmetric]
hoelzl@37489
   735
  unfolding columns_transpose[symmetric]
hoelzl@37489
   736
  unfolding matrix_right_invertible_span_columns
wenzelm@49644
   737
  ..
hoelzl@37489
   738
wenzelm@60420
   739
text \<open>The same result in terms of square matrices.\<close>
hoelzl@37489
   740
hoelzl@37489
   741
lemma matrix_left_right_inverse:
hoelzl@37489
   742
  fixes A A' :: "real ^'n^'n"
hoelzl@37489
   743
  shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
wenzelm@49644
   744
proof -
wenzelm@49644
   745
  { fix A A' :: "real ^'n^'n"
wenzelm@49644
   746
    assume AA': "A ** A' = mat 1"
hoelzl@37489
   747
    have sA: "surj (op *v A)"
hoelzl@37489
   748
      unfolding surj_def
hoelzl@37489
   749
      apply clarify
hoelzl@37489
   750
      apply (rule_tac x="(A' *v y)" in exI)
wenzelm@49644
   751
      apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
wenzelm@49644
   752
      done
hoelzl@37489
   753
    from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
hoelzl@37489
   754
    obtain f' :: "real ^'n \<Rightarrow> real ^'n"
hoelzl@37489
   755
      where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
hoelzl@37489
   756
    have th: "matrix f' ** A = mat 1"
wenzelm@49644
   757
      by (simp add: matrix_eq matrix_works[OF f'(1)]
wenzelm@49644
   758
          matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
hoelzl@37489
   759
    hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
wenzelm@49644
   760
    hence "matrix f' = A'"
wenzelm@49644
   761
      by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
hoelzl@37489
   762
    hence "matrix f' ** A = A' ** A" by simp
wenzelm@49644
   763
    hence "A' ** A = mat 1" by (simp add: th)
wenzelm@49644
   764
  }
hoelzl@37489
   765
  then show ?thesis by blast
hoelzl@37489
   766
qed
hoelzl@37489
   767
wenzelm@60420
   768
text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
hoelzl@37489
   769
hoelzl@37489
   770
definition "rowvector v = (\<chi> i j. (v$j))"
hoelzl@37489
   771
hoelzl@37489
   772
definition "columnvector v = (\<chi> i j. (v$i))"
hoelzl@37489
   773
wenzelm@49644
   774
lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
huffman@44136
   775
  by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
hoelzl@37489
   776
hoelzl@37489
   777
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
huffman@44136
   778
  by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
hoelzl@37489
   779
wenzelm@49644
   780
lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
hoelzl@37489
   781
  by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
hoelzl@37489
   782
wenzelm@49644
   783
lemma dot_matrix_product:
wenzelm@49644
   784
  "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
huffman@44136
   785
  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
hoelzl@37489
   786
hoelzl@37489
   787
lemma dot_matrix_vector_mul:
hoelzl@37489
   788
  fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
hoelzl@37489
   789
  shows "(A *v x) \<bullet> (B *v y) =
hoelzl@37489
   790
      (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
wenzelm@49644
   791
  unfolding dot_matrix_product transpose_columnvector[symmetric]
wenzelm@49644
   792
    dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
hoelzl@37489
   793
hoelzl@37489
   794
hoelzl@50526
   795
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in>UNIV}"
hoelzl@50526
   796
  by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
hoelzl@37489
   797
wenzelm@49644
   798
lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
hoelzl@50526
   799
  using Basis_le_infnorm[of "axis i 1" x]
hoelzl@50526
   800
  by (simp add: Basis_vec_def axis_eq_axis inner_axis)
hoelzl@37489
   801
wenzelm@49644
   802
lemma continuous_component: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
huffman@44647
   803
  unfolding continuous_def by (rule tendsto_vec_nth)
huffman@44213
   804
wenzelm@49644
   805
lemma continuous_on_component: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
huffman@44647
   806
  unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
huffman@44213
   807
hoelzl@37489
   808
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
huffman@44233
   809
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
huffman@44213
   810
hoelzl@37489
   811
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
wenzelm@49644
   812
  unfolding bounded_def
wenzelm@49644
   813
  apply clarify
wenzelm@49644
   814
  apply (rule_tac x="x $ i" in exI)
wenzelm@49644
   815
  apply (rule_tac x="e" in exI)
wenzelm@49644
   816
  apply clarify
wenzelm@49644
   817
  apply (rule order_trans [OF dist_vec_nth_le], simp)
wenzelm@49644
   818
  done
hoelzl@37489
   819
hoelzl@37489
   820
lemma compact_lemma_cart:
hoelzl@37489
   821
  fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
hoelzl@50998
   822
  assumes f: "bounded (range f)"
hoelzl@37489
   823
  shows "\<forall>d.
