src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
author huffman
Tue Mar 18 09:39:07 2014 -0700 (2014-03-18)
changeset 56196 32b7eafc5a52
parent 56193 c726ecfb22b6
child 56273 def3bbe6f2a5
permissions -rw-r--r--
remove unnecessary finiteness assumptions from lemmas about setsum
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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"
huffman@44136
   302
  unfolding scaleR_vec_def vector_scalar_mult_def by simp
hoelzl@37489
   303
hoelzl@37489
   304
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
hoelzl@37489
   305
  unfolding dist_norm scalar_mult_eq_scaleR
hoelzl@37489
   306
  unfolding scaleR_right_diff_distrib[symmetric] by simp
hoelzl@37489
   307
hoelzl@37489
   308
lemma setsum_component [simp]:
hoelzl@37489
   309
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
hoelzl@37489
   310
  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
wenzelm@49644
   311
proof (cases "finite S")
wenzelm@49644
   312
  case True
wenzelm@49644
   313
  then show ?thesis by induct simp_all
wenzelm@49644
   314
next
wenzelm@49644
   315
  case False
wenzelm@49644
   316
  then show ?thesis by simp
wenzelm@49644
   317
qed
hoelzl@37489
   318
hoelzl@37489
   319
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
huffman@44136
   320
  by (simp add: vec_eq_iff)
hoelzl@37489
   321
hoelzl@37489
   322
lemma setsum_cmul:
hoelzl@37489
   323
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
hoelzl@37489
   324
  shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
huffman@44136
   325
  by (simp add: vec_eq_iff setsum_right_distrib)
hoelzl@37489
   326
hoelzl@37489
   327
lemma setsum_norm_allsubsets_bound_cart:
hoelzl@37489
   328
  fixes f:: "'a \<Rightarrow> real ^'n"
hoelzl@37489
   329
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
hoelzl@37489
   330
  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
hoelzl@50526
   331
  using setsum_norm_allsubsets_bound[OF assms]
hoelzl@50526
   332
  by (simp add: DIM_cart Basis_real_def)
hoelzl@37489
   333
hoelzl@37489
   334
subsection {* Matrix operations *}
hoelzl@37489
   335
hoelzl@37489
   336
text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
hoelzl@37489
   337
wenzelm@49644
   338
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
wenzelm@49644
   339
    (infixl "**" 70)
hoelzl@37489
   340
  where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
hoelzl@37489
   341
wenzelm@49644
   342
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
wenzelm@49644
   343
    (infixl "*v" 70)
hoelzl@37489
   344
  where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
hoelzl@37489
   345
wenzelm@49644
   346
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
wenzelm@49644
   347
    (infixl "v*" 70)
hoelzl@37489
   348
  where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
hoelzl@37489
   349
hoelzl@37489
   350
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
hoelzl@37489
   351
definition transpose where 
hoelzl@37489
   352
  "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
hoelzl@37489
   353
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
hoelzl@37489
   354
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
hoelzl@37489
   355
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
hoelzl@37489
   356
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
hoelzl@37489
   357
hoelzl@37489
   358
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
hoelzl@37489
   359
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
hoelzl@37489
   360
  by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps)
hoelzl@37489
   361
hoelzl@37489
   362
lemma matrix_mul_lid:
hoelzl@37489
   363
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   364
  shows "mat 1 ** A = A"
hoelzl@37489
   365
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   366
  apply vector
wenzelm@49644
   367
  apply (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite]
wenzelm@49644
   368
    mult_1_left mult_zero_left if_True UNIV_I)
wenzelm@49644
   369
  done
hoelzl@37489
   370
hoelzl@37489
   371
hoelzl@37489
   372
lemma matrix_mul_rid:
hoelzl@37489
   373
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   374
  shows "A ** mat 1 = A"
hoelzl@37489
   375
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   376
  apply vector
wenzelm@49644
   377
  apply (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite]
wenzelm@49644
   378
    mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
wenzelm@49644
   379
  done
hoelzl@37489
   380
hoelzl@37489
   381
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
hoelzl@37489
   382
  apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
hoelzl@37489
   383
  apply (subst setsum_commute)
hoelzl@37489
   384
  apply simp
hoelzl@37489
   385
  done
hoelzl@37489
   386
hoelzl@37489
   387
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
wenzelm@49644
   388
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def
wenzelm@49644
   389
    setsum_right_distrib setsum_left_distrib mult_assoc)
hoelzl@37489
   390
  apply (subst setsum_commute)
hoelzl@37489
   391
  apply simp
hoelzl@37489
   392
  done
hoelzl@37489
   393
hoelzl@37489
   394
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
hoelzl@37489
   395
  apply (vector matrix_vector_mult_def mat_def)
wenzelm@49644
   396
  apply (simp add: if_distrib cond_application_beta setsum_delta' cong del: if_weak_cong)
wenzelm@49644
   397
  done
hoelzl@37489
   398
wenzelm@49644
   399
lemma matrix_transpose_mul:
wenzelm@49644
   400
    "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
huffman@44136
   401
  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute)
hoelzl@37489
   402
hoelzl@37489
   403
lemma matrix_eq:
hoelzl@37489
   404
  fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
hoelzl@37489
   405
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@37489
   406
  apply auto
huffman@44136
   407
  apply (subst vec_eq_iff)
hoelzl@37489
   408
  apply clarify
hoelzl@50526
   409
  apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
hoelzl@50526
   410
  apply (erule_tac x="axis ia 1" in allE)
hoelzl@37489
   411
  apply (erule_tac x="i" in allE)
hoelzl@50526
   412
  apply (auto simp add: if_distrib cond_application_beta axis_def
wenzelm@49644
   413
    setsum_delta[OF finite] cong del: if_weak_cong)
wenzelm@49644
   414
  done
hoelzl@37489
   415
wenzelm@49644
   416
lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
huffman@44136
   417
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   418
hoelzl@37489
   419
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
huffman@44136
   420
  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
hoelzl@37489
   421
  apply (subst setsum_commute)
wenzelm@49644
   422
  apply simp
wenzelm@49644
   423
  done
hoelzl@37489
   424
hoelzl@37489
   425
lemma transpose_mat: "transpose (mat n) = mat n"
hoelzl@37489
   426
  by (vector transpose_def mat_def)
hoelzl@37489
   427
hoelzl@37489
   428
lemma transpose_transpose: "transpose(transpose A) = A"
hoelzl@37489
   429
  by (vector transpose_def)
hoelzl@37489
   430
hoelzl@37489
   431
lemma row_transpose:
hoelzl@37489
   432
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   433
  shows "row i (transpose A) = column i A"
huffman@44136
   434
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   435
hoelzl@37489
   436
lemma column_transpose:
hoelzl@37489
   437
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   438
  shows "column i (transpose A) = row i A"
huffman@44136
   439
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   440
hoelzl@37489
   441
lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
wenzelm@49644
   442
  by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
