src/HOL/Library/Euclidean_Space.thy
author huffman
Thu Mar 19 01:29:19 2009 -0700 (2009-03-19)
changeset 30582 638b088bb840
parent 30549 d2d7874648bd
child 30655 88131f2807b6
child 30661 54858c8ad226
permissions -rw-r--r--
imported patch euclidean
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(* Title:      Library/Euclidean_Space
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   Author:     Amine Chaieb, University of Cambridge
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*)
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header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
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theory Euclidean_Space
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  imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Complex_Main
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  Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type
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  Inner_Product
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  uses ("normarith.ML")
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begin
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text{* Some common special cases.*}
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lemma forall_1: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
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  by (metis num1_eq_iff)
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lemma exhaust_2:
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  fixes x :: 2 shows "x = 1 \<or> x = 2"
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proof (induct x)
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  case (of_int z)
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  then have "0 <= z" and "z < 2" by simp_all
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  then have "z = 0 | z = 1" by arith
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  then show ?case by auto
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qed
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lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
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  by (metis exhaust_2)
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lemma exhaust_3:
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  fixes x :: 3 shows "x = 1 \<or> x = 2 \<or> x = 3"
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proof (induct x)
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  case (of_int z)
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  then have "0 <= z" and "z < 3" by simp_all
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  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
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  then show ?case by auto
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qed
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lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
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  by (metis exhaust_3)
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lemma UNIV_1: "UNIV = {1::1}"
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  by (auto simp add: num1_eq_iff)
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lemma UNIV_2: "UNIV = {1::2, 2::2}"
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  using exhaust_2 by auto
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lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
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  using exhaust_3 by auto
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lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
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  unfolding UNIV_1 by simp
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lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
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  unfolding UNIV_2 by simp
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lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
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  unfolding UNIV_3 by (simp add: add_ac)
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subsection{* Basic componentwise operations on vectors. *}
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instantiation "^" :: (plus,type) plus
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begin
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definition  vector_add_def : "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 "^" :: (times,type) times
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begin
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  definition vector_mult_def : "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 "^" :: (minus,type) minus begin
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  definition vector_minus_def : "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 "^" :: (uminus,type) uminus begin
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  definition vector_uminus_def : "uminus \<equiv> (\<lambda> x.  (\<chi> i. - (x$i)))"
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instance ..
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end
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instantiation "^" :: (zero,type) zero begin
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  definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)"
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instance ..
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end
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instantiation "^" :: (one,type) one begin
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  definition vector_one_def : "1 \<equiv> (\<chi> i. 1)"
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instance ..
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end
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instantiation "^" :: (ord,type) ord
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 begin
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definition vector_less_eq_def:
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  "less_eq (x :: 'a ^'b) y = (ALL i. x$i <= y$i)"
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definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i. x$i < y$i)"
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instance by (intro_classes)
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end
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instantiation "^" :: (scaleR, type) scaleR
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begin
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definition vector_scaleR_def: "scaleR = (\<lambda> r x.  (\<chi> i. scaleR r (x$i)))"
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instance ..
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end
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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" (infixr "*s" 75)
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  where "c *s x = (\<chi> i. c * (x$i))"
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text{* Constant Vectors *}
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definition "vec x = (\<chi> i. x)"
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text{* Dot products. *}
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definition dot :: "'a::{comm_monoid_add, times} ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" (infix "\<bullet>" 70) where
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  "x \<bullet> y = setsum (\<lambda>i. x$i * y$i) UNIV"
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lemma dot_1[simp]: "(x::'a::{comm_monoid_add, times}^1) \<bullet> y = (x$1) * (y$1)"
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  by (simp add: dot_def setsum_1)
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lemma dot_2[simp]: "(x::'a::{comm_monoid_add, times}^2) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2)"
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  by (simp add: dot_def setsum_2)
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lemma dot_3[simp]: "(x::'a::{comm_monoid_add, times}^3) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2) + (x$3) * (y$3)"
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  by (simp add: dot_def setsum_3)
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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 = HOL_basic_ss addsimps [@{thm dot_def}, @{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 = @{simpset} addsimps
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             [@{thm vector_add_def}, @{thm vector_mult_def},
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              @{thm vector_minus_def}, @{thm vector_uminus_def},
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              @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def},
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              @{thm vector_scaleR_def},
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              @{thm Cart_lambda_beta}, @{thm vector_scalar_mult_def}]
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 fun vector_arith_tac ths =
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   simp_tac ss1
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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 (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) 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 (ss2 addsimps ths)
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 in
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  Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths)))
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 end
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*} "Lifts trivial vector statements to real arith statements"
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lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def)
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lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def)
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text{* Obvious "component-pushing". *}
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lemma vec_component [simp]: "(vec x :: 'a ^ 'n)$i = x"
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  by (vector vec_def)
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lemma vector_add_component [simp]:
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  fixes x y :: "'a::{plus} ^ 'n"
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  shows "(x + y)$i = x$i + y$i"
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  by vector
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lemma vector_minus_component [simp]:
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  fixes x y :: "'a::{minus} ^ 'n"
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  shows "(x - y)$i = x$i - y$i"
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  by vector
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lemma vector_mult_component [simp]:
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  fixes x y :: "'a::{times} ^ 'n"
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  shows "(x * y)$i = x$i * y$i"
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  by vector
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lemma vector_smult_component [simp]:
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  fixes y :: "'a::{times} ^ 'n"
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  shows "(c *s y)$i = c * (y$i)"
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  by vector
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lemma vector_uminus_component [simp]:
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  fixes x :: "'a::{uminus} ^ 'n"
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  shows "(- x)$i = - (x$i)"
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  by vector
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lemma vector_scaleR_component [simp]:
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  fixes x :: "'a::scaleR ^ 'n"
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  shows "(scaleR r x)$i = scaleR r (x$i)"
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  by vector
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lemma cond_component: "(if b then x else y)$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 "^" :: (semigroup_add,type) semigroup_add
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  apply (intro_classes) by (vector add_assoc)
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instance "^" :: (monoid_add,type) monoid_add
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  apply (intro_classes) by vector+
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instance "^" :: (group_add,type) group_add
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  apply (intro_classes) by (vector algebra_simps)+
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instance "^" :: (ab_semigroup_add,type) ab_semigroup_add
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  apply (intro_classes) by (vector add_commute)
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instance "^" :: (comm_monoid_add,type) comm_monoid_add
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  apply (intro_classes) by vector
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instance "^" :: (ab_group_add,type) ab_group_add
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  apply (intro_classes) by vector+
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instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add
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  apply (intro_classes)
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  by (vector Cart_eq)+
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instance "^" :: (cancel_ab_semigroup_add,type) cancel_ab_semigroup_add
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  apply (intro_classes)
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  by (vector Cart_eq)
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instance "^" :: (real_vector, type) real_vector
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  by default (vector scaleR_left_distrib scaleR_right_distrib)+
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instance "^" :: (semigroup_mult,type) semigroup_mult
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  apply (intro_classes) by (vector mult_assoc)
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instance "^" :: (monoid_mult,type) monoid_mult
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  apply (intro_classes) by vector+
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instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult
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  apply (intro_classes) by (vector mult_commute)
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instance "^" :: (ab_semigroup_idem_mult,type) ab_semigroup_idem_mult
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  apply (intro_classes) by (vector mult_idem)
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instance "^" :: (comm_monoid_mult,type) comm_monoid_mult
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  apply (intro_classes) by vector
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fun vector_power :: "('a::{one,times} ^'n) \<Rightarrow> nat \<Rightarrow> 'a^'n" where
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  "vector_power x 0 = 1"
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  | "vector_power x (Suc n) = x * vector_power x n"
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instantiation "^" :: (recpower,type) recpower
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begin
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  definition vec_power_def: "op ^ \<equiv> vector_power"
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  instance
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  apply (intro_classes) by (simp_all add: vec_power_def)
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end
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instance "^" :: (semiring,type) semiring
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  apply (intro_classes) by (vector ring_simps)+
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instance "^" :: (semiring_0,type) semiring_0
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  apply (intro_classes) by (vector ring_simps)+
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instance "^" :: (semiring_1,type) semiring_1
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  apply (intro_classes) by vector
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instance "^" :: (comm_semiring,type) comm_semiring
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  apply (intro_classes) by (vector ring_simps)+
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instance "^" :: (comm_semiring_0,type) comm_semiring_0 by (intro_classes)
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instance "^" :: (cancel_comm_monoid_add, type) cancel_comm_monoid_add ..
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instance "^" :: (semiring_0_cancel,type) semiring_0_cancel by (intro_classes)
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instance "^" :: (comm_semiring_0_cancel,type) comm_semiring_0_cancel by (intro_classes)
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instance "^" :: (ring,type) ring by (intro_classes)
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instance "^" :: (semiring_1_cancel,type) semiring_1_cancel by (intro_classes)
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instance "^" :: (comm_semiring_1,type) comm_semiring_1 by (intro_classes)
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instance "^" :: (ring_1,type) ring_1 ..
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instance "^" :: (real_algebra,type) real_algebra
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  apply intro_classes
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  apply (simp_all add: vector_scaleR_def ring_simps)
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  apply vector
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  apply vector
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  done
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instance "^" :: (real_algebra_1,type) real_algebra_1 ..
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lemma of_nat_index:
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  "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
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  apply (induct n)
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  apply vector
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  apply vector
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  done
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lemma zero_index[simp]:
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  "(0 :: 'a::zero ^'n)$i = 0" by vector
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lemma one_index[simp]:
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  "(1 :: 'a::one ^'n)$i = 1" by vector
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lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \<noteq> 0"
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proof-
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  have "(1::'a) + of_nat n = 0 \<longleftrightarrow> of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp
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  also have "\<dots> \<longleftrightarrow> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff)
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  finally show ?thesis by simp
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qed
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instance "^" :: (semiring_char_0,type) semiring_char_0
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proof (intro_classes)
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  fix m n ::nat
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  show "(of_nat m :: 'a^'b) = of_nat n \<longleftrightarrow> m = n"
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    by (simp add: Cart_eq of_nat_index)
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qed
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instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes
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instance "^" :: (ring_char_0,type) ring_char_0 by intro_classes
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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 ring_simps)
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lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
chaieb@29842
   325
  by (vector ring_simps)
chaieb@29842
   326
lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
chaieb@29842
   327
lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
huffman@30489
   328
lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
chaieb@29842
   329
  by (vector ring_simps)
chaieb@29842
   330
lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
chaieb@29842
   331
lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
chaieb@29842
   332
lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
chaieb@29842
   333
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
huffman@30489
   334
lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
chaieb@29842
   335
  by (vector ring_simps)
chaieb@29842
   336
huffman@30489
   337
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
huffman@30582
   338
  by (simp add: Cart_eq)
chaieb@29842
   339
huffman@30040
   340
subsection {* Square root of sum of squares *}
huffman@30040
   341
huffman@30040
   342
definition
huffman@30040
   343
  "setL2 f A = sqrt (\<Sum>i\<in>A. (f i)\<twosuperior>)"
huffman@30040
   344
huffman@30040
   345
lemma setL2_cong:
huffman@30040
   346
  "\<lbrakk>A = B; \<And>x. x \<in> B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
huffman@30040
   347
  unfolding setL2_def by simp
huffman@30040
   348
huffman@30040
   349
lemma strong_setL2_cong:
huffman@30040
   350
  "\<lbrakk>A = B; \<And>x. x \<in> B =simp=> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
huffman@30040
   351
  unfolding setL2_def simp_implies_def by simp
huffman@30040
   352
huffman@30040
   353