hoelzl@37489
   824
        \<exists>l r. subseq r \<and>
hoelzl@37489
   825
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
hoelzl@37489
   826
proof
wenzelm@49644
   827
  fix d :: "'n set"
wenzelm@49644
   828
  have "finite d" by simp
hoelzl@37489
   829
  thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
hoelzl@37489
   830
      (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
wenzelm@49644
   831
  proof (induct d)
wenzelm@49644
   832
    case empty
wenzelm@49644
   833
    thus ?case unfolding subseq_def by auto
wenzelm@49644
   834
  next
wenzelm@49644
   835
    case (insert k d)
wenzelm@49644
   836
    obtain l1::"'a^'n" and r1 where r1:"subseq r1"
wenzelm@49644
   837
      and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
hoelzl@37489
   838
      using insert(3) by auto
wenzelm@60420
   839
    have s': "bounded ((\<lambda>x. x $ k) ` range f)" using \<open>bounded (range f)\<close>
hoelzl@50998
   840
      by (auto intro!: bounded_component_cart)
hoelzl@50998
   841
    have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` range f" by simp
hoelzl@50998
   842
    have "bounded (range (\<lambda>i. f (r1 i) $ k))"
hoelzl@50998
   843
      by (metis (lifting) bounded_subset image_subsetI f' s')
hoelzl@50998
   844
    then obtain l2 r2 where r2: "subseq r2"
wenzelm@49644
   845
      and lr2: "((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
hoelzl@50998
   846
      using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) $ k"] by (auto simp: o_def)
wenzelm@49644
   847
    def r \<equiv> "r1 \<circ> r2"
wenzelm@49644
   848
    have r: "subseq r"
hoelzl@37489
   849
      using r1 and r2 unfolding r_def o_def subseq_def by auto
hoelzl@37489
   850
    moreover
hoelzl@37489
   851
    def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
wenzelm@49644
   852
    { fix e :: real assume "e > 0"
wenzelm@60420
   853
      from lr1 \<open>e>0\<close> have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
wenzelm@49644
   854
        by blast
wenzelm@60420
   855
      from lr2 \<open>e>0\<close> have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially"
wenzelm@49644
   856
        by (rule tendstoD)
hoelzl@37489
   857
      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
hoelzl@37489
   858
        by (rule eventually_subseq)
hoelzl@37489
   859
      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@37489
   860
        using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
hoelzl@37489
   861
    }
hoelzl@37489
   862
    ultimately show ?case by auto
hoelzl@37489
   863
  qed
hoelzl@37489
   864
qed
hoelzl@37489
   865
huffman@44136
   866
instance vec :: (heine_borel, finite) heine_borel
hoelzl@37489
   867
proof
hoelzl@50998
   868
  fix f :: "nat \<Rightarrow> 'a ^ 'b"
hoelzl@50998
   869
  assume f: "bounded (range f)"
hoelzl@37489
   870
  then obtain l r where r: "subseq r"
wenzelm@49644
   871
      and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@50998
   872
    using compact_lemma_cart [OF f] by blast
hoelzl@37489
   873
  let ?d = "UNIV::'b set"
hoelzl@37489
   874
  { fix e::real assume "e>0"
hoelzl@37489
   875
    hence "0 < e / (real_of_nat (card ?d))"
wenzelm@49644
   876
      using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
hoelzl@37489
   877
    with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
hoelzl@37489
   878
      by simp
hoelzl@37489
   879
    moreover
wenzelm@49644
   880
    { fix n
wenzelm@49644
   881
      assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
hoelzl@37489
   882
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
huffman@44136
   883
        unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
hoelzl@37489
   884
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
