src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
author hoelzl
Tue, 18 Mar 2014 15:53:48 +0100
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cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
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header {*Instanciates the finite cartesian product of euclidean spaces as a euclidean space.*}
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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_Plus:
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  "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
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    (\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))"
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  unfolding Plus_def
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  by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)
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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 (rule setsum_Plus [OF finite finite])
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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{* Basic componentwise operations on vectors. *}
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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{* The ordering on one-dimensional vectors is linear. *}
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class cart_one =
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  assumes UNIV_one: "card (UNIV \<Colon> '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 default (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{* Constant Vectors *}
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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{* Also the scalar-vector multiplication. *}
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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 {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
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method_setup vector = {*
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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_addf} 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 => rtac @{thm setsum_cong2} i
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         ORELSE rtac @{thm setsum_0'} 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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*} "lift trivial vector statements to real arith statements"
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lemma vec_0[simp]: "vec 0 = 0" by (vector zero_vec_def)
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lemma vec_1[simp]: "vec 1 = 1" by (vector one_vec_def)
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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 vec_def)
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lemma vec_sub: "vec(x - y) = vec x - vec y" by (vector vec_def)
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lemma vec_cmul: "vec(c * x) = c *s vec x " by (vector vec_def)
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lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def)
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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{* Obvious "component-pushing". *}
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lemma vec_component [simp]: "vec x $ i = x"
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  by (vector vec_def)
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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 {* Some frequently useful arithmetic lemmas over vectors. *}
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instance vec :: (semigroup_mult, finite) semigroup_mult
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  by default (vector mult_assoc)
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instance vec :: (monoid_mult, finite) monoid_mult
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  by default vector+
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instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
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  by default (vector mult_commute)
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instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
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  by default vector
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instance vec :: (semiring, finite) semiring
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  by default (vector field_simps)+
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instance vec :: (semiring_0, finite) semiring_0
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  by default (vector field_simps)+
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instance vec :: (semiring_1, finite) semiring_1
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  by default vector
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instance vec :: (comm_semiring, finite) comm_semiring
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  by default (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 default (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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lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
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  unfolding dist_norm scalar_mult_eq_scaleR
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  unfolding scaleR_right_diff_distrib[symmetric] by simp
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lemma setsum_component [simp]:
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  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
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  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
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proof (cases "finite S")
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  case True
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  then show ?thesis by induct simp_all
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next
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  case False
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  then show ?thesis by simp
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qed
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lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
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  by (simp add: vec_eq_iff)
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lemma setsum_cmul:
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  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
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  shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
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  by (simp add: vec_eq_iff setsum_right_distrib)
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lemma setsum_norm_allsubsets_bound_cart:
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  fixes f:: "'a \<Rightarrow> real ^'n"
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  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
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  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
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  using setsum_norm_allsubsets_bound[OF assms]
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  by (simp add: DIM_cart Basis_real_def)
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subsection {* Matrix operations *}
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text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
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definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
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    (infixl "**" 70)
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  where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
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definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
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    (infixl "*v" 70)
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  where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
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definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
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    (infixl "v*" 70)
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  where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
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definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
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definition transpose where 
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  "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
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definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
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definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
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definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
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definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
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lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
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lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
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  by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps)
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lemma matrix_mul_lid:
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   363
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
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   364
  shows "mat 1 ** A = A"
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   365
  apply (simp add: matrix_matrix_mult_def mat_def)
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   366
  apply vector
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  apply (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite]
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    mult_1_left mult_zero_left if_True UNIV_I)
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   369
  done
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lemma matrix_mul_rid:
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  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
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  shows "A ** mat 1 = A"
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  apply (simp add: matrix_matrix_mult_def mat_def)
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  apply vector
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   377
  apply (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite]
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    mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
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   379
  done
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lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
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  apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
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   383
  apply (subst setsum_commute)
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   384
  apply simp
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   385
  done
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lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
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   388
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   389
    setsum_right_distrib setsum_left_distrib mult_assoc)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   390
  apply (subst setsum_commute)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   391
  apply simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   392
  done
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   393
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   394
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
   395
  apply (vector matrix_vector_mult_def mat_def)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   396
  apply (simp add: if_distrib cond_application_beta setsum_delta' cong del: if_weak_cong)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   397
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   398
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   399
lemma matrix_transpose_mul:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   400
    "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   401
  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
   402
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   403
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
   404
  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
   405
  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
   406
  apply auto
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   407
  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
   408
  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
   409
  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
   410
  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
   411
  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
   412
  apply (auto simp add: if_distrib cond_application_beta axis_def
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   413
    setsum_delta[OF finite] cong del: if_weak_cong)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   414
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   415
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   416
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
   417
  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
   418
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   419
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   420
  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   421