lemma setL2_infinite [simp]: "\<not> finite A \<Longrightarrow> setL2 f A = 0"
huffman@30040
   354
  unfolding setL2_def by simp
huffman@30040
   355
huffman@30040
   356
lemma setL2_empty [simp]: "setL2 f {} = 0"
huffman@30040
   357
  unfolding setL2_def by simp
huffman@30040
   358
huffman@30040
   359
lemma setL2_insert [simp]:
huffman@30040
   360
  "\<lbrakk>finite F; a \<notin> F\<rbrakk> \<Longrightarrow>
huffman@30040
   361
    setL2 f (insert a F) = sqrt ((f a)\<twosuperior> + (setL2 f F)\<twosuperior>)"
huffman@30040
   362
  unfolding setL2_def by (simp add: setsum_nonneg)
huffman@30040
   363
huffman@30040
   364
lemma setL2_nonneg [simp]: "0 \<le> setL2 f A"
huffman@30040
   365
  unfolding setL2_def by (simp add: setsum_nonneg)
huffman@30040
   366
huffman@30040
   367
lemma setL2_0': "\<forall>a\<in>A. f a = 0 \<Longrightarrow> setL2 f A = 0"
huffman@30040
   368
  unfolding setL2_def by simp
huffman@30040
   369
huffman@30040
   370
lemma setL2_mono:
huffman@30040
   371
  assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i"
huffman@30040
   372
  assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"
huffman@30040
   373
  shows "setL2 f K \<le> setL2 g K"
huffman@30040
   374
  unfolding setL2_def
huffman@30040
   375
  by (simp add: setsum_nonneg setsum_mono power_mono prems)
huffman@30040
   376
huffman@30040
   377
lemma setL2_right_distrib:
huffman@30040
   378
  "0 \<le> r \<Longrightarrow> r * setL2 f A = setL2 (\<lambda>x. r * f x) A"
huffman@30040
   379
  unfolding setL2_def
huffman@30040
   380
  apply (simp add: power_mult_distrib)
huffman@30040
   381
  apply (simp add: setsum_right_distrib [symmetric])
huffman@30040
   382
  apply (simp add: real_sqrt_mult setsum_nonneg)
huffman@30040
   383
  done
huffman@30040
   384
huffman@30040
   385
lemma setL2_left_distrib:
huffman@30040
   386
  "0 \<le> r \<Longrightarrow> setL2 f A * r = setL2 (\<lambda>x. f x * r) A"
huffman@30040
   387
  unfolding setL2_def
huffman@30040
   388
  apply (simp add: power_mult_distrib)
huffman@30040
   389
  apply (simp add: setsum_left_distrib [symmetric])
huffman@30040
   390
  apply (simp add: real_sqrt_mult setsum_nonneg)
huffman@30040
   391
  done
huffman@30040
   392
huffman@30040
   393
lemma setsum_nonneg_eq_0_iff:
huffman@30040
   394
  fixes f :: "'a \<Rightarrow> 'b::pordered_ab_group_add"
huffman@30040
   395
  shows "\<lbrakk>finite A; \<forall>x\<in>A. 0 \<le> f x\<rbrakk> \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
huffman@30040
   396
  apply (induct set: finite, simp)
huffman@30040
   397
  apply (simp add: add_nonneg_eq_0_iff setsum_nonneg)
huffman@30040
   398
  done
huffman@30040
   399
huffman@30040
   400
lemma setL2_eq_0_iff: "finite A \<Longrightarrow> setL2 f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
huffman@30040
   401
  unfolding setL2_def
huffman@30040
   402
  by (simp add: setsum_nonneg setsum_nonneg_eq_0_iff)
huffman@30040
   403
huffman@30040
   404
lemma setL2_triangle_ineq:
huffman@30040
   405
  shows "setL2 (\<lambda>i. f i + g i) A \<le> setL2 f A + setL2 g A"
huffman@30040
   406
proof (cases "finite A")
huffman@30040
   407
  case False
huffman@30040
   408
  thus ?thesis by simp
huffman@30040
   409
next
huffman@30040
   410
  case True
huffman@30040
   411
  thus ?thesis
huffman@30040
   412
  proof (induct set: finite)
huffman@30040
   413
    case empty
huffman@30040
   414
    show ?case by simp
huffman@30040
   415
  next
huffman@30040
   416
    case (insert x F)
huffman@30040
   417
    hence "sqrt ((f x + g x)\<twosuperior> + (setL2 (\<lambda>i. f i + g i) F)\<twosuperior>) \<le>
huffman@30040
   418
           sqrt ((f x + g x)\<twosuperior> + (setL2 f F + setL2 g F)\<twosuperior>)"
huffman@30040
   419
      by (intro real_sqrt_le_mono add_left_mono power_mono insert
huffman@30040
   420
                setL2_nonneg add_increasing zero_le_power2)
huffman@30040
   421
    also have
huffman@30040
   422
      "\<dots> \<le> sqrt ((f x)\<twosuperior> + (setL2 f F)\<twosuperior>) + sqrt ((g x)\<twosuperior> + (setL2 g F)\<twosuperior>)"
huffman@30040
   423
      by (rule real_sqrt_sum_squares_triangle_ineq)
huffman@30040
   424
    finally show ?case
huffman@30040
   425
      using insert by simp
huffman@30040
   426
  qed
huffman@30040
   427
qed
huffman@30040
   428
huffman@30040
   429
lemma sqrt_sum_squares_le_sum:
huffman@30040
   430
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) \<le> x + y"
huffman@30040
   431
  apply (rule power2_le_imp_le)
huffman@30040
   432
  apply (simp add: power2_sum)
huffman@30040
   433
  apply (simp add: mult_nonneg_nonneg)
huffman@30040
   434
  apply (simp add: add_nonneg_nonneg)
huffman@30040
   435
  done
huffman@30040
   436
huffman@30040
   437
lemma setL2_le_setsum [rule_format]:
huffman@30040
   438
  "(\<forall>i\<in>A. 0 \<le> f i) \<longrightarrow> setL2 f A \<le> setsum f A"
huffman@30040
   439
  apply (cases "finite A")
huffman@30040
   440
  apply (induct set: finite)
huffman@30040
   441
  apply simp
huffman@30040
   442
  apply clarsimp
huffman@30040
   443
  apply (erule order_trans [OF sqrt_sum_squares_le_sum])
huffman@30040
   444
  apply simp
huffman@30040
   445
  apply simp
huffman@30040
   446
  apply simp
huffman@30040
   447
  done
huffman@30040
   448
huffman@30040
   449
lemma sqrt_sum_squares_le_sum_abs: "sqrt (x\<twosuperior> + y\<twosuperior>) \<le> \<bar>x\<bar> + \<bar>y\<bar>"
huffman@30040
   450
  apply (rule power2_le_imp_le)
huffman@30040
   451
  apply (simp add: power2_sum)
huffman@30040
   452
  apply (simp add: mult_nonneg_nonneg)
huffman@30040
   453
  apply (simp add: add_nonneg_nonneg)
huffman@30040
   454
  done
huffman@30040
   455
huffman@30040
   456
lemma setL2_le_setsum_abs: "setL2 f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
huffman@30040
   457
  apply (cases "finite A")
huffman@30040
   458
  apply (induct set: finite)
huffman@30040
   459
  apply simp
huffman@30040
   460
  apply simp
huffman@30040
   461
  apply (rule order_trans [OF sqrt_sum_squares_le_sum_abs])
huffman@30040
   462
  apply simp
huffman@30040
   463
  apply simp
huffman@30040
   464
  done
huffman@30040
   465
huffman@30040
   466
lemma setL2_mult_ineq_lemma:
huffman@30040
   467
  fixes a b c d :: real
huffman@30040
   468
  shows "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
huffman@30040
   469
proof -
huffman@30040
   470
  have "0 \<le> (a * d - b * c)\<twosuperior>" by simp
huffman@30040
   471
  also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * d) * (b * c)"
huffman@30040
   472
    by (simp only: power2_diff power_mult_distrib)
huffman@30040
   473
  also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * c) * (b * d)"
huffman@30040
   474
    by simp
huffman@30040
   475
  finally show "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
huffman@30040
   476
    by simp
huffman@30040
   477
qed
huffman@30040
   478
huffman@30040
   479
lemma setL2_mult_ineq: "(\<Sum>i\<in>A. \<bar>f i\<bar> * \<bar>g i\<bar>) \<le> setL2 f A * setL2 g A"
huffman@30040
   480
  apply (cases "finite A")
huffman@30040
   481
  apply (induct set: finite)
huffman@30040
   482
  apply simp
huffman@30040
   483
  apply (rule power2_le_imp_le, simp)
huffman@30040
   484
  apply (rule order_trans)
huffman@30040
   485
  apply (rule power_mono)
huffman@30040
   486
  apply (erule add_left_mono)
huffman@30040
   487
  apply (simp add: add_nonneg_nonneg mult_nonneg_nonneg setsum_nonneg)
huffman@30040
   488
  apply (simp add: power2_sum)
huffman@30040
   489
  apply (simp add: power_mult_distrib)
huffman@30040
   490
  apply (simp add: right_distrib left_distrib)
huffman@30040
   491
  apply (rule ord_le_eq_trans)
huffman@30040
   492
  apply (rule setL2_mult_ineq_lemma)
huffman@30040
   493
  apply simp
huffman@30040
   494
  apply (intro mult_nonneg_nonneg setL2_nonneg)
huffman@30040
   495
  apply simp
huffman@30040
   496
  done
huffman@30040
   497
huffman@30040
   498
lemma member_le_setL2: "\<lbrakk>finite A; i \<in> A\<rbrakk> \<Longrightarrow> f i \<le> setL2 f A"
huffman@30040
   499
  apply (rule_tac s="insert i (A - {i})" and t="A" in subst)
huffman@30040
   500
  apply fast
huffman@30040
   501
  apply (subst setL2_insert)
huffman@30040
   502
  apply simp
huffman@30040
   503
  apply simp
huffman@30040
   504
  apply simp
huffman@30040
   505
  done
huffman@30040
   506
huffman@30040
   507
subsection {* Norms *}
huffman@30040
   508
huffman@30582
   509
instantiation "^" :: (real_normed_vector, finite) real_normed_vector
huffman@30040
   510
begin
huffman@30040
   511
huffman@30040
   512
definition vector_norm_def:
huffman@30582
   513
  "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) UNIV"
huffman@30040
   514
huffman@30040
   515
definition vector_sgn_def:
huffman@30040
   516
  "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
huffman@30040
   517
huffman@30040
   518
instance proof
huffman@30040
   519
  fix a :: real and x y :: "'a ^ 'b"
huffman@30040
   520
  show "0 \<le> norm x"
huffman@30040
   521
    unfolding vector_norm_def
huffman@30040
   522
    by (rule setL2_nonneg)
huffman@30040
   523
  show "norm x = 0 \<longleftrightarrow> x = 0"
huffman@30040
   524
    unfolding vector_norm_def
huffman@30040
   525
    by (simp add: setL2_eq_0_iff Cart_eq)
huffman@30040
   526
  show "norm (x + y) \<le> norm x + norm y"
huffman@30040
   527
    unfolding vector_norm_def
huffman@30040
   528
    apply (rule order_trans [OF _ setL2_triangle_ineq])
huffman@30582
   529
    apply (simp add: setL2_mono norm_triangle_ineq)
huffman@30040
   530
    done
huffman@30040
   531
  show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
huffman@30040
   532
    unfolding vector_norm_def
huffman@30582
   533
    by (simp add: norm_scaleR setL2_right_distrib)
huffman@30040
   534
  show "sgn x = scaleR (inverse (norm x)) x"
huffman@30040
   535
    by (rule vector_sgn_def)
huffman@30040
   536
qed
huffman@30040
   537
huffman@30040
   538
end
huffman@30040
   539
huffman@30045
   540
subsection {* Inner products *}
huffman@30045
   541
huffman@30582
   542
instantiation "^" :: (real_inner, finite) real_inner
huffman@30045
   543
begin
huffman@30045
   544
huffman@30045
   545
definition vector_inner_def:
huffman@30582
   546
  "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
huffman@30045
   547
huffman@30045
   548
instance proof
huffman@30045
   549
  fix r :: real and x y z :: "'a ^ 'b"
huffman@30045
   550
  show "inner x y = inner y x"
huffman@30045
   551
    unfolding vector_inner_def
huffman@30045
   552
    by (simp add: inner_commute)
huffman@30045
   553
  show "inner (x + y) z = inner x z + inner y z"
huffman@30045
   554
    unfolding vector_inner_def
huffman@30582
   555
    by (simp add: inner_left_distrib setsum_addf)
huffman@30045
   556
  show "inner (scaleR r x) y = r * inner x y"
huffman@30045
   557
    unfolding vector_inner_def
huffman@30582
   558
    by (simp add: inner_scaleR_left setsum_right_distrib)
huffman@30045
   559
  show "0 \<le> inner x x"
huffman@30045
   560
    unfolding vector_inner_def
huffman@30045
   561
    by (simp add: setsum_nonneg)
huffman@30045
   562
  show "inner x x = 0 \<longleftrightarrow> x = 0"
huffman@30045
   563
    unfolding vector_inner_def
huffman@30045
   564
    by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
huffman@30045
   565
  show "norm x = sqrt (inner x x)"
huffman@30045
   566
    unfolding vector_inner_def vector_norm_def setL2_def
huffman@30045
   567
    by (simp add: power2_norm_eq_inner)
huffman@30045
   568
qed
huffman@30045
   569
huffman@30045
   570
end
huffman@30045
   571
chaieb@29842
   572
subsection{* Properties of the dot product.  *}
chaieb@29842
   573
huffman@30489
   574
lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x"
chaieb@29842
   575
  by (vector mult_commute)
chaieb@29842
   576
lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \<bullet> z = (x \<bullet> z) + (y \<bullet> z)"
chaieb@29842
   577
  by (vector ring_simps)
huffman@30489
   578
lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)"
chaieb@29842
   579
  by (vector ring_simps)
huffman@30489
   580
lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)"
chaieb@29842
   581
  by (vector ring_simps)
huffman@30489
   582
lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)"
chaieb@29842
   583
  by (vector ring_simps)
chaieb@29842
   584
lemma dot_lmult: "(c *s x) \<bullet> y = (c::'a::ring) * (x \<bullet> y)" by (vector ring_simps)
chaieb@29842
   585
lemma dot_rmult: "x \<bullet> (c *s y) = (c::'a::comm_ring) * (x \<bullet> y)" by (vector ring_simps)
chaieb@29842
   586
lemma dot_lneg: "(-x) \<bullet> (y::'a::ring ^ 'n) = -(x \<bullet> y)" by vector
chaieb@29842
   587
lemma dot_rneg: "(x::'a::ring ^ 'n) \<bullet> (-y) = -(x \<bullet> y)" by vector
chaieb@29842
   588
lemma dot_lzero[simp]: "0 \<bullet> x = (0::'a::{comm_monoid_add, mult_zero})" by vector
chaieb@29842
   589
lemma dot_rzero[simp]: "x \<bullet> 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector
chaieb@29842
   590
lemma dot_pos_le[simp]: "(0::'a\<Colon>ordered_ring_strict) <= x \<bullet> x"
chaieb@29842
   591
  by (simp add: dot_def setsum_nonneg)
chaieb@29842
   592
chaieb@29842
   593
lemma setsum_squares_eq_0_iff: assumes fS: "finite F" and fp: "\<forall>x \<in> F. f x \<ge> (0 ::'a::pordered_ab_group_add)" shows "setsum f F = 0 \<longleftrightarrow> (ALL x:F. f x = 0)"
chaieb@29842
   594
using fS fp setsum_nonneg[OF fp]
chaieb@29842
   595
proof (induct set: finite)
chaieb@29842
   596
  case empty thus ?case by simp
chaieb@29842
   597
next
chaieb@29842
   598
  case (insert x F)
chaieb@29842
   599
  from insert.prems have Fx: "f x \<ge> 0" and Fp: "\<forall> a \<in> F. f a \<ge> 0" by simp_all
chaieb@29842
   600
  from insert.hyps Fp setsum_nonneg[OF Fp]
chaieb@29842
   601
  have h: "setsum f F = 0 \<longleftrightarrow> (\<forall>a \<in>F. f a = 0)" by metis
chaieb@29842
   602
  from sum_nonneg_eq_zero_iff[OF Fx  setsum_nonneg[OF Fp]] insert.hyps(1,2)
chaieb@29842
   603
  show ?case by (simp add: h)
chaieb@29842
   604
qed
chaieb@29842
   605
huffman@30582
   606
lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n::finite) = 0"
huffman@30582
   607
  by (simp add: dot_def setsum_squares_eq_0_iff Cart_eq)
huffman@30582
   608
huffman@30582
   609
lemma dot_pos_lt[simp]: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n::finite) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x]
huffman@30489
   610
  by (auto simp add: le_less)
chaieb@29842
   611
huffman@30040
   612
subsection{* The collapse of the general concepts to dimension one. *}
chaieb@29842
   613
chaieb@29842
   614
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
huffman@30582
   615
  by (simp add: Cart_eq forall_1)
chaieb@29842
   616
chaieb@29842
   617
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
chaieb@29842
   618
  apply auto
chaieb@29842
   619
  apply (erule_tac x= "x$1" in allE)
chaieb@29842
   620
  apply (simp only: vector_one[symmetric])
chaieb@29842
   621
  done
chaieb@29842
   622
huffman@30040
   623
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
huffman@30582
   624
  by (simp add: vector_norm_def UNIV_1)
huffman@30040
   625
huffman@30489
   626
lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
huffman@30040
   627
  by (simp add: norm_vector_1)
chaieb@29842
   628
chaieb@29842
   629
text{* Metric *}
chaieb@29842
   630
huffman@30040
   631
text {* FIXME: generalize to arbitrary @{text real_normed_vector} types *}
huffman@30582
   632
definition dist:: "real ^ 'n::finite \<Rightarrow> real ^ 'n \<Rightarrow> real" where
chaieb@29842
   633
  "dist x y = norm (x - y)"
chaieb@29842
   634
chaieb@29842
   635
lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
huffman@30582
   636
  by (auto simp add: norm_real dist_def)
chaieb@29842
   637
chaieb@29842
   638
subsection {* A connectedness or intermediate value lemma with several applications. *}
chaieb@29842
   639
chaieb@29842
   640
lemma connected_real_lemma:
huffman@30582
   641
  fixes f :: "real \<Rightarrow> real ^ 'n::finite"
chaieb@29842
   642
  assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
chaieb@29842
   643
  and dst: "\<And>e x. a <= x \<Longrightarrow> x <= b \<Longrightarrow> 0 < e ==> \<exists>d > 0. \<forall>y. abs(y - x) < d \<longrightarrow> dist(f y) (f x) < e"
chaieb@29842
   644
  and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
chaieb@29842
   645
  and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
chaieb@29842
   646
  and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
chaieb@29842
   647
  shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
chaieb@29842
   648
proof-
chaieb@29842
   649
  let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
huffman@30489
   650
  have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
huffman@30489
   651
  have Sub: "\<exists>y. isUb UNIV ?S y"
chaieb@29842
   652
    apply (rule exI[where x= b])
huffman@30489
   653
    using ab fb e12 by (auto simp add: isUb_def setle_def)
huffman@30489
   654
  from reals_complete[OF Se Sub] obtain l where
chaieb@29842
   655
    l: "isLub UNIV ?S l"by blast
chaieb@29842
   656
  have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
huffman@30489
   657
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
chaieb@29842
   658
    by (metis linorder_linear)
chaieb@29842
   659
  have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
chaieb@29842
   660
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
chaieb@29842
   661
    by (metis linorder_linear not_le)
chaieb@29842
   662
    have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
chaieb@29842
   663
    have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
chaieb@29842
   664
    have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo
chaieb@29842
   665
    {assume le2: "f l \<in> e2"
chaieb@29842
   666
      from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
chaieb@29842
   667
      hence lap: "l - a > 0" using alb by arith
huffman@30489
   668
      from e2[rule_format, OF le2] obtain e where
chaieb@29842
   669
	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
huffman@30489
   670
      from dst[OF alb e(1)] obtain d where
chaieb@29842
   671
	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
huffman@30489
   672
      have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1)
chaieb@29842
   673
	apply ferrack by arith
chaieb@29842
   674
      then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
chaieb@29842
   675
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
chaieb@29842
   676
      from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
chaieb@29842
   677
      moreover
chaieb@29842
   678
      have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
chaieb@29842
   679
      ultimately have False using e12 alb d' by auto}
chaieb@29842
   680
    moreover
chaieb@29842
   681
    {assume le1: "f l \<in> e1"
chaieb@29842
   682
    from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
chaieb@29842
   683
      hence blp: "b - l > 0" using alb by arith
huffman@30489
   684
      from e1[rule_format, OF le1] obtain e where
chaieb@29842
   685
	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
huffman@30489
   686
      from dst[OF alb e(1)] obtain d where
chaieb@29842
   687
	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
huffman@30489
   688
      have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo
chaieb@29842
   689
      then obtain d' where d': "d' > 0" "d' < d" by metis
chaieb@29842
   690
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
chaieb@29842
   691
      hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
chaieb@29842
   692
      with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
huffman@30489
   693
      with l d' have False
chaieb@29842
   694
	by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
chaieb@29842
   695
    ultimately show ?thesis using alb by metis
chaieb@29842
   696
qed
chaieb@29842
   697
huffman@29881
   698
text{* One immediately useful corollary is the existence of square roots! --- Should help to get rid of all the development of square-root for reals as a special case @{typ "real^1"} *}
chaieb@29842
   699
chaieb@29842
   700
lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
chaieb@29842
   701
proof-
huffman@30489
   702
  have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
chaieb@29842
   703
  thus ?thesis by (simp add: ring_simps power2_eq_square)
chaieb@29842
   704
qed
chaieb@29842
   705
chaieb@29842
   706
lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
chaieb@29842
   707
  using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_def, rule_format, of e x] apply (auto simp add: power2_eq_square)
chaieb@29842
   708
  apply (rule_tac x="s" in exI)
chaieb@29842
   709
  apply auto
chaieb@29842
   710
  apply (erule_tac x=y in allE)
chaieb@29842
   711
  apply auto
chaieb@29842
   712
  done
chaieb@29842
   713
chaieb@29842
   714
lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
chaieb@29842
   715
  using real_sqrt_le_iff[of x "y^2"] by simp
chaieb@29842
   716
chaieb@29842
   717
lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
chaieb@29842
   718
  using real_sqrt_le_mono[of "x^2" y] by simp
chaieb@29842
   719
chaieb@29842
   720
lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
chaieb@29842
   721
  using real_sqrt_less_mono[of "x^2" y] by simp
chaieb@29842
   722
huffman@30489
   723
lemma sqrt_even_pow2: assumes n: "even n"
chaieb@29842
   724
  shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
chaieb@29842
   725
proof-
huffman@30489
   726
  from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2
huffman@30489
   727
    by (auto simp add: nat_number)
chaieb@29842
   728
  from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
chaieb@29842
   729
    by (simp only: power_mult[symmetric] mult_commute)
huffman@30489
   730
  then show ?thesis  using m by simp
chaieb@29842
   731
qed
chaieb@29842
   732
chaieb@29842
   733
lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
chaieb@29842
   734
  apply (cases "x = 0", simp_all)
chaieb@29842
   735
  using sqrt_divide_self_eq[of x]
chaieb@29842
   736
  apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps)
chaieb@29842
   737
  done
chaieb@29842
   738
chaieb@29842
   739
text{* Hence derive more interesting properties of the norm. *}
chaieb@29842
   740
huffman@30582
   741
text {*
huffman@30582
   742
  This type-specific version is only here
huffman@30582
   743
  to make @{text normarith.ML} happy.