hoelzl@37489
   885
        by (rule setsum_strict_mono) (simp_all add: n)
hoelzl@37489
   886
      finally have "dist (f (r n)) l < e" by simp
hoelzl@37489
   887
    }
hoelzl@37489
   888
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
hoelzl@37489
   889
      by (rule eventually_elim1)
hoelzl@37489
   890
  }
wenzelm@49644
   891
  hence "((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
hoelzl@37489
   892
  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
hoelzl@37489
   893
qed
hoelzl@37489
   894
wenzelm@49644
   895
lemma interval_cart:
immler@54775
   896
  fixes a :: "real^'n"
immler@54775
   897
  shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
immler@56188
   898
    and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
immler@56188
   899
  by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
hoelzl@37489
   900
wenzelm@49644
   901
lemma mem_interval_cart:
immler@54775
   902
  fixes a :: "real^'n"
immler@54775
   903
  shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
immler@56188
   904
    and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
wenzelm@49644
   905
  using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
hoelzl@37489
   906
wenzelm@49644
   907
lemma interval_eq_empty_cart:
wenzelm@49644
   908
  fixes a :: "real^'n"
immler@54775
   909
  shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
immler@56188
   910
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
wenzelm@49644
   911
proof -
immler@54775
   912
  { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>box a b"
hoelzl@37489
   913
    hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto
hoelzl@37489
   914
    hence "a$i < b$i" by auto
wenzelm@49644
   915
    hence False using as by auto }
hoelzl@37489
   916
  moreover
hoelzl@37489
   917
  { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
hoelzl@37489
   918
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
   919
    { fix i
hoelzl@37489
   920
      have "a$i < b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
   921
      hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
hoelzl@37489
   922
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
   923
        by auto }
immler@54775
   924
    hence "box a b \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
hoelzl@37489
   925
  ultimately show ?th1 by blast
hoelzl@37489
   926
immler@56188
   927
  { fix i x assume as:"b$i < a$i" and x:"x\<in>cbox a b"
hoelzl@37489
   928
    hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto
hoelzl@37489
   929
    hence "a$i \<le> b$i" by auto
wenzelm@49644
   930
    hence False using as by auto }
hoelzl@37489
   931
  moreover
hoelzl@37489
   932
  { assume as:"\<forall>i. \<not> (b$i < a$i)"
hoelzl@37489
   933
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
   934
    { fix i
hoelzl@37489
   935
      have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
   936
      hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
hoelzl@37489
   937
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
   938
        by auto }
immler@56188
   939
    hence "cbox a b \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
hoelzl@37489
   940
  ultimately show ?th2 by blast
hoelzl@37489
   941
qed
hoelzl@37489
   942
wenzelm@49644
   943
lemma interval_ne_empty_cart:
wenzelm@49644
   944
  fixes a :: "real^'n"
immler@56188
   945
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
immler@54775
   946
    and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
hoelzl@37489
   947
  unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
hoelzl@37489
   948
    (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
   949
wenzelm@49644
   950
lemma subset_interval_imp_cart:
wenzelm@49644
   951
  fixes a :: "real^'n"