  apply (subst setsum_commute)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   422
  apply simp
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   423
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   424
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   425
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
   426
  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
   427
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   428
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
   429
  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
   430
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   431
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
   432
  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
   433
  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
   434
  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
   435
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   436
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
   437
  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
   438
  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
   439
  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
   440
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   441
lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   442
  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
   443
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   444
lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   445
  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
   446
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   447
text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   448
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   449
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
   450
  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
   451
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   452
lemma matrix_mult_vsum:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   453
  "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   454
  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   455
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   456
lemma vector_componentwise:
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   457
  "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   458
  by (simp add: axis_def if_distrib setsum_cases vec_eq_iff)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   459
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   460
lemma basis_expansion: "setsum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   461
  by (auto simp add: axis_def vec_eq_iff if_distrib setsum_cases cong del: if_weak_cong)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   462
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   463
lemma linear_componentwise:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   464
  fixes f:: "real ^'m \<Rightarrow> real ^ _"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   465
  assumes lf: "linear f"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   466
  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   467
proof -
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   468
  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
   469
  let ?N = "(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
   470
  have fM: "finite ?M" by simp
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
   471
  have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   472
    unfolding setsum_component by simp
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   473
  then show ?thesis
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
   474
    unfolding linear_setsum_mul[OF lf fM, 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
   475
    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
   476
    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
   477
    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
   478
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   479
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   480
text{* Inverse matrices  (not necessarily square) *}
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   481
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   482
definition
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   483
  "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
   484
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   485
definition
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   486
  "matrix_inv(A:: 'a::semiring_1^'n^'m) =
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   487
    (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
   488
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   489
text{* Correspondence between matrices and linear operators. *}
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   490
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   491
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
   492
  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
   493
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   494
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
   495
  by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   496
      field_simps setsum_right_distrib setsum_addf)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   497
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   498
lemma matrix_works:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   499
  assumes lf: "linear f"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   500
  shows "matrix f *v x = f (x::real ^ 'n)"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   501
  apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   502
  apply clarify
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   503
  apply (rule linear_componentwise[OF lf, symmetric])
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   504
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   505
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   506
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   507
  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
   508
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   509
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
   510
  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
   511
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   512
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
   513
  assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   514
    and lg: "linear (g::real^'m \<Rightarrow> real^_)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   515
  shows "matrix (g o f) = matrix g ** matrix f"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   516
  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
   517
  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
   518
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   519
lemma matrix_vector_column:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   520
  "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   521
  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
   522
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   523
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
   524
  apply (rule adjoint_unique)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   525
  apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   526
    setsum_left_distrib setsum_right_distrib)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   527
  apply (subst setsum_commute)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   528
  apply (auto simp add: mult_ac)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   529
  done
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   530
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   531
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
   532
  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
   533
  apply (subst matrix_vector_mul[OF lf])
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   534
  unfolding adjoint_matrix matrix_of_matrix_vector_mul
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   535
  apply rule
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   536
  done
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   537
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   538
44360
ea609ebdeebf section -> subsection
huffman
parents: 44282
diff changeset
   539
subsection {* lambda skolemization on cartesian products *}
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
(* 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
   542
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   543
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
37494
6e9f48cf6adf Make latex happy
hoelzl
parents: 37489
diff changeset
   544
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   545
proof -
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   546
  let ?S = "(UNIV :: 'n set)"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   547
  { assume H: "?rhs"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   548
    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
   549
  moreover
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   550
  { 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
   551
    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
   552
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   553
    { 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
   554
      from f have "P i (f i)" by metis
37494
6e9f48cf6adf Make latex happy
hoelzl
parents: 37489
diff changeset
   555
      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
   556
    }
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   557
    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
   558
    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
   559
  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
   560
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   561
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   562
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
   563
  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
   564
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   565
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
   566
  "(\<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
   567
  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
   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 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
   570
  "(\<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
   571
  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
   572
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   573
lemma matrix_left_invertible_injective:
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   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)"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   575
proof -
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   576
  { 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
   577
    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
   578
    hence "x = y"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   579
      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
   580
  moreover
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   581
  { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   582
    hence i: "inj (op *v A)" unfolding inj_on_def by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   583
    from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   584
    obtain g where g: "linear g" "g o op *v A = id" by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   585
    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
   586
      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
   587
      using g(2) by (simp add: fun_eq_iff)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   588
    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
   589
  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
   590
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   591
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   592
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
   593
  "(\<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
   594
  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
   595
  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
   596
  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
   597
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   598
lemma matrix_right_invertible_surjective:
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   599
  "(\<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
   600
proof -
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   601
  { fix B :: "real ^'m^'n"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   602
    assume AB: "A ** B = mat 1"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   603
    { 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
   604
      have "A *v (B *v x) = x"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   605
        by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   606
    hence "surj (op *v A)" unfolding surj_def by metis }
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   607
  moreover
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   608
  { assume sf: "surj (op *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
   609
    from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   610
    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   611
      by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   612
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   613
    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
   614
      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
   615
        matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
44165
d26a45f3c835 remove lemma stupid_ext
huffman
parents: 44140
diff changeset
   616
      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
   617
      .