huffman@30582
   744
*}
huffman@30582
   745
lemma norm_0: "norm (0::real ^ _) = 0"
huffman@30040
   746
  by (rule norm_zero)
huffman@30040
   747
chaieb@30263
   748
lemma norm_mul[simp]: "norm(a *s x) = abs(a) * norm x"
huffman@30040
   749
  by (simp add: vector_norm_def vector_component setL2_right_distrib
huffman@30040
   750
           abs_mult cong: strong_setL2_cong)
chaieb@29842
   751
lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
huffman@30040
   752
  by (simp add: vector_norm_def dot_def setL2_def power2_eq_square)
huffman@30040
   753
lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"
huffman@30040
   754
  by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
chaieb@29842
   755
lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
huffman@30040
   756
  by (simp add: real_vector_norm_def)
huffman@30582
   757
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n::finite)" by (metis norm_eq_zero)
chaieb@30263
   758
lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
chaieb@29842
   759
  by vector
chaieb@30263
   760
lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
chaieb@29842
   761
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
chaieb@30263
   762
lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
chaieb@29842
   763
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
chaieb@29842
   764
lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
chaieb@29842
   765
  by (metis vector_mul_lcancel)
chaieb@29842
   766
lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
chaieb@29842
   767
  by (metis vector_mul_rcancel)
huffman@30582
   768
lemma norm_cauchy_schwarz:
huffman@30582
   769
  fixes x y :: "real ^ 'n::finite"
huffman@30582
   770
  shows "x \<bullet> y <= norm x * norm y"
chaieb@29842
   771
proof-
chaieb@29842
   772
  {assume "norm x = 0"
huffman@30041
   773
    hence ?thesis by (simp add: dot_lzero dot_rzero)}
chaieb@29842
   774
  moreover
huffman@30489
   775
  {assume "norm y = 0"
huffman@30041
   776
    hence ?thesis by (simp add: dot_lzero dot_rzero)}
chaieb@29842
   777
  moreover
chaieb@29842
   778
  {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
chaieb@29842
   779
    let ?z = "norm y *s x - norm x *s y"
huffman@30041
   780
    from h have p: "norm x * norm y > 0" by (metis norm_ge_zero le_less zero_compare_simps)
chaieb@29842
   781
    from dot_pos_le[of ?z]
chaieb@29842
   782
    have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"
chaieb@29842
   783
      apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)
chaieb@29842
   784
      by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym)
chaieb@29842
   785
    hence "x\<bullet>y \<le> (norm x ^2 * norm y ^2) / (norm x * norm y)" using p
chaieb@29842
   786
      by (simp add: field_simps)
chaieb@29842
   787
    hence ?thesis using h by (simp add: power2_eq_square)}
chaieb@29842
   788
  ultimately show ?thesis by metis
chaieb@29842
   789
qed
chaieb@29842
   790
huffman@30582
   791
lemma norm_cauchy_schwarz_abs:
huffman@30582
   792
  fixes x y :: "real ^ 'n::finite"
huffman@30582
   793
  shows "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
chaieb@29842
   794
  using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
huffman@30041
   795
  by (simp add: real_abs_def dot_rneg)
chaieb@29842
   796
huffman@30582
   797
lemma norm_triangle_sub: "norm (x::real ^'n::finite) <= norm(y) + norm(x - y)"
huffman@30041
   798
  using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
huffman@30582
   799
lemma norm_triangle_le: "norm(x::real ^'n::finite) + norm y <= e ==> norm(x + y) <= e"
huffman@30041
   800
  by (metis order_trans norm_triangle_ineq)
huffman@30582
   801
lemma norm_triangle_lt: "norm(x::real ^'n::finite) + norm(y) < e ==> norm(x + y) < e"
huffman@30041
   802
  by (metis basic_trans_rules(21) norm_triangle_ineq)
chaieb@29842
   803
huffman@30582
   804
lemma setsum_delta:
huffman@30582
   805
  assumes fS: "finite S"
huffman@30582
   806
  shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
huffman@30582
   807
proof-
huffman@30582
   808
  let ?f = "(\<lambda>k. if k=a then b k else 0)"
huffman@30582
   809
  {assume a: "a \<notin> S"
huffman@30582
   810
    hence "\<forall> k\<in> S. ?f k = 0" by simp
huffman@30582
   811
    hence ?thesis  using a by simp}
huffman@30582
   812
  moreover
huffman@30582
   813
  {assume a: "a \<in> S"
huffman@30582
   814
    let ?A = "S - {a}"
huffman@30582
   815
    let ?B = "{a}"
huffman@30582
   816
    have eq: "S = ?A \<union> ?B" using a by blast
huffman@30582
   817
    have dj: "?A \<inter> ?B = {}" by simp
huffman@30582
   818
    from fS have fAB: "finite ?A" "finite ?B" by auto
huffman@30582
   819
    have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
huffman@30582
   820
      using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
huffman@30582
   821
      by simp
huffman@30582
   822
    then have ?thesis  using a by simp}
huffman@30582
   823
  ultimately show ?thesis by blast
huffman@30582
   824
qed
huffman@30582
   825
huffman@30582
   826
lemma component_le_norm: "\<bar>x$i\<bar> <= norm (x::real ^ 'n::finite)"
huffman@30040
   827
  apply (simp add: vector_norm_def)
huffman@30040
   828
  apply (rule member_le_setL2, simp_all)
huffman@30040
   829
  done
huffman@30040
   830
huffman@30582
   831
lemma norm_bound_component_le: "norm(x::real ^ 'n::finite) <= e
huffman@30582
   832
                ==> \<bar>x$i\<bar> <= e"
chaieb@29842
   833
  by (metis component_le_norm order_trans)
chaieb@29842
   834
huffman@30582
   835
lemma norm_bound_component_lt: "norm(x::real ^ 'n::finite) < e
huffman@30582
   836
                ==> \<bar>x$i\<bar> < e"
chaieb@29842
   837
  by (metis component_le_norm basic_trans_rules(21))
chaieb@29842
   838
huffman@30582
   839
lemma norm_le_l1: "norm (x:: real ^'n::finite) <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
huffman@30040
   840
  by (simp add: vector_norm_def setL2_le_setsum)
chaieb@29842
   841
huffman@30582
   842
lemma real_abs_norm: "\<bar>norm x\<bar> = norm (x :: real ^ _)"
huffman@30040
   843
  by (rule abs_norm_cancel)
huffman@30582
   844
lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n::finite) - norm y\<bar> <= norm(x - y)"
huffman@30040
   845
  by (rule norm_triangle_ineq3)
huffman@30582
   846
lemma norm_le: "norm(x::real ^ _) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
chaieb@29842
   847
  by (simp add: real_vector_norm_def)
huffman@30582
   848
lemma norm_lt: "norm(x::real ^ _) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
chaieb@29842
   849
  by (simp add: real_vector_norm_def)
huffman@30582
   850
lemma norm_eq: "norm (x::real ^ _) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
chaieb@29842
   851
  by (simp add: order_eq_iff norm_le)
huffman@30582
   852
lemma norm_eq_1: "norm(x::real ^ _) = 1 \<longleftrightarrow> x \<bullet> x = 1"
chaieb@29842
   853
  by (simp add: real_vector_norm_def)
chaieb@29842
   854
chaieb@29842
   855
text{* Squaring equations and inequalities involving norms.  *}
chaieb@29842
   856
chaieb@29842
   857
lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
huffman@30582
   858
  by (simp add: real_vector_norm_def)
chaieb@29842
   859
chaieb@29842
   860
lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
huffman@30040
   861
  by (auto simp add: real_vector_norm_def)
chaieb@29842
   862
chaieb@29842
   863
lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
chaieb@29842
   864
proof-
chaieb@29842
   865
  have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: ring_simps power2_eq_square)
chaieb@29842
   866
  also have "\<dots> \<longleftrightarrow> \<bar>x\<bar> \<le> \<bar>y\<bar>" apply (simp add: zero_compare_simps real_abs_def not_less) by arith
chaieb@29842
   867
finally show ?thesis ..
chaieb@29842
   868
qed
chaieb@29842
   869
chaieb@29842
   870
lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
huffman@30040
   871
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
huffman@30041
   872
  using norm_ge_zero[of x]
chaieb@29842
   873
  apply arith
chaieb@29842
   874
  done
chaieb@29842
   875
huffman@30489
   876
lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
huffman@30040
   877
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
huffman@30041
   878
  using norm_ge_zero[of x]
chaieb@29842
   879
  apply arith
chaieb@29842
   880
  done
chaieb@29842
   881
chaieb@29842
   882
lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
chaieb@29842
   883
  by (metis not_le norm_ge_square)
chaieb@29842
   884
lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
chaieb@29842
   885
  by (metis norm_le_square not_less)
chaieb@29842
   886
chaieb@29842
   887
text{* Dot product in terms of the norm rather than conversely. *}
chaieb@29842
   888
chaieb@29842
   889
lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
chaieb@29842
   890
  by (simp add: norm_pow_2 dot_ladd dot_radd dot_sym)
chaieb@29842
   891
chaieb@29842
   892
lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
chaieb@29842
   893
  by (simp add: norm_pow_2 dot_ladd dot_radd dot_lsub dot_rsub dot_sym)
chaieb@29842
   894
chaieb@29842
   895
chaieb@29842
   896
text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
chaieb@29842
   897
huffman@30582
   898
lemma vector_eq: "(x:: real ^ 'n::finite) = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y\<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@29842
   899
proof
chaieb@29842
   900
  assume "?lhs" then show ?rhs by simp
chaieb@29842
   901
next
chaieb@29842
   902
  assume ?rhs
chaieb@29842
   903
  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y\<bullet> y = 0" by simp
huffman@30489
   904
  hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
chaieb@29842
   905
    by (simp add: dot_rsub dot_lsub dot_sym)
chaieb@29842
   906
  then have "(x - y) \<bullet> (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub)
chaieb@29842
   907
  then show "x = y" by (simp add: dot_eq_0)
chaieb@29842
   908
qed
chaieb@29842
   909
chaieb@29842
   910
chaieb@29842
   911
subsection{* General linear decision procedure for normed spaces. *}
chaieb@29842
   912
chaieb@29842
   913
lemma norm_cmul_rule_thm: "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(c *s x)"
chaieb@29842
   914
  apply (clarsimp simp add: norm_mul)
chaieb@29842
   915
  apply (rule mult_mono1)
chaieb@29842
   916
  apply simp_all
chaieb@29842
   917
  done
chaieb@29842
   918
chaieb@30263
   919
  (* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
huffman@30582
   920
lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n::finite) \<Longrightarrow> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)"
chaieb@29842
   921
  apply (rule norm_triangle_le) by simp
chaieb@29842
   922
chaieb@29842
   923
lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"
chaieb@29842
   924
  by (simp add: ring_simps)
chaieb@29842
   925
chaieb@29842
   926
lemma pth_1: "(x::real^'n) == 1 *s x" by (simp only: vector_smult_lid)
chaieb@29842
   927
lemma pth_2: "x - (y::real^'n) == x + -y" by (atomize (full)) simp
chaieb@29842
   928
lemma pth_3: "(-x::real^'n) == -1 *s x" by vector
chaieb@29842
   929
lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+
chaieb@29842
   930
lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector
chaieb@29842
   931
lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps)
huffman@30489
   932
lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all
huffman@30489
   933
lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps)
chaieb@29842
   934
lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z"
chaieb@29842
   935
  "c *s x + (d *s x + z) == (c + d) *s x + z"
chaieb@29842
   936
  "(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+
chaieb@29842
   937
lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector
huffman@30489
   938
lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y"
chaieb@29842
   939
  "(c *s x + z) + d *s y == c *s x + (z + d *s y)"
chaieb@29842
   940
  "c *s x + (d *s y + z) == c *s x + (d *s y + z)"
chaieb@29842
   941
  "(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))"
chaieb@29842
   942
  by ((atomize (full)), vector)+
chaieb@29842
   943
lemma pth_c: "(c::real) *s x + d *s y == d *s y + c *s x"
chaieb@29842
   944
  "(c *s x + z) + d *s y == d *s y + (c *s x + z)"
chaieb@29842
   945
  "c *s x + (d *s y + z) == d *s y + (c *s x + z)"
chaieb@29842
   946
  "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+
chaieb@29842
   947
lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector
chaieb@29842
   948
huffman@30582
   949
lemma norm_imp_pos_and_ge: "norm (x::real ^ _) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
huffman@30041
   950
  by (atomize) (auto simp add: norm_ge_zero)
chaieb@29842
   951
chaieb@29842
   952
lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
chaieb@29842
   953
huffman@30489
   954
lemma norm_pths:
huffman@30582
   955
  "(x::real ^'n::finite) = y \<longleftrightarrow> norm (x - y) \<le> 0"
chaieb@29842
   956
  "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
huffman@30041
   957
  using norm_ge_zero[of "x - y"] by auto
chaieb@29842
   958
chaieb@29842
   959
use "normarith.ML"
chaieb@29842
   960
wenzelm@30549
   961
method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
chaieb@29842
   962
*} "Proves simple linear statements about vector norms"
chaieb@29842
   963
chaieb@29842
   964
chaieb@29842
   965
chaieb@29842
   966
text{* Hence more metric properties. *}
chaieb@29842
   967
chaieb@30263
   968
lemma dist_refl[simp]: "dist x x = 0" by norm
chaieb@29842
   969
chaieb@29842
   970
lemma dist_sym: "dist x y = dist y x"by norm
chaieb@29842
   971
chaieb@30263
   972
lemma dist_pos_le[simp]: "0 <= dist x y" by norm
chaieb@29842
   973
chaieb@29842
   974
lemma dist_triangle: "dist x z <= dist x y + dist y z" by norm
chaieb@29842
   975
chaieb@29842
   976
lemma dist_triangle_alt: "dist y z <= dist x y + dist x z" by norm
chaieb@29842
   977
chaieb@30263
   978
lemma dist_eq_0[simp]: "dist x y = 0 \<longleftrightarrow> x = y" by norm
chaieb@29842
   979
huffman@30489
   980
lemma dist_pos_lt: "x \<noteq> y ==> 0 < dist x y" by norm
huffman@30489
   981
lemma dist_nz:  "x \<noteq> y \<longleftrightarrow> 0 < dist x y" by norm
huffman@30489
   982
huffman@30489
   983
lemma dist_triangle_le: "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e" by norm
huffman@30489
   984
huffman@30489
   985
lemma dist_triangle_lt: "dist x z + dist y z < e ==> dist x y < e" by norm
huffman@30489
   986
huffman@30489
   987
lemma dist_triangle_half_l: "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 ==> dist x1 x2 < e" by norm
huffman@30489
   988
huffman@30489
   989
lemma dist_triangle_half_r: "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 ==> dist x1 x2 < e" by norm
chaieb@29842
   990
chaieb@29842
   991
lemma dist_triangle_add: "dist (x + y) (x' + y') <= dist x x' + dist y y'"
huffman@30489
   992
  by norm
huffman@30489
   993
huffman@30489
   994
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
huffman@30489
   995
  unfolding dist_def vector_ssub_ldistrib[symmetric] norm_mul ..