hoelzl@37489
   443
wenzelm@49644
   444
lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
wenzelm@49644
   445
  by (metis transpose_transpose rows_transpose)
hoelzl@37489
   446
hoelzl@37489
   447
text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
hoelzl@37489
   448
hoelzl@37489
   449
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
huffman@44136
   450
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   451
wenzelm@49644
   452
lemma matrix_mult_vsum:
wenzelm@49644
   453
  "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
huffman@44136
   454
  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
hoelzl@37489
   455
hoelzl@37489
   456
lemma vector_componentwise:
hoelzl@50526
   457
  "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
hoelzl@50526
   458
  by (simp add: axis_def if_distrib setsum_cases vec_eq_iff)
hoelzl@50526
   459
hoelzl@50526
   460
lemma basis_expansion: "setsum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
hoelzl@50526
   461
  by (auto simp add: axis_def vec_eq_iff if_distrib setsum_cases cong del: if_weak_cong)
hoelzl@37489
   462
hoelzl@37489
   463
lemma linear_componentwise:
hoelzl@37489
   464
  fixes f:: "real ^'m \<Rightarrow> real ^ _"
hoelzl@37489
   465
  assumes lf: "linear f"
hoelzl@50526
   466
  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
wenzelm@49644
   467
proof -
hoelzl@37489
   468
  let ?M = "(UNIV :: 'm set)"
hoelzl@37489
   469
  let ?N = "(UNIV :: 'n set)"
hoelzl@50526
   470
  have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
hoelzl@50526
   471
    unfolding setsum_component by simp
wenzelm@49644
   472
  then show ?thesis
huffman@56196
   473
    unfolding linear_setsum_mul[OF lf, symmetric]
hoelzl@50526
   474
    unfolding scalar_mult_eq_scaleR[symmetric]
hoelzl@50526
   475
    unfolding basis_expansion
hoelzl@50526
   476
    by simp
hoelzl@37489
   477
qed
hoelzl@37489
   478
hoelzl@37489
   479
text{* Inverse matrices  (not necessarily square) *}
hoelzl@37489
   480
wenzelm@49644
   481
definition
wenzelm@49644
   482
  "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   483
wenzelm@49644
   484
definition
wenzelm@49644
   485
  "matrix_inv(A:: 'a::semiring_1^'n^'m) =
wenzelm@49644
   486
    (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   487
hoelzl@37489
   488
text{* Correspondence between matrices and linear operators. *}
hoelzl@37489
   489
wenzelm@49644
   490
definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
hoelzl@50526
   491
  where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
hoelzl@37489
   492
hoelzl@37489
   493
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
huffman@53600
   494
  by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
wenzelm@49644
   495
      field_simps setsum_right_distrib setsum_addf)
hoelzl@37489
   496
wenzelm@49644
   497
lemma matrix_works:
wenzelm@49644
   498
  assumes lf: "linear f"
wenzelm@49644
   499
  shows "matrix f *v x = f (x::real ^ 'n)"
wenzelm@49644
   500
  apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
wenzelm@49644
   501
  apply clarify
wenzelm@49644
   502
  apply (rule linear_componentwise[OF lf, symmetric])
wenzelm@49644
   503
  done
hoelzl@37489
   504
wenzelm@49644
   505
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
wenzelm@49644
   506
  by (simp add: ext matrix_works)
hoelzl@37489
   507
hoelzl@37489
   508
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
hoelzl@37489
   509
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
hoelzl@37489
   510
hoelzl@37489
   511
lemma matrix_compose:
hoelzl@37489
   512
  assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
wenzelm@49644
   513
    and lg: "linear (g::real^'m \<Rightarrow> real^_)"
hoelzl@37489
   514
  shows "matrix (g o f) = matrix g ** matrix f"
hoelzl@37489
   515
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
wenzelm@49644
   516
  by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
hoelzl@37489
   517
wenzelm@49644
   518
lemma matrix_vector_column:
wenzelm@49644
   519
  "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
huffman@44136
   520
  by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult_commute)
hoelzl@37489
   521
hoelzl@37489
   522
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
hoelzl@37489
   523
  apply (rule adjoint_unique)
wenzelm@49644
   524
  apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
wenzelm@49644
   525
    setsum_left_distrib setsum_right_distrib)
hoelzl@37489
   526
  apply (subst setsum_commute)
hoelzl@37489
   527
  apply (auto simp add: mult_ac)
hoelzl@37489
   528
  done
hoelzl@37489
   529
hoelzl@37489
   530
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
hoelzl@37489
   531
  shows "matrix(adjoint f) = transpose(matrix f)"
hoelzl@37489
   532
  apply (subst matrix_vector_mul[OF lf])
wenzelm@49644
   533
  unfolding adjoint_matrix matrix_of_matrix_vector_mul
wenzelm@49644
   534
  apply rule
wenzelm@49644
   535
  done
wenzelm@49644
   536
hoelzl@37489
   537
huffman@44360
   538
subsection {* lambda skolemization on cartesian products *}
hoelzl@37489
   539
hoelzl@37489
   540
(* FIXME: rename do choice_cart *)
hoelzl@37489
   541
hoelzl@37489
   542
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
hoelzl@37494
   543
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   544
proof -
hoelzl@37489
   545
  let ?S = "(UNIV :: 'n set)"
wenzelm@49644
   546
  { assume H: "?rhs"
wenzelm@49644
   547
    then have ?lhs by auto }
hoelzl@37489
   548
  moreover
wenzelm@49644
   549
  { assume H: "?lhs"
hoelzl@37489
   550
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
hoelzl@37489
   551
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
wenzelm@49644
   552
    { fix i
hoelzl@37489
   553
      from f have "P i (f i)" by metis
hoelzl@37494
   554
      then have "P i (?x $ i)" by auto
hoelzl@37489
   555
    }
hoelzl@37489
   556
    hence "\<forall>i. P i (?x$i)" by metis
hoelzl@37489
   557
    hence ?rhs by metis }
hoelzl@37489
   558
  ultimately show ?thesis by metis
hoelzl@37489
   559
qed
hoelzl@37489
   560
hoelzl@37489
   561
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
hoelzl@50526
   562
  unfolding inner_simps scalar_mult_eq_scaleR by auto
hoelzl@37489
   563
hoelzl@37489
   564
lemma left_invertible_transpose:
hoelzl@37489
   565
  "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
hoelzl@37489
   566
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   567
hoelzl@37489
   568
lemma right_invertible_transpose:
hoelzl@37489
   569
  "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
hoelzl@37489
   570
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   571
hoelzl@37489
   572
lemma matrix_left_invertible_injective:
wenzelm@49644
   573
  "(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
wenzelm@49644
   574
proof -
wenzelm@49644
   575
  { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
hoelzl@37489
   576
    from xy have "B*v (A *v x) = B *v (A*v y)" by simp
hoelzl@37489
   577
    hence "x = y"
wenzelm@49644
   578
      unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
hoelzl@37489
   579
  moreover
wenzelm@49644
   580
  { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
hoelzl@37489
   581
    hence i: "inj (op *v A)" unfolding inj_on_def by auto
hoelzl@37489
   582
    from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
hoelzl@37489
   583
    obtain g where g: "linear g" "g o op *v A = id" by blast
hoelzl@37489
   584
    have "matrix g ** A = mat 1"
hoelzl@37489
   585
      unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   586
      using g(2) by (simp add: fun_eq_iff)
wenzelm@49644
   587
    then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
hoelzl@37489
   588