immler@56188
   952
  shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
immler@56188
   953
    and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
immler@56188
   954
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
immler@54775
   955
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b"
hoelzl@37489
   956
  unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
hoelzl@37489
   957
  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
   958
wenzelm@49644
   959
lemma interval_sing:
wenzelm@49644
   960
  fixes a :: "'a::linorder^'n"
wenzelm@49644
   961
  shows "{a .. a} = {a} \<and> {a<..<a} = {}"
wenzelm@49644
   962
  apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
wenzelm@49644
   963
  done
hoelzl@37489
   964
wenzelm@49644
   965
lemma subset_interval_cart:
wenzelm@49644
   966
  fixes a :: "real^'n"
immler@56188
   967
  shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
immler@56188
   968
    and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
immler@56188
   969
    and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
immler@54775
   970
    and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
immler@56188
   971
  using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
   972
wenzelm@49644
   973
lemma disjoint_interval_cart:
wenzelm@49644
   974
  fixes a::"real^'n"
immler@56188
   975
  shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
immler@56188
   976
    and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
immler@56188
   977
    and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
immler@54775
   978
    and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
hoelzl@50526
   979
  using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
   980
wenzelm@49644
   981
lemma inter_interval_cart:
immler@54775
   982
  fixes a :: "real^'n"
immler@56188
   983
  shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
immler@56188
   984
  unfolding inter_interval
immler@56188
   985
  by (auto simp: mem_box less_eq_vec_def)
immler@56188
   986
    (auto simp: Basis_vec_def inner_axis)
hoelzl@37489
   987
wenzelm@49644
   988
lemma closed_interval_left_cart:
wenzelm@49644
   989
  fixes b :: "real^'n"
hoelzl@37489
   990
  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
huffman@44233
   991
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
hoelzl@37489
   992
wenzelm@49644
   993
lemma closed_interval_right_cart:
wenzelm@49644
   994
  fixes a::"real^'n"
hoelzl@37489
   995
  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
huffman@44233
   996
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
hoelzl@37489
   997
wenzelm@49644
   998
lemma is_interval_cart:
wenzelm@49644
   999
  "is_interval (s::(real^'n) set) \<longleftrightarrow>
wenzelm@49644
  1000
    (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
hoelzl@50526
  1001
  by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
hoelzl@37489
  1002
wenzelm@49644
  1003
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
huffman@44233
  1004
  by (simp add: closed_Collect_le)
hoelzl@37489
  1005
wenzelm@49644
  1006
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
huffman@44233
  1007
  by (simp add: closed_Collect_le)
hoelzl@37489
  1008
wenzelm@49644
  1009
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
wenzelm@49644
  1010
  by (simp add: open_Collect_less)
wenzelm@49644
  1011
wenzelm@49644
  1012
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
huffman@44233
  1013
  by (simp add: open_Collect_less)
hoelzl@37489
  1014
wenzelm@49644
  1015
lemma Lim_component_le_cart:
wenzelm@49644
  1016
  fixes f :: "'a \<Rightarrow> real^'n"
hoelzl@50526
  1017
  assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"
hoelzl@37489
  1018
  shows "l$i \<le> b"