huffman@30489
   996
huffman@30489
   997
lemma dist_triangle_add_half: " dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 ==> dist(x + y) (x' + y') < e" by norm
huffman@30489
   998
huffman@30489
   999
lemma dist_le_0[simp]: "dist x y <= 0 \<longleftrightarrow> x = y" by norm
chaieb@29842
  1000
huffman@30582
  1001
lemma setsum_component [simp]:
huffman@30582
  1002
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
huffman@30582
  1003
  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
huffman@30582
  1004
  by (cases "finite S", induct S set: finite, simp_all)
huffman@30582
  1005
chaieb@29842
  1006
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
huffman@30582
  1007
  by (simp add: Cart_eq)
chaieb@29842
  1008
huffman@30489
  1009
lemma setsum_clauses:
chaieb@29842
  1010
  shows "setsum f {} = 0"
chaieb@29842
  1011
  and "finite S \<Longrightarrow> setsum f (insert x S) =
chaieb@29842
  1012
                 (if x \<in> S then setsum f S else f x + setsum f S)"
chaieb@29842
  1013
  by (auto simp add: insert_absorb)
chaieb@29842
  1014
huffman@30489
  1015
lemma setsum_cmul:
chaieb@29842
  1016
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
chaieb@29842
  1017
  shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
huffman@30582
  1018
  by (simp add: Cart_eq setsum_right_distrib)
chaieb@29842
  1019
huffman@30489
  1020
lemma setsum_norm:
chaieb@29842
  1021
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
chaieb@29842
  1022
  assumes fS: "finite S"
chaieb@29842
  1023
  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
chaieb@29842
  1024
proof(induct rule: finite_induct[OF fS])
huffman@30041
  1025
  case 1 thus ?case by simp
chaieb@29842
  1026
next
chaieb@29842
  1027
  case (2 x S)
chaieb@29842
  1028
  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
chaieb@29842
  1029
  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
chaieb@29842
  1030
    using "2.hyps" by simp
chaieb@29842
  1031
  finally  show ?case  using "2.hyps" by simp
chaieb@29842
  1032
qed
chaieb@29842
  1033
huffman@30489
  1034
lemma real_setsum_norm:
huffman@30582
  1035
  fixes f :: "'a \<Rightarrow> real ^'n::finite"
chaieb@29842
  1036
  assumes fS: "finite S"
chaieb@29842
  1037
  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
chaieb@29842
  1038
proof(induct rule: finite_induct[OF fS])
huffman@30040
  1039
  case 1 thus ?case by simp
chaieb@29842
  1040
next
chaieb@29842
  1041
  case (2 x S)
huffman@30040
  1042
  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
chaieb@29842
  1043
  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
chaieb@29842
  1044
    using "2.hyps" by simp
chaieb@29842
  1045
  finally  show ?case  using "2.hyps" by simp
chaieb@29842
  1046
qed
chaieb@29842
  1047
huffman@30489
  1048
lemma setsum_norm_le:
chaieb@29842
  1049
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
chaieb@29842
  1050
  assumes fS: "finite S"
chaieb@29842
  1051
  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
chaieb@29842
  1052
  shows "norm (setsum f S) \<le> setsum g S"
chaieb@29842
  1053
proof-
huffman@30489
  1054
  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
chaieb@29842
  1055
    by - (rule setsum_mono, simp)
chaieb@29842
  1056
  then show ?thesis using setsum_norm[OF fS, of f] fg
chaieb@29842
  1057
    by arith
chaieb@29842
  1058
qed
chaieb@29842
  1059
huffman@30489
  1060
lemma real_setsum_norm_le:
huffman@30582
  1061
  fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
chaieb@29842
  1062
  assumes fS: "finite S"
chaieb@29842
  1063
  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
chaieb@29842
  1064
  shows "norm (setsum f S) \<le> setsum g S"
chaieb@29842
  1065
proof-
huffman@30489
  1066
  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
chaieb@29842
  1067
    by - (rule setsum_mono, simp)
chaieb@29842
  1068
  then show ?thesis using real_setsum_norm[OF fS, of f] fg
chaieb@29842
  1069
    by arith
chaieb@29842
  1070
qed
chaieb@29842
  1071
chaieb@29842
  1072
lemma setsum_norm_bound:
chaieb@29842
  1073
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
chaieb@29842
  1074
  assumes fS: "finite S"
chaieb@29842
  1075
  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
chaieb@29842
  1076
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
chaieb@29842
  1077
  using setsum_norm_le[OF fS K] setsum_constant[symmetric]
chaieb@29842
  1078
  by simp
chaieb@29842
  1079
chaieb@29842
  1080
lemma real_setsum_norm_bound:
huffman@30582
  1081
  fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
chaieb@29842
  1082
  assumes fS: "finite S"
chaieb@29842
  1083
  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
chaieb@29842
  1084
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
chaieb@29842
  1085
  using real_setsum_norm_le[OF fS K] setsum_constant[symmetric]
chaieb@29842
  1086
  by simp
chaieb@29842
  1087
chaieb@29842
  1088
lemma setsum_vmul:
chaieb@29842
  1089
  fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector,semiring, mult_zero}"
chaieb@29842
  1090
  assumes fS: "finite S"
chaieb@29842
  1091
  shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
chaieb@29842
  1092
proof(induct rule: finite_induct[OF fS])
chaieb@29842
  1093
  case 1 then show ?case by (simp add: vector_smult_lzero)
chaieb@29842
  1094
next
chaieb@29842
  1095
  case (2 x F)
huffman@30489
  1096
  from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v"
chaieb@29842
  1097
    by simp
huffman@30489
  1098
  also have "\<dots> = f x *s v + setsum f F *s v"
chaieb@29842
  1099
    by (simp add: vector_sadd_rdistrib)
chaieb@29842
  1100
  also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp
chaieb@29842
  1101
  finally show ?case .
chaieb@29842
  1102
qed
chaieb@29842
  1103
chaieb@29842
  1104
(* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \<Rightarrow> real ^'n"]  ---
chaieb@29842
  1105
 Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *)
chaieb@29842
  1106
chaieb@29842
  1107
lemma setsum_add_split: assumes mn: "(m::nat) \<le> n + 1"
chaieb@29842
  1108
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
chaieb@29842
  1109
proof-
chaieb@29842
  1110
  let ?A = "{m .. n}"
chaieb@29842
  1111
  let ?B = "{n + 1 .. n + p}"
huffman@30489
  1112
  have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto
chaieb@29842
  1113
  have d: "?A \<inter> ?B = {}" by auto
chaieb@29842
  1114
  from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto
chaieb@29842
  1115
qed
chaieb@29842
  1116
chaieb@29842
  1117
lemma setsum_natinterval_left:
huffman@30489
  1118
  assumes mn: "(m::nat) <= n"
chaieb@29842
  1119
  shows "setsum f {m..n} = f m + setsum f {m + 1..n}"
chaieb@29842
  1120
proof-
chaieb@29842
  1121
  from mn have "{m .. n} = insert m {m+1 .. n}" by auto
chaieb@29842
  1122
  then show ?thesis by auto
chaieb@29842
  1123
qed
chaieb@29842
  1124
huffman@30489
  1125
lemma setsum_natinterval_difff:
chaieb@29842
  1126
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
chaieb@29842
  1127
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
chaieb@29842
  1128
          (if m <= n then f m - f(n + 1) else 0)"
chaieb@29842
  1129
by (induct n, auto simp add: ring_simps not_le le_Suc_eq)
chaieb@29842
  1130
chaieb@29842
  1131
lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
chaieb@29842
  1132
chaieb@29842
  1133
lemma setsum_setsum_restrict:
chaieb@29842
  1134
  "finite S \<Longrightarrow> finite T \<Longrightarrow> setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y\<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"
chaieb@29842
  1135
  apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
chaieb@29842
  1136
  by (rule setsum_commute)
chaieb@29842
  1137
chaieb@29842
  1138
lemma setsum_image_gen: assumes fS: "finite S"
chaieb@29842
  1139
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
chaieb@29842
  1140
proof-
chaieb@29842
  1141
  {fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto}
chaieb@29842
  1142
  note th0 = this
huffman@30489
  1143
  have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
huffman@30489
  1144
    apply (rule setsum_cong2)
chaieb@29842
  1145
    by (simp add: th0)
chaieb@29842
  1146
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
chaieb@29842
  1147
    apply (rule setsum_setsum_restrict[OF fS])
chaieb@29842
  1148
    by (rule finite_imageI[OF fS])
chaieb@29842
  1149
  finally show ?thesis .
chaieb@29842
  1150
qed
chaieb@29842
  1151
chaieb@29842
  1152
    (* FIXME: Here too need stupid finiteness assumption on T!!! *)
chaieb@29842
  1153
lemma setsum_group:
chaieb@29842
  1154
  assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
chaieb@29842
  1155
  shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
huffman@30489
  1156
chaieb@29842
  1157
apply (subst setsum_image_gen[OF fS, of g f])
chaieb@30263
  1158
apply (rule setsum_mono_zero_right[OF fT fST])
chaieb@29842
  1159
by (auto intro: setsum_0')
chaieb@29842
  1160
chaieb@29842
  1161
lemma vsum_norm_allsubsets_bound:
huffman@30582
  1162
  fixes f:: "'a \<Rightarrow> real ^'n::finite"
huffman@30489
  1163
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
huffman@30582
  1164
  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
chaieb@29842
  1165
proof-
huffman@30582
  1166
  let ?d = "real CARD('n)"
chaieb@29842
  1167
  let ?nf = "\<lambda>x. norm (f x)"
huffman@30582
  1168
  let ?U = "UNIV :: 'n set"
chaieb@29842
  1169
  have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x $ i\<bar>) P) ?U"
chaieb@29842
  1170
    by (rule setsum_commute)
chaieb@29842
  1171
  have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
chaieb@29842
  1172
  have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
chaieb@29842
  1173
    apply (rule setsum_mono)
chaieb@29842
  1174
    by (rule norm_le_l1)
chaieb@29842
  1175
  also have "\<dots> \<le> 2 * ?d * e"
chaieb@29842
  1176
    unfolding th0 th1
chaieb@29842
  1177
  proof(rule setsum_bounded)
chaieb@29842
  1178
    fix i assume i: "i \<in> ?U"
chaieb@29842
  1179
    let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
chaieb@29842
  1180
    let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
chaieb@29842
  1181
    have thp: "P = ?Pp \<union> ?Pn" by auto
chaieb@29842
  1182
    have thp0: "?Pp \<inter> ?Pn ={}" by auto
chaieb@29842
  1183
    have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
chaieb@29842
  1184
    have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
huffman@30582
  1185
      using component_le_norm[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
huffman@30582
  1186
      by (auto intro: abs_le_D1)
chaieb@29842
  1187
    have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
huffman@30582
  1188
      using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
huffman@30582
  1189
      by (auto simp add: setsum_negf intro: abs_le_D1)
huffman@30489
  1190
    have "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn"
chaieb@29842
  1191
      apply (subst thp)
huffman@30489
  1192
      apply (rule setsum_Un_zero)
chaieb@29842
  1193
      using fP thp0 by auto
chaieb@29842
  1194
    also have "\<dots> \<le> 2*e" using Pne Ppe by arith
chaieb@29842
  1195
    finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
chaieb@29842
  1196
  qed
chaieb@29842
  1197
  finally show ?thesis .
chaieb@29842
  1198
qed
chaieb@29842
  1199
chaieb@29842
  1200
lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> (y::'a::{comm_ring}^'n) = setsum (\<lambda>x. f x \<bullet> y) S "
chaieb@30263
  1201
  by (induct rule: finite_induct, auto simp add: dot_lzero dot_ladd dot_radd)
chaieb@29842
  1202
chaieb@29842
  1203
lemma dot_rsum: "finite S \<Longrightarrow> (y::'a::{comm_ring}^'n) \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
chaieb@29842
  1204
  by (induct rule: finite_induct, auto simp add: dot_rzero dot_radd)
chaieb@29842
  1205
chaieb@29842
  1206
subsection{* Basis vectors in coordinate directions. *}
chaieb@29842
  1207
chaieb@29842
  1208
chaieb@29842
  1209
definition "basis k = (\<chi> i. if i = k then 1 else 0)"
chaieb@29842
  1210
huffman@30582
  1211
lemma basis_component [simp]: "basis k $ i = (if k=i then 1 else 0)"
huffman@30582
  1212
  unfolding basis_def by simp
huffman@30582
  1213
huffman@30489
  1214
lemma delta_mult_idempotent:
chaieb@29842
  1215
  "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" by (cases "k=a", auto)
chaieb@29842
  1216
chaieb@29842
  1217
lemma norm_basis:
huffman@30582
  1218
  shows "norm (basis k :: real ^'n::finite) = 1"
chaieb@29842
  1219
  apply (simp add: basis_def real_vector_norm_def dot_def)
chaieb@29842
  1220
  apply (vector delta_mult_idempotent)
huffman@30582
  1221
  using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"]
chaieb@29842
  1222
  apply auto
chaieb@29842
  1223
  done
chaieb@29842
  1224
huffman@30582
  1225
lemma norm_basis_1: "norm(basis 1 :: real ^'n::{finite,one}) = 1"
huffman@30582
  1226
  by (rule norm_basis)
huffman@30582
  1227
huffman@30582
  1228
lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n::finite). norm x = c"
huffman@30582
  1229
  apply (rule exI[where x="c *s basis arbitrary"])
huffman@30582
  1230
  by (simp only: norm_mul norm_basis)
chaieb@29842
  1231
huffman@30489
  1232
lemma vector_choose_dist: assumes e: "0 <= e"
huffman@30582
  1233
  shows "\<exists>(y::real^'n::finite). dist x y = e"
chaieb@29842
  1234
proof-
chaieb@29842
  1235
  from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"
chaieb@29842
  1236
    by blast
chaieb@29842
  1237
  then have "dist x (x - c) = e" by (simp add: dist_def)
chaieb@29842
  1238
  then show ?thesis by blast
chaieb@29842
  1239
qed
chaieb@29842
  1240
huffman@30582
  1241
lemma basis_inj: "inj (basis :: 'n \<Rightarrow> real ^'n::finite)"
huffman@30582
  1242
  by (simp add: inj_on_def Cart_eq)
chaieb@29842
  1243
chaieb@29842
  1244
lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
chaieb@29842
  1245
  by auto
chaieb@29842
  1246
chaieb@29842
  1247
lemma basis_expansion:
huffman@30582
  1248
  "setsum (\<lambda>i. (x$i) *s basis i) UNIV = (x::('a::ring_1) ^'n::finite)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
huffman@30582
  1249
  by (auto simp add: Cart_eq cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
chaieb@29842
  1250
huffman@30489
  1251
lemma basis_expansion_unique:
huffman@30582
  1252
  "setsum (\<lambda>i. f i *s basis i) UNIV = (x::('a::comm_ring_1) ^'n::finite) \<longleftrightarrow> (\<forall>i. f i = x$i)"
huffman@30582
  1253
  by (simp add: Cart_eq setsum_delta cond_value_iff cong del: if_weak_cong)
chaieb@29842
  1254
chaieb@29842
  1255
lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
chaieb@29842
  1256
  by auto
chaieb@29842
  1257
chaieb@29842
  1258
lemma dot_basis:
huffman@30582
  1259
  shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i :: 'a^'n::finite) = (x$i :: 'a::semiring_1)"
huffman@30582
  1260
  by (auto simp add: dot_def basis_def cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)
huffman@30582
  1261
huffman@30582
  1262
lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
huffman@30582
  1263
  by (auto simp add: Cart_eq)
chaieb@29842
  1264
huffman@30489
  1265
lemma basis_nonzero:
chaieb@29842
  1266
  shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
huffman@30582
  1267
  by (simp add: basis_eq_0)
huffman@30582
  1268
huffman@30582
  1269
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = (z::'a::semiring_1^'n::finite)"
chaieb@29842
  1270
  apply (auto simp add: Cart_eq dot_basis)
chaieb@29842
  1271
  apply (erule_tac x="basis i" in allE)
chaieb@29842
  1272
  apply (simp add: dot_basis)
chaieb@29842
  1273
  apply (subgoal_tac "y = z")
chaieb@29842
  1274
  apply simp
huffman@30582
  1275
  apply (simp add: Cart_eq)
chaieb@29842
  1276
  done
chaieb@29842
  1277
huffman@30582
  1278
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = (y::'a::semiring_1^'n::finite)"
chaieb@29842
  1279
  apply (auto simp add: Cart_eq dot_basis)
chaieb@29842
  1280
  apply (erule_tac x="basis i" in allE)
chaieb@29842
  1281
  apply (simp add: dot_basis)
chaieb@29842
  1282
  apply (subgoal_tac "x = y")
chaieb@29842
  1283
  apply simp
huffman@30582
  1284
  apply (simp add: Cart_eq)
chaieb@29842
  1285
  done
chaieb@29842
  1286
chaieb@29842
  1287
subsection{* Orthogonality. *}
chaieb@29842
  1288
chaieb@29842
  1289
definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
chaieb@29842
  1290
chaieb@29842
  1291
lemma orthogonal_basis:
huffman@30582
  1292
  shows "orthogonal (basis i :: 'a^'n::finite) x \<longleftrightarrow> x$i = (0::'a::ring_1)"
huffman@30582
  1293
  by (auto simp add: orthogonal_def dot_def basis_def cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
chaieb@29842
  1294
chaieb@29842
  1295
lemma orthogonal_basis_basis:
huffman@30582
  1296
  shows "orthogonal (basis i :: 'a::ring_1^'n::finite) (basis j) \<longleftrightarrow> i \<noteq> j"
huffman@30582
  1297
  unfolding orthogonal_basis[of i] basis_component[of j] by simp
chaieb@29842
  1298
chaieb@29842
  1299
  (* FIXME : Maybe some of these require less than comm_ring, but not all*)
chaieb@29842
  1300
lemma orthogonal_clauses:
chaieb@29842
  1301
  "orthogonal a (0::'a::comm_ring ^'n)"
chaieb@29842
  1302
  "orthogonal a x ==> orthogonal a (c *s x)"
chaieb@29842
  1303
  "orthogonal a x ==> orthogonal a (-x)"
chaieb@29842
  1304
  "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x + y)"
chaieb@29842
  1305
  "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x - y)"
chaieb@29842
  1306
  "orthogonal 0 a"
chaieb@29842
  1307
  "orthogonal x a ==> orthogonal (c *s x) a"
chaieb@29842
  1308
  "orthogonal x a ==> orthogonal (-x) a"
chaieb@29842
  1309
  "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x + y) a"
chaieb@29842
  1310
  "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x - y) a"
chaieb@29842
  1311
  unfolding orthogonal_def dot_rneg dot_rmult dot_radd dot_rsub
chaieb@29842
  1312
  dot_lzero dot_rzero dot_lneg dot_lmult dot_ladd dot_lsub
chaieb@29842
  1313
  by simp_all
chaieb@29842
  1314
chaieb@29842
  1315
lemma orthogonal_commute: "orthogonal (x::'a::{ab_semigroup_mult,comm_monoid_add} ^'n)y \<longleftrightarrow> orthogonal y x"
chaieb@29842
  1316
  by (simp add: orthogonal_def dot_sym)
chaieb@29842
  1317
chaieb@29842
  1318
subsection{* Explicit vector construction from lists. *}
chaieb@29842
  1319
huffman@30582
  1320
primrec from_nat :: "nat \<Rightarrow> 'a::{monoid_add,one}"
huffman@30582
  1321
where "from_nat 0 = 0" | "from_nat (Suc n) = 1 + from_nat n"
huffman@30582
  1322
huffman@30582
  1323
lemma from_nat [simp]: "from_nat = of_nat"
huffman@30582
  1324
by (rule ext, induct_tac x, simp_all)
huffman@30582
  1325
huffman@30582
  1326
primrec
huffman@30582
  1327
  list_fun :: "nat \<Rightarrow> _ list \<Rightarrow> _ \<Rightarrow> _"
huffman@30582
  1328
where
huffman@30582
  1329
  "list_fun n [] = (\<lambda>x. 0)"
huffman@30582
  1330
| "list_fun n (x # xs) = fun_upd (list_fun (Suc n) xs) (from_nat n) x"
huffman@30582
  1331
huffman@30582
  1332
definition "vector l = (\<chi> i. list_fun 1 l i)"
huffman@30582
  1333
(*definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"*)
chaieb@29842
  1334
chaieb@29842
  1335
lemma vector_1: "(vector[x]) $1 = x"
huffman@30582
  1336
  unfolding vector_def by simp
chaieb@29842
  1337
chaieb@29842
  1338
lemma vector_2:
chaieb@29842
  1339
 "(vector[x,y]) $1 = x"
chaieb@29842
  1340
 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
huffman@30582
  1341
  unfolding vector_def by simp_all
chaieb@29842
  1342
chaieb@29842
  1343
lemma vector_3:
chaieb@29842
  1344
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
chaieb@29842
  1345
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
chaieb@29842
  1346
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
huffman@30582
  1347
  unfolding vector_def by simp_all
chaieb@29842
  1348
chaieb@29842
  1349
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
chaieb@29842
  1350
  apply auto
chaieb@29842
  1351
  apply (erule_tac x="v$1" in allE)
chaieb@29842
  1352
  apply (subgoal_tac "vector [v$1] = v")
chaieb@29842
  1353
  apply simp
huffman@30582
  1354
  apply (vector vector_def)
huffman@30582
  1355
  apply (simp add: forall_1)
huffman@30582
  1356
  done
chaieb@29842
  1357
chaieb@29842
  1358
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
chaieb@29842
  1359
  apply auto
chaieb@29842
  1360
  apply (erule_tac x="v$1" in allE)
chaieb@29842
  1361
  apply (erule_tac x="v$2" in allE)
chaieb@29842
  1362
  apply (subgoal_tac "vector [v$1, v$2] = v")
chaieb@29842
  1363
  apply simp
huffman@30582
  1364
  apply (vector vector_def)
huffman@30582
  1365
  apply (simp add: forall_2)
chaieb@29842
  1366
  done
chaieb@29842
  1367
chaieb@29842
  1368
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
chaieb@29842
  1369
  apply auto
chaieb@29842
  1370
  apply (erule_tac x="v$1" in allE)
chaieb@29842
  1371
  apply (erule_tac x="v$2" in allE)
chaieb@29842
  1372
  apply (erule_tac x="v$3" in allE)
chaieb@29842
  1373
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
chaieb@29842
  1374
  apply simp
huffman@30582
  1375
  apply (vector vector_def)
huffman@30582
  1376
  apply (simp add: forall_3)
chaieb@29842
  1377
  done
chaieb@29842
  1378
chaieb@29842
  1379
subsection{* Linear functions. *}
chaieb@29842
  1380
chaieb@29842
  1381
definition "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *s x) = c *s f x)"
chaieb@29842
  1382
chaieb@29842
  1383
lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. (c::'a::comm_semiring) *s f x)"
huffman@30582
  1384
  by (vector linear_def Cart_eq ring_simps)
chaieb@29842
  1385
chaieb@29842
  1386
lemma linear_compose_neg: "linear (f :: 'a ^'n \<Rightarrow> 'a::comm_ring ^'m) ==> linear (\<lambda>x. -(f(x)))" by (vector linear_def Cart_eq)
chaieb@29842
  1387
chaieb@29842
  1388
lemma linear_compose_add: "linear (f :: 'a ^'n \<Rightarrow> 'a::semiring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
chaieb@29842
  1389
  by (vector linear_def Cart_eq ring_simps)
chaieb@29842
  1390
chaieb@29842
  1391
lemma linear_compose_sub: "linear (f :: 'a ^'n \<Rightarrow> 'a::ring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
chaieb@29842
  1392
  by (vector linear_def Cart_eq ring_simps)
chaieb@29842
  1393
chaieb@29842
  1394
lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
chaieb@29842
  1395
  by (simp add: linear_def)
chaieb@29842
  1396
chaieb@29842
  1397
lemma linear_id: "linear id" by (simp add: linear_def id_def)
chaieb@29842
  1398
chaieb@29842
  1399
lemma linear_zero: "linear (\<lambda>x. 0::'a::semiring_1 ^ 'n)" by (simp add: linear_def)
chaieb@29842
  1400
chaieb@29842
  1401
lemma linear_compose_setsum:
chaieb@29842
  1402
  assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a :: 'a::semiring_1 ^ 'n \<Rightarrow> 'a ^ 'm)"
chaieb@29842
  1403
  shows "linear(\<lambda>x. setsum (\<lambda>a. f a x :: 'a::semiring_1 ^'m) S)"
chaieb@29842
  1404
  using lS
chaieb@29842
  1405
  apply (induct rule: finite_induct[OF fS])
chaieb@29842
  1406
  by (auto simp add: linear_zero intro: linear_compose_add)
chaieb@29842
  1407
chaieb@29842
  1408
lemma linear_vmul_component:
chaieb@29842
  1409
  fixes f:: "'a::semiring_1^'m \<Rightarrow> 'a^'n"
huffman@30582
  1410
  assumes lf: "linear f"
chaieb@29842
  1411
  shows "linear (\<lambda>x. f x $ k *s v)"
huffman@30582
  1412
  using lf
chaieb@29842
  1413
  apply (auto simp add: linear_def )
chaieb@29842
  1414
  by (vector ring_simps)+
chaieb@29842
  1415
chaieb@29842
  1416
lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)"
chaieb@29842
  1417
  unfolding linear_def
chaieb@29842
  1418
  apply clarsimp
chaieb@29842
  1419
  apply (erule allE[where x="0::'a"])
chaieb@29842
  1420
  apply simp
chaieb@29842
  1421
  done
chaieb@29842
  1422
chaieb@29842
  1423
lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def)
chaieb@29842
  1424
chaieb@29842
  1425
lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \<Rightarrow> _) ==> f (-x) = - f x"
chaieb@29842
  1426
  unfolding vector_sneg_minus1
huffman@30489
  1427
  using linear_cmul[of f] by auto
huffman@30489
  1428
huffman@30489
  1429
lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
chaieb@29842
  1430
chaieb@29842
  1431
lemma linear_sub: "linear (f::'a::ring_1 ^'n \<Rightarrow> _) ==> f(x - y) = f x - f y"
chaieb@29842
  1432
  by (simp add: diff_def linear_add linear_neg)
chaieb@29842
  1433
huffman@30489
  1434
lemma linear_setsum:
chaieb@29842
  1435
  fixes f:: "'a::semiring_1^'n \<Rightarrow> _"
chaieb@29842
  1436
  assumes lf: "linear f" and fS: "finite S"
chaieb@29842
  1437
  shows "f (setsum g S) = setsum (f o g) S"
chaieb@29842
  1438
proof (induct rule: finite_induct[OF fS])
chaieb@29842
  1439
  case 1 thus ?case by (simp add: linear_0[OF lf])
chaieb@29842
  1440
next
chaieb@29842
  1441
  case (2 x F)
chaieb@29842
  1442
  have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
chaieb@29842
  1443
    by simp
chaieb@29842
  1444
  also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
chaieb@29842
  1445
  also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
chaieb@29842
  1446
  finally show ?case .
chaieb@29842
  1447
qed
chaieb@29842
  1448
chaieb@29842
  1449
lemma linear_setsum_mul:
chaieb@29842
  1450
  fixes f:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m"
chaieb@29842
  1451
  assumes lf: "linear f" and fS: "finite S"
chaieb@29842
  1452
  shows "f (setsum (\<lambda>i. c i *s v i) S) = setsum (\<lambda>i. c i *s f (v i)) S"
chaieb@29842
  1453
  using linear_setsum[OF lf fS, of "\<lambda>i. c i *s v i" , unfolded o_def]
huffman@30489
  1454
  linear_cmul[OF lf] by simp
chaieb@29842
  1455
chaieb@29842
  1456
lemma linear_injective_0:
chaieb@29842
  1457
  assumes lf: "linear (f:: 'a::ring_1 ^ 'n \<Rightarrow> _)"
chaieb@29842
  1458
  shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
chaieb@29842
  1459
proof-
chaieb@29842
  1460
  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
chaieb@29842
  1461
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
huffman@30489
  1462
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
chaieb@29842
  1463
    by (simp add: linear_sub[OF lf])
chaieb@29842
  1464
  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
chaieb@29842
  1465
  finally show ?thesis .
chaieb@29842
  1466
qed
chaieb@29842
  1467
chaieb@29842
  1468
lemma linear_bounded:
huffman@30582
  1469
  fixes f:: "real ^'m::finite \<Rightarrow> real ^'n::finite"
chaieb@29842
  1470
  assumes lf: "linear f"
chaieb@29842
  1471
  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
chaieb@29842
  1472
proof-
huffman@30582
  1473
  let ?S = "UNIV:: 'm set"
chaieb@29842
  1474
  let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
chaieb@29842
  1475
  have fS: "finite ?S" by simp
chaieb@29842
  1476
  {fix x:: "real ^ 'm"
huffman@30582
  1477
    let ?g = "(\<lambda>i. (x$i) *s (basis i) :: real ^ 'm)"
chaieb@29842
  1478
    have "norm (f x) = norm (f (setsum (\<lambda>i. (x$i) *s (basis i)) ?S))"
chaieb@29842
  1479
      by (simp only:  basis_expansion)
chaieb@29842
  1480
    also have "\<dots> = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)"
chaieb@29842
  1481
      using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf]
chaieb@29842
  1482
      by auto
chaieb@29842
  1483
    finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)" .
chaieb@29842
  1484
    {fix i assume i: "i \<in> ?S"
huffman@30582
  1485
      from component_le_norm[of x i]
chaieb@29842
  1486
      have "norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x"
chaieb@29842
  1487
      unfolding norm_mul
chaieb@29842
  1488
      apply (simp only: mult_commute)
chaieb@29842
  1489
      apply (rule mult_mono)
huffman@30041
  1490
      by (auto simp add: ring_simps norm_ge_zero) }
chaieb@29842
  1491
    then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
chaieb@29842
  1492
    from real_setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (f (basis i))", OF th]
chaieb@29842
  1493
    have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
chaieb@29842
  1494
  then show ?thesis by blast
chaieb@29842
  1495
qed
chaieb@29842
  1496
chaieb@29842
  1497
lemma linear_bounded_pos:
huffman@30582
  1498
  fixes f:: "real ^'n::finite \<Rightarrow> real ^ 'm::finite"
chaieb@29842
  1499
  assumes lf: "linear f"
chaieb@29842
  1500
  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
chaieb@29842
  1501
proof-
huffman@30489
  1502
  from linear_bounded[OF lf] obtain B where
chaieb@29842
  1503
    B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
chaieb@29842
  1504
  let ?K = "\<bar>B\<bar> + 1"
chaieb@29842
  1505
  have Kp: "?K > 0" by arith
chaieb@29842
  1506
    {assume C: "B < 0"
huffman@30041
  1507
      have "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff)
chaieb@29842
  1508
      with C have "B * norm (1:: real ^ 'n) < 0"
chaieb@29842
  1509
	by (simp add: zero_compare_simps)
huffman@30041
  1510
      with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp
chaieb@29842
  1511
    }
chaieb@29842
  1512
    then have Bp: "B \<ge> 0" by ferrack
chaieb@29842
  1513
    {fix x::"real ^ 'n"
chaieb@29842
  1514
      have "norm (f x) \<le> ?K *  norm x"
huffman@30041
  1515
      using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
huffman@30040
  1516
      apply (auto simp add: ring_simps split add: abs_split)
huffman@30040
  1517
      apply (erule order_trans, simp)
huffman@30040
  1518
      done
chaieb@29842
  1519
  }
chaieb@29842
  1520
  then show ?thesis using Kp by blast
chaieb@29842
  1521
qed
chaieb@29842
  1522
chaieb@29842
  1523
subsection{* Bilinear functions. *}
chaieb@29842
  1524
chaieb@29842
  1525
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
chaieb@29842
  1526
chaieb@29842
  1527
lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
chaieb@29842
  1528
  by (simp add: bilinear_def linear_def)
chaieb@29842
  1529
lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
chaieb@29842
  1530
  by (simp add: bilinear_def linear_def)
chaieb@29842
  1531
chaieb@29842
  1532
lemma bilinear_lmul: "bilinear h ==> h (c *s x) y = c *s (h x y)"
chaieb@29842
  1533
  by (simp add: bilinear_def linear_def)
chaieb@29842
  1534
chaieb@29842
  1535
lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)"
chaieb@29842
  1536
  by (simp add: bilinear_def linear_def)
chaieb@29842
  1537
chaieb@29842
  1538
lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)"
chaieb@29842
  1539
  by (simp only: vector_sneg_minus1 bilinear_lmul)
chaieb@29842
  1540
chaieb@29842
  1541
lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y"
chaieb@29842
  1542
  by (simp only: vector_sneg_minus1 bilinear_rmul)
chaieb@29842
  1543
chaieb@29842
  1544
lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
chaieb@29842
  1545
  using add_imp_eq[of x y 0] by auto
huffman@30489
  1546
huffman@30489
  1547
lemma bilinear_lzero:
chaieb@29842
  1548
  fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h 0 x = 0"
huffman@30489
  1549
  using bilinear_ladd[OF bh, of 0 0 x]
chaieb@29842
  1550
    by (simp add: eq_add_iff ring_simps)
chaieb@29842
  1551
huffman@30489
  1552
lemma bilinear_rzero:
chaieb@29842
  1553
  fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h x 0 = 0"
huffman@30489
  1554
  using bilinear_radd[OF bh, of x 0 0 ]
chaieb@29842
  1555
    by (simp add: eq_add_iff ring_simps)
chaieb@29842
  1556
chaieb@29842
  1557
lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ 'n)) z = h x z - h y z"
chaieb@29842
  1558
  by (simp  add: diff_def bilinear_ladd bilinear_lneg)
chaieb@29842
  1559
chaieb@29842
  1560
lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ 'n)) = h z x - h z y"
chaieb@29842
  1561
  by (simp  add: diff_def bilinear_radd bilinear_rneg)
chaieb@29842
  1562
chaieb@29842
  1563
lemma bilinear_setsum:
chaieb@29842
  1564
  fixes h:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m \<Rightarrow> 'a ^ 'k"
chaieb@29842
  1565
  assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
chaieb@29842
  1566
  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
huffman@30489
  1567
proof-
chaieb@29842
  1568
  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
chaieb@29842
  1569
    apply (rule linear_setsum[unfolded o_def])
chaieb@29842
  1570
    using bh fS by (auto simp add: bilinear_def)
chaieb@29842
  1571
  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
chaieb@29842
  1572
    apply (rule setsum_cong, simp)
chaieb@29842
  1573
    apply (rule linear_setsum[unfolded o_def])
chaieb@29842
  1574
    using bh fT by (auto simp add: bilinear_def)
chaieb@29842
  1575
  finally show ?thesis unfolding setsum_cartesian_product .
chaieb@29842
  1576
qed
chaieb@29842
  1577
chaieb@29842
  1578
lemma bilinear_bounded:
huffman@30582
  1579
  fixes h:: "real ^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real ^ 'k::finite"
chaieb@29842
  1580
  assumes bh: "bilinear h"
chaieb@29842
  1581
  shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
huffman@30489
  1582
proof-
huffman@30582
  1583
  let ?M = "UNIV :: 'm set"
huffman@30582
  1584
  let ?N = "UNIV :: 'n set"
chaieb@29842
  1585
  let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
chaieb@29842
  1586
  have fM: "finite ?M" and fN: "finite ?N" by simp_all
chaieb@29842
  1587
  {fix x:: "real ^ 'm" and  y :: "real^'n"
chaieb@29842
  1588
    have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$i) *s basis i) ?M) (setsum (\<lambda>i. (y$i) *s basis i) ?N))" unfolding basis_expansion ..
chaieb@29842
  1589
    also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$i) *s basis i) ((y$j) *s basis j)) (?M \<times> ?N))"  unfolding bilinear_setsum[OF bh fM fN] ..
chaieb@29842
  1590
    finally have th: "norm (h x y) = \<dots>" .
chaieb@29842
  1591
    have "norm (h x y) \<le> ?B * norm x * norm y"
chaieb@29842
  1592
      apply (simp add: setsum_left_distrib th)
chaieb@29842
  1593
      apply (rule real_setsum_norm_le)
chaieb@29842
  1594
      using fN fM
chaieb@29842
  1595
      apply simp
chaieb@29842
  1596
      apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)
chaieb@29842
  1597
      apply (rule mult_mono)
huffman@30041
  1598
      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
chaieb@29842
  1599
      apply (rule mult_mono)
huffman@30041
  1600
      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
chaieb@29842
  1601
      done}
chaieb@29842
  1602
  then show ?thesis by metis
chaieb@29842
  1603
qed
chaieb@29842
  1604
chaieb@29842
  1605
lemma bilinear_bounded_pos:
huffman@30582
  1606
  fixes h:: "real ^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real ^ 'k::finite"
chaieb@29842
  1607
  assumes bh: "bilinear h"
chaieb@29842
  1608
  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
chaieb@29842
  1609
proof-
huffman@30489
  1610
  from bilinear_bounded[OF bh] obtain B where
chaieb@29842
  1611
    B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
chaieb@29842
  1612
  let ?K = "\<bar>B\<bar> + 1"
chaieb@29842
  1613
  have Kp: "?K > 0" by arith
chaieb@29842
  1614
  have KB: "B < ?K" by arith
chaieb@29842
  1615
  {fix x::"real ^'m" and y :: "real ^'n"
chaieb@29842
  1616
    from KB Kp
chaieb@29842
  1617
    have "B * norm x * norm y \<le> ?K * norm x * norm y"
huffman@30489
  1618
      apply -
chaieb@29842
  1619
      apply (rule mult_right_mono, rule mult_right_mono)
huffman@30041
  1620
      by (auto simp add: norm_ge_zero)
chaieb@29842
  1621
    then have "norm (h x y) \<le> ?K * norm x * norm y"
huffman@30489
  1622
      using B[rule_format, of x y] by simp}
chaieb@29842
  1623
  with Kp show ?thesis by blast
chaieb@29842
  1624
qed
chaieb@29842
  1625
chaieb@29842
  1626
subsection{* Adjoints. *}
chaieb@29842
  1627
chaieb@29842
  1628
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
chaieb@29842
  1629
chaieb@29842
  1630
lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
chaieb@29842
  1631
chaieb@29842
  1632
lemma adjoint_works_lemma:
huffman@30582
  1633
  fixes f:: "'a::ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
chaieb@29842
  1634
  assumes lf: "linear f"
chaieb@29842
  1635
  shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
chaieb@29842
  1636
proof-
huffman@30582
  1637
  let ?N = "UNIV :: 'n set"
huffman@30582
  1638
  let ?M = "UNIV :: 'm set"
chaieb@29842
  1639
  have fN: "finite ?N" by simp
chaieb@29842
  1640
  have fM: "finite ?M" by simp
chaieb@29842
  1641
  {fix y:: "'a ^ 'm"
chaieb@29842
  1642
    let ?w = "(\<chi> i. (f (basis i) \<bullet> y)) :: 'a ^ 'n"
chaieb@29842
  1643
    {fix x
chaieb@29842
  1644
      have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *s basis i) ?N) \<bullet> y"
chaieb@29842
  1645
	by (simp only: basis_expansion)
chaieb@29842
  1646
      also have "\<dots> = (setsum (\<lambda>i. (x$i) *s f (basis i)) ?N) \<bullet> y"
huffman@30489
  1647
	unfolding linear_setsum[OF lf fN]
chaieb@29842
  1648
	by (simp add: linear_cmul[OF lf])
chaieb@29842
  1649
      finally have "f x \<bullet> y = x \<bullet> ?w"
chaieb@29842
  1650
	apply (simp only: )
huffman@30582
  1651
	apply (simp add: dot_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] ring_simps)
chaieb@29842
  1652
	done}
chaieb@29842
  1653
  }
huffman@30489
  1654
  then show ?thesis unfolding adjoint_def
chaieb@29842
  1655
    some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
chaieb@29842
  1656
    using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
chaieb@29842
  1657
    by metis
chaieb@29842
  1658
qed
chaieb@29842
  1659
chaieb@29842
  1660
lemma adjoint_works:
huffman@30582
  1661
  fixes f:: "'a::ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
chaieb@29842
  1662
  assumes lf: "linear f"
chaieb@29842
  1663
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
chaieb@29842
  1664
  using adjoint_works_lemma[OF lf] by metis
chaieb@29842
  1665
chaieb@29842
  1666
chaieb@29842
  1667
lemma adjoint_linear:
huffman@30582
  1668
  fixes f :: "'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
chaieb@29842
  1669
  assumes lf: "linear f"
chaieb@29842
  1670
  shows "linear (adjoint f)"
chaieb@29842
  1671
  by (simp add: linear_def vector_eq_ldot[symmetric] dot_radd dot_rmult adjoint_works[OF lf])
chaieb@29842
  1672
chaieb@29842
  1673
lemma adjoint_clauses:
huffman@30582
  1674
  fixes f:: "'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
chaieb@29842
  1675
  assumes lf: "linear f"
chaieb@29842
  1676
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
chaieb@29842
  1677
  and "adjoint f y \<bullet> x = y \<bullet> f x"
chaieb@29842
  1678
  by (simp_all add: adjoint_works[OF lf] dot_sym )
chaieb@29842
  1679
chaieb@29842
  1680
lemma adjoint_adjoint:
huffman@30582
  1681
  fixes f:: "'a::comm_ring_1 ^ 'n::finite \<Rightarrow> 'a ^ 'm::finite"
chaieb@29842
  1682
  assumes lf: "linear f"
chaieb@29842
  1683
  shows "adjoint (adjoint f) = f"
chaieb@29842
  1684
  apply (rule ext)
chaieb@29842
  1685
  by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf])
chaieb@29842
  1686
chaieb@29842
  1687
lemma adjoint_unique:
huffman@30582
  1688
  fixes f:: "'a::comm_ring_1 ^ 'n::finite \<Rightarrow> 'a ^ 'm::finite"
chaieb@29842
  1689
  assumes lf: "linear f" and u: "\<forall>x y. f' x \<bullet> y = x \<bullet> f y"
chaieb@29842
  1690
  shows "f' = adjoint f"
chaieb@29842
  1691
  apply (rule ext)
chaieb@29842
  1692
  using u
chaieb@29842
  1693
  by (simp add: vector_eq_rdot[symmetric] adjoint_clauses[OF lf])
chaieb@29842
  1694
huffman@29881
  1695
text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
chaieb@29842
  1696
chaieb@29842
  1697
consts generic_mult :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" (infixr "\<star>" 75)
chaieb@29842
  1698
huffman@30489
  1699
defs (overloaded)
huffman@30582
  1700
matrix_matrix_mult_def: "(m:: ('a::semiring_1) ^'n^'m) \<star> (m' :: 'a ^'p^'n) \<equiv> (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
chaieb@29842
  1701
huffman@30489
  1702
abbreviation
chaieb@29842
  1703
  matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)
chaieb@29842
  1704
  where "m ** m' == m\<star> m'"
chaieb@29842
  1705
huffman@30489
  1706
defs (overloaded)
huffman@30582
  1707
  matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
chaieb@29842
  1708
huffman@30489
  1709
abbreviation
chaieb@29842
  1710
  matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)
huffman@30489
  1711
  where
chaieb@29842
  1712
  "m *v v == m \<star> v"
chaieb@29842
  1713
huffman@30489
  1714
defs (overloaded)
huffman@30582
  1715
  vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (x$i)) (UNIV :: 'm set)) :: 'a^'n"
chaieb@29842
  1716
huffman@30489
  1717
abbreviation
chaieb@29842
  1718
  vactor_matrix_mult' :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)
huffman@30489
  1719
  where
chaieb@29842
  1720
  "v v* m == v \<star> m"
chaieb@29842
  1721
huffman@30582
  1722
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
chaieb@29842
  1723
definition "(transp::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
huffman@30582
  1724
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
huffman@30582
  1725
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
huffman@30582
  1726
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
huffman@30582
  1727
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
chaieb@29842
  1728
chaieb@29842
  1729
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
chaieb@29842
  1730
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \<star> B) + (A \<star> C)"
chaieb@29842
  1731
  by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps)
chaieb@29842
  1732
huffman@30489
  1733
lemma setsum_delta':
huffman@30489
  1734
  assumes fS: "finite S" shows
huffman@30489
  1735
  "setsum (\<lambda>k. if a = k then b k else 0) S =
chaieb@29842
  1736
     (if a\<in> S then b a else 0)"
huffman@30489
  1737
  using setsum_delta[OF fS, of a b, symmetric]
chaieb@29842
  1738
  by (auto intro: setsum_cong)
chaieb@29842
  1739
huffman@30582
  1740
lemma matrix_mul_lid:
huffman@30582
  1741
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n::finite"
huffman@30582
  1742
  shows "mat 1 ** A = A"
chaieb@29842
  1743
  apply (simp add: matrix_matrix_mult_def mat_def)
chaieb@29842
  1744
  apply vector
huffman@30582
  1745
  by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite]  mult_1_left mult_zero_left if_True UNIV_I)
huffman@30582
  1746
huffman@30582
  1747
huffman@30582
  1748
lemma matrix_mul_rid:
huffman@30582
  1749
  fixes A :: "'a::semiring_1 ^ 'm::finite ^ 'n"
huffman@30582
  1750
  shows "A ** mat 1 = A"
chaieb@29842
  1751
  apply (simp add: matrix_matrix_mult_def mat_def)
chaieb@29842
  1752
  apply vector
huffman@30582
  1753
  by (auto simp only: cond_value_iff cond_application_beta setsum_delta[OF finite]  mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
chaieb@29842
  1754
chaieb@29842
  1755
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
chaieb@29842
  1756
  apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
chaieb@29842
  1757
  apply (subst setsum_commute)
chaieb@29842
  1758
  apply simp
chaieb@29842
  1759
  done
chaieb@29842
  1760
chaieb@29842
  1761
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
chaieb@29842
  1762
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
chaieb@29842
  1763
  apply (subst setsum_commute)
chaieb@29842
  1764
  apply simp
chaieb@29842
  1765
  done
chaieb@29842
  1766
huffman@30582
  1767
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n::finite)"
chaieb@29842
  1768
  apply (vector matrix_vector_mult_def mat_def)
huffman@30489
  1769
  by (simp add: cond_value_iff cond_application_beta
chaieb@29842
  1770
    setsum_delta' cong del: if_weak_cong)
chaieb@29842
  1771
chaieb@29842
  1772
lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)"
huffman@30582
  1773
  by (simp add: matrix_matrix_mult_def transp_def Cart_eq mult_commute)
huffman@30582
  1774
huffman@30582
  1775
lemma matrix_eq:
huffman@30582
  1776
  fixes A B :: "'a::semiring_1 ^ 'n::finite ^ 'm"
huffman@30582
  1777
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@29842
  1778
  apply auto
chaieb@29842
  1779
  apply (subst Cart_eq)
chaieb@29842
  1780
  apply clarify
huffman@30582
  1781
  apply (clarsimp simp add: matrix_vector_mult_def basis_def cond_value_iff cond_application_beta Cart_eq cong del: if_weak_cong)
chaieb@29842
  1782
  apply (erule_tac x="basis ia" in allE)
huffman@30582
  1783
  apply (erule_tac x="i" in allE)
huffman@30582
  1784
  by (auto simp add: basis_def cond_value_iff cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong)
chaieb@29842
  1785
huffman@30489
  1786
lemma matrix_vector_mul_component:
chaieb@29842
  1787
  shows "((A::'a::semiring_1^'n'^'m) *v x)$k = (A$k) \<bullet> x"
huffman@30582
  1788
  by (simp add: matrix_vector_mult_def dot_def)
chaieb@29842
  1789
chaieb@29842
  1790
lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) v* A) \<bullet> y = x \<bullet> (A *v y)"
huffman@30582
  1791
  apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
chaieb@29842
  1792
  apply (subst setsum_commute)
chaieb@29842
  1793
  by simp
chaieb@29842
  1794
chaieb@29842
  1795
lemma transp_mat: "transp (mat n) = mat n"
chaieb@29842
  1796
  by (vector transp_def mat_def)
chaieb@29842
  1797
chaieb@29842
  1798
lemma transp_transp: "transp(transp A) = A"
chaieb@29842
  1799
  by (vector transp_def)
chaieb@29842
  1800
huffman@30489
  1801
lemma row_transp:
chaieb@29842
  1802
  fixes A:: "'a::semiring_1^'n^'m"
chaieb@29842
  1803
  shows "row i (transp A) = column i A"
huffman@30582
  1804
  by (simp add: row_def column_def transp_def Cart_eq)
chaieb@29842
  1805
chaieb@29842
  1806
lemma column_transp:
chaieb@29842
  1807
  fixes A:: "'a::semiring_1^'n^'m"
chaieb@29842
  1808
  shows "column i (transp A) = row i A"
huffman@30582
  1809
  by (simp add: row_def column_def transp_def Cart_eq)
chaieb@29842
  1810
chaieb@29842
  1811
lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A"
huffman@30582
  1812
by (auto simp add: rows_def columns_def row_transp intro: set_ext)
chaieb@29842
  1813
chaieb@29842
  1814
lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp)
chaieb@29842
  1815
chaieb@29842
  1816
text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
chaieb@29842
  1817
chaieb@29842
  1818
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
chaieb@29842
  1819
  by (simp add: matrix_vector_mult_def dot_def)
chaieb@29842
  1820
huffman@30582
  1821
lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
huffman@30582
  1822
  by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute)
chaieb@29842
  1823
chaieb@29842
  1824
lemma vector_componentwise:
huffman@30582
  1825
  "(x::'a::ring_1^'n::finite) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) (UNIV :: 'n set))"
chaieb@29842
  1826
  apply (subst basis_expansion[symmetric])
huffman@30582
  1827
  by (vector Cart_eq setsum_component)
chaieb@29842
  1828
chaieb@29842
  1829
lemma linear_componentwise:
huffman@30582
  1830
  fixes f:: "'a::ring_1 ^ 'm::finite \<Rightarrow> 'a ^ 'n"
huffman@30582
  1831
  assumes lf: "linear f"
huffman@30582
  1832
  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
chaieb@29842
  1833
proof-
huffman@30582
  1834
  let ?M = "(UNIV :: 'm set)"
huffman@30582
  1835
  let ?N = "(UNIV :: 'n set)"
chaieb@29842
  1836
  have fM: "finite ?M" by simp
chaieb@29842
  1837
  have "?rhs = (setsum (\<lambda>i.(x$i) *s f (basis i) ) ?M)$j"
huffman@30582
  1838
    unfolding vector_smult_component[symmetric]
huffman@30582
  1839
    unfolding setsum_component[of "(\<lambda>i.(x$i) *s f (basis i :: 'a^'m))" ?M]
chaieb@29842
  1840
    ..
chaieb@29842
  1841
  then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion ..
chaieb@29842
  1842
qed
chaieb@29842
  1843
chaieb@29842
  1844
text{* Inverse matrices  (not necessarily square) *}
chaieb@29842
  1845
chaieb@29842
  1846
definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
chaieb@29842
  1847
chaieb@29842
  1848
definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
chaieb@29842
  1849
        (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
chaieb@29842
  1850
chaieb@29842
  1851
text{* Correspondence between matrices and linear operators. *}
chaieb@29842
  1852
chaieb@29842
  1853
definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
chaieb@29842
  1854
where "matrix f = (\<chi> i j. (f(basis j))$i)"
chaieb@29842
  1855
chaieb@29842
  1856
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::'a::comm_semiring_1 ^ 'n))"
huffman@30582
  1857
  by (simp add: linear_def matrix_vector_mult_def Cart_eq ring_simps setsum_right_distrib setsum_addf)
huffman@30582
  1858
huffman@30582
  1859
lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n::finite)"
huffman@30582
  1860
apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute)
chaieb@29842
  1861
apply clarify
chaieb@29842
  1862
apply (rule linear_componentwise[OF lf, symmetric])
chaieb@29842
  1863
done
chaieb@29842
  1864
huffman@30582
  1865
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::'a::comm_ring_1 ^ 'n::finite))" by (simp add: ext matrix_works)
huffman@30582
  1866
huffman@30582
  1867
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n::finite)) = A"
chaieb@29842
  1868
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
chaieb@29842
  1869
huffman@30489
  1870
lemma matrix_compose:
huffman@30582
  1871
  assumes lf: "linear (f::'a::comm_ring_1^'n::finite \<Rightarrow> 'a^'m::finite)"
huffman@30582
  1872
  and lg: "linear (g::'a::comm_ring_1^'m::finite \<Rightarrow> 'a^'k)"
chaieb@29842
  1873
  shows "matrix (g o f) = matrix g ** matrix f"
chaieb@29842
  1874
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
chaieb@29842
  1875
  by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
chaieb@29842
  1876
huffman@30582
  1877
lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s ((transp A)$i)) (UNIV:: 'n set)"
huffman@30582
  1878
  by (simp add: matrix_vector_mult_def transp_def Cart_eq mult_commute)
huffman@30582
  1879
huffman@30582
  1880
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::'a::comm_ring_1^'n::finite^'m::finite) *v x) = (\<lambda>x. transp A *v x)"
chaieb@29842
  1881
  apply (rule adjoint_unique[symmetric])
chaieb@29842
  1882
  apply (rule matrix_vector_mul_linear)
huffman@30582
  1883
  apply (simp add: transp_def dot_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
chaieb@29842
  1884
  apply (subst setsum_commute)
chaieb@29842
  1885
  apply (auto simp add: mult_ac)
chaieb@29842
  1886
  done
chaieb@29842
  1887
huffman@30582
  1888
lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n::finite \<Rightarrow> 'a ^ 'm::finite)"
chaieb@29842
  1889
  shows "matrix(adjoint f) = transp(matrix f)"
chaieb@29842
  1890
  apply (subst matrix_vector_mul[OF lf])
chaieb@29842
  1891
  unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
chaieb@29842
  1892
chaieb@29842
  1893
subsection{* Interlude: Some properties of real sets *}
chaieb@29842
  1894
chaieb@29842
  1895
lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
chaieb@29842
  1896
  shows "\<forall>n \<ge> m. d n < e m"
chaieb@29842
  1897
  using prems apply auto
chaieb@29842
  1898
  apply (erule_tac x="n" in allE)
chaieb@29842
  1899
  apply (erule_tac x="n" in allE)
chaieb@29842
  1900
  apply auto
chaieb@29842
  1901
  done
chaieb@29842
  1902
chaieb@29842
  1903
huffman@30489
  1904
lemma real_convex_bound_lt:
chaieb@29842
  1905
  assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"
huffman@30489
  1906
  and uv: "u + v = 1"
chaieb@29842
  1907
  shows "u * x + v * y < a"
chaieb@29842
  1908
proof-
chaieb@29842
  1909
  have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
chaieb@29842
  1910
  have "a = a * (u + v)" unfolding uv  by simp
chaieb@29842
  1911
  hence th: "u * a + v * a = a" by (simp add: ring_simps)
chaieb@29842
  1912
  from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_compare_simps)
chaieb@29842
  1913
  from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_compare_simps)
chaieb@29842
  1914
  from xa ya u v have "u * x + v * y < u * a + v * a"
chaieb@29842
  1915
    apply (cases "u = 0", simp_all add: uv')
chaieb@29842
  1916
    apply(rule mult_strict_left_mono)
chaieb@29842
  1917
    using uv' apply simp_all
huffman@30489
  1918
chaieb@29842
  1919
    apply (rule add_less_le_mono)
chaieb@29842
  1920
    apply(rule mult_strict_left_mono)
chaieb@29842
  1921
    apply simp_all
chaieb@29842
  1922
    apply (rule mult_left_mono)
chaieb@29842
  1923
    apply simp_all
chaieb@29842
  1924
    done
chaieb@29842
  1925
  thus ?thesis unfolding th .
chaieb@29842
  1926
qed
chaieb@29842
  1927
huffman@30489
  1928
lemma real_convex_bound_le:
chaieb@29842
  1929
  assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"
huffman@30489
  1930
  and uv: "u + v = 1"
chaieb@29842
  1931
  shows "u * x + v * y \<le> a"
chaieb@29842
  1932
proof-
chaieb@29842
  1933
  from xa ya u v have "u * x + v * y \<le> u * a + v * a" by (simp add: add_mono mult_left_mono)
chaieb@29842
  1934
  also have "\<dots> \<le> (u + v) * a" by (simp add: ring_simps)
chaieb@29842
  1935
  finally show ?thesis unfolding uv by simp
chaieb@29842
  1936
qed
chaieb@29842
  1937
chaieb@29842
  1938
lemma infinite_enumerate: assumes fS: "infinite S"
chaieb@29842
  1939
  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
chaieb@29842
  1940
unfolding subseq_def
chaieb@29842
  1941
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
chaieb@29842
  1942
chaieb@29842
  1943
lemma approachable_lt_le: "(\<exists>(d::real)>0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
chaieb@29842
  1944
apply auto
chaieb@29842
  1945
apply (rule_tac x="d/2" in exI)
chaieb@29842
  1946
apply auto
chaieb@29842
  1947
done
chaieb@29842
  1948
chaieb@29842
  1949
huffman@30489
  1950
lemma triangle_lemma:
chaieb@29842
  1951
  assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
chaieb@29842
  1952
  shows "x <= y + z"
chaieb@29842
  1953
proof-
chaieb@29842
  1954
  have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y  by (simp add: zero_compare_simps)
chaieb@29842
  1955
  with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square ring_simps)
chaieb@29842
  1956
  from y z have yz: "y + z \<ge> 0" by arith
chaieb@29842
  1957
  from power2_le_imp_le[OF th yz] show ?thesis .
chaieb@29842
  1958
qed
chaieb@29842
  1959
chaieb@29842
  1960
huffman@30582
  1961
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
huffman@30582
  1962
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@29842
  1963
proof-
huffman@30582
  1964
  let ?S = "(UNIV :: 'n set)"
chaieb@29842
  1965
  {assume H: "?rhs"
chaieb@29842
  1966
    then have ?lhs by auto}
chaieb@29842
  1967
  moreover
chaieb@29842
  1968
  {assume H: "?lhs"
huffman@30582
  1969
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
chaieb@29842
  1970
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
huffman@30582
  1971
    {fix i
huffman@30582
  1972
      from f have "P i (f i)" by metis
huffman@30582
  1973
      then have "P i (?x$i)" by auto
chaieb@29842
  1974
    }
huffman@30582
  1975
    hence "\<forall>i. P i (?x$i)" by metis
chaieb@29842
  1976
    hence ?rhs by metis }
chaieb@29842
  1977
  ultimately show ?thesis by metis
huffman@30489
  1978
qed
chaieb@29842
  1979
chaieb@29842
  1980
(* Supremum and infimum of real sets *)
chaieb@29842
  1981
chaieb@29842
  1982
chaieb@29842
  1983
definition rsup:: "real set \<Rightarrow> real" where
chaieb@29842
  1984
  "rsup S = (SOME a. isLub UNIV S a)"
chaieb@29842
  1985
chaieb@29842
  1986
lemma rsup_alt: "rsup S = (SOME a. (\<forall>x \<in> S. x \<le> a) \<and> (\<forall>b. (\<forall>x \<in> S. x \<le> b) \<longrightarrow> a \<le> b))"  by (auto simp  add: isLub_def rsup_def leastP_def isUb_def setle_def setge_def)
chaieb@29842
  1987
chaieb@29842
  1988
lemma rsup: assumes Se: "S \<noteq> {}" and b: "\<exists>b. S *<= b"
chaieb@29842
  1989
  shows "isLub UNIV S (rsup S)"
chaieb@29842
  1990
using Se b
chaieb@29842
  1991
unfolding rsup_def
chaieb@29842
  1992
apply clarify
chaieb@29842
  1993
apply (rule someI_ex)
chaieb@29842
  1994
apply (rule reals_complete)
chaieb@29842
  1995
by (auto simp add: isUb_def setle_def)
chaieb@29842
  1996
chaieb@29842
  1997
lemma rsup_le: assumes Se: "S \<noteq> {}" and Sb: "S *<= b" shows "rsup S \<le> b"
chaieb@29842
  1998
proof-
chaieb@29842
  1999
  from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def)
huffman@30489
  2000
  from rsup[OF Se] Sb have "isLub UNIV S (rsup S)"  by blast
chaieb@29842
  2001
  then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def)
chaieb@29842
  2002
qed
chaieb@29842
  2003
chaieb@29842
  2004
lemma rsup_finite_Max: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2005
  shows "rsup S = Max S"
chaieb@29842
  2006
using fS Se
chaieb@29842
  2007
proof-
chaieb@29842
  2008
  let ?m = "Max S"
chaieb@29842
  2009
  from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis
chaieb@29842
  2010
  with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def)
huffman@30489
  2011
  from Max_in[OF fS Se] lub have mrS: "?m \<le> rsup S"
chaieb@29842
  2012
    by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)
huffman@30489
  2013
  moreover
chaieb@29842
  2014
  have "rsup S \<le> ?m" using Sm lub
chaieb@29842
  2015
    by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
huffman@30489
  2016
  ultimately  show ?thesis by arith
chaieb@29842
  2017
qed
chaieb@29842
  2018
chaieb@29842
  2019
lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2020
  shows "rsup S \<in> S"
chaieb@29842
  2021
  using rsup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis
chaieb@29842
  2022
chaieb@29842
  2023
lemma rsup_finite_Ub: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2024
  shows "isUb S S (rsup S)"
chaieb@29842
  2025
  using rsup_finite_Max[OF fS Se] rsup_finite_in[OF fS Se] Max_ge[OF fS]
chaieb@29842
  2026
  unfolding isUb_def setle_def by metis
chaieb@29842
  2027
chaieb@29842
  2028
lemma rsup_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2029
  shows "a \<le> rsup S \<longleftrightarrow> (\<exists> x \<in> S. a \<le> x)"
chaieb@29842
  2030
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
chaieb@29842
  2031
chaieb@29842
  2032
lemma rsup_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2033
  shows "a \<ge> rsup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)"
chaieb@29842
  2034
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
chaieb@29842
  2035
chaieb@29842
  2036
lemma rsup_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2037
  shows "a < rsup S \<longleftrightarrow> (\<exists> x \<in> S. a < x)"
chaieb@29842
  2038
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
chaieb@29842
  2039
chaieb@29842
  2040
lemma rsup_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2041
  shows "a > rsup S \<longleftrightarrow> (\<forall> x \<in> S. a > x)"
chaieb@29842
  2042
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
chaieb@29842
  2043
chaieb@29842
  2044
lemma rsup_unique: assumes b: "S *<= b" and S: "\<forall>b' < b. \<exists>x \<in> S. b' < x"
chaieb@29842
  2045
  shows "rsup S = b"
huffman@30489
  2046
using b S
chaieb@29842
  2047
unfolding setle_def rsup_alt
chaieb@29842
  2048
apply -
chaieb@29842
  2049
apply (rule some_equality)
chaieb@29842
  2050
apply (metis  linorder_not_le order_eq_iff[symmetric])+
chaieb@29842
  2051
done
chaieb@29842
  2052
chaieb@29842
  2053
lemma rsup_le_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. T *<= b) \<Longrightarrow> rsup S \<le> rsup T"
chaieb@29842
  2054
  apply (rule rsup_le)
chaieb@29842
  2055
  apply simp
chaieb@29842
  2056
  using rsup[of T] by (auto simp add: isLub_def leastP_def setge_def setle_def isUb_def)
chaieb@29842
  2057
chaieb@29842
  2058
lemma isUb_def': "isUb R S = (\<lambda>x. S *<= x \<and> x \<in> R)"
chaieb@29842
  2059
  apply (rule ext)
chaieb@29842
  2060
  by (metis isUb_def)
chaieb@29842
  2061
chaieb@29842
  2062
lemma UNIV_trivial: "UNIV x" using UNIV_I[of x] by (metis mem_def)
chaieb@29842
  2063
lemma rsup_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
chaieb@29842
  2064
  shows "a \<le> rsup S \<and> rsup S \<le> b"
chaieb@29842
  2065
proof-
chaieb@29842
  2066
  from rsup[OF Se] u have lub: "isLub UNIV S (rsup S)" by blast
chaieb@29842
  2067
  hence b: "rsup S \<le> b" using u by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
chaieb@29842
  2068
  from Se obtain y where y: "y \<in> S" by blast
chaieb@29842
  2069
  from lub l have "a \<le> rsup S" apply (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
chaieb@29842
  2070
    apply (erule ballE[where x=y])
chaieb@29842
  2071
    apply (erule ballE[where x=y])
chaieb@29842
  2072
    apply arith
chaieb@29842
  2073
    using y apply auto
chaieb@29842
  2074
    done
chaieb@29842
  2075
  with b show ?thesis by blast
chaieb@29842
  2076
qed
chaieb@29842
  2077
chaieb@29842
  2078
lemma rsup_abs_le: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rsup S\<bar> \<le> a"
chaieb@29842
  2079
  unfolding abs_le_interval_iff  using rsup_bounds[of S "-a" a]
chaieb@29842
  2080
  by (auto simp add: setge_def setle_def)
chaieb@29842
  2081
chaieb@29842
  2082
lemma rsup_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rsup S - l\<bar> \<le> e"
chaieb@29842
  2083
proof-
chaieb@29842
  2084
  have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
huffman@30489
  2085
  show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th
chaieb@29842
  2086
    by  (auto simp add: setge_def setle_def)
chaieb@29842
  2087
qed
chaieb@29842
  2088
chaieb@29842
  2089
definition rinf:: "real set \<Rightarrow> real" where
chaieb@29842
  2090
  "rinf S = (SOME a. isGlb UNIV S a)"
chaieb@29842
  2091
chaieb@29842
  2092
lemma rinf_alt: "rinf S = (SOME a. (\<forall>x \<in> S. x \<ge> a) \<and> (\<forall>b. (\<forall>x \<in> S. x \<ge> b) \<longrightarrow> a \<ge> b))"  by (auto simp  add: isGlb_def rinf_def greatestP_def isLb_def setle_def setge_def)
chaieb@29842
  2093
chaieb@29842
  2094
lemma reals_complete_Glb: assumes Se: "\<exists>x. x \<in> S" and lb: "\<exists> y. isLb UNIV S y"
chaieb@29842
  2095
  shows "\<exists>(t::real). isGlb UNIV S t"
chaieb@29842
  2096
proof-
chaieb@29842
  2097
  let ?M = "uminus ` S"
chaieb@29842
  2098
  from lb have th: "\<exists>y. isUb UNIV ?M y" apply (auto simp add: isUb_def isLb_def setle_def setge_def)
chaieb@29842
  2099
    by (rule_tac x="-y" in exI, auto)
chaieb@29842
  2100
  from Se have Me: "\<exists>x. x \<in> ?M" by blast
chaieb@29842
  2101
  from reals_complete[OF Me th] obtain t where t: "isLub UNIV ?M t" by blast
chaieb@29842
  2102
  have "isGlb UNIV S (- t)" using t
chaieb@29842
  2103
    apply (auto simp add: isLub_def isGlb_def leastP_def greatestP_def setle_def setge_def isUb_def isLb_def)
chaieb@29842
  2104
    apply (erule_tac x="-y" in allE)
chaieb@29842
  2105
    apply auto
chaieb@29842
  2106
    done
chaieb@29842
  2107
  then show ?thesis by metis
chaieb@29842
  2108
qed
chaieb@29842
  2109
chaieb@29842
  2110
lemma rinf: assumes Se: "S \<noteq> {}" and b: "\<exists>b. b <=* S"
chaieb@29842
  2111
  shows "isGlb UNIV S (rinf S)"
chaieb@29842
  2112
using Se b
chaieb@29842
  2113
unfolding rinf_def
chaieb@29842
  2114
apply clarify
chaieb@29842
  2115
apply (rule someI_ex)
chaieb@29842
  2116
apply (rule reals_complete_Glb)
chaieb@29842
  2117
apply (auto simp add: isLb_def setle_def setge_def)
chaieb@29842
  2118
done
chaieb@29842
  2119
chaieb@29842
  2120
lemma rinf_ge: assumes Se: "S \<noteq> {}" and Sb: "b <=* S" shows "rinf S \<ge> b"
chaieb@29842
  2121
proof-
chaieb@29842
  2122
  from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def)
huffman@30489
  2123
  from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)"  by blast
chaieb@29842
  2124
  then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def)
chaieb@29842
  2125
qed
chaieb@29842
  2126
chaieb@29842
  2127
lemma rinf_finite_Min: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2128
  shows "rinf S = Min S"
chaieb@29842
  2129
using fS Se
chaieb@29842
  2130
proof-
chaieb@29842
  2131
  let ?m = "Min S"
chaieb@29842
  2132
  from Min_le[OF fS] have Sm: "\<forall> x\<in> S. x \<ge> ?m" by metis
chaieb@29842
  2133
  with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def)
huffman@30489
  2134
  from Min_in[OF fS Se] glb have mrS: "?m \<ge> rinf S"
chaieb@29842
  2135
    by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def)
huffman@30489
  2136
  moreover
chaieb@29842
  2137
  have "rinf S \<ge> ?m" using Sm glb
chaieb@29842
  2138
    by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def)
huffman@30489
  2139
  ultimately  show ?thesis by arith
chaieb@29842
  2140
qed
chaieb@29842
  2141
chaieb@29842
  2142
lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2143
  shows "rinf S \<in> S"
chaieb@29842
  2144
  using rinf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis
chaieb@29842
  2145
chaieb@29842
  2146
lemma rinf_finite_Lb: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2147
  shows "isLb S S (rinf S)"
chaieb@29842
  2148
  using rinf_finite_Min[OF fS Se] rinf_finite_in[OF fS Se] Min_le[OF fS]
chaieb@29842
  2149
  unfolding isLb_def setge_def by metis
chaieb@29842
  2150
chaieb@29842
  2151
lemma rinf_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2152
  shows "a \<le> rinf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)"
chaieb@29842
  2153
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
chaieb@29842
  2154
chaieb@29842
  2155
lemma rinf_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2156
  shows "a \<ge> rinf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)"
chaieb@29842
  2157
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
chaieb@29842
  2158
chaieb@29842
  2159
lemma rinf_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2160
  shows "a < rinf S \<longleftrightarrow> (\<forall> x \<in> S. a < x)"
chaieb@29842
  2161
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
chaieb@29842
  2162
chaieb@29842
  2163
lemma rinf_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2164
  shows "a > rinf S \<longleftrightarrow> (\<exists> x \<in> S. a > x)"
chaieb@29842
  2165
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
chaieb@29842
  2166
chaieb@29842
  2167
lemma rinf_unique: assumes b: "b <=* S" and S: "\<forall>b' > b. \<exists>x \<in> S. b' > x"
chaieb@29842
  2168
  shows "rinf S = b"
huffman@30489
  2169
using b S
chaieb@29842
  2170
unfolding setge_def rinf_alt
chaieb@29842
  2171
apply -
chaieb@29842
  2172
apply (rule some_equality)
chaieb@29842
  2173
apply (metis  linorder_not_le order_eq_iff[symmetric])+
chaieb@29842
  2174
done
chaieb@29842
  2175
chaieb@29842
  2176
lemma rinf_ge_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. b <=* T) \<Longrightarrow> rinf S >= rinf T"
chaieb@29842
  2177
  apply (rule rinf_ge)
chaieb@29842
  2178
  apply simp
chaieb@29842
  2179
  using rinf[of T] by (auto simp add: isGlb_def greatestP_def setge_def setle_def isLb_def)
chaieb@29842
  2180
chaieb@29842
  2181
lemma isLb_def': "isLb R S = (\<lambda>x. x <=* S \<and> x \<in> R)"
chaieb@29842
  2182
  apply (rule ext)
chaieb@29842
  2183
  by (metis isLb_def)
chaieb@29842
  2184
chaieb@29842
  2185
lemma rinf_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
chaieb@29842
  2186
  shows "a \<le> rinf S \<and> rinf S \<le> b"
chaieb@29842
  2187
proof-
chaieb@29842
  2188
  from rinf[OF Se] l have lub: "isGlb UNIV S (rinf S)" by blast
chaieb@29842
  2189
  hence b: "a \<le> rinf S" using l by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
chaieb@29842
  2190
  from Se obtain y where y: "y \<in> S" by blast
chaieb@29842
  2191
  from lub u have "b \<ge> rinf S" apply (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
chaieb@29842
  2192
    apply (erule ballE[where x=y])
chaieb@29842
  2193
    apply (erule ballE[where x=y])
chaieb@29842
  2194
    apply arith
chaieb@29842
  2195
    using y apply auto
chaieb@29842
  2196
    done
chaieb@29842
  2197
  with b show ?thesis by blast
chaieb@29842
  2198
qed
chaieb@29842
  2199
chaieb@29842
  2200
lemma rinf_abs_ge: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rinf S\<bar> \<le> a"
chaieb@29842
  2201
  unfolding abs_le_interval_iff  using rinf_bounds[of S "-a" a]
chaieb@29842
  2202
  by (auto simp add: setge_def setle_def)
chaieb@29842
  2203
chaieb@29842
  2204
lemma rinf_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rinf S - l\<bar> \<le> e"
chaieb@29842
  2205
proof-
chaieb@29842
  2206
  have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
huffman@30489
  2207
  show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th
chaieb@29842
  2208
    by  (auto simp add: setge_def setle_def)
chaieb@29842
  2209
qed
chaieb@29842
  2210
chaieb@29842
  2211
chaieb@29842
  2212
chaieb@29842
  2213
subsection{* Operator norm. *}
chaieb@29842
  2214
chaieb@29842
  2215
definition "onorm f = rsup {norm (f x)| x. norm x = 1}"
chaieb@29842
  2216
chaieb@29842
  2217
lemma norm_bound_generalize:
huffman@30582
  2218
  fixes f:: "real ^'n::finite \<Rightarrow> real^'m::finite"
chaieb@29842
  2219
  assumes lf: "linear f"
chaieb@29842
  2220
  shows "(\<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> b) \<longleftrightarrow> (\<forall>x. norm (f x) \<le> b * norm x)" (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@29842
  2221
proof-
chaieb@29842
  2222
  {assume H: ?rhs
chaieb@29842
  2223
    {fix x :: "real^'n" assume x: "norm x = 1"
chaieb@29842
  2224
      from H[rule_format, of x] x have "norm (f x) \<le> b" by simp}
chaieb@29842
  2225
    then have ?lhs by blast }
chaieb@29842
  2226
chaieb@29842
  2227
  moreover
chaieb@29842
  2228
  {assume H: ?lhs
huffman@30582
  2229
    from H[rule_format, of "basis arbitrary"]
huffman@30582
  2230
    have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis arbitrary)"]
huffman@30040
  2231
      by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])
chaieb@29842
  2232
    {fix x :: "real ^'n"
chaieb@29842
  2233
      {assume "x = 0"
huffman@30041
  2234
	then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}
chaieb@29842
  2235
      moreover
chaieb@29842
  2236
      {assume x0: "x \<noteq> 0"
huffman@30041
  2237
	hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
chaieb@29842
  2238
	let ?c = "1/ norm x"
huffman@30040
  2239
	have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul)
chaieb@29842
  2240
	with H have "norm (f(?c*s x)) \<le> b" by blast
huffman@30489
  2241
	hence "?c * norm (f x) \<le> b"
chaieb@29842
  2242
	  by (simp add: linear_cmul[OF lf] norm_mul)
huffman@30489
  2243
	hence "norm (f x) \<le> b * norm x"
huffman@30041
  2244
	  using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
chaieb@29842
  2245
      ultimately have "norm (f x) \<le> b * norm x" by blast}
chaieb@29842
  2246
    then have ?rhs by blast}
chaieb@29842
  2247
  ultimately show ?thesis by blast
chaieb@29842
  2248
qed
chaieb@29842
  2249
chaieb@29842
  2250
lemma onorm:
huffman@30582
  2251
  fixes f:: "real ^'n::finite \<Rightarrow> real ^'m::finite"
chaieb@29842
  2252
  assumes lf: "linear f"
chaieb@29842
  2253
  shows "norm (f x) <= onorm f * norm x"
chaieb@29842
  2254
  and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
chaieb@29842
  2255
proof-
chaieb@29842
  2256
  {
chaieb@29842
  2257
    let ?S = "{norm (f x) |x. norm x = 1}"
huffman@30582
  2258
    have Se: "?S \<noteq> {}" using  norm_basis by auto
huffman@30489
  2259
    from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
chaieb@29842
  2260
      unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
chaieb@29842
  2261
    {from rsup[OF Se b, unfolded onorm_def[symmetric]]
huffman@30489
  2262
      show "norm (f x) <= onorm f * norm x"
huffman@30489
  2263
	apply -
chaieb@29842
  2264
	apply (rule spec[where x = x])
chaieb@29842
  2265
	unfolding norm_bound_generalize[OF lf, symmetric]
chaieb@29842
  2266
	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
chaieb@29842
  2267
    {
huffman@30489
  2268
      show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
chaieb@29842
  2269
	using rsup[OF Se b, unfolded onorm_def[symmetric]]
chaieb@29842
  2270
	unfolding norm_bound_generalize[OF lf, symmetric]
chaieb@29842
  2271
	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
chaieb@29842
  2272
  }
chaieb@29842
  2273
qed
chaieb@29842
  2274
huffman@30582
  2275
lemma onorm_pos_le: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" shows "0 <= onorm f"
huffman@30582
  2276
  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis arbitrary"], unfolded norm_basis] by simp
huffman@30582
  2277
huffman@30582
  2278
lemma onorm_eq_0: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)"
chaieb@29842
  2279
  shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
chaieb@29842
  2280
  using onorm[OF lf]
huffman@30041
  2281
  apply (auto simp add: onorm_pos_le)
chaieb@29842
  2282
  apply atomize
chaieb@29842
  2283
  apply (erule allE[where x="0::real"])
chaieb@29842
  2284
  using onorm_pos_le[OF lf]
chaieb@29842
  2285
  apply arith
chaieb@29842
  2286
  done
chaieb@29842
  2287
huffman@30582
  2288
lemma onorm_const: "onorm(\<lambda>x::real^'n::finite. (y::real ^ 'm::finite)) = norm y"
chaieb@29842
  2289
proof-
chaieb@29842
  2290
  let ?f = "\<lambda>x::real^'n. (y::real ^ 'm)"
chaieb@29842
  2291
  have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
chaieb@29842
  2292
    by(auto intro: vector_choose_size set_ext)
chaieb@29842
  2293
  show ?thesis
chaieb@29842
  2294
    unfolding onorm_def th
chaieb@29842
  2295
    apply (rule rsup_unique) by (simp_all  add: setle_def)
chaieb@29842
  2296
qed
chaieb@29842
  2297
huffman@30582
  2298
lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n::finite \<Rightarrow> real ^'m::finite)"
chaieb@29842
  2299
  shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"
chaieb@29842
  2300
  unfolding onorm_eq_0[OF lf, symmetric]
chaieb@29842
  2301
  using onorm_pos_le[OF lf] by arith
chaieb@29842
  2302
chaieb@29842
  2303
lemma onorm_compose:
huffman@30582
  2304
  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)"
huffman@30582
  2305
  and lg: "linear (g::real^'k::finite \<Rightarrow> real^'n::finite)"
chaieb@29842
  2306
  shows "onorm (f o g) <= onorm f * onorm g"
chaieb@29842
  2307
  apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
chaieb@29842
  2308
  unfolding o_def
chaieb@29842
  2309
  apply (subst mult_assoc)
chaieb@29842
  2310
  apply (rule order_trans)
chaieb@29842
  2311
  apply (rule onorm(1)[OF lf])
chaieb@29842
  2312
  apply (rule mult_mono1)
chaieb@29842
  2313
  apply (rule onorm(1)[OF lg])
chaieb@29842
  2314
  apply (rule onorm_pos_le[OF lf])
chaieb@29842
  2315
  done
chaieb@29842
  2316
huffman@30582
  2317
lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real^'m::finite)"
chaieb@29842
  2318
  shows "onorm (\<lambda>x. - f x) \<le> onorm f"
chaieb@29842
  2319
  using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
huffman@30041
  2320
  unfolding norm_minus_cancel by metis
chaieb@29842
  2321
huffman@30582
  2322
lemma onorm_neg: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real^'m::finite)"
chaieb@29842
  2323
  shows "onorm (\<lambda>x. - f x) = onorm f"
chaieb@29842
  2324
  using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]]
chaieb@29842
  2325
  by simp
chaieb@29842
  2326
chaieb@29842
  2327
lemma onorm_triangle:
huffman@30582
  2328
  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" and lg: "linear g"
chaieb@29842
  2329
  shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
chaieb@29842
  2330
  apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
chaieb@29842
  2331
  apply (rule order_trans)
huffman@30041
  2332
  apply (rule norm_triangle_ineq)
chaieb@29842
  2333
  apply (simp add: distrib)
chaieb@29842
  2334
  apply (rule add_mono)
chaieb@29842
  2335
  apply (rule onorm(1)[OF lf])
chaieb@29842
  2336
  apply (rule onorm(1)[OF lg])
chaieb@29842
  2337
  done
chaieb@29842
  2338
huffman@30582
  2339
lemma onorm_triangle_le: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
chaieb@29842
  2340
  \<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"
chaieb@29842
  2341
  apply (rule order_trans)
chaieb@29842
  2342
  apply (rule onorm_triangle)
chaieb@29842
  2343
  apply assumption+
chaieb@29842
  2344
  done
chaieb@29842
  2345
huffman@30582
  2346
lemma onorm_triangle_lt: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
chaieb@29842
  2347
  ==> onorm(\<lambda>x. f x + g x) < e"
chaieb@29842
  2348
  apply (rule order_le_less_trans)
chaieb@29842
  2349
  apply (rule onorm_triangle)
chaieb@29842
  2350
  by assumption+
chaieb@29842
  2351
chaieb@29842
  2352
(* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)
chaieb@29842
  2353
chaieb@29842
  2354
definition vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x = (\<chi> i. x)"
chaieb@29842
  2355
huffman@30489
  2356
definition dest_vec1:: "'a ^1 \<Rightarrow> 'a"
chaieb@29842
  2357
  where "dest_vec1 x = (x$1)"
chaieb@29842
  2358
chaieb@29842
  2359
lemma vec1_component[simp]: "(vec1 x)$1 = x"
chaieb@29842
  2360
  by (simp add: vec1_def)
chaieb@29842
  2361
chaieb@29842
  2362
lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
huffman@30582
  2363
  by (simp_all add: vec1_def dest_vec1_def Cart_eq forall_1)
chaieb@29842
  2364
chaieb@29842
  2365
lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))" by (metis vec1_dest_vec1)
chaieb@29842
  2366
huffman@30489
  2367
lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))" by (metis vec1_dest_vec1)
chaieb@29842
  2368
chaieb@29842
  2369
lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))"  by (metis vec1_dest_vec1)
chaieb@29842
  2370
chaieb@29842
  2371
lemma exists_dest_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(dest_vec1 x))"by (metis vec1_dest_vec1)
chaieb@29842
  2372
chaieb@29842
  2373
lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
chaieb@29842
  2374
chaieb@29842
  2375
lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
chaieb@29842
  2376
chaieb@29842
  2377
lemma vec1_in_image_vec1: "vec1 x \<in> (vec1 ` S) \<longleftrightarrow> x \<in> S" by auto
chaieb@29842
  2378
chaieb@29842
  2379
lemma vec1_vec: "vec1 x = vec x" by (vector vec1_def)
chaieb@29842
  2380
chaieb@29842
  2381
lemma vec1_add: "vec1(x + y) = vec1 x + vec1 y" by (vector vec1_def)
chaieb@29842
  2382
lemma vec1_sub: "vec1(x - y) = vec1 x - vec1 y" by (vector vec1_def)
chaieb@29842
  2383
lemma vec1_cmul: "vec1(c* x) = c *s vec1 x " by (vector vec1_def)
chaieb@29842
  2384
lemma vec1_neg: "vec1(- x) = - vec1 x " by (vector vec1_def)
chaieb@29842
  2385
chaieb@29842
  2386
lemma vec1_setsum: assumes fS: "finite S"
chaieb@29842
  2387
  shows "vec1(setsum f S) = setsum (vec1 o f) S"
chaieb@29842
  2388
  apply (induct rule: finite_induct[OF fS])
chaieb@29842
  2389
  apply (simp add: vec1_vec)
chaieb@29842
  2390
  apply (auto simp add: vec1_add)
chaieb@29842
  2391
  done
chaieb@29842
  2392
chaieb@29842
  2393
lemma dest_vec1_lambda: "dest_vec1(\<chi> i. x i) = x 1"
chaieb@29842
  2394
  by (simp add: dest_vec1_def)
chaieb@29842
  2395
chaieb@29842
  2396
lemma dest_vec1_vec: "dest_vec1(vec x) = x"
chaieb@29842
  2397
  by (simp add: vec1_vec[symmetric])
chaieb@29842
  2398
chaieb@29842
  2399
lemma dest_vec1_add: "dest_vec1(x + y) = dest_vec1 x + dest_vec1 y"
chaieb@29842
  2400
 by (metis vec1_dest_vec1 vec1_add)
chaieb@29842
  2401
chaieb@29842
  2402
lemma dest_vec1_sub: "dest_vec1(x - y) = dest_vec1 x - dest_vec1 y"
chaieb@29842
  2403
 by (metis vec1_dest_vec1 vec1_sub)
chaieb@29842
  2404
chaieb@29842
  2405
lemma dest_vec1_cmul: "dest_vec1(c*sx) = c * dest_vec1 x"
chaieb@29842
  2406
 by (metis vec1_dest_vec1 vec1_cmul)
chaieb@29842
  2407
chaieb@29842
  2408
lemma dest_vec1_neg: "dest_vec1(- x) = - dest_vec1 x"
chaieb@29842
  2409
 by (metis vec1_dest_vec1 vec1_neg)
chaieb@29842
  2410
chaieb@29842
  2411
lemma dest_vec1_0[simp]: "dest_vec1 0 = 0" by (metis vec_0 dest_vec1_vec)
chaieb@29842
  2412
chaieb@29842
  2413
lemma dest_vec1_sum: assumes fS: "finite S"