  ultimately show ?thesis by blast
hoelzl@37489
   589
qed
hoelzl@37489
   590
hoelzl@37489
   591
lemma matrix_left_invertible_ker:
hoelzl@37489
   592
  "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
hoelzl@37489
   593
  unfolding matrix_left_invertible_injective
hoelzl@37489
   594
  using linear_injective_0[OF matrix_vector_mul_linear, of A]
hoelzl@37489
   595
  by (simp add: inj_on_def)
hoelzl@37489
   596
hoelzl@37489
   597
lemma matrix_right_invertible_surjective:
wenzelm@49644
   598
  "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
wenzelm@49644
   599
proof -
wenzelm@49644
   600
  { fix B :: "real ^'m^'n"
wenzelm@49644
   601
    assume AB: "A ** B = mat 1"
wenzelm@49644
   602
    { fix x :: "real ^ 'm"
hoelzl@37489
   603
      have "A *v (B *v x) = x"
wenzelm@49644
   604
        by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
hoelzl@37489
   605
    hence "surj (op *v A)" unfolding surj_def by metis }
hoelzl@37489
   606
  moreover
wenzelm@49644
   607
  { assume sf: "surj (op *v A)"
hoelzl@37489
   608
    from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
hoelzl@37489
   609
    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
hoelzl@37489
   610
      by blast
hoelzl@37489
   611
hoelzl@37489
   612
    have "A ** (matrix g) = mat 1"
hoelzl@37489
   613
      unfolding matrix_eq  matrix_vector_mul_lid
hoelzl@37489
   614
        matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   615
      using g(2) unfolding o_def fun_eq_iff id_def
hoelzl@37489
   616
      .
hoelzl@37489
   617
    hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
hoelzl@37489
   618
  }
hoelzl@37489
   619
  ultimately show ?thesis unfolding surj_def by blast
hoelzl@37489
   620
qed
hoelzl@37489
   621
hoelzl@37489
   622
lemma matrix_left_invertible_independent_columns:
hoelzl@37489
   623
  fixes A :: "real^'n^'m"
wenzelm@49644
   624
  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
wenzelm@49644
   625
      (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
wenzelm@49644
   626
    (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   627
proof -
hoelzl@37489
   628
  let ?U = "UNIV :: 'n set"
wenzelm@49644
   629
  { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
wenzelm@49644
   630
    { fix c i
wenzelm@49644
   631
      assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
hoelzl@37489
   632
      let ?x = "\<chi> i. c i"
hoelzl@37489
   633
      have th0:"A *v ?x = 0"
hoelzl@37489
   634
        using c
huffman@44136
   635
        unfolding matrix_mult_vsum vec_eq_iff
hoelzl@37489
   636
        by auto
hoelzl@37489
   637
      from k[rule_format, OF th0] i
huffman@44136
   638
      have "c i = 0" by (vector vec_eq_iff)}
wenzelm@49644
   639
    hence ?rhs by blast }
hoelzl@37489
   640
  moreover
wenzelm@49644
   641
  { assume H: ?rhs
wenzelm@49644
   642
    { fix x assume x: "A *v x = 0"
hoelzl@37489
   643
      let ?c = "\<lambda>i. ((x$i ):: real)"
hoelzl@37489
   644
      from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
wenzelm@49644
   645
      have "x = 0" by vector }
wenzelm@49644
   646
  }
hoelzl@37489
   647
  ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
hoelzl@37489
   648
qed
hoelzl@37489
   649
hoelzl@37489
   650
lemma matrix_right_invertible_independent_rows:
hoelzl@37489
   651
  fixes A :: "real^'n^'m"
wenzelm@49644
   652
  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
wenzelm@49644
   653
    (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
hoelzl@37489
   654
  unfolding left_invertible_transpose[symmetric]
hoelzl@37489
   655
    matrix_left_invertible_independent_columns
hoelzl@37489
   656
  by (simp add: column_transpose)
hoelzl@37489
   657
hoelzl@37489
   658
lemma matrix_right_invertible_span_columns:
wenzelm@49644
   659
  "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
wenzelm@49644
   660
    span (columns A) = UNIV" (is "?lhs = ?rhs")
wenzelm@49644
   661
proof -
hoelzl@37489
   662
  let ?U = "UNIV :: 'm set"
hoelzl@37489
   663
  have fU: "finite ?U" by simp
hoelzl@37489
   664
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
hoelzl@37489
   665
    unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
wenzelm@49644
   666
    apply (subst eq_commute)
wenzelm@49644
   667
    apply rule
wenzelm@49644
   668
    done
hoelzl@37489
   669
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
wenzelm@49644
   670
  { assume h: ?lhs
wenzelm@49644
   671
    { fix x:: "real ^'n"
wenzelm@49644
   672
      from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
wenzelm@49644
   673
        where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
wenzelm@49644
   674
      have "x \<in> span (columns A)"
wenzelm@49644
   675
        unfolding y[symmetric]
huffman@56196
   676
        apply (rule span_setsum)
wenzelm@49644
   677
        apply clarify
hoelzl@50526
   678
        unfolding scalar_mult_eq_scaleR
wenzelm@49644
   679
        apply (rule span_mul)
wenzelm@49644
   680
        apply (rule span_superset)
wenzelm@49644
   681
        unfolding columns_def
wenzelm@49644
   682
        apply blast
wenzelm@49644
   683
        done
wenzelm@49644
   684
    }
wenzelm@49644
   685
    then have ?rhs unfolding rhseq by blast }
hoelzl@37489
   686
  moreover
wenzelm@49644
   687
  { assume h:?rhs
hoelzl@37489
   688
    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
wenzelm@49644
   689
    { fix y
wenzelm@49644
   690
      have "?P y"
hoelzl@50526
   691
      proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
hoelzl@37489
   692
        show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
hoelzl@37489
   693
          by (rule exI[where x=0], simp)
hoelzl@37489
   694
      next
wenzelm@49644
   695
        fix c y1 y2
wenzelm@49644
   696
        assume y1: "y1 \<in> columns A" and y2: "?P y2"
hoelzl@37489
   697
        from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
hoelzl@37489
   698
          unfolding columns_def by blast
hoelzl@37489
   699
        from y2 obtain x:: "real ^'m" where
hoelzl@37489
   700
          x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
hoelzl@37489
   701
        let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
hoelzl@37489
   702
        show "?P (c*s y1 + y2)"
webertj@49962
   703
        proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)
wenzelm@49644
   704
          fix j
wenzelm@49644
   705
          have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
   706
              else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
wenzelm@49644
   707
            using i(1) by (simp add: field_simps)
wenzelm@49644
   708
          have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
   709
              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)$j))) ?U"
wenzelm@49644
   710
            apply (rule setsum_cong[OF refl])
wenzelm@49644
   711
            using th apply blast
wenzelm@49644
   712
            done
wenzelm@49644
   713
          also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
wenzelm@49644
   714
            by (simp add: setsum_addf)
wenzelm@49644
   715
          also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
wenzelm@49644
   716
            unfolding setsum_delta[OF fU]
wenzelm@49644
   717
            using i(1) by simp
wenzelm@49644
   718
          finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
   719
            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
wenzelm@49644
   720
        qed
wenzelm@49644
   721
      next
wenzelm@49644
   722
        show "y \<in> span (columns A)"
wenzelm@49644
   723
          unfolding h by blast
wenzelm@49644
   724
      qed
wenzelm@49644
   725
    }
wenzelm@49644
   726
    then have ?lhs unfolding lhseq ..