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   618
    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
   619
  }
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   620
  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
   621
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   622
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   623
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
   624
  fixes A :: "real^'n^'m"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   625
  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   626
      (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   627
    (is "?lhs \<longleftrightarrow> ?rhs")
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   628
proof -
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   629
  let ?U = "UNIV :: 'n set"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   630
  { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   631
    { fix c i
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   632
      assume c: "setsum (\<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
   633
      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
   634
      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
   635
        using c
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   636
        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
   637
        by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   638
      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
   639
      have "c i = 0" by (vector vec_eq_iff)}
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   640
    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
   641
  moreover
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   642
  { assume H: ?rhs
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   643
    { 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
   644
      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
   645
      from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   646
      have "x = 0" by vector }
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   647
  }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   648
  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
   649
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   650
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   651
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
   652
  fixes A :: "real^'n^'m"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   653
  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   654
    (\<forall>c. setsum (\<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
   655
  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
   656
    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
   657
  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
   658
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   659
lemma matrix_right_invertible_span_columns:
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   660
  "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   661
    span (columns A) = UNIV" (is "?lhs = ?rhs")
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   662
proof -
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   663
  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
   664
  have fU: "finite ?U" by simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   665
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   666
    unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   667
    apply (subst eq_commute)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   668
    apply rule
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   669
    done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   670
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   671
  { assume h: ?lhs
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   672
    { fix x:: "real ^'n"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   673
      from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   674
        where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   675
      have "x \<in> span (columns A)"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   676
        unfolding y[symmetric]
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   677
        apply (rule span_setsum[OF fU])
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   678
        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
   679
        unfolding scalar_mult_eq_scaleR
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   680
        apply (rule span_mul)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   681
        apply (rule span_superset)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   682
        unfolding columns_def
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   683
        apply blast
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   684
        done
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   685
    }
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   686
    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
   687
  moreover
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   688
  { assume h:?rhs
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   689
    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   690
    { fix y
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   691
      have "?P y"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   692
      proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   693
        show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   694
          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
   695
      next
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   696
        fix c y1 y2
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   697
        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
   698
        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
   699
          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
   700
        from y2 obtain x:: "real ^'m" where
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   701
          x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   702
        let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   703
        show "?P (c*s y1 + y2)"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 49644
diff changeset
   704
        proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   705
          fix j
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   706
          have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   707
              else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   708
            using i(1) by (simp add: field_simps)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   709
          have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   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)$j))) ?U"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   711
            apply (rule setsum_cong[OF refl])
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   712
            using th apply blast
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   713
            done
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   714
          also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   715
            by (simp add: setsum_addf)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   716
          also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   717
            unfolding setsum_delta[OF fU]
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   718
            using i(1) by simp
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   719
          finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   720
            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   721
        qed
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   722
      next
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   723
        show "y \<in> span (columns A)"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   724
          unfolding h by blast
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   725
      qed
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   726
    }
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   727
    then have ?lhs unfolding lhseq ..
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   728
  }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   729
  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
   730
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   731
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   732
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
   733
  "(\<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
   734
  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
   735
  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
   736
  unfolding matrix_right_invertible_span_columns
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   737
  ..