hoelzl@50526
  1019
  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
hoelzl@37489
  1020
wenzelm@49644
  1021
lemma Lim_component_ge_cart:
wenzelm@49644
  1022
  fixes f :: "'a \<Rightarrow> real^'n"
hoelzl@37489
  1023
  assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
hoelzl@37489
  1024
  shows "b \<le> l$i"
hoelzl@50526
  1025
  by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
hoelzl@37489
  1026
wenzelm@49644
  1027
lemma Lim_component_eq_cart:
wenzelm@49644
  1028
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@49644
  1029
  assumes net: "(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
hoelzl@37489
  1030
  shows "l$i = b"
wenzelm@49644
  1031
  using ev[unfolded order_eq_iff eventually_conj_iff] and
wenzelm@49644
  1032
    Lim_component_ge_cart[OF net, of b i] and
hoelzl@37489
  1033
    Lim_component_le_cart[OF net, of i b] by auto
hoelzl@37489
  1034
wenzelm@49644
  1035
lemma connected_ivt_component_cart:
wenzelm@49644
  1036
  fixes x :: "real^'n"
wenzelm@49644
  1037
  shows "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s.  z$k = a)"
hoelzl@50526
  1038
  using connected_ivt_hyperplane[of s x y "axis k 1" a]
hoelzl@50526
  1039
  by (auto simp add: inner_axis inner_commute)
hoelzl@37489
  1040
wenzelm@49644
  1041
lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
hoelzl@37489
  1042
  unfolding subspace_def by auto
hoelzl@37489
  1043
hoelzl@37489
  1044
lemma closed_substandard_cart:
huffman@44213
  1045
  "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
  1046
proof -
huffman@44213
  1047
  { fix i::'n
huffman@44213
  1048
    have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
  1049
      by (cases "P i") (simp_all add: closed_Collect_eq) }
huffman@44213
  1050
  thus ?thesis
huffman@44213
  1051
    unfolding Collect_all_eq by (simp add: closed_INT)
hoelzl@37489
  1052
qed
hoelzl@37489
  1053
wenzelm@49644
  1054
lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
wenzelm@49644
  1055
  (is "dim ?A = _")
wenzelm@49644
  1056
proof -
hoelzl@50526
  1057
  let ?a = "\<lambda>x. axis x 1 :: real^'n"
hoelzl@50526
  1058
  have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
hoelzl@50526
  1059
    by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
hoelzl@50526
  1060
  have "?a ` d \<subseteq> Basis"
hoelzl@50526
  1061
    by (auto simp: Basis_vec_def)
wenzelm@49644
  1062
  thus ?thesis
hoelzl@50526
  1063
    using dim_substandard[of "?a ` d"] card_image[of ?a d]
hoelzl@50526
  1064
    by (auto simp: axis_eq_axis inj_on_def *)
hoelzl@37489
  1065
qed
hoelzl@37489
  1066
hoelzl@37489
  1067
lemma affinity_inverses:
hoelzl@37489
  1068
  assumes m0: "m \<noteq> (0::'a::field)"
hoelzl@37489
  1069
  shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
hoelzl@37489
  1070
  "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
hoelzl@37489
  1071
  using m0
haftmann@54230
  1072
  apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
haftmann@54230
  1073
  apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
wenzelm@49644
  1074
  done
hoelzl@37489
  1075
hoelzl@37489
  1076
lemma vector_affinity_eq:
hoelzl@37489
  1077
  assumes m0: "(m::'a::field) \<noteq> 0"
hoelzl@37489
  1078
  shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
hoelzl@37489
  1079
proof
hoelzl@37489
  1080
  assume h: "m *s x + c = y"
hoelzl@37489
  1081
  hence "m *s x = y - c" by (simp add: field_simps)
hoelzl@37489
  1082
  hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
hoelzl@37489
  1083
  then show "x = inverse m *s y + - (inverse m *s c)"
hoelzl@37489
  1084
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1085
next
hoelzl@37489
  1086
  assume h: "x = inverse m *s y + - (inverse m *s c)"
haftmann@54230
  1087
  show "m *s x + c = y" unfolding h