wenzelm@49644
   727
  }
hoelzl@37489
   728
  ultimately show ?thesis by blast
hoelzl@37489
   729
qed
hoelzl@37489
   730
hoelzl@37489
   731
lemma matrix_left_invertible_span_rows:
hoelzl@37489
   732
  "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
hoelzl@37489
   733
  unfolding right_invertible_transpose[symmetric]
hoelzl@37489
   734
  unfolding columns_transpose[symmetric]
hoelzl@37489
   735
  unfolding matrix_right_invertible_span_columns
wenzelm@49644
   736
  ..
hoelzl@37489
   737
hoelzl@37489
   738
text {* The same result in terms of square matrices. *}
hoelzl@37489
   739
hoelzl@37489
   740
lemma matrix_left_right_inverse:
hoelzl@37489
   741
  fixes A A' :: "real ^'n^'n"
hoelzl@37489
   742
  shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
wenzelm@49644
   743
proof -
wenzelm@49644
   744
  { fix A A' :: "real ^'n^'n"
wenzelm@49644
   745
    assume AA': "A ** A' = mat 1"
hoelzl@37489
   746
    have sA: "surj (op *v A)"
hoelzl@37489
   747
      unfolding surj_def
hoelzl@37489
   748
      apply clarify
hoelzl@37489
   749
      apply (rule_tac x="(A' *v y)" in exI)
wenzelm@49644
   750
      apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
wenzelm@49644
   751
      done
hoelzl@37489
   752
    from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
hoelzl@37489
   753
    obtain f' :: "real ^'n \<Rightarrow> real ^'n"
hoelzl@37489
   754
      where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
hoelzl@37489
   755
    have th: "matrix f' ** A = mat 1"
wenzelm@49644
   756
      by (simp add: matrix_eq matrix_works[OF f'(1)]
wenzelm@49644
   757
          matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
hoelzl@37489
   758
    hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
wenzelm@49644
   759
    hence "matrix f' = A'"
wenzelm@49644
   760
      by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
hoelzl@37489
   761
    hence "matrix f' ** A = A' ** A" by simp
wenzelm@49644
   762
    hence "A' ** A = mat 1" by (simp add: th)
wenzelm@49644
   763
  }
hoelzl@37489
   764
  then show ?thesis by blast
hoelzl@37489
   765
qed
hoelzl@37489
   766
hoelzl@37489
   767
text {* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *}
hoelzl@37489
   768
hoelzl@37489
   769
definition "rowvector v = (\<chi> i j. (v$j))"
hoelzl@37489
   770
hoelzl@37489
   771
definition "columnvector v = (\<chi> i j. (v$i))"
hoelzl@37489
   772
wenzelm@49644
   773
lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
huffman@44136
   774
  by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
hoelzl@37489
   775
hoelzl@37489
   776
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
huffman@44136
   777
  by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
hoelzl@37489
   778
wenzelm@49644
   779
lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
hoelzl@37489
   780
  by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
hoelzl@37489
   781
wenzelm@49644
   782
lemma dot_matrix_product:
wenzelm@49644
   783
  "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
huffman@44136
   784
  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
hoelzl@37489
   785
hoelzl@37489
   786
lemma dot_matrix_vector_mul:
hoelzl@37489
   787
  fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
hoelzl@37489
   788
  shows "(A *v x) \<bullet> (B *v y) =
hoelzl@37489
   789
      (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
wenzelm@49644
   790
  unfolding dot_matrix_product transpose_columnvector[symmetric]
wenzelm@49644
   791
    dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
hoelzl@37489
   792
hoelzl@37489
   793
hoelzl@50526
   794
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in>UNIV}"
hoelzl@50526
   795
  by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
hoelzl@37489
   796
wenzelm@49644
   797
lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
hoelzl@50526
   798
  using Basis_le_infnorm[of "axis i 1" x]
hoelzl@50526
   799
  by (simp add: Basis_vec_def axis_eq_axis inner_axis)
hoelzl@37489
   800
wenzelm@49644
   801
lemma continuous_component: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
huffman@44647
   802
  unfolding continuous_def by (rule tendsto_vec_nth)
huffman@44213
   803
wenzelm@49644
   804
lemma continuous_on_component: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
huffman@44647
   805
  unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
huffman@44213
   806
hoelzl@37489
   807
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
huffman@44233
   808
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
huffman@44213
   809
hoelzl@37489
   810
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
wenzelm@49644
   811
  unfolding bounded_def
wenzelm@49644
   812
  apply clarify
wenzelm@49644
   813
  apply (rule_tac x="x $ i" in exI)
wenzelm@49644
   814
  apply (rule_tac x="e" in exI)
wenzelm@49644
   815
  apply clarify
wenzelm@49644
   816
  apply (rule order_trans [OF dist_vec_nth_le], simp)
wenzelm@49644
   817
  done
hoelzl@37489
   818
hoelzl@37489
   819
lemma compact_lemma_cart:
hoelzl@37489
   820
  fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
hoelzl@50998
   821
  assumes f: "bounded (range f)"
hoelzl@37489
   822
  shows "\<forall>d.
hoelzl@37489
   823
        \<exists>l r. subseq r \<and>
hoelzl@37489
   824
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
hoelzl@37489
   825
proof
wenzelm@49644
   826
  fix d :: "'n set"
wenzelm@49644
   827
  have "finite d" by simp
hoelzl@37489
   828
  thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
hoelzl@37489
   829
      (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
wenzelm@49644
   830
  proof (induct d)
wenzelm@49644
   831
    case empty
wenzelm@49644
   832
    thus ?case unfolding subseq_def by auto
wenzelm@49644
   833
  next
wenzelm@49644
   834
    case (insert k d)
wenzelm@49644
   835
    obtain l1::"'a^'n" and r1 where r1:"subseq r1"
wenzelm@49644
   836
      and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
hoelzl@37489
   837
      using insert(3) by auto
hoelzl@50998
   838
    have s': "bounded ((\<lambda>x. x $ k) ` range f)" using `bounded (range f)`
hoelzl@50998
   839
      by (auto intro!: bounded_component_cart)
hoelzl@50998
   840
    have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` range f" by simp
hoelzl@50998
   841
    have "bounded (range (\<lambda>i. f (r1 i) $ k))"
hoelzl@50998
   842
      by (metis (lifting) bounded_subset image_subsetI f' s')
hoelzl@50998
   843
    then obtain l2 r2 where r2: "subseq r2"
wenzelm@49644
   844
      and lr2: "((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
hoelzl@50998
   845
      using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) $ k"] by (auto simp: o_def)
wenzelm@49644
   846
    def r \<equiv> "r1 \<circ> r2"
wenzelm@49644
   847
    have r: "subseq r"
hoelzl@37489
   848
      using r1 and r2 unfolding r_def o_def subseq_def by auto
hoelzl@37489
   849
    moreover
hoelzl@37489
   850
    def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
wenzelm@49644
   851
    { fix e :: real assume "e > 0"
wenzelm@49644
   852
      from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
wenzelm@49644
   853
        by blast
wenzelm@49644
   854
      from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially"
wenzelm@49644
   855
        by (rule tendstoD)
hoelzl@37489
   856
      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
hoelzl@37489
   857
        by (rule eventually_subseq)
hoelzl@37489
   858
      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@37489
   859
        using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
hoelzl@37489
   860
    }
hoelzl@37489
   861
    ultimately show ?case by auto
hoelzl@37489
   862
  qed
hoelzl@37489
   863
qed
hoelzl@37489
   864
huffman@44136
   865
instance vec :: (heine_borel, finite) heine_borel
hoelzl@37489
   866
proof
hoelzl@50998
   867
  fix f :: "nat \<Rightarrow> 'a ^ 'b"
hoelzl@50998
   868
  assume f: "bounded (range f)"
hoelzl@37489
   869
  then obtain l r where r: "subseq r"
wenzelm@49644
   870
      and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@50998
   871
    using compact_lemma_cart [OF f] by blast
hoelzl@37489
   872
  let ?d = "UNIV::'b set"
hoelzl@37489
   873
  { fix e::real assume "e>0"
hoelzl@37489
   874
    hence "0 < e / (real_of_nat (card ?d))"
wenzelm@49644
   875
      using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
hoelzl@37489
   876
    with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
hoelzl@37489
   877
      by simp
hoelzl@37489
   878
    moreover
wenzelm@49644
   879
    { fix n
wenzelm@49644
   880
      assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
hoelzl@37489
   881
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
huffman@44136
   882
        unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
hoelzl@37489
   883
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
hoelzl@37489
   884
        by (rule setsum_strict_mono) (simp_all add: n)
hoelzl@37489
   885
      finally have "dist (f (r n)) l < e" by simp
hoelzl@37489
   886
    }
hoelzl@37489
   887
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
hoelzl@37489
   888
      by (rule eventually_elim1)
hoelzl@37489
   889
  }
wenzelm@49644
   890
  hence "((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
hoelzl@37489
   891
  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
hoelzl@37489
   892
qed
hoelzl@37489
   893
wenzelm@49644
   894
lemma interval_cart:
immler@54775
   895
  fixes a :: "real^'n"
immler@54775
   896
  shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
immler@56188
   897
    and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
immler@56188
   898
  by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
hoelzl@37489
   899
wenzelm@49644
   900
lemma mem_interval_cart:
immler@54775
   901
  fixes a :: "real^'n"
immler@54775
   902
  shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
immler@56188
   903
    and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
wenzelm@49644
   904
  using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
hoelzl@37489
   905
wenzelm@49644
   906
lemma interval_eq_empty_cart:
wenzelm@49644
   907
  fixes a :: "real^'n"
immler@54775
   908
  shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
immler@56188
   909
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
wenzelm@49644
   910
proof -
immler@54775
   911
  { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>box a b"
hoelzl@37489
   912
    hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto
hoelzl@37489
   913
    hence "a$i < b$i" by auto
wenzelm@49644
   914
    hence False using as by auto }
hoelzl@37489
   915
  moreover
hoelzl@37489
   916
  { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
hoelzl@37489
   917
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
   918
    { fix i
hoelzl@37489
   919
      have "a$i < b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
   920
      hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
hoelzl@37489
   921
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
   922
        by auto }
immler@54775
   923
    hence "box a b \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
hoelzl@37489
   924
  ultimately show ?th1 by blast
hoelzl@37489
   925
immler@56188
   926
  { fix i x assume as:"b$i < a$i" and x:"x\<in>cbox a b"
hoelzl@37489
   927
    hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto
hoelzl@37489
   928
    hence "a$i \<le> b$i" by auto
wenzelm@49644
   929
    hence False using as by auto }
hoelzl@37489
   930
  moreover
hoelzl@37489
   931
  { assume as:"\<forall>i. \<not> (b$i < a$i)"
hoelzl@37489
   932
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
   933
    { fix i
hoelzl@37489
   934
      have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
   935
      hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
hoelzl@37489
   936
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
   937
        by auto }
immler@56188
   938
    hence "cbox a b \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
hoelzl@37489
   939
  ultimately show ?th2 by blast
hoelzl@37489
   940
qed
hoelzl@37489
   941
wenzelm@49644
   942
lemma interval_ne_empty_cart:
wenzelm@49644
   943
  fixes a :: "real^'n"
immler@56188
   944
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
immler@54775
   945
    and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
hoelzl@37489
   946
  unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
hoelzl@37489
   947
    (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
   948
wenzelm@49644
   949
lemma subset_interval_imp_cart:
wenzelm@49644
   950
  fixes a :: "real^'n"
immler@56188
   951
  shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
immler@56188
   952
    and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
immler@56188
   953
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
immler@54775
   954
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b"
hoelzl@37489
   955
  unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
hoelzl@37489
   956
  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
   957
wenzelm@49644
   958
lemma interval_sing:
wenzelm@49644
   959
  fixes a :: "'a::linorder^'n"
wenzelm@49644
   960
  shows "{a .. a} = {a} \<and> {a<..<a} = {}"
wenzelm@49644
   961
  apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
wenzelm@49644
   962
  done
hoelzl@37489
   963
wenzelm@49644
   964
lemma subset_interval_cart:
wenzelm@49644
   965
  fixes a :: "real^'n"
immler@56188
   966
  shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
immler@56188
   967
    and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
immler@56188
   968
    and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
immler@54775
   969
    and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
immler@56188
   970
  using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
   971
wenzelm@49644
   972
lemma disjoint_interval_cart:
wenzelm@49644
   973
  fixes a::"real^'n"
immler@56188
   974
  shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
immler@56188
   975
    and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
immler@56188
   976
    and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
immler@54775
   977
    and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
hoelzl@50526
   978
  using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
   979
wenzelm@49644
   980
lemma inter_interval_cart:
immler@54775
   981
  fixes a :: "real^'n"
immler@56188
   982
  shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
immler@56188
   983
  unfolding inter_interval
immler@56188
   984
  by (auto simp: mem_box less_eq_vec_def)
immler@56188
   985
    (auto simp: Basis_vec_def inner_axis)
hoelzl@37489
   986
wenzelm@49644
   987
lemma closed_interval_left_cart:
wenzelm@49644
   988
  fixes b :: "real^'n"
hoelzl@37489
   989
  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
huffman@44233
   990
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
hoelzl@37489
   991
wenzelm@49644
   992
lemma closed_interval_right_cart:
wenzelm@49644
   993
  fixes a::"real^'n"
hoelzl@37489
   994
  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
huffman@44233
   995
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
hoelzl@37489
   996
wenzelm@49644
   997
lemma is_interval_cart:
wenzelm@49644
   998
  "is_interval (s::(real^'n) set) \<longleftrightarrow>
wenzelm@49644
   999
    (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
hoelzl@50526
  1000
  by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
hoelzl@37489
  1001
wenzelm@49644
  1002
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
huffman@44233
  1003
  by (simp add: closed_Collect_le)
hoelzl@37489
  1004
wenzelm@49644
  1005
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
huffman@44233
  1006
  by (simp add: closed_Collect_le)
hoelzl@37489
  1007
wenzelm@49644
  1008
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
wenzelm@49644
  1009
  by (simp add: open_Collect_less)
wenzelm@49644
  1010
wenzelm@49644
  1011
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
huffman@44233
  1012
  by (simp add: open_Collect_less)
hoelzl@37489
  1013
wenzelm@49644
  1014
lemma Lim_component_le_cart:
wenzelm@49644
  1015
  fixes f :: "'a \<Rightarrow> real^'n"
hoelzl@50526
  1016
  assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"
hoelzl@37489
  1017
  shows "l$i \<le> b"
hoelzl@50526
  1018
  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
hoelzl@37489
  1019
wenzelm@49644
  1020
lemma Lim_component_ge_cart:
wenzelm@49644
  1021
  fixes f :: "'a \<Rightarrow> real^'n"
hoelzl@37489
  1022
  assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
hoelzl@37489
  1023
  shows "b \<le> l$i"
hoelzl@50526
  1024
  by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
hoelzl@37489
  1025
wenzelm@49644
  1026
lemma Lim_component_eq_cart:
wenzelm@49644
  1027
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@49644
  1028
  assumes net: "(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
hoelzl@37489
  1029
  shows "l$i = b"
wenzelm@49644
  1030
  using ev[unfolded order_eq_iff eventually_conj_iff] and
wenzelm@49644
  1031
    Lim_component_ge_cart[OF net, of b i] and
hoelzl@37489
  1032
    Lim_component_le_cart[OF net, of i b] by auto
hoelzl@37489
  1033
wenzelm@49644
  1034
lemma connected_ivt_component_cart:
wenzelm@49644
  1035
  fixes x :: "real^'n"
wenzelm@49644
  1036
  shows "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s.  z$k = a)"
hoelzl@50526
  1037
  using connected_ivt_hyperplane[of s x y "axis k 1" a]
hoelzl@50526
  1038
  by (auto simp add: inner_axis inner_commute)
hoelzl@37489
  1039
wenzelm@49644
  1040
lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
hoelzl@37489
  1041
  unfolding subspace_def by auto
hoelzl@37489
  1042
hoelzl@37489
  1043
lemma closed_substandard_cart:
huffman@44213
  1044
  "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
  1045
proof -
huffman@44213
  1046
  { fix i::'n
huffman@44213
  1047
    have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
  1048
      by (cases "P i") (simp_all add: closed_Collect_eq) }
huffman@44213
  1049
  thus ?thesis
huffman@44213
  1050
    unfolding Collect_all_eq by (simp add: closed_INT)
hoelzl@37489
  1051
qed
hoelzl@37489
  1052
wenzelm@49644
  1053
lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
wenzelm@49644
  1054
  (is "dim ?A = _")
wenzelm@49644
  1055
proof -
hoelzl@50526
  1056
  let ?a = "\<lambda>x. axis x 1 :: real^'n"
hoelzl@50526
  1057
  have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
hoelzl@50526
  1058
    by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
hoelzl@50526
  1059
  have "?a ` d \<subseteq> Basis"
hoelzl@50526
  1060
    by (auto simp: Basis_vec_def)
wenzelm@49644
  1061
  thus ?thesis
hoelzl@50526
  1062
    using dim_substandard[of "?a ` d"] card_image[of ?a d]
hoelzl@50526
  1063
    by (auto simp: axis_eq_axis inj_on_def *)
hoelzl@37489
  1064
qed
hoelzl@37489
  1065
hoelzl@37489
  1066
lemma affinity_inverses:
hoelzl@37489
  1067
  assumes m0: "m \<noteq> (0::'a::field)"
hoelzl@37489
  1068
  shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
hoelzl@37489
  1069
  "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
hoelzl@37489
  1070
  using m0
haftmann@54230
  1071
  apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
haftmann@54230
  1072
  apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
wenzelm@49644
  1073
  done
hoelzl@37489
  1074
hoelzl@37489
  1075
lemma vector_affinity_eq:
hoelzl@37489
  1076
  assumes m0: "(m::'a::field) \<noteq> 0"
hoelzl@37489
  1077
  shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
hoelzl@37489
  1078
proof
hoelzl@37489
  1079
  assume h: "m *s x + c = y"
hoelzl@37489
  1080
  hence "m *s x = y - c" by (simp add: field_simps)
hoelzl@37489
  1081
  hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
hoelzl@37489
  1082
  then show "x = inverse m *s y + - (inverse m *s c)"
hoelzl@37489
  1083
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1084
next
hoelzl@37489
  1085
  assume h: "x = inverse m *s y + - (inverse m *s c)"
haftmann@54230
  1086
  show "m *s x + c = y" unfolding h
hoelzl@37489
  1087
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1088
qed
hoelzl@37489
  1089
hoelzl@37489
  1090
lemma vector_eq_affinity:
wenzelm@49644
  1091
    "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
hoelzl@37489
  1092
  using vector_affinity_eq[where m=m and x=x and y=y and c=c]
hoelzl@37489
  1093
  by metis
hoelzl@37489
  1094
hoelzl@50526
  1095
lemma vector_cart:
hoelzl@50526
  1096
  fixes f :: "real^'n \<Rightarrow> real"
hoelzl@50526
  1097
  shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
hoelzl@50526
  1098
  unfolding euclidean_eq_iff[where 'a="real^'n"]
hoelzl@50526
  1099
  by simp (simp add: Basis_vec_def inner_axis)
hoelzl@50526
  1100
  
hoelzl@50526
  1101
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
hoelzl@50526
  1102
  by (rule vector_cart)
wenzelm@49644
  1103
huffman@44360
  1104
subsection "Convex Euclidean Space"
hoelzl@37489
  1105
hoelzl@50526
  1106
lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
hoelzl@50526
  1107
  using const_vector_cart[of 1] by (simp add: one_vec_def)
hoelzl@37489
  1108
hoelzl@37489
  1109
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
hoelzl@37489
  1110
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
hoelzl@37489
  1111
hoelzl@50526
  1112
lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
hoelzl@37489
  1113
hoelzl@37489
  1114
lemma convex_box_cart:
hoelzl@37489
  1115
  assumes "\<And>i. convex {x. P i x}"
hoelzl@37489
  1116
  shows "convex {x. \<forall>i. P i (x$i)}"
hoelzl@37489
  1117
  using assms unfolding convex_def by auto
hoelzl@37489
  1118
hoelzl@37489
  1119
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
hoelzl@37489
  1120
  by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)
hoelzl@37489
  1121
hoelzl@37489
  1122
lemma unit_interval_convex_hull_cart:
immler@56188
  1123
  "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}"
immler@56188
  1124
  unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
hoelzl@50526
  1125
  by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
hoelzl@37489
  1126
hoelzl@37489
  1127
lemma cube_convex_hull_cart:
wenzelm@49644
  1128
  assumes "0 < d"
wenzelm@49644
  1129
  obtains s::"(real^'n) set"
immler@56188
  1130
    where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
wenzelm@49644
  1131
proof -
wenzelm@55522
  1132
  from assms obtain s where "finite s"
immler@56188
  1133
    and "cbox (x - setsum (op *\<^sub>R d) Basis) (x + setsum (op *\<^sub>R d) Basis) = convex hull s"
wenzelm@55522
  1134
    by (rule cube_convex_hull)
wenzelm@55522
  1135
  with that[of s] show thesis
wenzelm@55522
  1136
    by (simp add: const_vector_cart)
hoelzl@37489
  1137
qed
hoelzl@37489
  1138
hoelzl@37489
  1139
hoelzl@37489
  1140
subsection "Derivative"
hoelzl@37489
  1141
wenzelm@49644
  1142
lemma differentiable_at_imp_differentiable_on:
wenzelm@49644
  1143
  "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
hoelzl@51641
  1144
  by (metis differentiable_at_withinI differentiable_on_def)
hoelzl@37489
  1145
hoelzl@37489
  1146
definition "jacobian f net = matrix(frechet_derivative f net)"
hoelzl@37489
  1147
wenzelm@49644
  1148
lemma jacobian_works:
wenzelm@49644
  1149
  "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
wenzelm@49644
  1150
    (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
wenzelm@49644
  1151
  apply rule
wenzelm@49644
  1152
  unfolding jacobian_def
wenzelm@49644
  1153
  apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
wenzelm@49644
  1154
  apply (rule differentiableI)
wenzelm@49644
  1155
  apply assumption
wenzelm@49644
  1156
  unfolding frechet_derivative_works
wenzelm@49644
  1157
  apply assumption
wenzelm@49644
  1158
  done
hoelzl@37489
  1159
hoelzl@37489
  1160
wenzelm@49644
  1161
subsection {* Component of the differential must be zero if it exists at a local
wenzelm@49644
  1162
  maximum or minimum for that corresponding component. *}
hoelzl@37489
  1163
hoelzl@50526
  1164
lemma differential_zero_maxmin_cart:
wenzelm@49644
  1165
  fixes f::"real^'a \<Rightarrow> real^'b"
wenzelm@49644
  1166
  assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
hoelzl@50526
  1167
    "f differentiable (at x)"
hoelzl@50526
  1168
  shows "jacobian f (at x) $ k = 0"
hoelzl@50526
  1169
  using differential_zero_maxmin_component[of "axis k 1" e x f] assms
hoelzl@50526
  1170
    vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
hoelzl@50526
  1171
  by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
wenzelm@49644
  1172
hoelzl@37494
  1173
subsection {* Lemmas for working on @{typ "real^1"} *}
hoelzl@37489
  1174
hoelzl@37489
  1175
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
wenzelm@49644
  1176
  by (metis (full_types) num1_eq_iff)
hoelzl@37489
  1177
hoelzl@37489
  1178
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
wenzelm@49644
  1179
  by auto (metis (full_types) num1_eq_iff)
hoelzl@37489
  1180
hoelzl@37489
  1181
lemma exhaust_2:
wenzelm@49644
  1182
  fixes x :: 2
wenzelm@49644
  1183
  shows "x = 1 \<or> x = 2"
hoelzl@37489
  1184
proof (induct x)
hoelzl@37489
  1185
  case (of_int z)
hoelzl@37489
  1186
  then have "0 <= z" and "z < 2" by simp_all
hoelzl@37489
  1187
  then have "z = 0 | z = 1" by arith
hoelzl@37489
  1188
  then show ?case by auto
hoelzl@37489
  1189
qed
hoelzl@37489
  1190
hoelzl@37489
  1191
lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
hoelzl@37489
  1192
  by (metis exhaust_2)
hoelzl@37489
  1193
hoelzl@37489
  1194
lemma exhaust_3:
wenzelm@49644
  1195
  fixes x :: 3
wenzelm@49644
  1196
  shows "x = 1 \<or> x = 2 \<or> x = 3"
hoelzl@37489
  1197
proof (induct x)
hoelzl@37489
  1198
  case (of_int z)
hoelzl@37489
  1199
  then have "0 <= z" and "z < 3" by simp_all
hoelzl@37489
  1200
  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
hoelzl@37489
  1201
  then show ?case by auto
hoelzl@37489
  1202
qed
hoelzl@37489
  1203
hoelzl@37489
  1204
lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
hoelzl@37489
  1205
  by (metis exhaust_3)
hoelzl@37489
  1206
hoelzl@37489
  1207
lemma UNIV_1 [simp]: "UNIV = {1::1}"
hoelzl@37489
  1208
  by (auto simp add: num1_eq_iff)
hoelzl@37489
  1209
hoelzl@37489
  1210
lemma UNIV_2: "UNIV = {1::2, 2::2}"
hoelzl@37489
  1211
  using exhaust_2 by auto
hoelzl@37489
  1212
hoelzl@37489
  1213
lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
hoelzl@37489
  1214
  using exhaust_3 by auto
hoelzl@37489
  1215
hoelzl@37489
  1216
lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
hoelzl@37489
  1217
  unfolding UNIV_1 by simp
hoelzl@37489
  1218
hoelzl@37489
  1219
lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
hoelzl@37489
  1220
  unfolding UNIV_2 by simp
hoelzl@37489
  1221
hoelzl@37489
  1222
lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
hoelzl@37489
  1223
  unfolding UNIV_3 by (simp add: add_ac)
hoelzl@37489
  1224
wenzelm@49644
  1225
instantiation num1 :: cart_one
wenzelm@49644
  1226
begin
wenzelm@49644
  1227
wenzelm@49644
  1228
instance
wenzelm@49644
  1229
proof
hoelzl@37489
  1230
  show "CARD(1) = Suc 0" by auto
wenzelm@49644
  1231
qed
wenzelm@49644
  1232
wenzelm@49644
  1233
end
hoelzl@37489
  1234
hoelzl@37489
  1235
subsection{* The collapse of the general concepts to dimension one. *}
hoelzl@37489
  1236
hoelzl@37489
  1237
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
huffman@44136
  1238
  by (simp add: vec_eq_iff)
hoelzl@37489
  1239
hoelzl@37489
  1240
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
hoelzl@37489
  1241
  apply auto
hoelzl@37489
  1242
  apply (erule_tac x= "x$1" in allE)
hoelzl@37489
  1243
  apply (simp only: vector_one[symmetric])
hoelzl@37489
  1244
  done
hoelzl@37489
  1245
hoelzl@37489
  1246
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
huffman@44136
  1247
  by (simp add: norm_vec_def)
hoelzl@37489
  1248
hoelzl@37489
  1249
lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
hoelzl@37489
  1250
  by (simp add: norm_vector_1)
hoelzl@37489
  1251
hoelzl@37489
  1252
lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
hoelzl@37489
  1253
  by (auto simp add: norm_real dist_norm)
hoelzl@37489
  1254
wenzelm@49644
  1255
hoelzl@37489
  1256
subsection{* Explicit vector construction from lists. *}
hoelzl@37489
  1257
hoelzl@43995
  1258
definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
hoelzl@37489
  1259
hoelzl@37489
  1260
lemma vector_1: "(vector[x]) $1 = x"
hoelzl@37489
  1261
  unfolding vector_def by simp
hoelzl@37489
  1262
hoelzl@37489
  1263
lemma vector_2:
hoelzl@37489
  1264
 "(vector[x,y]) $1 = x"
hoelzl@37489
  1265
 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
hoelzl@37489
  1266
  unfolding vector_def by simp_all
hoelzl@37489
  1267
hoelzl@37489
  1268
lemma vector_3:
hoelzl@37489
  1269
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
hoelzl@37489
  1270
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
hoelzl@37489
  1271
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
hoelzl@37489
  1272
  unfolding vector_def by simp_all
hoelzl@37489
  1273
hoelzl@37489
  1274
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
hoelzl@37489
  1275
  apply auto
hoelzl@37489
  1276
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1277
  apply (subgoal_tac "vector [v$1] = v")
hoelzl@37489
  1278
  apply simp
hoelzl@37489
  1279
  apply (vector vector_def)
hoelzl@37489
  1280
  apply simp
hoelzl@37489
  1281
  done
hoelzl@37489
  1282
hoelzl@37489
  1283
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
hoelzl@37489
  1284
  apply auto
hoelzl@37489
  1285
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1286
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1287
  apply (subgoal_tac "vector [v$1, v$2] = v")
hoelzl@37489
  1288
  apply simp
hoelzl@37489
  1289
  apply (vector vector_def)
hoelzl@37489
  1290
  apply (simp add: forall_2)
hoelzl@37489
  1291
  done
hoelzl@37489
  1292
hoelzl@37489
  1293
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
hoelzl@37489
  1294
  apply auto
hoelzl@37489
  1295
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1296
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1297
  apply (erule_tac x="v$3" in allE)
hoelzl@37489
  1298
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
hoelzl@37489
  1299
  apply simp
hoelzl@37489
  1300
  apply (vector vector_def)
hoelzl@37489
  1301
  apply (simp add: forall_3)
hoelzl@37489
  1302
  done
hoelzl@37489
  1303
hoelzl@37489
  1304
lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
wenzelm@49644
  1305
  apply (rule bounded_linearI[where K=1])
hoelzl@37489
  1306
  using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
hoelzl@37489
  1307
wenzelm@49644
  1308
lemma integral_component_eq_cart[simp]:
immler@56188
  1309
  fixes f :: "'n::euclidean_space \<Rightarrow> real^'m"
wenzelm@49644
  1310
  assumes "f integrable_on s"
wenzelm@49644
  1311
  shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"
hoelzl@37489
  1312
  using integral_linear[OF assms(1) bounded_linear_component_cart,unfolded o_def] .
hoelzl@37489
  1313
hoelzl@37489
  1314
lemma interval_split_cart:
hoelzl@37489
  1315
  "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
immler@56188
  1316
  "cbox a b \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
wenzelm@49644
  1317
  apply (rule_tac[!] set_eqI)
immler@56188
  1318
  unfolding Int_iff mem_interval_cart mem_Collect_eq interval_cbox_cart
wenzelm@49644
  1319
  unfolding vec_lambda_beta
wenzelm@49644
  1320
  by auto
hoelzl@37489
  1321
hoelzl@50526
  1322
lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "\<forall>i. a$i < b$i \<and> u$i < v$i" 
hoelzl@50526
  1323
  shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
hoelzl@50526
  1324
  using assms by (intro interval_bij_bij) (auto simp: Basis_vec_def inner_axis)
hoelzl@37489
  1325
hoelzl@37489
  1326
end