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   738
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   739
text {* The same result in terms of square matrices. *}
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   740
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   741
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
   742
  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
   743
  shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   744
proof -
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   745
  { fix A A' :: "real ^'n^'n"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   746
    assume AA': "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
   747
    have sA: "surj (op *v A)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   748
      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
   749
      apply clarify
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   750
      apply (rule_tac x="(A' *v y)" in exI)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   751
      apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   752
      done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   753
    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
   754
    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
   755
      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
   756
    have th: "matrix f' ** A = mat 1"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   757
      by (simp add: matrix_eq matrix_works[OF f'(1)]
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   758
          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
   759
    hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   760
    hence "matrix f' = A'"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   761
      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
   762
    hence "matrix f' ** A = A' ** A" by simp
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   763
    hence "A' ** A = mat 1" by (simp add: th)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   764
  }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   765
  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
   766
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   767
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   768
text {* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *}
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   769
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   770
definition "rowvector v = (\<chi> i j. (v$j))"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   771
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   772
definition "columnvector v = (\<chi> i j. (v$i))"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   773
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   774
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
   775
  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
   776
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   777
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
   778
  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
   779
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   780
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
   781
  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
   782
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   783
lemma dot_matrix_product:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   784
  "(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
   785
  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
   786
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   787
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
   788
  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
   789
  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
   790
      (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   791
  unfolding dot_matrix_product transpose_columnvector[symmetric]
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   792
    dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   793
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   794
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   795
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in>UNIV}"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   796
  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
   797
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   798
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
   799
  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
   800
  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
   801
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   802
lemma continuous_component: "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
   803
  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
   804
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   805
lemma continuous_on_component: "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
   806
  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
   807
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   808
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44219
diff changeset
   809
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
44213
6fb54701a11b add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents: 44211
diff changeset
   810
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   811
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   812
  unfolding bounded_def
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   813
  apply clarify
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   814
  apply (rule_tac x="x $ i" in exI)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   815
  apply (rule_tac x="e" in exI)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   816
  apply clarify
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   817
  apply (rule order_trans [OF dist_vec_nth_le], simp)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   818
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   819
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   820
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
   821
  fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
50998
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   822
  assumes f: "bounded (range f)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   823
  shows "\<forall>d.
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   824
        \<exists>l r. subseq r \<and>
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   825
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   826
proof
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   827
  fix d :: "'n set"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   828
  have "finite 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
   829
  thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   830
      (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   831
  proof (induct d)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   832
    case empty
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   833
    thus ?case unfolding subseq_def by auto
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   834
  next
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   835
    case (insert k d)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   836
    obtain l1::"'a^'n" and r1 where r1:"subseq r1"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   837
      and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   838
      using insert(3) by auto
50998
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   839
    have s': "bounded ((\<lambda>x. x $ k) ` range f)" using `bounded (range f)`
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   840
      by (auto intro!: bounded_component_cart)
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   841
    have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` range f" by simp
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   842
    have "bounded (range (\<lambda>i. f (r1 i) $ k))"
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   843
      by (metis (lifting) bounded_subset image_subsetI f' s')
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   844
    then obtain l2 r2 where r2: "subseq r2"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   845
      and lr2: "((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
50998
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   846
      using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) $ k"] by (auto simp: o_def)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   847
    def r \<equiv> "r1 \<circ> r2"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   848
    have r: "subseq r"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   849
      using r1 and r2 unfolding r_def o_def subseq_def by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   850
    moreover
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   851
    def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   852
    { fix e :: real assume "e > 0"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   853
      from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   854
        by blast
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   855
      from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   856
        by (rule tendstoD)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   857
      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   858
        by (rule eventually_subseq)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   859
      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   860
        using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   861
    }
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   862
    ultimately show ?case by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   863
  qed
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
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   866
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
   867
proof
50998
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   868
  fix f :: "nat \<Rightarrow> 'a ^ 'b"
501200635659 simplify heine_borel type class
hoelzl
parents: 50526
diff changeset
   869
  assume f: "bounded (range f)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   870
  then obtain l r where r: "subseq r"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   871
      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
   872
    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
   873
  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
   874
  { 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
   875