hoelzl@37489
  1088
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1089
qed
hoelzl@37489
  1090
hoelzl@37489
  1091
lemma vector_eq_affinity:
wenzelm@49644
  1092
    "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
hoelzl@37489
  1093
  using vector_affinity_eq[where m=m and x=x and y=y and c=c]
hoelzl@37489
  1094
  by metis
hoelzl@37489
  1095
hoelzl@50526
  1096
lemma vector_cart:
hoelzl@50526
  1097
  fixes f :: "real^'n \<Rightarrow> real"
hoelzl@50526
  1098
  shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
hoelzl@50526
  1099
  unfolding euclidean_eq_iff[where 'a="real^'n"]
hoelzl@50526
  1100
  by simp (simp add: Basis_vec_def inner_axis)
hoelzl@50526
  1101
  
hoelzl@50526
  1102
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
hoelzl@50526
  1103
  by (rule vector_cart)
wenzelm@49644
  1104
huffman@44360
  1105
subsection "Convex Euclidean Space"
hoelzl@37489
  1106
hoelzl@50526
  1107
lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
hoelzl@50526
  1108
  using const_vector_cart[of 1] by (simp add: one_vec_def)
hoelzl@37489
  1109
hoelzl@37489
  1110
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
hoelzl@37489
  1111
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
hoelzl@37489
  1112
hoelzl@50526
  1113
lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
hoelzl@37489
  1114
hoelzl@37489
  1115
lemma convex_box_cart:
hoelzl@37489
  1116
  assumes "\<And>i. convex {x. P i x}"
hoelzl@37489
  1117
  shows "convex {x. \<forall>i. P i (x$i)}"
hoelzl@37489
  1118
  using assms unfolding convex_def by auto
hoelzl@37489
  1119
hoelzl@37489
  1120
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
hoelzl@37489
  1121
  by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)
hoelzl@37489
  1122
hoelzl@37489
  1123
lemma unit_interval_convex_hull_cart:
immler@56188
  1124
  "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}"
immler@56188
  1125
  unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
hoelzl@50526
  1126
  by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
hoelzl@37489
  1127
hoelzl@37489
  1128
lemma cube_convex_hull_cart:
wenzelm@49644
  1129
  assumes "0 < d"
wenzelm@49644
  1130
  obtains s::"(real^'n) set"
immler@56188
  1131
    where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
wenzelm@49644
  1132
proof -
wenzelm@55522
  1133
  from assms obtain s where "finite s"
immler@56188
  1134
    and "cbox (x - setsum (op *\<^sub>R d) Basis) (x + setsum (op *\<^sub>R d) Basis) = convex hull s"
wenzelm@55522
  1135
    by (rule cube_convex_hull)
wenzelm@55522
  1136
  with that[of s] show thesis
wenzelm@55522
  1137
    by (simp add: const_vector_cart)
hoelzl@37489
  1138
qed
hoelzl@37489
  1139
hoelzl@37489
  1140
hoelzl@37489
  1141
subsection "Derivative"
hoelzl@37489
  1142
hoelzl@37489
  1143
definition "jacobian f net = matrix(frechet_derivative f net)"
hoelzl@37489
  1144
wenzelm@49644
  1145
lemma jacobian_works:
wenzelm@49644
  1146
  "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
wenzelm@49644
  1147
    (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
wenzelm@49644
  1148
  apply rule
wenzelm@49644
  1149
  unfolding jacobian_def
wenzelm@49644
  1150
  apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
wenzelm@49644
  1151
  apply (rule differentiableI)
wenzelm@49644
  1152
  apply assumption
wenzelm@49644
  1153
  unfolding frechet_derivative_works
wenzelm@49644
  1154
  apply assumption
wenzelm@49644
  1155
  done
hoelzl@37489
  1156
hoelzl@37489
  1157
wenzelm@60420
  1158
subsection \<open>Component of the differential must be zero if it exists at a local
wenzelm@60420
  1159
  maximum or minimum for that corresponding component.\<close>
hoelzl@37489
  1160
hoelzl@50526
  1161
lemma differential_zero_maxmin_cart:
wenzelm@49644
  1162
  fixes f::"real^'a \<Rightarrow> real^'b"
wenzelm@49644
  1163
  assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
hoelzl@50526
  1164
    "f differentiable (at x)"
hoelzl@50526
  1165
  shows "jacobian f (at x) $ k = 0"
hoelzl@50526
  1166
  using differential_zero_maxmin_component[of "axis k 1" e x f] assms
hoelzl@50526
  1167
    vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
hoelzl@50526
  1168
  by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
wenzelm@49644
  1169
wenzelm@60420
  1170
subsection \<open>Lemmas for working on @{typ "real^1"}\<close>
hoelzl@37489
  1171
hoelzl@37489
  1172
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
wenzelm@49644
  1173
  by (metis (full_types) num1_eq_iff)
hoelzl@37489
  1174
hoelzl@37489
  1175
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
wenzelm@49644
  1176
  by auto (metis (full_types) num1_eq_iff)
hoelzl@37489
  1177
hoelzl@37489
  1178
lemma exhaust_2:
wenzelm@49644
  1179
  fixes x :: 2
wenzelm@49644
  1180
  shows "x = 1 \<or> x = 2"
hoelzl@37489
  1181
proof (induct x)
hoelzl@37489
  1182
  case (of_int z)
hoelzl@37489
  1183
  then have "0 <= z" and "z < 2" by simp_all
hoelzl@37489
  1184
  then have "z = 0 | z = 1" by arith
hoelzl@37489
  1185
  then show ?case by auto
hoelzl@37489
  1186
qed
hoelzl@37489
  1187
hoelzl@37489
  1188
lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
hoelzl@37489
  1189
  by (metis exhaust_2)
hoelzl@37489
  1190
hoelzl@37489
  1191
lemma exhaust_3:
wenzelm@49644
  1192
  fixes x :: 3
wenzelm@49644
  1193
  shows "x = 1 \<or> x = 2 \<or> x = 3"
hoelzl@37489
  1194
proof (induct x)
hoelzl@37489
  1195
  case (of_int z)
hoelzl@37489
  1196
  then have "0 <= z" and "z < 3" by simp_all
hoelzl@37489
  1197
  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
hoelzl@37489
  1198
  then show ?case by auto
hoelzl@37489
  1199
qed
hoelzl@37489
  1200
hoelzl@37489
  1201
lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
hoelzl@37489
  1202
  by (metis exhaust_3)
hoelzl@37489
  1203
hoelzl@37489
  1204
lemma UNIV_1 [simp]: "UNIV = {1::1}"
hoelzl@37489
  1205
  by (auto simp add: num1_eq_iff)
hoelzl@37489
  1206
hoelzl@37489
  1207
lemma UNIV_2: "UNIV = {1::2, 2::2}"
hoelzl@37489
  1208
  using exhaust_2 by auto
hoelzl@37489
  1209
hoelzl@37489
  1210
lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
hoelzl@37489
  1211
  using exhaust_3 by auto
hoelzl@37489
  1212
hoelzl@37489
  1213
lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
hoelzl@37489
  1214
  unfolding UNIV_1 by simp
hoelzl@37489
  1215
hoelzl@37489
  1216
lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
hoelzl@37489
  1217
  unfolding UNIV_2 by simp
hoelzl@37489
  1218
hoelzl@37489
  1219
lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
haftmann@57514
  1220
  unfolding UNIV_3 by (simp add: ac_simps)
hoelzl@37489
  1221
wenzelm@49644
  1222
instantiation num1 :: cart_one
wenzelm@49644
  1223
begin
wenzelm@49644
  1224
wenzelm@49644
  1225
instance
wenzelm@49644
  1226
proof
hoelzl@37489
  1227
  show "CARD(1) = Suc 0" by auto
wenzelm@49644
  1228
qed
wenzelm@49644
  1229
wenzelm@49644
  1230
end
hoelzl@37489
  1231
wenzelm@60420
  1232
subsection\<open>The collapse of the general concepts to dimension one.\<close>
hoelzl@37489
  1233
hoelzl@37489
  1234
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
huffman@44136
  1235
  by (simp add: vec_eq_iff)
hoelzl@37489
  1236
hoelzl@37489
  1237
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
hoelzl@37489
  1238
  apply auto
hoelzl@37489
  1239
  apply (erule_tac x= "x$1" in allE)
hoelzl@37489
  1240
  apply (simp only: vector_one[symmetric])
hoelzl@37489
  1241
  done
hoelzl@37489
  1242
hoelzl@37489
  1243
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
huffman@44136
  1244
  by (simp add: norm_vec_def)
hoelzl@37489
  1245
hoelzl@37489
  1246
lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
hoelzl@37489
  1247
  by (simp add: norm_vector_1)
hoelzl@37489
  1248
hoelzl@37489
  1249
lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
hoelzl@37489
  1250
  by (auto simp add: norm_real dist_norm)
hoelzl@37489
  1251
wenzelm@49644
  1252
wenzelm@60420
  1253
subsection\<open>Explicit vector construction from lists.\<close>
hoelzl@37489
  1254
hoelzl@43995
  1255
definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
hoelzl@37489
  1256
hoelzl@37489
  1257
lemma vector_1: "(vector[x]) $1 = x"
hoelzl@37489
  1258
  unfolding vector_def by simp
hoelzl@37489
  1259
hoelzl@37489
  1260
lemma vector_2:
hoelzl@37489
  1261
 "(vector[x,y]) $1 = x"
hoelzl@37489
  1262
 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
hoelzl@37489
  1263
  unfolding vector_def by simp_all
hoelzl@37489
  1264
hoelzl@37489
  1265
lemma vector_3:
hoelzl@37489
  1266
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
hoelzl@37489
  1267
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
hoelzl@37489
  1268
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
hoelzl@37489
  1269
  unfolding vector_def by simp_all
hoelzl@37489
  1270
hoelzl@37489
  1271
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
hoelzl@37489
  1272
  apply auto
hoelzl@37489
  1273
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1274
  apply (subgoal_tac "vector [v$1] = v")
hoelzl@37489
  1275
  apply simp
hoelzl@37489
  1276
  apply (vector vector_def)
hoelzl@37489
  1277
  apply simp
hoelzl@37489
  1278
  done
hoelzl@37489
  1279
hoelzl@37489
  1280
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
hoelzl@37489
  1281
  apply auto
hoelzl@37489
  1282
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1283
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1284
  apply (subgoal_tac "vector [v$1, v$2] = v")
hoelzl@37489
  1285
  apply simp
hoelzl@37489
  1286
  apply (vector vector_def)
hoelzl@37489
  1287
  apply (simp add: forall_2)
hoelzl@37489
  1288
  done
hoelzl@37489
  1289
hoelzl@37489
  1290
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
hoelzl@37489
  1291
  apply auto
hoelzl@37489
  1292
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1293
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1294
  apply (erule_tac x="v$3" in allE)
hoelzl@37489
  1295
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
hoelzl@37489
  1296
  apply simp
hoelzl@37489
  1297
  apply (vector vector_def)
hoelzl@37489
  1298
  apply (simp add: forall_3)
hoelzl@37489
  1299
  done
hoelzl@37489
  1300
hoelzl@37489
  1301
lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
wenzelm@49644
  1302
  apply (rule bounded_linearI[where K=1])
hoelzl@37489
  1303
  using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
hoelzl@37489
  1304
wenzelm@49644
  1305
lemma integral_component_eq_cart[simp]:
immler@56188
  1306
  fixes f :: "'n::euclidean_space \<Rightarrow> real^'m"
wenzelm@49644
  1307
  assumes "f integrable_on s"
wenzelm@49644
  1308
  shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"
hoelzl@37489
  1309
  using integral_linear[OF assms(1) bounded_linear_component_cart,unfolded o_def] .
hoelzl@37489
  1310
hoelzl@37489
  1311
lemma interval_split_cart:
hoelzl@37489
  1312
  "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
immler@56188
  1313
  "cbox a b \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
wenzelm@49644
  1314
  apply (rule_tac[!] set_eqI)
immler@56188
  1315
  unfolding Int_iff mem_interval_cart mem_Collect_eq interval_cbox_cart
wenzelm@49644
  1316
  unfolding vec_lambda_beta
wenzelm@49644
  1317
  by auto
hoelzl@37489
  1318
hoelzl@37489
  1319
end