    hence "0 < e / (real_of_nat (card ?d))"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   876
      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
   877
    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
   878
      by simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   879
    moreover
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   880
    { fix n
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   881
      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
   882
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   883
        unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   884
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   885
        by (rule setsum_strict_mono) (simp_all add: n)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   886
      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
   887
    }
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   888
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   889
      by (rule eventually_elim1)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   890
  }
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   891
  hence "((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff 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
   892
  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   893
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   894
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   895
lemma interval_cart:
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   896
  fixes a :: "real^'n"
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   897
  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
   898
    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
   899
  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
   900
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   901
lemma mem_interval_cart:
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   902
  fixes a :: "real^'n"
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   903
  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
   904
    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
   905
  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
   906
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   907
lemma interval_eq_empty_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   908
  fixes a :: "real^'n"
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   909
  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
   910
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   911
proof -
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   912
  { 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
   913
    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
   914
    hence "a$i < b$i" by auto
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   915
    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
   916
  moreover
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   917
  { 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
   918
    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
   919
    { fix i
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   920
      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
   921
      hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   922
        unfolding vector_smult_component and vector_add_component
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   923
        by auto }
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   924
    hence "box a b \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   925
  ultimately show ?th1 by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   926
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   927
  { fix i x assume as:"b$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
   928
    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
   929
    hence "a$i \<le> b$i" by auto
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   930
    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
   931
  moreover
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   932
  { 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
   933
    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
   934
    { fix i
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   935
      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
   936
      hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   937
        unfolding vector_smult_component and vector_add_component
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   938
        by auto }
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   939
    hence "cbox a b \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   940
  ultimately show ?th2 by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   941
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   942
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   943
lemma interval_ne_empty_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   944
  fixes a :: "real^'n"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   945
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   946
    and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   947
  unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   948
    (* BH: Why doesn't just "auto" work here? *)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   949
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   950
lemma subset_interval_imp_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   951
  fixes a :: "real^'n"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   952
  shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   953
    and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   954
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   955
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   956
  unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   957
  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   958
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   959
lemma interval_sing:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   960
  fixes a :: "'a::linorder^'n"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   961
  shows "{a .. a} = {a} \<and> {a<..<a} = {}"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   962
  apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   963
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   964
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   965
lemma subset_interval_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   966
  fixes a :: "real^'n"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   967
  shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   968
    and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   969
    and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   970
    and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   971
  using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   972
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   973
lemma disjoint_interval_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   974
  fixes a::"real^'n"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   975
  shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   976
    and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   977
    and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   978
    and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
   979
  using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   980
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   981
lemma inter_interval_cart:
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   982
  fixes a :: "real^'n"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   983
  shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   984
  unfolding inter_interval
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   985
  by (auto simp: mem_box less_eq_vec_def)
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
   986
    (auto simp: Basis_vec_def inner_axis)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   987
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   988
lemma closed_interval_left_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   989
  fixes b :: "real^'n"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   990
  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44219
diff changeset
   991
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   992
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   993
lemma closed_interval_right_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   994
  fixes a::"real^'n"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   995
  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44219
diff changeset
   996
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
   997
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   998
lemma is_interval_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
   999
  "is_interval (s::(real^'n) set) \<longleftrightarrow>
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1000
    (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1001
  by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1002
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1003
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44219
diff changeset
  1004
  by (simp add: closed_Collect_le)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1005
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1006
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44219
diff changeset
  1007
  by (simp add: closed_Collect_le)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1008
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1009
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1010
  by (simp add: open_Collect_less)
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1011
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1012
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44219
diff changeset
  1013
  by (simp add: open_Collect_less)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1014
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1015
lemma Lim_component_le_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1016
  fixes f :: "'a \<Rightarrow> 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
  1017
  assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1018
  shows "l$i \<le> b"
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
  1019
  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1020
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1021
lemma Lim_component_ge_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1022
  fixes f :: "'a \<Rightarrow> 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
  1023
  assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1024
  shows "b \<le> l$i"
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
  1025
  by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1026
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1027
lemma Lim_component_eq_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1028
  fixes f :: "'a \<Rightarrow> real^'n"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1029
  assumes net: "(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1030
  shows "l$i = b"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1031
  using ev[unfolded order_eq_iff eventually_conj_iff] and
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1032
    Lim_component_ge_cart[OF net, of b i] and
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1033
    Lim_component_le_cart[OF net, of i b] by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1034
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1035
lemma connected_ivt_component_cart:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1036
  fixes x :: "real^'n"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  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$k = a)"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1038
  using connected_ivt_hyperplane[of s x y "axis k 1" a]
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1039
  by (auto simp add: inner_axis inner_commute)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1040
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1041
lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$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
  1042
  unfolding subspace_def by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1043
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1044
lemma closed_substandard_cart:
44213
6fb54701a11b add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents: 44211
diff changeset
  1045
  "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1046
proof -
44213
6fb54701a11b add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents: 44211
diff changeset
  1047
  { fix i::'n
6fb54701a11b add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents: 44211
diff changeset
  1048
    have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1049
      by (cases "P i") (simp_all add: closed_Collect_eq) }
44213
6fb54701a11b add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents: 44211
diff changeset
  1050
  thus ?thesis
6fb54701a11b add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents: 44211
diff changeset
  1051
    unfolding Collect_all_eq by (simp add: closed_INT)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1052
qed
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 dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1055
  (is "dim ?A = _")
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1056
proof -
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
  1057
  let ?a = "\<lambda>x. axis x 1 :: real^'n"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1058
  have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1059
    by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1060
  have "?a ` d \<subseteq> Basis"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1061
    by (auto simp: Basis_vec_def)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1062
  thus ?thesis
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
  1063
    using dim_substandard[of "?a ` d"] card_image[of ?a d]
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1064
    by (auto simp: axis_eq_axis inj_on_def *)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1065
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1066
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1067
lemma affinity_inverses:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1068
  assumes m0: "m \<noteq> (0::'a::field)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1069
  shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1070
  "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1071
  using m0
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53600
diff changeset
  1072
  apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53600
diff changeset
  1073
  apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1074
  done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1075
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1076
lemma vector_affinity_eq:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1077
  assumes m0: "(m::'a::field) \<noteq> 0"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1078
  shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1079
proof
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1080
  assume h: "m *s x + c = y"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1081
  hence "m *s x = y - c" by (simp add: field_simps)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1082
  hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1083
  then show "x = inverse m *s y + - (inverse m *s c)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1084
    using m0 by (simp add: vector_smult_assoc 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
  1085
next
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1086
  assume h: "x = inverse m *s y + - (inverse m *s c)"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53600
diff changeset
  1087
  show "m *s x + c = y" unfolding h
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1088
    using m0 by (simp add: vector_smult_assoc 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
  1089
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1090
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1091
lemma vector_eq_affinity:
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1092
    "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1093
  using vector_affinity_eq[where m=m and x=x and y=y and c=c]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1094
  by metis
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1095
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
  1096
lemma vector_cart:
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1097
  fixes f :: "real^'n \<Rightarrow> real"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1098
  shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1099
  unfolding euclidean_eq_iff[where 'a="real^'n"]
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1100
  by simp (simp add: Basis_vec_def inner_axis)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1101
  
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1102
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1103
  by (rule vector_cart)
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1104
44360
ea609ebdeebf section -> subsection
huffman
parents: 44282
diff changeset
  1105
subsection "Convex Euclidean Space"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1106
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
  1107
lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49962
diff changeset
  1108
  using const_vector_cart[of 1] by (simp add: one_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
  1109
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1110
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1111
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1112
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
  1113
lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1114
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1115
lemma convex_box_cart:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1116
  assumes "\<And>i. convex {x. P i x}"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1117
  shows "convex {x. \<forall>i. P i (x$i)}"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1118
  using assms unfolding convex_def by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1119
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1120
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1121
  by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1122
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1123
lemma unit_interval_convex_hull_cart:
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1124
  "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}"
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1125
  unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[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
  1126
  by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp 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
  1127
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1128
lemma cube_convex_hull_cart:
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1129
  assumes "0 < d"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1130
  obtains s::"(real^'n) set"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1131
    where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1132
proof -
55522
23d2cbac6dce tuned proofs;
wenzelm
parents: 54776
diff changeset
  1133
  from assms obtain s where "finite s"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 55522
diff changeset
  1134
    and "cbox (x - setsum (op *\<^sub>R d) Basis) (x + setsum (op *\<^sub>R d) Basis) = convex hull s"
55522
23d2cbac6dce tuned proofs;
wenzelm
parents: 54776
diff changeset
  1135
    by (rule cube_convex_hull)
23d2cbac6dce tuned proofs;
wenzelm
parents: 54776
diff changeset
  1136
  with that[of s] show thesis
23d2cbac6dce tuned proofs;
wenzelm
parents: 54776
diff changeset
  1137
    by (simp add: const_vector_cart)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1138
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1139
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1140
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1141
subsection "Derivative"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1142
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1143
lemma differentiable_at_imp_differentiable_on:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1144
  "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51489
diff changeset
  1145
  by (metis differentiable_at_withinI differentiable_on_def)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1146
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1147
definition "jacobian f net = matrix(frechet_derivative f net)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff changeset
  1148
49644
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1149
lemma jacobian_works:
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1150
  "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1151
    (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1152
  apply rule
343bfcbad2ec tuned proofs;
wenzelm
parents: 49197
diff changeset
  1153
  unfolding jacobian_def
343bfcbad2ec tuned proofs;
wenzelm
parents: