src/HOL/Library/Euclidean_Space.thy
author huffman
Thu Jun 04 14:32:00 2009 -0700 (2009-06-04)
changeset 31445 c8a474a919a7
parent 31417 c12b25b7f015
child 31492 5400beeddb55
permissions -rw-r--r--
generalize norm method to work over class real_normed_vector
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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
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  Complex_Main "~~/src/HOL/Decision_Procs/Dense_Linear_Order"
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  Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type
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  Inner_Product
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uses "positivstellensatz.ML" ("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" (infixl "*s" 70)
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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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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"
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  by (vector ring_simps)
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lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
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lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
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lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
chaieb@29842
   323
  by (vector ring_simps)
chaieb@29842
   324
lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
chaieb@29842
   325
lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
chaieb@29842
   326
lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
chaieb@29842
   327
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
huffman@30489
   328
lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
chaieb@29842
   329
  by (vector ring_simps)
chaieb@29842
   330
huffman@30489
   331
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
huffman@30582
   332
  by (simp add: Cart_eq)
chaieb@29842
   333
huffman@30040
   334
subsection {* Square root of sum of squares *}
huffman@30040
   335
huffman@30040
   336
definition
huffman@30040
   337
  "setL2 f A = sqrt (\<Sum>i\<in>A. (f i)\<twosuperior>)"
huffman@30040
   338
huffman@30040
   339
lemma setL2_cong:
huffman@30040
   340
  "\<lbrakk>A = B; \<And>x. x \<in> B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
huffman@30040
   341
  unfolding setL2_def by simp
huffman@30040
   342
huffman@30040
   343
lemma strong_setL2_cong:
huffman@30040
   344
  "\<lbrakk>A = B; \<And>x. x \<in> B =simp=> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
huffman@30040
   345
  unfolding setL2_def simp_implies_def by simp
huffman@30040
   346
huffman@30040
   347
lemma setL2_infinite [simp]: "\<not> finite A \<Longrightarrow> setL2 f A = 0"
huffman@30040
   348
  unfolding setL2_def by simp
huffman@30040
   349
huffman@30040
   350
lemma setL2_empty [simp]: "setL2 f {} = 0"
huffman@30040
   351
  unfolding setL2_def by simp
huffman@30040
   352
huffman@30040
   353
lemma setL2_insert [simp]:
huffman@30040
   354
  "\<lbrakk>finite F; a \<notin> F\<rbrakk> \<Longrightarrow>
huffman@30040
   355
    setL2 f (insert a F) = sqrt ((f a)\<twosuperior> + (setL2 f F)\<twosuperior>)"
huffman@30040
   356
  unfolding setL2_def by (simp add: setsum_nonneg)
huffman@30040
   357
huffman@30040
   358
lemma setL2_nonneg [simp]: "0 \<le> setL2 f A"
huffman@30040
   359
  unfolding setL2_def by (simp add: setsum_nonneg)
huffman@30040
   360
huffman@30040
   361
lemma setL2_0': "\<forall>a\<in>A. f a = 0 \<Longrightarrow> setL2 f A = 0"
huffman@30040
   362
  unfolding setL2_def by simp
huffman@30040
   363
huffman@30040
   364
lemma setL2_mono:
huffman@30040
   365
  assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i"
huffman@30040
   366
  assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"
huffman@30040
   367
  shows "setL2 f K \<le> setL2 g K"
huffman@30040
   368
  unfolding setL2_def
huffman@30040
   369
  by (simp add: setsum_nonneg setsum_mono power_mono prems)
huffman@30040
   370
huffman@30040
   371
lemma setL2_right_distrib:
huffman@30040
   372
  "0 \<le> r \<Longrightarrow> r * setL2 f A = setL2 (\<lambda>x. r * f x) A"
huffman@30040
   373
  unfolding setL2_def
huffman@30040
   374
  apply (simp add: power_mult_distrib)
huffman@30040
   375
  apply (simp add: setsum_right_distrib [symmetric])
huffman@30040
   376
  apply (simp add: real_sqrt_mult setsum_nonneg)
huffman@30040
   377
  done
huffman@30040
   378
huffman@30040
   379
lemma setL2_left_distrib:
huffman@30040
   380
  "0 \<le> r \<Longrightarrow> setL2 f A * r = setL2 (\<lambda>x. f x * r) A"
huffman@30040
   381
  unfolding setL2_def
huffman@30040
   382
  apply (simp add: power_mult_distrib)
huffman@30040
   383
  apply (simp add: setsum_left_distrib [symmetric])
huffman@30040
   384
  apply (simp add: real_sqrt_mult setsum_nonneg)
huffman@30040
   385
  done
huffman@30040
   386
huffman@30040
   387
lemma setsum_nonneg_eq_0_iff:
huffman@30040
   388
  fixes f :: "'a \<Rightarrow> 'b::pordered_ab_group_add"
huffman@30040
   389
  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
   390
  apply (induct set: finite, simp)
huffman@30040
   391
  apply (simp add: add_nonneg_eq_0_iff setsum_nonneg)
huffman@30040
   392
  done
huffman@30040
   393
huffman@30040
   394
lemma setL2_eq_0_iff: "finite A \<Longrightarrow> setL2 f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
huffman@30040
   395
  unfolding setL2_def
huffman@30040
   396
  by (simp add: setsum_nonneg setsum_nonneg_eq_0_iff)
huffman@30040
   397
huffman@30040
   398
lemma setL2_triangle_ineq:
huffman@30040
   399
  shows "setL2 (\<lambda>i. f i + g i) A \<le> setL2 f A + setL2 g A"
huffman@30040
   400
proof (cases "finite A")
huffman@30040
   401
  case False
huffman@30040
   402
  thus ?thesis by simp
huffman@30040
   403
next
huffman@30040
   404
  case True
huffman@30040
   405
  thus ?thesis
huffman@30040
   406
  proof (induct set: finite)
huffman@30040
   407
    case empty
huffman@30040
   408
    show ?case by simp
huffman@30040
   409
  next
huffman@30040
   410
    case (insert x F)
huffman@30040
   411
    hence "sqrt ((f x + g x)\<twosuperior> + (setL2 (\<lambda>i. f i + g i) F)\<twosuperior>) \<le>
huffman@30040
   412
           sqrt ((f x + g x)\<twosuperior> + (setL2 f F + setL2 g F)\<twosuperior>)"
huffman@30040
   413
      by (intro real_sqrt_le_mono add_left_mono power_mono insert
huffman@30040
   414
                setL2_nonneg add_increasing zero_le_power2)
huffman@30040
   415
    also have
huffman@30040
   416
      "\<dots> \<le> sqrt ((f x)\<twosuperior> + (setL2 f F)\<twosuperior>) + sqrt ((g x)\<twosuperior> + (setL2 g F)\<twosuperior>)"
huffman@30040
   417
      by (rule real_sqrt_sum_squares_triangle_ineq)
huffman@30040
   418
    finally show ?case
huffman@30040
   419
      using insert by simp
huffman@30040
   420
  qed
huffman@30040
   421
qed
huffman@30040
   422
huffman@30040
   423
lemma sqrt_sum_squares_le_sum:
huffman@30040
   424
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) \<le> x + y"
huffman@30040
   425
  apply (rule power2_le_imp_le)
huffman@30040
   426
  apply (simp add: power2_sum)
huffman@30040
   427
  apply (simp add: mult_nonneg_nonneg)
huffman@30040
   428
  apply (simp add: add_nonneg_nonneg)
huffman@30040
   429
  done
huffman@30040
   430
huffman@30040
   431
lemma setL2_le_setsum [rule_format]:
huffman@30040
   432
  "(\<forall>i\<in>A. 0 \<le> f i) \<longrightarrow> setL2 f A \<le> setsum f A"
huffman@30040
   433
  apply (cases "finite A")
huffman@30040
   434
  apply (induct set: finite)
huffman@30040
   435
  apply simp
huffman@30040
   436
  apply clarsimp
huffman@30040
   437
  apply (erule order_trans [OF sqrt_sum_squares_le_sum])
huffman@30040
   438
  apply simp
huffman@30040
   439
  apply simp
huffman@30040
   440
  apply simp
huffman@30040
   441
  done
huffman@30040
   442
huffman@30040
   443
lemma sqrt_sum_squares_le_sum_abs: "sqrt (x\<twosuperior> + y\<twosuperior>) \<le> \<bar>x\<bar> + \<bar>y\<bar>"
huffman@30040
   444
  apply (rule power2_le_imp_le)
huffman@30040
   445
  apply (simp add: power2_sum)
huffman@30040
   446
  apply (simp add: mult_nonneg_nonneg)
huffman@30040
   447
  apply (simp add: add_nonneg_nonneg)
huffman@30040
   448
  done
huffman@30040
   449
huffman@30040
   450
lemma setL2_le_setsum_abs: "setL2 f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
huffman@30040
   451
  apply (cases "finite A")
huffman@30040
   452
  apply (induct set: finite)
huffman@30040
   453
  apply simp
huffman@30040
   454
  apply simp
huffman@30040
   455
  apply (rule order_trans [OF sqrt_sum_squares_le_sum_abs])
huffman@30040
   456
  apply simp
huffman@30040
   457
  apply simp
huffman@30040
   458
  done
huffman@30040
   459
huffman@30040
   460
lemma setL2_mult_ineq_lemma:
huffman@30040
   461
  fixes a b c d :: real
huffman@30040
   462
  shows "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
huffman@30040
   463
proof -
huffman@30040
   464
  have "0 \<le> (a * d - b * c)\<twosuperior>" by simp
huffman@30040
   465
  also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * d) * (b * c)"
huffman@30040
   466
    by (simp only: power2_diff power_mult_distrib)
huffman@30040
   467
  also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * c) * (b * d)"
huffman@30040
   468
    by simp
huffman@30040
   469
  finally show "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
huffman@30040
   470
    by simp
huffman@30040
   471
qed
huffman@30040
   472
huffman@30040
   473
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
   474
  apply (cases "finite A")
huffman@30040
   475
  apply (induct set: finite)
huffman@30040
   476
  apply simp
huffman@30040
   477
  apply (rule power2_le_imp_le, simp)
huffman@30040
   478
  apply (rule order_trans)
huffman@30040
   479
  apply (rule power_mono)
huffman@30040
   480
  apply (erule add_left_mono)
huffman@30040
   481
  apply (simp add: add_nonneg_nonneg mult_nonneg_nonneg setsum_nonneg)
huffman@30040
   482
  apply (simp add: power2_sum)
huffman@30040
   483
  apply (simp add: power_mult_distrib)
huffman@30040
   484
  apply (simp add: right_distrib left_distrib)
huffman@30040
   485
  apply (rule ord_le_eq_trans)
huffman@30040
   486
  apply (rule setL2_mult_ineq_lemma)
huffman@30040
   487
  apply simp
huffman@30040
   488
  apply (intro mult_nonneg_nonneg setL2_nonneg)
huffman@30040
   489
  apply simp
huffman@30040
   490
  done
huffman@30040
   491
huffman@30040
   492
lemma member_le_setL2: "\<lbrakk>finite A; i \<in> A\<rbrakk> \<Longrightarrow> f i \<le> setL2 f A"
huffman@30040
   493
  apply (rule_tac s="insert i (A - {i})" and t="A" in subst)
huffman@30040
   494
  apply fast
huffman@30040
   495
  apply (subst setL2_insert)
huffman@30040
   496
  apply simp
huffman@30040
   497
  apply simp
huffman@30040
   498
  apply simp
huffman@30040
   499
  done
huffman@30040
   500
huffman@31344
   501
subsection {* Metric *}
huffman@31344
   502
huffman@31344
   503
instantiation "^" :: (metric_space, finite) metric_space
huffman@31344
   504
begin
huffman@31344
   505
huffman@31344
   506
definition dist_vector_def:
huffman@31344
   507
  "dist (x::'a^'b) (y::'a^'b) = setL2 (\<lambda>i. dist (x$i) (y$i)) UNIV"
huffman@31344
   508
huffman@31417
   509
definition topo_vector_def:
huffman@31417
   510
  "topo = {S::('a ^ 'b) set. \<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S}"
huffman@31416
   511
huffman@31344
   512
instance proof
huffman@31344
   513
  fix x y :: "'a ^ 'b"
huffman@31344
   514
  show "dist x y = 0 \<longleftrightarrow> x = y"
huffman@31344
   515
    unfolding dist_vector_def
huffman@31344
   516
    by (simp add: setL2_eq_0_iff Cart_eq)
huffman@31344
   517
next
huffman@31344
   518
  fix x y z :: "'a ^ 'b"
huffman@31344
   519
  show "dist x y \<le> dist x z + dist y z"
huffman@31344
   520
    unfolding dist_vector_def
huffman@31344
   521
    apply (rule order_trans [OF _ setL2_triangle_ineq])
huffman@31344
   522
    apply (simp add: setL2_mono dist_triangle2)
huffman@31344
   523
    done
huffman@31416
   524
next
huffman@31417
   525
  show "topo = {S::('a ^ 'b) set. \<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S}"
huffman@31417
   526
    by (rule topo_vector_def)
huffman@31344
   527
qed
huffman@31344
   528
huffman@31344
   529
end
huffman@31344
   530
huffman@31389
   531
lemma dist_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
huffman@31389
   532
unfolding dist_vector_def
huffman@31389
   533
by (rule member_le_setL2) simp_all
huffman@31389
   534
huffman@31389
   535
lemma tendsto_Cart_nth:
huffman@31389
   536
  fixes X :: "'a \<Rightarrow> 'b::metric_space ^ 'n::finite"
huffman@31389
   537
  assumes "tendsto (\<lambda>n. X n) a net"
huffman@31389
   538
  shows "tendsto (\<lambda>n. X n $ i) (a $ i) net"
huffman@31389
   539
proof (rule tendstoI)
huffman@31389
   540
  fix e :: real assume "0 < e"
huffman@31389
   541
  with assms have "eventually (\<lambda>n. dist (X n) a < e) net"
huffman@31389
   542
    by (rule tendstoD)
huffman@31389
   543
  thus "eventually (\<lambda>n. dist (X n $ i) (a $ i) < e) net"
huffman@31389
   544
  proof (rule eventually_elim1)
huffman@31389
   545
    fix n :: 'a
huffman@31389
   546
    have "dist (X n $ i) (a $ i) \<le> dist (X n) a"
huffman@31389
   547
      by (rule dist_nth_le)
huffman@31389
   548
    also assume "dist (X n) a < e"
huffman@31389
   549
    finally show "dist (X n $ i) (a $ i) < e" .
huffman@31389
   550
  qed
huffman@31389
   551
qed
huffman@31389
   552
huffman@31389
   553
lemma LIMSEQ_Cart_nth:
huffman@31389
   554
  "(X ----> a) \<Longrightarrow> (\<lambda>n. X n $ i) ----> a $ i"
huffman@31389
   555
unfolding LIMSEQ_conv_tendsto by (rule tendsto_Cart_nth)
huffman@31389
   556
huffman@31389
   557
lemma LIM_Cart_nth:
huffman@31389
   558
  "(f -- x --> y) \<Longrightarrow> (\<lambda>x. f x $ i) -- x --> y $ i"
huffman@31389
   559
unfolding LIM_conv_tendsto by (rule tendsto_Cart_nth)
huffman@31389
   560
huffman@31389
   561
lemma Cauchy_Cart_nth:
huffman@31406
   562
  "Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n $ i)"
huffman@31406
   563
unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_nth_le])
huffman@31389
   564
huffman@31389
   565
lemma LIMSEQ_vector:
huffman@31389
   566
  fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n::finite"
huffman@31389
   567
  assumes X: "\<And>i. (\<lambda>n. X n $ i) ----> (a $ i)"
huffman@31389
   568
  shows "X ----> a"
huffman@31389
   569
proof (rule metric_LIMSEQ_I)
huffman@31389
   570
  fix r :: real assume "0 < r"
huffman@31389
   571
  then have "0 < r / of_nat CARD('n)" (is "0 < ?s")
huffman@31389
   572
    by (simp add: divide_pos_pos)
huffman@31389
   573
  def N \<equiv> "\<lambda>i. LEAST N. \<forall>n\<ge>N. dist (X n $ i) (a $ i) < ?s"
huffman@31389
   574
  def M \<equiv> "Max (range N)"
huffman@31389
   575
  have "\<And>i. \<exists>N. \<forall>n\<ge>N. dist (X n $ i) (a $ i) < ?s"
huffman@31389
   576
    using X `0 < ?s` by (rule metric_LIMSEQ_D)
huffman@31389
   577
  hence "\<And>i. \<forall>n\<ge>N i. dist (X n $ i) (a $ i) < ?s"
huffman@31389
   578
    unfolding N_def by (rule LeastI_ex)
huffman@31389
   579
  hence M: "\<And>i. \<forall>n\<ge>M. dist (X n $ i) (a $ i) < ?s"
huffman@31389
   580
    unfolding M_def by simp
huffman@31389
   581
  {
huffman@31389
   582
    fix n :: nat assume "M \<le> n"
huffman@31389
   583
    have "dist (X n) a = setL2 (\<lambda>i. dist (X n $ i) (a $ i)) UNIV"
huffman@31389
   584
      unfolding dist_vector_def ..
huffman@31389
   585
    also have "\<dots> \<le> setsum (\<lambda>i. dist (X n $ i) (a $ i)) UNIV"
huffman@31389
   586
      by (rule setL2_le_setsum [OF zero_le_dist])
huffman@31389
   587
    also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
huffman@31389
   588
      by (rule setsum_strict_mono, simp_all add: M `M \<le> n`)
huffman@31389
   589
    also have "\<dots> = r"
huffman@31389
   590
      by simp
huffman@31389
   591
    finally have "dist (X n) a < r" .
huffman@31389
   592
  }
huffman@31389
   593
  hence "\<forall>n\<ge>M. dist (X n) a < r"
huffman@31389
   594
    by simp
huffman@31389
   595
  then show "\<exists>M. \<forall>n\<ge>M. dist (X n) a < r" ..
huffman@31389
   596
qed
huffman@31389
   597
huffman@31389
   598
lemma Cauchy_vector:
huffman@31389
   599
  fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n::finite"
huffman@31389
   600
  assumes X: "\<And>i. Cauchy (\<lambda>n. X n $ i)"
huffman@31389
   601
  shows "Cauchy (\<lambda>n. X n)"
huffman@31389
   602
proof (rule metric_CauchyI)
huffman@31389
   603
  fix r :: real assume "0 < r"
huffman@31389
   604
  then have "0 < r / of_nat CARD('n)" (is "0 < ?s")
huffman@31389
   605
    by (simp add: divide_pos_pos)
huffman@31389
   606
  def N \<equiv> "\<lambda>i. LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"
huffman@31389
   607
  def M \<equiv> "Max (range N)"
huffman@31389
   608
  have "\<And>i. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"
huffman@31389
   609
    using X `0 < ?s` by (rule metric_CauchyD)
huffman@31389
   610
  hence "\<And>i. \<forall>m\<ge>N i. \<forall>n\<ge>N i. dist (X m $ i) (X n $ i) < ?s"
huffman@31389
   611
    unfolding N_def by (rule LeastI_ex)
huffman@31389
   612
  hence M: "\<And>i. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m $ i) (X n $ i) < ?s"
huffman@31389
   613
    unfolding M_def by simp
huffman@31389
   614
  {
huffman@31389
   615
    fix m n :: nat
huffman@31389
   616
    assume "M \<le> m" "M \<le> n"
huffman@31389
   617
    have "dist (X m) (X n) = setL2 (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
huffman@31389
   618
      unfolding dist_vector_def ..
huffman@31389
   619
    also have "\<dots> \<le> setsum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
huffman@31389
   620
      by (rule setL2_le_setsum [OF zero_le_dist])
huffman@31389
   621
    also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
huffman@31389
   622
      by (rule setsum_strict_mono, simp_all add: M `M \<le> m` `M \<le> n`)
huffman@31389
   623
    also have "\<dots> = r"
huffman@31389
   624
      by simp
huffman@31389
   625
    finally have "dist (X m) (X n) < r" .
huffman@31389
   626
  }
huffman@31389
   627
  hence "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r"
huffman@31389
   628
    by simp
huffman@31389
   629
  then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
huffman@31389
   630
qed
huffman@31389
   631
huffman@31406
   632
instance "^" :: (complete_space, finite) complete_space
huffman@31406
   633
proof
huffman@31406
   634
  fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
huffman@31406
   635
  have "\<And>i. (\<lambda>n. X n $ i) ----> lim (\<lambda>n. X n $ i)"
huffman@31406
   636
    using Cauchy_Cart_nth [OF `Cauchy X`]
huffman@31406
   637
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@31406
   638
  hence "X ----> Cart_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
huffman@31406
   639
    by (simp add: LIMSEQ_vector)
huffman@31406
   640
  then show "convergent X"
huffman@31406
   641
    by (rule convergentI)
huffman@31406
   642
qed
huffman@31406
   643
huffman@30040
   644
subsection {* Norms *}
huffman@30040
   645
huffman@30582
   646
instantiation "^" :: (real_normed_vector, finite) real_normed_vector
huffman@30040
   647
begin
huffman@30040
   648
huffman@30040
   649
definition vector_norm_def:
huffman@30582
   650
  "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) UNIV"
huffman@30040
   651
huffman@30040
   652
definition vector_sgn_def:
huffman@30040
   653
  "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
huffman@30040
   654
huffman@30040
   655
instance proof
huffman@30040
   656
  fix a :: real and x y :: "'a ^ 'b"
huffman@30040
   657
  show "0 \<le> norm x"
huffman@30040
   658
    unfolding vector_norm_def
huffman@30040
   659
    by (rule setL2_nonneg)
huffman@30040
   660
  show "norm x = 0 \<longleftrightarrow> x = 0"
huffman@30040
   661
    unfolding vector_norm_def
huffman@30040
   662
    by (simp add: setL2_eq_0_iff Cart_eq)
huffman@30040
   663
  show "norm (x + y) \<le> norm x + norm y"
huffman@30040
   664
    unfolding vector_norm_def
huffman@30040
   665
    apply (rule order_trans [OF _ setL2_triangle_ineq])
huffman@30582
   666
    apply (simp add: setL2_mono norm_triangle_ineq)
huffman@30040
   667
    done
huffman@30040
   668
  show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
huffman@30040
   669
    unfolding vector_norm_def
huffman@30582
   670
    by (simp add: norm_scaleR setL2_right_distrib)
huffman@30040
   671
  show "sgn x = scaleR (inverse (norm x)) x"
huffman@30040
   672
    by (rule vector_sgn_def)
huffman@31289
   673
  show "dist x y = norm (x - y)"
huffman@31344
   674
    unfolding dist_vector_def vector_norm_def
huffman@31344
   675
    by (simp add: dist_norm)
huffman@30040
   676
qed
huffman@30040
   677
huffman@30040
   678
end
huffman@30040
   679
huffman@31389
   680
lemma norm_nth_le: "norm (x $ i) \<le> norm x"
huffman@31389
   681
unfolding vector_norm_def
huffman@31389
   682
by (rule member_le_setL2) simp_all
huffman@31389
   683
huffman@31389
   684
interpretation Cart_nth: bounded_linear "\<lambda>x. x $ i"
huffman@31389
   685
apply default
huffman@31389
   686
apply (rule vector_add_component)
huffman@31389
   687
apply (rule vector_scaleR_component)
huffman@31389
   688
apply (rule_tac x="1" in exI, simp add: norm_nth_le)
huffman@31389
   689
done
huffman@31389
   690
huffman@31406
   691
instance "^" :: (banach, finite) banach ..
huffman@31406
   692
huffman@30045
   693
subsection {* Inner products *}
huffman@30045
   694
huffman@30582
   695
instantiation "^" :: (real_inner, finite) real_inner
huffman@30045
   696
begin
huffman@30045
   697
huffman@30045
   698
definition vector_inner_def:
huffman@30582
   699
  "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
huffman@30045
   700
huffman@30045
   701
instance proof
huffman@30045
   702
  fix r :: real and x y z :: "'a ^ 'b"
huffman@30045
   703
  show "inner x y = inner y x"
huffman@30045
   704
    unfolding vector_inner_def
huffman@30045
   705
    by (simp add: inner_commute)
huffman@30045
   706
  show "inner (x + y) z = inner x z + inner y z"
huffman@30045
   707
    unfolding vector_inner_def
huffman@30582
   708
    by (simp add: inner_left_distrib setsum_addf)
huffman@30045
   709
  show "inner (scaleR r x) y = r * inner x y"
huffman@30045
   710
    unfolding vector_inner_def
huffman@30582
   711
    by (simp add: inner_scaleR_left setsum_right_distrib)
huffman@30045
   712
  show "0 \<le> inner x x"
huffman@30045
   713
    unfolding vector_inner_def
huffman@30045
   714
    by (simp add: setsum_nonneg)
huffman@30045
   715
  show "inner x x = 0 \<longleftrightarrow> x = 0"
huffman@30045
   716
    unfolding vector_inner_def
huffman@30045
   717
    by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
huffman@30045
   718
  show "norm x = sqrt (inner x x)"
huffman@30045
   719
    unfolding vector_inner_def vector_norm_def setL2_def
huffman@30045
   720
    by (simp add: power2_norm_eq_inner)
huffman@30045
   721
qed
huffman@30045
   722
huffman@30045
   723
end
huffman@30045
   724
chaieb@29842
   725
subsection{* Properties of the dot product.  *}
chaieb@29842
   726
huffman@30489
   727
lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x"
chaieb@29842
   728
  by (vector mult_commute)
chaieb@29842
   729
lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \<bullet> z = (x \<bullet> z) + (y \<bullet> z)"
chaieb@29842
   730
  by (vector ring_simps)
huffman@30489
   731
lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)"
chaieb@29842
   732
  by (vector ring_simps)
huffman@30489
   733
lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)"
chaieb@29842
   734
  by (vector ring_simps)
huffman@30489
   735
lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)"
chaieb@29842
   736
  by (vector ring_simps)
chaieb@29842
   737
lemma dot_lmult: "(c *s x) \<bullet> y = (c::'a::ring) * (x \<bullet> y)" by (vector ring_simps)
chaieb@29842
   738
lemma dot_rmult: "x \<bullet> (c *s y) = (c::'a::comm_ring) * (x \<bullet> y)" by (vector ring_simps)
chaieb@29842
   739
lemma dot_lneg: "(-x) \<bullet> (y::'a::ring ^ 'n) = -(x \<bullet> y)" by vector
chaieb@29842
   740
lemma dot_rneg: "(x::'a::ring ^ 'n) \<bullet> (-y) = -(x \<bullet> y)" by vector
chaieb@29842
   741
lemma dot_lzero[simp]: "0 \<bullet> x = (0::'a::{comm_monoid_add, mult_zero})" by vector
chaieb@29842
   742
lemma dot_rzero[simp]: "x \<bullet> 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector
chaieb@29842
   743
lemma dot_pos_le[simp]: "(0::'a\<Colon>ordered_ring_strict) <= x \<bullet> x"
chaieb@29842
   744
  by (simp add: dot_def setsum_nonneg)
chaieb@29842
   745
chaieb@29842
   746
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
   747
using fS fp setsum_nonneg[OF fp]
chaieb@29842
   748
proof (induct set: finite)
chaieb@29842
   749
  case empty thus ?case by simp
chaieb@29842
   750
next
chaieb@29842
   751
  case (insert x F)
chaieb@29842
   752
  from insert.prems have Fx: "f x \<ge> 0" and Fp: "\<forall> a \<in> F. f a \<ge> 0" by simp_all
chaieb@29842
   753
  from insert.hyps Fp setsum_nonneg[OF Fp]
chaieb@29842
   754
  have h: "setsum f F = 0 \<longleftrightarrow> (\<forall>a \<in>F. f a = 0)" by metis
haftmann@31034
   755
  from add_nonneg_eq_0_iff[OF Fx  setsum_nonneg[OF Fp]] insert.hyps(1,2)
chaieb@29842
   756
  show ?case by (simp add: h)
chaieb@29842
   757
qed
chaieb@29842
   758
huffman@30582
   759
lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n::finite) = 0"
huffman@30582
   760
  by (simp add: dot_def setsum_squares_eq_0_iff Cart_eq)
huffman@30582
   761
huffman@30582
   762
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
   763
  by (auto simp add: le_less)
chaieb@29842
   764
huffman@30040
   765
subsection{* The collapse of the general concepts to dimension one. *}
chaieb@29842
   766
chaieb@29842
   767
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
huffman@30582
   768
  by (simp add: Cart_eq forall_1)
chaieb@29842
   769
chaieb@29842
   770
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
chaieb@29842
   771
  apply auto
chaieb@29842
   772
  apply (erule_tac x= "x$1" in allE)
chaieb@29842
   773
  apply (simp only: vector_one[symmetric])
chaieb@29842
   774
  done
chaieb@29842
   775
huffman@30040
   776
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
huffman@30582
   777
  by (simp add: vector_norm_def UNIV_1)
huffman@30040
   778
huffman@30489
   779
lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
huffman@30040
   780
  by (simp add: norm_vector_1)
chaieb@29842
   781
chaieb@29842
   782
lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
huffman@31289
   783
  by (auto simp add: norm_real dist_norm)
chaieb@29842
   784
chaieb@29842
   785
subsection {* A connectedness or intermediate value lemma with several applications. *}
chaieb@29842
   786
chaieb@29842
   787
lemma connected_real_lemma:
huffman@30582
   788
  fixes f :: "real \<Rightarrow> real ^ 'n::finite"
chaieb@29842
   789
  assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
chaieb@29842
   790
  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
   791
  and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
chaieb@29842
   792
  and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
chaieb@29842
   793
  and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
chaieb@29842
   794
  shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
chaieb@29842
   795
proof-
chaieb@29842
   796
  let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
huffman@30489
   797
  have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
huffman@30489
   798
  have Sub: "\<exists>y. isUb UNIV ?S y"
chaieb@29842
   799
    apply (rule exI[where x= b])
huffman@30489
   800
    using ab fb e12 by (auto simp add: isUb_def setle_def)
huffman@30489
   801
  from reals_complete[OF Se Sub] obtain l where
chaieb@29842
   802
    l: "isLub UNIV ?S l"by blast
chaieb@29842
   803
  have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
huffman@30489
   804
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
chaieb@29842
   805
    by (metis linorder_linear)
chaieb@29842
   806
  have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
chaieb@29842
   807
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
chaieb@29842
   808
    by (metis linorder_linear not_le)
chaieb@29842
   809
    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
   810
    have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
chaieb@29842
   811
    have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo
chaieb@29842
   812
    {assume le2: "f l \<in> e2"
chaieb@29842
   813
      from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
chaieb@29842
   814
      hence lap: "l - a > 0" using alb by arith
huffman@30489
   815
      from e2[rule_format, OF le2] obtain e where
chaieb@29842
   816
	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
huffman@30489
   817
      from dst[OF alb e(1)] obtain d where
chaieb@29842
   818
	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
huffman@30489
   819
      have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1)
chaieb@29842
   820
	apply ferrack by arith
chaieb@29842
   821
      then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
chaieb@29842
   822
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
chaieb@29842
   823
      from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
chaieb@29842
   824
      moreover
chaieb@29842
   825
      have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
chaieb@29842
   826
      ultimately have False using e12 alb d' by auto}
chaieb@29842
   827
    moreover
chaieb@29842
   828
    {assume le1: "f l \<in> e1"
chaieb@29842
   829
    from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
chaieb@29842
   830
      hence blp: "b - l > 0" using alb by arith
huffman@30489
   831
      from e1[rule_format, OF le1] obtain e where
chaieb@29842
   832
	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
huffman@30489
   833
      from dst[OF alb e(1)] obtain d where
chaieb@29842
   834
	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
huffman@30489
   835
      have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo
chaieb@29842
   836
      then obtain d' where d': "d' > 0" "d' < d" by metis
chaieb@29842
   837
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
chaieb@29842
   838
      hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
chaieb@29842
   839
      with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
huffman@30489
   840
      with l d' have False
chaieb@29842
   841
	by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
chaieb@29842
   842
    ultimately show ?thesis using alb by metis
chaieb@29842
   843
qed
chaieb@29842
   844
huffman@29881
   845
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
   846
chaieb@29842
   847
lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
chaieb@29842
   848
proof-
huffman@30489
   849
  have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
chaieb@29842
   850
  thus ?thesis by (simp add: ring_simps power2_eq_square)
chaieb@29842
   851
qed
chaieb@29842
   852
chaieb@29842
   853
lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
huffman@31340
   854
  using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_eq, rule_format, of e x] apply (auto simp add: power2_eq_square)
chaieb@29842
   855
  apply (rule_tac x="s" in exI)
chaieb@29842
   856
  apply auto
chaieb@29842
   857
  apply (erule_tac x=y in allE)
chaieb@29842
   858
  apply auto
chaieb@29842
   859
  done
chaieb@29842
   860
chaieb@29842
   861
lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
chaieb@29842
   862
  using real_sqrt_le_iff[of x "y^2"] by simp
chaieb@29842
   863
chaieb@29842
   864
lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
chaieb@29842
   865
  using real_sqrt_le_mono[of "x^2" y] by simp
chaieb@29842
   866
chaieb@29842
   867
lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
chaieb@29842
   868
  using real_sqrt_less_mono[of "x^2" y] by simp
chaieb@29842
   869
huffman@30489
   870
lemma sqrt_even_pow2: assumes n: "even n"
chaieb@29842
   871
  shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
chaieb@29842
   872
proof-
huffman@30489
   873
  from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2
huffman@30489
   874
    by (auto simp add: nat_number)
chaieb@29842
   875
  from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
chaieb@29842
   876
    by (simp only: power_mult[symmetric] mult_commute)
huffman@30489
   877
  then show ?thesis  using m by simp
chaieb@29842
   878
qed
chaieb@29842
   879
chaieb@29842
   880
lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
chaieb@29842
   881
  apply (cases "x = 0", simp_all)
chaieb@29842
   882
  using sqrt_divide_self_eq[of x]
chaieb@29842
   883
  apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps)
chaieb@29842
   884
  done
chaieb@29842
   885
chaieb@29842
   886
text{* Hence derive more interesting properties of the norm. *}
chaieb@29842
   887
huffman@30582
   888
text {*
huffman@30582
   889
  This type-specific version is only here
huffman@30582
   890
  to make @{text normarith.ML} happy.
huffman@30582
   891
*}
huffman@30582
   892
lemma norm_0: "norm (0::real ^ _) = 0"
huffman@30040
   893
  by (rule norm_zero)
huffman@30040
   894
chaieb@30263
   895
lemma norm_mul[simp]: "norm(a *s x) = abs(a) * norm x"
huffman@30040
   896
  by (simp add: vector_norm_def vector_component setL2_right_distrib
huffman@30040
   897
           abs_mult cong: strong_setL2_cong)
chaieb@29842
   898
lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
huffman@30040
   899
  by (simp add: vector_norm_def dot_def setL2_def power2_eq_square)
huffman@30040
   900
lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"
huffman@30040
   901
  by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
chaieb@29842
   902
lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
huffman@30040
   903
  by (simp add: real_vector_norm_def)
huffman@30582
   904
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n::finite)" by (metis norm_eq_zero)
chaieb@30263
   905
lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
chaieb@29842
   906
  by vector
chaieb@30263
   907
lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
chaieb@29842
   908
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
chaieb@30263
   909
lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
chaieb@29842
   910
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
chaieb@29842
   911
lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
chaieb@29842
   912
  by (metis vector_mul_lcancel)
chaieb@29842
   913
lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
chaieb@29842
   914
  by (metis vector_mul_rcancel)
huffman@30582
   915
lemma norm_cauchy_schwarz:
huffman@30582
   916
  fixes x y :: "real ^ 'n::finite"
huffman@30582
   917
  shows "x \<bullet> y <= norm x * norm y"
chaieb@29842
   918
proof-
chaieb@29842
   919
  {assume "norm x = 0"
huffman@30041
   920
    hence ?thesis by (simp add: dot_lzero dot_rzero)}
chaieb@29842
   921
  moreover
huffman@30489
   922
  {assume "norm y = 0"
huffman@30041
   923
    hence ?thesis by (simp add: dot_lzero dot_rzero)}
chaieb@29842
   924
  moreover
chaieb@29842
   925
  {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
chaieb@29842
   926
    let ?z = "norm y *s x - norm x *s y"
huffman@30041
   927
    from h have p: "norm x * norm y > 0" by (metis norm_ge_zero le_less zero_compare_simps)
chaieb@29842
   928
    from dot_pos_le[of ?z]
chaieb@29842
   929
    have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"
chaieb@29842
   930
      apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)
chaieb@29842
   931
      by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym)
chaieb@29842
   932
    hence "x\<bullet>y \<le> (norm x ^2 * norm y ^2) / (norm x * norm y)" using p
chaieb@29842
   933
      by (simp add: field_simps)
chaieb@29842
   934
    hence ?thesis using h by (simp add: power2_eq_square)}
chaieb@29842
   935
  ultimately show ?thesis by metis
chaieb@29842
   936
qed
chaieb@29842
   937
huffman@30582
   938
lemma norm_cauchy_schwarz_abs:
huffman@30582
   939
  fixes x y :: "real ^ 'n::finite"
huffman@30582
   940
  shows "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
chaieb@29842
   941
  using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
huffman@30041
   942
  by (simp add: real_abs_def dot_rneg)
chaieb@29842
   943
huffman@31398
   944
lemma norm_triangle_sub:
huffman@31398
   945
  fixes x y :: "'a::real_normed_vector"
huffman@31398
   946
  shows "norm x \<le> norm y  + norm (x - y)"
huffman@30041
   947
  using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
huffman@31398
   948
huffman@30582
   949
lemma norm_triangle_le: "norm(x::real ^'n::finite) + norm y <= e ==> norm(x + y) <= e"
huffman@30041
   950
  by (metis order_trans norm_triangle_ineq)
huffman@30582
   951
lemma norm_triangle_lt: "norm(x::real ^'n::finite) + norm(y) < e ==> norm(x + y) < e"
huffman@30041
   952
  by (metis basic_trans_rules(21) norm_triangle_ineq)
chaieb@29842
   953
huffman@30582
   954
lemma setsum_delta:
huffman@30582
   955
  assumes fS: "finite S"
huffman@30582
   956
  shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
huffman@30582
   957
proof-
huffman@30582
   958
  let ?f = "(\<lambda>k. if k=a then b k else 0)"
huffman@30582
   959
  {assume a: "a \<notin> S"
huffman@30582
   960
    hence "\<forall> k\<in> S. ?f k = 0" by simp
huffman@30582
   961
    hence ?thesis  using a by simp}
huffman@30582
   962
  moreover
huffman@30582
   963
  {assume a: "a \<in> S"
huffman@30582
   964
    let ?A = "S - {a}"
huffman@30582
   965
    let ?B = "{a}"
huffman@30582
   966
    have eq: "S = ?A \<union> ?B" using a by blast
huffman@30582
   967
    have dj: "?A \<inter> ?B = {}" by simp
huffman@30582
   968
    from fS have fAB: "finite ?A" "finite ?B" by auto
huffman@30582
   969
    have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
huffman@30582
   970
      using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
huffman@30582
   971
      by simp
huffman@30582
   972
    then have ?thesis  using a by simp}
huffman@30582
   973
  ultimately show ?thesis by blast
huffman@30582
   974
qed
huffman@30582
   975
huffman@30582
   976
lemma component_le_norm: "\<bar>x$i\<bar> <= norm (x::real ^ 'n::finite)"
huffman@30040
   977
  apply (simp add: vector_norm_def)
huffman@30040
   978
  apply (rule member_le_setL2, simp_all)
huffman@30040
   979
  done
huffman@30040
   980
huffman@30582
   981
lemma norm_bound_component_le: "norm(x::real ^ 'n::finite) <= e
huffman@30582
   982
                ==> \<bar>x$i\<bar> <= e"
chaieb@29842
   983
  by (metis component_le_norm order_trans)
chaieb@29842
   984
huffman@30582
   985
lemma norm_bound_component_lt: "norm(x::real ^ 'n::finite) < e
huffman@30582
   986
                ==> \<bar>x$i\<bar> < e"
chaieb@29842
   987
  by (metis component_le_norm basic_trans_rules(21))
chaieb@29842
   988
huffman@30582
   989
lemma norm_le_l1: "norm (x:: real ^'n::finite) <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
huffman@30040
   990
  by (simp add: vector_norm_def setL2_le_setsum)
chaieb@29842
   991
huffman@30582
   992
lemma real_abs_norm: "\<bar>norm x\<bar> = norm (x :: real ^ _)"
huffman@30040
   993
  by (rule abs_norm_cancel)
huffman@30582
   994
lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n::finite) - norm y\<bar> <= norm(x - y)"
huffman@30040
   995
  by (rule norm_triangle_ineq3)
huffman@30582
   996
lemma norm_le: "norm(x::real ^ _) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
chaieb@29842
   997
  by (simp add: real_vector_norm_def)
huffman@30582
   998
lemma norm_lt: "norm(x::real ^ _) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
chaieb@29842
   999
  by (simp add: real_vector_norm_def)
huffman@30582
  1000
lemma norm_eq: "norm (x::real ^ _) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
chaieb@29842
  1001
  by (simp add: order_eq_iff norm_le)
huffman@30582
  1002
lemma norm_eq_1: "norm(x::real ^ _) = 1 \<longleftrightarrow> x \<bullet> x = 1"
chaieb@29842
  1003
  by (simp add: real_vector_norm_def)
chaieb@29842
  1004
chaieb@29842
  1005
text{* Squaring equations and inequalities involving norms.  *}
chaieb@29842
  1006
chaieb@29842
  1007
lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
huffman@30582
  1008
  by (simp add: real_vector_norm_def)
chaieb@29842
  1009
chaieb@29842
  1010
lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
huffman@30040
  1011
  by (auto simp add: real_vector_norm_def)
chaieb@29842
  1012
chaieb@29842
  1013
lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
chaieb@29842
  1014
proof-
chaieb@29842
  1015
  have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: ring_simps power2_eq_square)
chaieb@29842
  1016
  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
  1017
finally show ?thesis ..
chaieb@29842
  1018
qed
chaieb@29842
  1019
chaieb@29842
  1020
lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
huffman@30040
  1021
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
huffman@30041
  1022
  using norm_ge_zero[of x]
chaieb@29842
  1023
  apply arith
chaieb@29842
  1024
  done
chaieb@29842
  1025
huffman@30489
  1026
lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
huffman@30040
  1027
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
huffman@30041
  1028
  using norm_ge_zero[of x]
chaieb@29842
  1029
  apply arith
chaieb@29842
  1030
  done
chaieb@29842
  1031
chaieb@29842
  1032
lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
chaieb@29842
  1033
  by (metis not_le norm_ge_square)
chaieb@29842
  1034
lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
chaieb@29842
  1035
  by (metis norm_le_square not_less)
chaieb@29842
  1036
chaieb@29842
  1037
text{* Dot product in terms of the norm rather than conversely. *}
chaieb@29842
  1038
chaieb@29842
  1039
lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
chaieb@29842
  1040
  by (simp add: norm_pow_2 dot_ladd dot_radd dot_sym)
chaieb@29842
  1041
chaieb@29842
  1042
lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
chaieb@29842
  1043
  by (simp add: norm_pow_2 dot_ladd dot_radd dot_lsub dot_rsub dot_sym)
chaieb@29842
  1044
chaieb@29842
  1045
chaieb@29842
  1046
text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
chaieb@29842
  1047
huffman@30582
  1048
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
  1049
proof
chaieb@29842
  1050
  assume "?lhs" then show ?rhs by simp
chaieb@29842
  1051
next
chaieb@29842
  1052
  assume ?rhs
chaieb@29842
  1053
  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y\<bullet> y = 0" by simp
huffman@30489
  1054
  hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
chaieb@29842
  1055
    by (simp add: dot_rsub dot_lsub dot_sym)
chaieb@29842
  1056
  then have "(x - y) \<bullet> (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub)
chaieb@29842
  1057
  then show "x = y" by (simp add: dot_eq_0)
chaieb@29842
  1058
qed
chaieb@29842
  1059
chaieb@29842
  1060
chaieb@29842
  1061
subsection{* General linear decision procedure for normed spaces. *}
chaieb@29842
  1062
huffman@31445
  1063
lemma norm_cmul_rule_thm:
huffman@31445
  1064
  fixes x :: "'a::real_normed_vector"
huffman@31445
  1065
  shows "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(scaleR c x)"
huffman@31445
  1066
  unfolding norm_scaleR
huffman@31445
  1067
  apply (erule mult_mono1)
huffman@31445
  1068
  apply simp
chaieb@29842
  1069
  done
chaieb@29842
  1070
chaieb@30263
  1071
  (* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
huffman@31445
  1072
lemma norm_add_rule_thm:
huffman@31445
  1073
  fixes x1 x2 :: "'a::real_normed_vector"
huffman@31445
  1074
  shows "norm x1 \<le> b1 \<Longrightarrow> norm x2 \<le> b2 \<Longrightarrow> norm (x1 + x2) \<le> b1 + b2"
huffman@31445
  1075
  by (rule order_trans [OF norm_triangle_ineq add_mono])
chaieb@29842
  1076
chaieb@29842
  1077
lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"
chaieb@29842
  1078
  by (simp add: ring_simps)
chaieb@29842
  1079
huffman@31445
  1080
lemma pth_1:
huffman@31445
  1081
  fixes x :: "'a::real_normed_vector"
huffman@31445
  1082
  shows "x == scaleR 1 x" by simp
huffman@31445
  1083
huffman@31445
  1084
lemma pth_2:
huffman@31445
  1085
  fixes x :: "'a::real_normed_vector"
huffman@31445
  1086
  shows "x - y == x + -y" by (atomize (full)) simp
huffman@31445
  1087
huffman@31445
  1088
lemma pth_3:
huffman@31445
  1089
  fixes x :: "'a::real_normed_vector"
huffman@31445
  1090
  shows "- x == scaleR (-1) x" by simp
huffman@31445
  1091
huffman@31445
  1092
lemma pth_4:
huffman@31445
  1093
  fixes x :: "'a::real_normed_vector"
huffman@31445
  1094
  shows "scaleR 0 x == 0" and "scaleR c 0 = (0::'a)" by simp_all
huffman@31445
  1095
huffman@31445
  1096
lemma pth_5:
huffman@31445
  1097
  fixes x :: "'a::real_normed_vector"
huffman@31445
  1098
  shows "scaleR c (scaleR d x) == scaleR (c * d) x" by simp
huffman@31445
  1099
huffman@31445
  1100
lemma pth_6:
huffman@31445
  1101
  fixes x :: "'a::real_normed_vector"
huffman@31445
  1102
  shows "scaleR c (x + y) == scaleR c x + scaleR c y"
huffman@31445
  1103
  by (simp add: scaleR_right_distrib)
huffman@31445
  1104
huffman@31445
  1105
lemma pth_7:
huffman@31445
  1106
  fixes x :: "'a::real_normed_vector"
huffman@31445
  1107
  shows "0 + x == x" and "x + 0 == x" by simp_all
huffman@31445
  1108
huffman@31445
  1109
lemma pth_8:
huffman@31445
  1110
  fixes x :: "'a::real_normed_vector"
huffman@31445
  1111
  shows "scaleR c x + scaleR d x == scaleR (c + d) x"
huffman@31445
  1112
  by (simp add: scaleR_left_distrib)
huffman@31445
  1113
huffman@31445
  1114
lemma pth_9:
huffman@31445
  1115
  fixes x :: "'a::real_normed_vector" shows
huffman@31445
  1116
  "(scaleR c x + z) + scaleR d x == scaleR (c + d) x + z"
huffman@31445
  1117
  "scaleR c x + (scaleR d x + z) == scaleR (c + d) x + z"
huffman@31445
  1118
  "(scaleR c x + w) + (scaleR d x + z) == scaleR (c + d) x + (w + z)"
huffman@31445
  1119
  by (simp_all add: algebra_simps)
huffman@31445
  1120
huffman@31445
  1121
lemma pth_a:
huffman@31445
  1122
  fixes x :: "'a::real_normed_vector"
huffman@31445
  1123
  shows "scaleR 0 x + y == y" by simp
huffman@31445
  1124
huffman@31445
  1125
lemma pth_b:
huffman@31445
  1126
  fixes x :: "'a::real_normed_vector" shows
huffman@31445
  1127
  "scaleR c x + scaleR d y == scaleR c x + scaleR d y"
huffman@31445
  1128
  "(scaleR c x + z) + scaleR d y == scaleR c x + (z + scaleR d y)"
huffman@31445
  1129
  "scaleR c x + (scaleR d y + z) == scaleR c x + (scaleR d y + z)"
huffman@31445
  1130
  "(scaleR c x + w) + (scaleR d y + z) == scaleR c x + (w + (scaleR d y + z))"
huffman@31445
  1131
  by (simp_all add: algebra_simps)
huffman@31445
  1132
huffman@31445
  1133
lemma pth_c:
huffman@31445
  1134
  fixes x :: "'a::real_normed_vector" shows
huffman@31445
  1135
  "scaleR c x + scaleR d y == scaleR d y + scaleR c x"
huffman@31445
  1136
  "(scaleR c x + z) + scaleR d y == scaleR d y + (scaleR c x + z)"
huffman@31445
  1137
  "scaleR c x + (scaleR d y + z) == scaleR d y + (scaleR c x + z)"
huffman@31445
  1138
  "(scaleR c x + w) + (scaleR d y + z) == scaleR d y + ((scaleR c x + w) + z)"
huffman@31445
  1139
  by (simp_all add: algebra_simps)
huffman@31445
  1140
huffman@31445
  1141
lemma pth_d:
huffman@31445
  1142
  fixes x :: "'a::real_normed_vector"
huffman@31445
  1143
  shows "x + 0 == x" by simp
huffman@31445
  1144
huffman@31445
  1145
lemma norm_imp_pos_and_ge:
huffman@31445
  1146
  fixes x :: "'a::real_normed_vector"
huffman@31445
  1147
  shows "norm x == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
huffman@31445
  1148
  by atomize auto
chaieb@29842
  1149
chaieb@29842
  1150
lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
chaieb@29842
  1151
huffman@30489
  1152
lemma norm_pths:
huffman@31445
  1153
  fixes x :: "'a::real_normed_vector" shows
huffman@31445
  1154
  "x = y \<longleftrightarrow> norm (x - y) \<le> 0"
chaieb@29842
  1155
  "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
huffman@30041
  1156
  using norm_ge_zero[of "x - y"] by auto
chaieb@29842
  1157
huffman@31344
  1158
lemma vector_dist_norm:
huffman@31445
  1159
  fixes x :: "'a::real_normed_vector"
huffman@31344
  1160
  shows "dist x y = norm (x - y)"
huffman@31344
  1161
  by (rule dist_norm)
huffman@31344
  1162
chaieb@29842
  1163
use "normarith.ML"
chaieb@29842
  1164
wenzelm@30549
  1165
method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
chaieb@29842
  1166
*} "Proves simple linear statements about vector norms"
chaieb@29842
  1167
chaieb@29842
  1168
chaieb@29842
  1169
text{* Hence more metric properties. *}
chaieb@29842
  1170
huffman@31289
  1171
lemma dist_triangle_alt:
huffman@31289
  1172
  fixes x y z :: "'a::metric_space"
huffman@31289
  1173
  shows "dist y z <= dist x y + dist x z"
huffman@31285
  1174
using dist_triangle [of y z x] by (simp add: dist_commute)
huffman@31285
  1175
huffman@31289
  1176
lemma dist_pos_lt:
huffman@31289
  1177
  fixes x y :: "'a::metric_space"
huffman@31289
  1178
  shows "x \<noteq> y ==> 0 < dist x y"
huffman@31289
  1179
by (simp add: zero_less_dist_iff)
huffman@31289
  1180
huffman@31289
  1181
lemma dist_nz:
huffman@31289
  1182
  fixes x y :: "'a::metric_space"
huffman@31289
  1183
  shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
huffman@31289
  1184
by (simp add: zero_less_dist_iff)
huffman@31289
  1185
huffman@31289
  1186
lemma dist_triangle_le:
huffman@31289
  1187
  fixes x y z :: "'a::metric_space"
huffman@31289
  1188
  shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
huffman@31285
  1189
by (rule order_trans [OF dist_triangle2])
huffman@31285
  1190
huffman@31289
  1191
lemma dist_triangle_lt:
huffman@31289
  1192
  fixes x y z :: "'a::metric_space"
huffman@31289
  1193
  shows "dist x z + dist y z < e ==> dist x y < e"
huffman@31285
  1194
by (rule le_less_trans [OF dist_triangle2])
huffman@31285
  1195
huffman@31285
  1196
lemma dist_triangle_half_l:
huffman@31289
  1197
  fixes x1 x2 y :: "'a::metric_space"
huffman@31289
  1198
  shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
huffman@31285
  1199
by (rule dist_triangle_lt [where z=y], simp)
huffman@31285
  1200
huffman@31285
  1201
lemma dist_triangle_half_r:
huffman@31289
  1202
  fixes x1 x2 y :: "'a::metric_space"
huffman@31289
  1203
  shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
huffman@31285
  1204
by (rule dist_triangle_half_l, simp_all add: dist_commute)
chaieb@29842
  1205
huffman@31289
  1206
lemma dist_triangle_add:
huffman@31289
  1207
  fixes x y x' y' :: "'a::real_normed_vector"
huffman@31289
  1208
  shows "dist (x + y) (x' + y') <= dist x x' + dist y y'"
huffman@31445
  1209
  by norm
huffman@30489
  1210
huffman@30489
  1211
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
huffman@31289
  1212
  unfolding dist_norm vector_ssub_ldistrib[symmetric] norm_mul ..
huffman@30489
  1213
huffman@31285
  1214
lemma dist_triangle_add_half:
huffman@31289
  1215
  fixes x x' y y' :: "'a::real_normed_vector"
huffman@31289
  1216
  shows "dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 \<Longrightarrow> dist(x + y) (x' + y') < e"
huffman@31445
  1217
  by norm
chaieb@29842
  1218
huffman@30582
  1219
lemma setsum_component [simp]:
huffman@30582
  1220
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
huffman@30582
  1221
  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
huffman@30582
  1222
  by (cases "finite S", induct S set: finite, simp_all)
huffman@30582
  1223
chaieb@29842
  1224
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
huffman@30582
  1225
  by (simp add: Cart_eq)
chaieb@29842
  1226
huffman@30489
  1227
lemma setsum_clauses:
chaieb@29842
  1228
  shows "setsum f {} = 0"
chaieb@29842
  1229
  and "finite S \<Longrightarrow> setsum f (insert x S) =
chaieb@29842
  1230
                 (if x \<in> S then setsum f S else f x + setsum f S)"
chaieb@29842
  1231
  by (auto simp add: insert_absorb)
chaieb@29842
  1232
huffman@30489
  1233
lemma setsum_cmul:
chaieb@29842
  1234
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
chaieb@29842
  1235
  shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
huffman@30582
  1236
  by (simp add: Cart_eq setsum_right_distrib)
chaieb@29842
  1237
huffman@30489
  1238
lemma setsum_norm:
chaieb@29842
  1239
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
chaieb@29842
  1240
  assumes fS: "finite S"
chaieb@29842
  1241
  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
chaieb@29842
  1242
proof(induct rule: finite_induct[OF fS])
huffman@30041
  1243
  case 1 thus ?case by simp
chaieb@29842
  1244
next
chaieb@29842
  1245
  case (2 x S)
chaieb@29842
  1246
  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
  1247
  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
chaieb@29842
  1248
    using "2.hyps" by simp
chaieb@29842
  1249
  finally  show ?case  using "2.hyps" by simp
chaieb@29842
  1250
qed
chaieb@29842
  1251
huffman@30489
  1252
lemma real_setsum_norm:
huffman@30582
  1253
  fixes f :: "'a \<Rightarrow> real ^'n::finite"
chaieb@29842
  1254
  assumes fS: "finite S"
chaieb@29842
  1255
  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
chaieb@29842
  1256
proof(induct rule: finite_induct[OF fS])
huffman@30040
  1257
  case 1 thus ?case by simp
chaieb@29842
  1258
next
chaieb@29842
  1259
  case (2 x S)
huffman@30040
  1260
  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
  1261
  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
chaieb@29842
  1262
    using "2.hyps" by simp
chaieb@29842
  1263
  finally  show ?case  using "2.hyps" by simp
chaieb@29842
  1264
qed
chaieb@29842
  1265
huffman@30489
  1266
lemma setsum_norm_le:
chaieb@29842
  1267
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
chaieb@29842
  1268
  assumes fS: "finite S"
chaieb@29842
  1269
  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
chaieb@29842
  1270
  shows "norm (setsum f S) \<le> setsum g S"
chaieb@29842
  1271
proof-
huffman@30489
  1272
  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
chaieb@29842
  1273
    by - (rule setsum_mono, simp)
chaieb@29842
  1274
  then show ?thesis using setsum_norm[OF fS, of f] fg
chaieb@29842
  1275
    by arith
chaieb@29842
  1276
qed
chaieb@29842
  1277
huffman@30489
  1278
lemma real_setsum_norm_le:
huffman@30582
  1279
  fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
chaieb@29842
  1280
  assumes fS: "finite S"
chaieb@29842
  1281
  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
chaieb@29842
  1282
  shows "norm (setsum f S) \<le> setsum g S"
chaieb@29842
  1283
proof-
huffman@30489
  1284
  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
chaieb@29842
  1285
    by - (rule setsum_mono, simp)
chaieb@29842
  1286
  then show ?thesis using real_setsum_norm[OF fS, of f] fg
chaieb@29842
  1287
    by arith
chaieb@29842
  1288
qed
chaieb@29842
  1289
chaieb@29842
  1290
lemma setsum_norm_bound:
chaieb@29842
  1291
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
chaieb@29842
  1292
  assumes fS: "finite S"
chaieb@29842
  1293
  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
chaieb@29842
  1294
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
chaieb@29842
  1295
  using setsum_norm_le[OF fS K] setsum_constant[symmetric]
chaieb@29842
  1296
  by simp
chaieb@29842
  1297
chaieb@29842
  1298
lemma real_setsum_norm_bound:
huffman@30582
  1299
  fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
chaieb@29842
  1300
  assumes fS: "finite S"
chaieb@29842
  1301
  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
chaieb@29842
  1302
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
chaieb@29842
  1303
  using real_setsum_norm_le[OF fS K] setsum_constant[symmetric]
chaieb@29842
  1304
  by simp
chaieb@29842
  1305
chaieb@29842
  1306
lemma setsum_vmul:
chaieb@29842
  1307
  fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector,semiring, mult_zero}"
chaieb@29842
  1308
  assumes fS: "finite S"
chaieb@29842
  1309
  shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
chaieb@29842
  1310
proof(induct rule: finite_induct[OF fS])
chaieb@29842
  1311
  case 1 then show ?case by (simp add: vector_smult_lzero)
chaieb@29842
  1312
next
chaieb@29842
  1313
  case (2 x F)
huffman@30489
  1314
  from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v"
chaieb@29842
  1315
    by simp
huffman@30489
  1316
  also have "\<dots> = f x *s v + setsum f F *s v"
chaieb@29842
  1317
    by (simp add: vector_sadd_rdistrib)
chaieb@29842
  1318
  also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp
chaieb@29842
  1319
  finally show ?case .
chaieb@29842
  1320
qed
chaieb@29842
  1321
chaieb@29842
  1322
(* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \<Rightarrow> real ^'n"]  ---
chaieb@29842
  1323
 Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *)
chaieb@29842
  1324
chaieb@29842
  1325
lemma setsum_add_split: assumes mn: "(m::nat) \<le> n + 1"
chaieb@29842
  1326
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
chaieb@29842
  1327
proof-
chaieb@29842
  1328
  let ?A = "{m .. n}"
chaieb@29842
  1329
  let ?B = "{n + 1 .. n + p}"
huffman@30489
  1330
  have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto
chaieb@29842
  1331
  have d: "?A \<inter> ?B = {}" by auto
chaieb@29842
  1332
  from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto
chaieb@29842
  1333
qed
chaieb@29842
  1334
chaieb@29842
  1335
lemma setsum_natinterval_left:
huffman@30489
  1336
  assumes mn: "(m::nat) <= n"
chaieb@29842
  1337
  shows "setsum f {m..n} = f m + setsum f {m + 1..n}"
chaieb@29842
  1338
proof-
chaieb@29842
  1339
  from mn have "{m .. n} = insert m {m+1 .. n}" by auto
chaieb@29842
  1340
  then show ?thesis by auto
chaieb@29842
  1341
qed
chaieb@29842
  1342
huffman@30489
  1343
lemma setsum_natinterval_difff:
chaieb@29842
  1344
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
chaieb@29842
  1345
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
chaieb@29842
  1346
          (if m <= n then f m - f(n + 1) else 0)"
chaieb@29842
  1347
by (induct n, auto simp add: ring_simps not_le le_Suc_eq)
chaieb@29842
  1348
chaieb@29842
  1349
lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
chaieb@29842
  1350
chaieb@29842
  1351
lemma setsum_setsum_restrict:
chaieb@29842
  1352
  "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
  1353
  apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
chaieb@29842
  1354
  by (rule setsum_commute)
chaieb@29842
  1355
chaieb@29842
  1356
lemma setsum_image_gen: assumes fS: "finite S"
chaieb@29842
  1357
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
chaieb@29842
  1358
proof-
chaieb@29842
  1359
  {fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto}
chaieb@29842
  1360
  note th0 = this
huffman@30489
  1361
  have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
huffman@30489
  1362
    apply (rule setsum_cong2)
chaieb@29842
  1363
    by (simp add: th0)
chaieb@29842
  1364
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
chaieb@29842
  1365
    apply (rule setsum_setsum_restrict[OF fS])
chaieb@29842
  1366
    by (rule finite_imageI[OF fS])
chaieb@29842
  1367
  finally show ?thesis .
chaieb@29842
  1368
qed
chaieb@29842
  1369
chaieb@29842
  1370
    (* FIXME: Here too need stupid finiteness assumption on T!!! *)
chaieb@29842
  1371
lemma setsum_group:
chaieb@29842
  1372
  assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
chaieb@29842
  1373
  shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
huffman@30489
  1374
chaieb@29842
  1375
apply (subst setsum_image_gen[OF fS, of g f])
chaieb@30263
  1376
apply (rule setsum_mono_zero_right[OF fT fST])
chaieb@29842
  1377
by (auto intro: setsum_0')
chaieb@29842
  1378
chaieb@29842
  1379
lemma vsum_norm_allsubsets_bound:
huffman@30582
  1380
  fixes f:: "'a \<Rightarrow> real ^'n::finite"
huffman@30489
  1381
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
huffman@30582
  1382
  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
chaieb@29842
  1383
proof-
huffman@30582
  1384
  let ?d = "real CARD('n)"
chaieb@29842
  1385
  let ?nf = "\<lambda>x. norm (f x)"
huffman@30582
  1386
  let ?U = "UNIV :: 'n set"
chaieb@29842
  1387
  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
  1388
    by (rule setsum_commute)
chaieb@29842
  1389
  have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
chaieb@29842
  1390
  have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
chaieb@29842
  1391
    apply (rule setsum_mono)
chaieb@29842
  1392
    by (rule norm_le_l1)
chaieb@29842
  1393
  also have "\<dots> \<le> 2 * ?d * e"
chaieb@29842
  1394
    unfolding th0 th1
chaieb@29842
  1395
  proof(rule setsum_bounded)
chaieb@29842
  1396
    fix i assume i: "i \<in> ?U"
chaieb@29842
  1397
    let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
chaieb@29842
  1398
    let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
chaieb@29842
  1399
    have thp: "P = ?Pp \<union> ?Pn" by auto
chaieb@29842
  1400
    have thp0: "?Pp \<inter> ?Pn ={}" by auto
chaieb@29842
  1401
    have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
chaieb@29842
  1402
    have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
huffman@30582
  1403
      using component_le_norm[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
huffman@30582
  1404
      by (auto intro: abs_le_D1)
chaieb@29842
  1405
    have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
huffman@30582
  1406
      using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
huffman@30582
  1407
      by (auto simp add: setsum_negf intro: abs_le_D1)
huffman@30489
  1408
    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
  1409
      apply (subst thp)
huffman@30489
  1410
      apply (rule setsum_Un_zero)
chaieb@29842
  1411
      using fP thp0 by auto
chaieb@29842
  1412
    also have "\<dots> \<le> 2*e" using Pne Ppe by arith
chaieb@29842
  1413
    finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
chaieb@29842
  1414
  qed
chaieb@29842
  1415
  finally show ?thesis .
chaieb@29842
  1416
qed
chaieb@29842
  1417
chaieb@29842
  1418
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
  1419
  by (induct rule: finite_induct, auto simp add: dot_lzero dot_ladd dot_radd)
chaieb@29842
  1420
chaieb@29842
  1421
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
  1422
  by (induct rule: finite_induct, auto simp add: dot_rzero dot_radd)
chaieb@29842
  1423
chaieb@29842
  1424
subsection{* Basis vectors in coordinate directions. *}
chaieb@29842
  1425
chaieb@29842
  1426
chaieb@29842
  1427
definition "basis k = (\<chi> i. if i = k then 1 else 0)"
chaieb@29842
  1428
huffman@30582
  1429
lemma basis_component [simp]: "basis k $ i = (if k=i then 1 else 0)"
huffman@30582
  1430
  unfolding basis_def by simp
huffman@30582
  1431
huffman@30489
  1432
lemma delta_mult_idempotent:
chaieb@29842
  1433
  "(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
  1434
chaieb@29842
  1435
lemma norm_basis:
huffman@30582
  1436
  shows "norm (basis k :: real ^'n::finite) = 1"
chaieb@29842
  1437
  apply (simp add: basis_def real_vector_norm_def dot_def)
chaieb@29842
  1438
  apply (vector delta_mult_idempotent)
huffman@30582
  1439
  using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"]
chaieb@29842
  1440
  apply auto
chaieb@29842
  1441
  done
chaieb@29842
  1442
huffman@30582
  1443
lemma norm_basis_1: "norm(basis 1 :: real ^'n::{finite,one}) = 1"
huffman@30582
  1444
  by (rule norm_basis)
huffman@30582
  1445
huffman@30582
  1446
lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n::finite). norm x = c"
huffman@30582
  1447
  apply (rule exI[where x="c *s basis arbitrary"])
huffman@30582
  1448
  by (simp only: norm_mul norm_basis)
chaieb@29842
  1449
huffman@30489
  1450
lemma vector_choose_dist: assumes e: "0 <= e"
huffman@30582
  1451
  shows "\<exists>(y::real^'n::finite). dist x y = e"
chaieb@29842
  1452
proof-
chaieb@29842
  1453
  from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"
chaieb@29842
  1454
    by blast
huffman@31289
  1455
  then have "dist x (x - c) = e" by (simp add: dist_norm)
chaieb@29842
  1456
  then show ?thesis by blast
chaieb@29842
  1457
qed
chaieb@29842
  1458
huffman@30582
  1459
lemma basis_inj: "inj (basis :: 'n \<Rightarrow> real ^'n::finite)"
huffman@30582
  1460
  by (simp add: inj_on_def Cart_eq)
chaieb@29842
  1461
chaieb@29842
  1462
lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
chaieb@29842
  1463
  by auto
chaieb@29842
  1464
chaieb@29842
  1465
lemma basis_expansion:
huffman@30582
  1466
  "setsum (\<lambda>i. (x$i) *s basis i) UNIV = (x::('a::ring_1) ^'n::finite)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
huffman@30582
  1467
  by (auto simp add: Cart_eq cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
chaieb@29842
  1468
huffman@30489
  1469
lemma basis_expansion_unique:
huffman@30582
  1470
  "setsum (\<lambda>i. f i *s basis i) UNIV = (x::('a::comm_ring_1) ^'n::finite) \<longleftrightarrow> (\<forall>i. f i = x$i)"
huffman@30582
  1471
  by (simp add: Cart_eq setsum_delta cond_value_iff cong del: if_weak_cong)
chaieb@29842
  1472
chaieb@29842
  1473
lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
chaieb@29842
  1474
  by auto
chaieb@29842
  1475
chaieb@29842
  1476
lemma dot_basis:
huffman@30582
  1477
  shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i :: 'a^'n::finite) = (x$i :: 'a::semiring_1)"
huffman@30582
  1478
  by (auto simp add: dot_def basis_def cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)
huffman@30582
  1479
huffman@30582
  1480
lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
huffman@30582
  1481
  by (auto simp add: Cart_eq)
chaieb@29842
  1482
huffman@30489
  1483
lemma basis_nonzero:
chaieb@29842
  1484
  shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
huffman@30582
  1485
  by (simp add: basis_eq_0)
huffman@30582
  1486
huffman@30582
  1487
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = (z::'a::semiring_1^'n::finite)"
chaieb@29842
  1488
  apply (auto simp add: Cart_eq dot_basis)
chaieb@29842
  1489
  apply (erule_tac x="basis i" in allE)
chaieb@29842
  1490
  apply (simp add: dot_basis)
chaieb@29842
  1491
  apply (subgoal_tac "y = z")
chaieb@29842
  1492
  apply simp
huffman@30582
  1493
  apply (simp add: Cart_eq)
chaieb@29842
  1494
  done
chaieb@29842
  1495
huffman@30582
  1496
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = (y::'a::semiring_1^'n::finite)"
chaieb@29842
  1497
  apply (auto simp add: Cart_eq dot_basis)
chaieb@29842
  1498
  apply (erule_tac x="basis i" in allE)
chaieb@29842
  1499
  apply (simp add: dot_basis)
chaieb@29842
  1500
  apply (subgoal_tac "x = y")
chaieb@29842
  1501
  apply simp
huffman@30582
  1502
  apply (simp add: Cart_eq)
chaieb@29842
  1503
  done
chaieb@29842
  1504
chaieb@29842
  1505
subsection{* Orthogonality. *}
chaieb@29842
  1506
chaieb@29842
  1507
definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
chaieb@29842
  1508
chaieb@29842
  1509
lemma orthogonal_basis:
huffman@30582
  1510
  shows "orthogonal (basis i :: 'a^'n::finite) x \<longleftrightarrow> x$i = (0::'a::ring_1)"
huffman@30582
  1511
  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
  1512
chaieb@29842
  1513
lemma orthogonal_basis_basis:
huffman@30582
  1514
  shows "orthogonal (basis i :: 'a::ring_1^'n::finite) (basis j) \<longleftrightarrow> i \<noteq> j"
huffman@30582
  1515
  unfolding orthogonal_basis[of i] basis_component[of j] by simp
chaieb@29842
  1516
chaieb@29842
  1517
  (* FIXME : Maybe some of these require less than comm_ring, but not all*)
chaieb@29842
  1518
lemma orthogonal_clauses:
chaieb@29842
  1519
  "orthogonal a (0::'a::comm_ring ^'n)"
chaieb@29842
  1520
  "orthogonal a x ==> orthogonal a (c *s x)"
chaieb@29842
  1521
  "orthogonal a x ==> orthogonal a (-x)"
chaieb@29842
  1522
  "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x + y)"
chaieb@29842
  1523
  "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x - y)"
chaieb@29842
  1524
  "orthogonal 0 a"
chaieb@29842
  1525
  "orthogonal x a ==> orthogonal (c *s x) a"
chaieb@29842
  1526
  "orthogonal x a ==> orthogonal (-x) a"
chaieb@29842
  1527
  "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x + y) a"
chaieb@29842
  1528
  "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x - y) a"
chaieb@29842
  1529
  unfolding orthogonal_def dot_rneg dot_rmult dot_radd dot_rsub
chaieb@29842
  1530
  dot_lzero dot_rzero dot_lneg dot_lmult dot_ladd dot_lsub
chaieb@29842
  1531
  by simp_all
chaieb@29842
  1532
chaieb@29842
  1533
lemma orthogonal_commute: "orthogonal (x::'a::{ab_semigroup_mult,comm_monoid_add} ^'n)y \<longleftrightarrow> orthogonal y x"
chaieb@29842
  1534
  by (simp add: orthogonal_def dot_sym)
chaieb@29842
  1535
chaieb@29842
  1536
subsection{* Explicit vector construction from lists. *}
chaieb@29842
  1537
huffman@30582
  1538
primrec from_nat :: "nat \<Rightarrow> 'a::{monoid_add,one}"
huffman@30582
  1539
where "from_nat 0 = 0" | "from_nat (Suc n) = 1 + from_nat n"
huffman@30582
  1540
huffman@30582
  1541
lemma from_nat [simp]: "from_nat = of_nat"
huffman@30582
  1542
by (rule ext, induct_tac x, simp_all)
huffman@30582
  1543
huffman@30582
  1544
primrec
huffman@30582
  1545
  list_fun :: "nat \<Rightarrow> _ list \<Rightarrow> _ \<Rightarrow> _"
huffman@30582
  1546
where
huffman@30582
  1547
  "list_fun n [] = (\<lambda>x. 0)"
huffman@30582
  1548
| "list_fun n (x # xs) = fun_upd (list_fun (Suc n) xs) (from_nat n) x"
huffman@30582
  1549
huffman@30582
  1550
definition "vector l = (\<chi> i. list_fun 1 l i)"
huffman@30582
  1551
(*definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"*)
chaieb@29842
  1552
chaieb@29842
  1553
lemma vector_1: "(vector[x]) $1 = x"
huffman@30582
  1554
  unfolding vector_def by simp
chaieb@29842
  1555
chaieb@29842
  1556
lemma vector_2:
chaieb@29842
  1557
 "(vector[x,y]) $1 = x"
chaieb@29842
  1558
 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
huffman@30582
  1559
  unfolding vector_def by simp_all
chaieb@29842
  1560
chaieb@29842
  1561
lemma vector_3:
chaieb@29842
  1562
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
chaieb@29842
  1563
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
chaieb@29842
  1564
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
huffman@30582
  1565
  unfolding vector_def by simp_all
chaieb@29842
  1566
chaieb@29842
  1567
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
chaieb@29842
  1568
  apply auto
chaieb@29842
  1569
  apply (erule_tac x="v$1" in allE)
chaieb@29842
  1570
  apply (subgoal_tac "vector [v$1] = v")
chaieb@29842
  1571
  apply simp
huffman@30582
  1572
  apply (vector vector_def)
huffman@30582
  1573
  apply (simp add: forall_1)
huffman@30582
  1574
  done
chaieb@29842
  1575
chaieb@29842
  1576
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
chaieb@29842
  1577
  apply auto
chaieb@29842
  1578
  apply (erule_tac x="v$1" in allE)
chaieb@29842
  1579
  apply (erule_tac x="v$2" in allE)
chaieb@29842
  1580
  apply (subgoal_tac "vector [v$1, v$2] = v")
chaieb@29842
  1581
  apply simp
huffman@30582
  1582
  apply (vector vector_def)
huffman@30582
  1583
  apply (simp add: forall_2)
chaieb@29842
  1584
  done
chaieb@29842
  1585
chaieb@29842
  1586
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
chaieb@29842
  1587
  apply auto
chaieb@29842
  1588
  apply (erule_tac x="v$1" in allE)
chaieb@29842
  1589
  apply (erule_tac x="v$2" in allE)
chaieb@29842
  1590
  apply (erule_tac x="v$3" in allE)
chaieb@29842
  1591
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
chaieb@29842
  1592
  apply simp
huffman@30582
  1593
  apply (vector vector_def)
huffman@30582
  1594
  apply (simp add: forall_3)
chaieb@29842
  1595
  done
chaieb@29842
  1596
chaieb@29842
  1597
subsection{* Linear functions. *}
chaieb@29842
  1598
chaieb@29842
  1599
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
  1600
chaieb@29842
  1601
lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. (c::'a::comm_semiring) *s f x)"
huffman@30582
  1602
  by (vector linear_def Cart_eq ring_simps)
chaieb@29842
  1603
chaieb@29842
  1604
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
  1605
chaieb@29842
  1606
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
  1607
  by (vector linear_def Cart_eq ring_simps)
chaieb@29842
  1608
chaieb@29842
  1609
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
  1610
  by (vector linear_def Cart_eq ring_simps)
chaieb@29842
  1611
chaieb@29842
  1612
lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
chaieb@29842
  1613
  by (simp add: linear_def)
chaieb@29842
  1614
chaieb@29842
  1615
lemma linear_id: "linear id" by (simp add: linear_def id_def)
chaieb@29842
  1616
chaieb@29842
  1617
lemma linear_zero: "linear (\<lambda>x. 0::'a::semiring_1 ^ 'n)" by (simp add: linear_def)
chaieb@29842
  1618
chaieb@29842
  1619
lemma linear_compose_setsum:
chaieb@29842
  1620
  assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a :: 'a::semiring_1 ^ 'n \<Rightarrow> 'a ^ 'm)"
chaieb@29842
  1621
  shows "linear(\<lambda>x. setsum (\<lambda>a. f a x :: 'a::semiring_1 ^'m) S)"
chaieb@29842
  1622
  using lS
chaieb@29842
  1623
  apply (induct rule: finite_induct[OF fS])
chaieb@29842
  1624
  by (auto simp add: linear_zero intro: linear_compose_add)
chaieb@29842
  1625
chaieb@29842
  1626
lemma linear_vmul_component:
chaieb@29842
  1627
  fixes f:: "'a::semiring_1^'m \<Rightarrow> 'a^'n"
huffman@30582
  1628
  assumes lf: "linear f"
chaieb@29842
  1629
  shows "linear (\<lambda>x. f x $ k *s v)"
huffman@30582
  1630
  using lf
chaieb@29842
  1631
  apply (auto simp add: linear_def )
chaieb@29842
  1632
  by (vector ring_simps)+
chaieb@29842
  1633
chaieb@29842
  1634
lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)"
chaieb@29842
  1635
  unfolding linear_def
chaieb@29842
  1636
  apply clarsimp
chaieb@29842
  1637
  apply (erule allE[where x="0::'a"])
chaieb@29842
  1638
  apply simp
chaieb@29842
  1639
  done
chaieb@29842
  1640
chaieb@29842
  1641
lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def)
chaieb@29842
  1642
chaieb@29842
  1643
lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \<Rightarrow> _) ==> f (-x) = - f x"
chaieb@29842
  1644
  unfolding vector_sneg_minus1
huffman@30489
  1645
  using linear_cmul[of f] by auto
huffman@30489
  1646
huffman@30489
  1647
lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
chaieb@29842
  1648
chaieb@29842
  1649
lemma linear_sub: "linear (f::'a::ring_1 ^'n \<Rightarrow> _) ==> f(x - y) = f x - f y"
chaieb@29842
  1650
  by (simp add: diff_def linear_add linear_neg)
chaieb@29842
  1651
huffman@30489
  1652
lemma linear_setsum:
chaieb@29842
  1653
  fixes f:: "'a::semiring_1^'n \<Rightarrow> _"
chaieb@29842
  1654
  assumes lf: "linear f" and fS: "finite S"
chaieb@29842
  1655
  shows "f (setsum g S) = setsum (f o g) S"
chaieb@29842
  1656
proof (induct rule: finite_induct[OF fS])
chaieb@29842
  1657
  case 1 thus ?case by (simp add: linear_0[OF lf])
chaieb@29842
  1658
next
chaieb@29842
  1659
  case (2 x F)
chaieb@29842
  1660
  have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
chaieb@29842
  1661
    by simp
chaieb@29842
  1662
  also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
chaieb@29842
  1663
  also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
chaieb@29842
  1664
  finally show ?case .
chaieb@29842
  1665
qed
chaieb@29842
  1666
chaieb@29842
  1667
lemma linear_setsum_mul:
chaieb@29842
  1668
  fixes f:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m"
chaieb@29842
  1669
  assumes lf: "linear f" and fS: "finite S"
chaieb@29842
  1670
  shows "f (setsum (\<lambda>i. c i *s v i) S) = setsum (\<lambda>i. c i *s f (v i)) S"
chaieb@29842
  1671
  using linear_setsum[OF lf fS, of "\<lambda>i. c i *s v i" , unfolded o_def]
huffman@30489
  1672
  linear_cmul[OF lf] by simp
chaieb@29842
  1673
chaieb@29842
  1674
lemma linear_injective_0:
chaieb@29842
  1675
  assumes lf: "linear (f:: 'a::ring_1 ^ 'n \<Rightarrow> _)"
chaieb@29842
  1676
  shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
chaieb@29842
  1677
proof-
chaieb@29842
  1678
  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
chaieb@29842
  1679
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
huffman@30489
  1680
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
chaieb@29842
  1681
    by (simp add: linear_sub[OF lf])
chaieb@29842
  1682
  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
chaieb@29842
  1683
  finally show ?thesis .
chaieb@29842
  1684
qed
chaieb@29842
  1685
chaieb@29842
  1686
lemma linear_bounded:
huffman@30582
  1687
  fixes f:: "real ^'m::finite \<Rightarrow> real ^'n::finite"
chaieb@29842
  1688
  assumes lf: "linear f"
chaieb@29842
  1689
  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
chaieb@29842
  1690
proof-
huffman@30582
  1691
  let ?S = "UNIV:: 'm set"
chaieb@29842
  1692
  let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
chaieb@29842
  1693
  have fS: "finite ?S" by simp
chaieb@29842
  1694
  {fix x:: "real ^ 'm"
huffman@30582
  1695
    let ?g = "(\<lambda>i. (x$i) *s (basis i) :: real ^ 'm)"
chaieb@29842
  1696
    have "norm (f x) = norm (f (setsum (\<lambda>i. (x$i) *s (basis i)) ?S))"
chaieb@29842
  1697
      by (simp only:  basis_expansion)
chaieb@29842
  1698
    also have "\<dots> = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)"
chaieb@29842
  1699
      using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf]
chaieb@29842
  1700
      by auto
chaieb@29842
  1701
    finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)" .
chaieb@29842
  1702
    {fix i assume i: "i \<in> ?S"
huffman@30582
  1703
      from component_le_norm[of x i]
chaieb@29842
  1704
      have "norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x"
chaieb@29842
  1705
      unfolding norm_mul
chaieb@29842
  1706
      apply (simp only: mult_commute)
chaieb@29842
  1707
      apply (rule mult_mono)
huffman@30041
  1708
      by (auto simp add: ring_simps norm_ge_zero) }
chaieb@29842
  1709
    then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
chaieb@29842
  1710
    from real_setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (f (basis i))", OF th]
chaieb@29842
  1711
    have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
chaieb@29842
  1712
  then show ?thesis by blast
chaieb@29842
  1713
qed
chaieb@29842
  1714
chaieb@29842
  1715
lemma linear_bounded_pos:
huffman@30582
  1716
  fixes f:: "real ^'n::finite \<Rightarrow> real ^ 'm::finite"
chaieb@29842
  1717
  assumes lf: "linear f"
chaieb@29842
  1718
  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
chaieb@29842
  1719
proof-
huffman@30489
  1720
  from linear_bounded[OF lf] obtain B where
chaieb@29842
  1721
    B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
chaieb@29842
  1722
  let ?K = "\<bar>B\<bar> + 1"
chaieb@29842
  1723
  have Kp: "?K > 0" by arith
chaieb@29842
  1724
    {assume C: "B < 0"
huffman@30041
  1725
      have "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff)
chaieb@29842
  1726
      with C have "B * norm (1:: real ^ 'n) < 0"
chaieb@29842
  1727
	by (simp add: zero_compare_simps)
huffman@30041
  1728
      with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp
chaieb@29842
  1729
    }
chaieb@29842
  1730
    then have Bp: "B \<ge> 0" by ferrack
chaieb@29842
  1731
    {fix x::"real ^ 'n"
chaieb@29842
  1732
      have "norm (f x) \<le> ?K *  norm x"
huffman@30041
  1733
      using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
huffman@30040
  1734
      apply (auto simp add: ring_simps split add: abs_split)
huffman@30040
  1735
      apply (erule order_trans, simp)
huffman@30040
  1736
      done
chaieb@29842
  1737
  }
chaieb@29842
  1738
  then show ?thesis using Kp by blast
chaieb@29842
  1739
qed
chaieb@29842
  1740
chaieb@29842
  1741
subsection{* Bilinear functions. *}
chaieb@29842
  1742
chaieb@29842
  1743
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
chaieb@29842
  1744
chaieb@29842
  1745
lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
chaieb@29842
  1746
  by (simp add: bilinear_def linear_def)
chaieb@29842
  1747
lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
chaieb@29842
  1748
  by (simp add: bilinear_def linear_def)
chaieb@29842
  1749
chaieb@29842
  1750
lemma bilinear_lmul: "bilinear h ==> h (c *s x) y = c *s (h x y)"
chaieb@29842
  1751
  by (simp add: bilinear_def linear_def)
chaieb@29842
  1752
chaieb@29842
  1753
lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)"
chaieb@29842
  1754
  by (simp add: bilinear_def linear_def)
chaieb@29842
  1755
chaieb@29842
  1756
lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)"
chaieb@29842
  1757
  by (simp only: vector_sneg_minus1 bilinear_lmul)
chaieb@29842
  1758
chaieb@29842
  1759
lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y"
chaieb@29842
  1760
  by (simp only: vector_sneg_minus1 bilinear_rmul)
chaieb@29842
  1761
chaieb@29842
  1762
lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
chaieb@29842
  1763
  using add_imp_eq[of x y 0] by auto
huffman@30489
  1764
huffman@30489
  1765
lemma bilinear_lzero:
chaieb@29842
  1766
  fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h 0 x = 0"
huffman@30489
  1767
  using bilinear_ladd[OF bh, of 0 0 x]
chaieb@29842
  1768
    by (simp add: eq_add_iff ring_simps)
chaieb@29842
  1769
huffman@30489
  1770
lemma bilinear_rzero:
chaieb@29842
  1771
  fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h x 0 = 0"
huffman@30489
  1772
  using bilinear_radd[OF bh, of x 0 0 ]
chaieb@29842
  1773
    by (simp add: eq_add_iff ring_simps)
chaieb@29842
  1774
chaieb@29842
  1775
lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ 'n)) z = h x z - h y z"
chaieb@29842
  1776
  by (simp  add: diff_def bilinear_ladd bilinear_lneg)
chaieb@29842
  1777
chaieb@29842
  1778
lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ 'n)) = h z x - h z y"
chaieb@29842
  1779
  by (simp  add: diff_def bilinear_radd bilinear_rneg)
chaieb@29842
  1780
chaieb@29842
  1781
lemma bilinear_setsum:
chaieb@29842
  1782
  fixes h:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m \<Rightarrow> 'a ^ 'k"
chaieb@29842
  1783
  assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
chaieb@29842
  1784
  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
huffman@30489
  1785
proof-
chaieb@29842
  1786
  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
chaieb@29842
  1787
    apply (rule linear_setsum[unfolded o_def])
chaieb@29842
  1788
    using bh fS by (auto simp add: bilinear_def)
chaieb@29842
  1789
  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
chaieb@29842
  1790
    apply (rule setsum_cong, simp)
chaieb@29842
  1791
    apply (rule linear_setsum[unfolded o_def])
chaieb@29842
  1792
    using bh fT by (auto simp add: bilinear_def)
chaieb@29842
  1793
  finally show ?thesis unfolding setsum_cartesian_product .
chaieb@29842
  1794
qed
chaieb@29842
  1795
chaieb@29842
  1796
lemma bilinear_bounded:
huffman@30582
  1797
  fixes h:: "real ^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real ^ 'k::finite"
chaieb@29842
  1798
  assumes bh: "bilinear h"
chaieb@29842
  1799
  shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
huffman@30489
  1800
proof-
huffman@30582
  1801
  let ?M = "UNIV :: 'm set"
huffman@30582
  1802
  let ?N = "UNIV :: 'n set"
chaieb@29842
  1803
  let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
chaieb@29842
  1804
  have fM: "finite ?M" and fN: "finite ?N" by simp_all
chaieb@29842
  1805
  {fix x:: "real ^ 'm" and  y :: "real^'n"
chaieb@29842
  1806
    have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$i) *s basis i) ?M) (setsum (\<lambda>i. (y$i) *s basis i) ?N))" unfolding basis_expansion ..
chaieb@29842
  1807
    also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$i) *s basis i) ((y$j) *s basis j)) (?M \<times> ?N))"  unfolding bilinear_setsum[OF bh fM fN] ..
chaieb@29842
  1808
    finally have th: "norm (h x y) = \<dots>" .
chaieb@29842
  1809
    have "norm (h x y) \<le> ?B * norm x * norm y"
chaieb@29842
  1810
      apply (simp add: setsum_left_distrib th)
chaieb@29842
  1811
      apply (rule real_setsum_norm_le)
chaieb@29842
  1812
      using fN fM
chaieb@29842
  1813
      apply simp
chaieb@29842
  1814
      apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)
chaieb@29842
  1815
      apply (rule mult_mono)
huffman@30041
  1816
      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
chaieb@29842
  1817
      apply (rule mult_mono)
huffman@30041
  1818
      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
chaieb@29842
  1819
      done}
chaieb@29842
  1820
  then show ?thesis by metis
chaieb@29842
  1821
qed
chaieb@29842
  1822
chaieb@29842
  1823
lemma bilinear_bounded_pos:
huffman@30582
  1824
  fixes h:: "real ^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real ^ 'k::finite"
chaieb@29842
  1825
  assumes bh: "bilinear h"
chaieb@29842
  1826
  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
chaieb@29842
  1827
proof-
huffman@30489
  1828
  from bilinear_bounded[OF bh] obtain B where
chaieb@29842
  1829
    B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
chaieb@29842
  1830
  let ?K = "\<bar>B\<bar> + 1"
chaieb@29842
  1831
  have Kp: "?K > 0" by arith
chaieb@29842
  1832
  have KB: "B < ?K" by arith
chaieb@29842
  1833
  {fix x::"real ^'m" and y :: "real ^'n"
chaieb@29842
  1834
    from KB Kp
chaieb@29842
  1835
    have "B * norm x * norm y \<le> ?K * norm x * norm y"
huffman@30489
  1836
      apply -
chaieb@29842
  1837
      apply (rule mult_right_mono, rule mult_right_mono)
huffman@30041
  1838
      by (auto simp add: norm_ge_zero)
chaieb@29842
  1839
    then have "norm (h x y) \<le> ?K * norm x * norm y"
huffman@30489
  1840
      using B[rule_format, of x y] by simp}
chaieb@29842
  1841
  with Kp show ?thesis by blast
chaieb@29842
  1842
qed
chaieb@29842
  1843
chaieb@29842
  1844
subsection{* Adjoints. *}
chaieb@29842
  1845
chaieb@29842
  1846
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
chaieb@29842
  1847
chaieb@29842
  1848
lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
chaieb@29842
  1849
chaieb@29842
  1850
lemma adjoint_works_lemma:
huffman@30582
  1851
  fixes f:: "'a::ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
chaieb@29842
  1852
  assumes lf: "linear f"
chaieb@29842
  1853
  shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
chaieb@29842
  1854
proof-
huffman@30582
  1855
  let ?N = "UNIV :: 'n set"
huffman@30582
  1856
  let ?M = "UNIV :: 'm set"
chaieb@29842
  1857
  have fN: "finite ?N" by simp
chaieb@29842
  1858
  have fM: "finite ?M" by simp
chaieb@29842
  1859
  {fix y:: "'a ^ 'm"
chaieb@29842
  1860
    let ?w = "(\<chi> i. (f (basis i) \<bullet> y)) :: 'a ^ 'n"
chaieb@29842
  1861
    {fix x
chaieb@29842
  1862
      have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *s basis i) ?N) \<bullet> y"
chaieb@29842
  1863
	by (simp only: basis_expansion)
chaieb@29842
  1864
      also have "\<dots> = (setsum (\<lambda>i. (x$i) *s f (basis i)) ?N) \<bullet> y"
huffman@30489
  1865
	unfolding linear_setsum[OF lf fN]
chaieb@29842
  1866
	by (simp add: linear_cmul[OF lf])
chaieb@29842
  1867
      finally have "f x \<bullet> y = x \<bullet> ?w"
chaieb@29842
  1868
	apply (simp only: )
huffman@30582
  1869
	apply (simp add: dot_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] ring_simps)
chaieb@29842
  1870
	done}
chaieb@29842
  1871
  }
huffman@30489
  1872
  then show ?thesis unfolding adjoint_def
chaieb@29842
  1873
    some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
chaieb@29842
  1874
    using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
chaieb@29842
  1875
    by metis
chaieb@29842
  1876
qed
chaieb@29842
  1877
chaieb@29842
  1878
lemma adjoint_works:
huffman@30582
  1879
  fixes f:: "'a::ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
chaieb@29842
  1880
  assumes lf: "linear f"
chaieb@29842
  1881
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
chaieb@29842
  1882
  using adjoint_works_lemma[OF lf] by metis
chaieb@29842
  1883
chaieb@29842
  1884
chaieb@29842
  1885
lemma adjoint_linear:
huffman@30582
  1886
  fixes f :: "'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
chaieb@29842
  1887
  assumes lf: "linear f"
chaieb@29842
  1888
  shows "linear (adjoint f)"
chaieb@29842
  1889
  by (simp add: linear_def vector_eq_ldot[symmetric] dot_radd dot_rmult adjoint_works[OF lf])
chaieb@29842
  1890
chaieb@29842
  1891
lemma adjoint_clauses:
huffman@30582
  1892
  fixes f:: "'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
chaieb@29842
  1893
  assumes lf: "linear f"
chaieb@29842
  1894
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
chaieb@29842
  1895
  and "adjoint f y \<bullet> x = y \<bullet> f x"
chaieb@29842
  1896
  by (simp_all add: adjoint_works[OF lf] dot_sym )
chaieb@29842
  1897
chaieb@29842
  1898
lemma adjoint_adjoint:
huffman@30582
  1899
  fixes f:: "'a::comm_ring_1 ^ 'n::finite \<Rightarrow> 'a ^ 'm::finite"
chaieb@29842
  1900
  assumes lf: "linear f"
chaieb@29842
  1901
  shows "adjoint (adjoint f) = f"
chaieb@29842
  1902
  apply (rule ext)
chaieb@29842
  1903
  by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf])
chaieb@29842
  1904
chaieb@29842
  1905
lemma adjoint_unique:
huffman@30582
  1906
  fixes f:: "'a::comm_ring_1 ^ 'n::finite \<Rightarrow> 'a ^ 'm::finite"
chaieb@29842
  1907
  assumes lf: "linear f" and u: "\<forall>x y. f' x \<bullet> y = x \<bullet> f y"
chaieb@29842
  1908
  shows "f' = adjoint f"
chaieb@29842
  1909
  apply (rule ext)
chaieb@29842
  1910
  using u
chaieb@29842
  1911
  by (simp add: vector_eq_rdot[symmetric] adjoint_clauses[OF lf])
chaieb@29842
  1912
huffman@29881
  1913
text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
chaieb@29842
  1914
chaieb@29842
  1915
consts generic_mult :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" (infixr "\<star>" 75)
chaieb@29842
  1916
huffman@30489
  1917
defs (overloaded)
huffman@30582
  1918
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)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
chaieb@29842
  1919
huffman@30489
  1920
abbreviation
chaieb@29842
  1921
  matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)
chaieb@29842
  1922
  where "m ** m' == m\<star> m'"
chaieb@29842
  1923
huffman@30489
  1924
defs (overloaded)
huffman@30582
  1925
  matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
chaieb@29842
  1926
huffman@30489
  1927
abbreviation
chaieb@29842
  1928
  matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)
huffman@30489
  1929
  where
chaieb@29842
  1930
  "m *v v == m \<star> v"
chaieb@29842
  1931
huffman@30489
  1932
defs (overloaded)
huffman@30582
  1933
  vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (x$i)) (UNIV :: 'm set)) :: 'a^'n"
chaieb@29842
  1934
huffman@30489
  1935
abbreviation
chaieb@29842
  1936
  vactor_matrix_mult' :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)
huffman@30489
  1937
  where
chaieb@29842
  1938
  "v v* m == v \<star> m"
chaieb@29842
  1939
huffman@30582
  1940
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
chaieb@29842
  1941
definition "(transp::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
huffman@30582
  1942
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
huffman@30582
  1943
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
huffman@30582
  1944
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
huffman@30582
  1945
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
chaieb@29842
  1946
chaieb@29842
  1947
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
chaieb@29842
  1948
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \<star> B) + (A \<star> C)"
chaieb@29842
  1949
  by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps)
chaieb@29842
  1950
huffman@30489
  1951
lemma setsum_delta':
huffman@30489
  1952
  assumes fS: "finite S" shows
huffman@30489
  1953
  "setsum (\<lambda>k. if a = k then b k else 0) S =
chaieb@29842
  1954
     (if a\<in> S then b a else 0)"
huffman@30489
  1955
  using setsum_delta[OF fS, of a b, symmetric]
chaieb@29842
  1956
  by (auto intro: setsum_cong)
chaieb@29842
  1957
huffman@30582
  1958
lemma matrix_mul_lid:
huffman@30582
  1959
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n::finite"
huffman@30582
  1960
  shows "mat 1 ** A = A"
chaieb@29842
  1961
  apply (simp add: matrix_matrix_mult_def mat_def)
chaieb@29842
  1962
  apply vector
huffman@30582
  1963
  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
  1964
huffman@30582
  1965
huffman@30582
  1966
lemma matrix_mul_rid:
huffman@30582
  1967
  fixes A :: "'a::semiring_1 ^ 'm::finite ^ 'n"
huffman@30582
  1968
  shows "A ** mat 1 = A"
chaieb@29842
  1969
  apply (simp add: matrix_matrix_mult_def mat_def)
chaieb@29842
  1970
  apply vector
huffman@30582
  1971
  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
  1972
chaieb@29842
  1973
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
chaieb@29842
  1974
  apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
chaieb@29842
  1975
  apply (subst setsum_commute)
chaieb@29842
  1976
  apply simp
chaieb@29842
  1977
  done
chaieb@29842
  1978
chaieb@29842
  1979
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
chaieb@29842
  1980
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
chaieb@29842
  1981
  apply (subst setsum_commute)
chaieb@29842
  1982
  apply simp
chaieb@29842
  1983
  done
chaieb@29842
  1984
huffman@30582
  1985
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n::finite)"
chaieb@29842
  1986
  apply (vector matrix_vector_mult_def mat_def)
huffman@30489
  1987
  by (simp add: cond_value_iff cond_application_beta
chaieb@29842
  1988
    setsum_delta' cong del: if_weak_cong)
chaieb@29842
  1989
chaieb@29842
  1990
lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)"
huffman@30582
  1991
  by (simp add: matrix_matrix_mult_def transp_def Cart_eq mult_commute)
huffman@30582
  1992
huffman@30582
  1993
lemma matrix_eq:
huffman@30582
  1994
  fixes A B :: "'a::semiring_1 ^ 'n::finite ^ 'm"
huffman@30582
  1995
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@29842
  1996
  apply auto
chaieb@29842
  1997
  apply (subst Cart_eq)
chaieb@29842
  1998
  apply clarify
huffman@30582
  1999
  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
  2000
  apply (erule_tac x="basis ia" in allE)
huffman@30582
  2001
  apply (erule_tac x="i" in allE)
huffman@30582
  2002
  by (auto simp add: basis_def cond_value_iff cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong)
chaieb@29842
  2003
huffman@30489
  2004
lemma matrix_vector_mul_component:
chaieb@29842
  2005
  shows "((A::'a::semiring_1^'n'^'m) *v x)$k = (A$k) \<bullet> x"
huffman@30582
  2006
  by (simp add: matrix_vector_mult_def dot_def)
chaieb@29842
  2007
chaieb@29842
  2008
lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) v* A) \<bullet> y = x \<bullet> (A *v y)"
huffman@30582
  2009
  apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
chaieb@29842
  2010
  apply (subst setsum_commute)
chaieb@29842
  2011
  by simp
chaieb@29842
  2012
chaieb@29842
  2013
lemma transp_mat: "transp (mat n) = mat n"
chaieb@29842
  2014
  by (vector transp_def mat_def)
chaieb@29842
  2015
chaieb@29842
  2016
lemma transp_transp: "transp(transp A) = A"
chaieb@29842
  2017
  by (vector transp_def)
chaieb@29842
  2018
huffman@30489
  2019
lemma row_transp:
chaieb@29842
  2020
  fixes A:: "'a::semiring_1^'n^'m"
chaieb@29842
  2021
  shows "row i (transp A) = column i A"
huffman@30582
  2022
  by (simp add: row_def column_def transp_def Cart_eq)
chaieb@29842
  2023
chaieb@29842
  2024
lemma column_transp:
chaieb@29842
  2025
  fixes A:: "'a::semiring_1^'n^'m"
chaieb@29842
  2026
  shows "column i (transp A) = row i A"
huffman@30582
  2027
  by (simp add: row_def column_def transp_def Cart_eq)
chaieb@29842
  2028
chaieb@29842
  2029
lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A"
huffman@30582
  2030
by (auto simp add: rows_def columns_def row_transp intro: set_ext)
chaieb@29842
  2031
chaieb@29842
  2032
lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp)
chaieb@29842
  2033
chaieb@29842
  2034
text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
chaieb@29842
  2035
chaieb@29842
  2036
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
chaieb@29842
  2037
  by (simp add: matrix_vector_mult_def dot_def)
chaieb@29842
  2038
huffman@30582
  2039
lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
huffman@30582
  2040
  by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute)
chaieb@29842
  2041
chaieb@29842
  2042
lemma vector_componentwise:
huffman@30582
  2043
  "(x::'a::ring_1^'n::finite) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) (UNIV :: 'n set))"
chaieb@29842
  2044
  apply (subst basis_expansion[symmetric])
huffman@30582
  2045
  by (vector Cart_eq setsum_component)
chaieb@29842
  2046
chaieb@29842
  2047
lemma linear_componentwise:
huffman@30582
  2048
  fixes f:: "'a::ring_1 ^ 'm::finite \<Rightarrow> 'a ^ 'n"
huffman@30582
  2049
  assumes lf: "linear f"
huffman@30582
  2050
  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
chaieb@29842
  2051
proof-
huffman@30582
  2052
  let ?M = "(UNIV :: 'm set)"
huffman@30582
  2053
  let ?N = "(UNIV :: 'n set)"
chaieb@29842
  2054
  have fM: "finite ?M" by simp
chaieb@29842
  2055
  have "?rhs = (setsum (\<lambda>i.(x$i) *s f (basis i) ) ?M)$j"
huffman@30582
  2056
    unfolding vector_smult_component[symmetric]
huffman@30582
  2057
    unfolding setsum_component[of "(\<lambda>i.(x$i) *s f (basis i :: 'a^'m))" ?M]
chaieb@29842
  2058
    ..
chaieb@29842
  2059
  then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion ..
chaieb@29842
  2060
qed
chaieb@29842
  2061
chaieb@29842
  2062
text{* Inverse matrices  (not necessarily square) *}
chaieb@29842
  2063
chaieb@29842
  2064
definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
chaieb@29842
  2065
chaieb@29842
  2066
definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
chaieb@29842
  2067
        (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
chaieb@29842
  2068
chaieb@29842
  2069
text{* Correspondence between matrices and linear operators. *}
chaieb@29842
  2070
chaieb@29842
  2071
definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
chaieb@29842
  2072
where "matrix f = (\<chi> i j. (f(basis j))$i)"
chaieb@29842
  2073
chaieb@29842
  2074
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::'a::comm_semiring_1 ^ 'n))"
huffman@30582
  2075
  by (simp add: linear_def matrix_vector_mult_def Cart_eq ring_simps setsum_right_distrib setsum_addf)
huffman@30582
  2076
huffman@30582
  2077
lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n::finite)"
huffman@30582
  2078
apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute)
chaieb@29842
  2079
apply clarify
chaieb@29842
  2080
apply (rule linear_componentwise[OF lf, symmetric])
chaieb@29842
  2081
done
chaieb@29842
  2082
huffman@30582
  2083
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
  2084
huffman@30582
  2085
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n::finite)) = A"
chaieb@29842
  2086
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
chaieb@29842
  2087
huffman@30489
  2088
lemma matrix_compose:
huffman@30582
  2089
  assumes lf: "linear (f::'a::comm_ring_1^'n::finite \<Rightarrow> 'a^'m::finite)"
huffman@30582
  2090
  and lg: "linear (g::'a::comm_ring_1^'m::finite \<Rightarrow> 'a^'k)"
chaieb@29842
  2091
  shows "matrix (g o f) = matrix g ** matrix f"
chaieb@29842
  2092
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
chaieb@29842
  2093
  by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
chaieb@29842
  2094
huffman@30582
  2095
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
  2096
  by (simp add: matrix_vector_mult_def transp_def Cart_eq mult_commute)
huffman@30582
  2097
huffman@30582
  2098
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
  2099
  apply (rule adjoint_unique[symmetric])
chaieb@29842
  2100
  apply (rule matrix_vector_mul_linear)
huffman@30582
  2101
  apply (simp add: transp_def dot_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
chaieb@29842
  2102
  apply (subst setsum_commute)
chaieb@29842
  2103
  apply (auto simp add: mult_ac)
chaieb@29842
  2104
  done
chaieb@29842
  2105
huffman@30582
  2106
lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n::finite \<Rightarrow> 'a ^ 'm::finite)"
chaieb@29842
  2107
  shows "matrix(adjoint f) = transp(matrix f)"
chaieb@29842
  2108
  apply (subst matrix_vector_mul[OF lf])
chaieb@29842
  2109
  unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
chaieb@29842
  2110
chaieb@29842
  2111
subsection{* Interlude: Some properties of real sets *}
chaieb@29842
  2112
chaieb@29842
  2113
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
  2114
  shows "\<forall>n \<ge> m. d n < e m"
chaieb@29842
  2115
  using prems apply auto
chaieb@29842
  2116
  apply (erule_tac x="n" in allE)
chaieb@29842
  2117
  apply (erule_tac x="n" in allE)
chaieb@29842
  2118
  apply auto
chaieb@29842
  2119
  done
chaieb@29842
  2120
chaieb@29842
  2121
huffman@30489
  2122
lemma real_convex_bound_lt:
chaieb@29842
  2123
  assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"
huffman@30489
  2124
  and uv: "u + v = 1"
chaieb@29842
  2125
  shows "u * x + v * y < a"
chaieb@29842
  2126
proof-
chaieb@29842
  2127
  have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
chaieb@29842
  2128
  have "a = a * (u + v)" unfolding uv  by simp
chaieb@29842
  2129
  hence th: "u * a + v * a = a" by (simp add: ring_simps)
chaieb@29842
  2130
  from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_compare_simps)
chaieb@29842
  2131
  from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_compare_simps)
chaieb@29842
  2132
  from xa ya u v have "u * x + v * y < u * a + v * a"
chaieb@29842
  2133
    apply (cases "u = 0", simp_all add: uv')
chaieb@29842
  2134
    apply(rule mult_strict_left_mono)
chaieb@29842
  2135
    using uv' apply simp_all
huffman@30489
  2136
chaieb@29842
  2137
    apply (rule add_less_le_mono)
chaieb@29842
  2138
    apply(rule mult_strict_left_mono)
chaieb@29842
  2139
    apply simp_all
chaieb@29842
  2140
    apply (rule mult_left_mono)
chaieb@29842
  2141
    apply simp_all
chaieb@29842
  2142
    done
chaieb@29842
  2143
  thus ?thesis unfolding th .
chaieb@29842
  2144
qed
chaieb@29842
  2145
huffman@30489
  2146
lemma real_convex_bound_le:
chaieb@29842
  2147
  assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"
huffman@30489
  2148
  and uv: "u + v = 1"
chaieb@29842
  2149
  shows "u * x + v * y \<le> a"
chaieb@29842
  2150
proof-
chaieb@29842
  2151
  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
  2152
  also have "\<dots> \<le> (u + v) * a" by (simp add: ring_simps)
chaieb@29842
  2153
  finally show ?thesis unfolding uv by simp
chaieb@29842
  2154
qed
chaieb@29842
  2155
chaieb@29842
  2156
lemma infinite_enumerate: assumes fS: "infinite S"
chaieb@29842
  2157
  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
chaieb@29842
  2158
unfolding subseq_def
chaieb@29842
  2159
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
chaieb@29842
  2160
chaieb@29842
  2161
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
  2162
apply auto
chaieb@29842
  2163
apply (rule_tac x="d/2" in exI)
chaieb@29842
  2164
apply auto
chaieb@29842
  2165
done
chaieb@29842
  2166
chaieb@29842
  2167
huffman@30489
  2168
lemma triangle_lemma:
chaieb@29842
  2169
  assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
chaieb@29842
  2170
  shows "x <= y + z"
chaieb@29842
  2171
proof-
chaieb@29842
  2172
  have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y  by (simp add: zero_compare_simps)
chaieb@29842
  2173
  with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square ring_simps)
chaieb@29842
  2174
  from y z have yz: "y + z \<ge> 0" by arith
chaieb@29842
  2175
  from power2_le_imp_le[OF th yz] show ?thesis .
chaieb@29842
  2176
qed
chaieb@29842
  2177
chaieb@29842
  2178
huffman@30582
  2179
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
huffman@30582
  2180
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@29842
  2181
proof-
huffman@30582
  2182
  let ?S = "(UNIV :: 'n set)"
chaieb@29842
  2183
  {assume H: "?rhs"
chaieb@29842
  2184
    then have ?lhs by auto}
chaieb@29842
  2185
  moreover
chaieb@29842
  2186
  {assume H: "?lhs"
huffman@30582
  2187
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
chaieb@29842
  2188
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
huffman@30582
  2189
    {fix i
huffman@30582
  2190
      from f have "P i (f i)" by metis
huffman@30582
  2191
      then have "P i (?x$i)" by auto
chaieb@29842
  2192
    }
huffman@30582
  2193
    hence "\<forall>i. P i (?x$i)" by metis
chaieb@29842
  2194
    hence ?rhs by metis }
chaieb@29842
  2195
  ultimately show ?thesis by metis
huffman@30489
  2196
qed
chaieb@29842
  2197
chaieb@29842
  2198
(* Supremum and infimum of real sets *)
chaieb@29842
  2199
chaieb@29842
  2200
chaieb@29842
  2201
definition rsup:: "real set \<Rightarrow> real" where
chaieb@29842
  2202
  "rsup S = (SOME a. isLub UNIV S a)"
chaieb@29842
  2203
chaieb@29842
  2204
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
  2205
chaieb@29842
  2206
lemma rsup: assumes Se: "S \<noteq> {}" and b: "\<exists>b. S *<= b"
chaieb@29842
  2207
  shows "isLub UNIV S (rsup S)"
chaieb@29842
  2208
using Se b
chaieb@29842
  2209
unfolding rsup_def
chaieb@29842
  2210
apply clarify
chaieb@29842
  2211
apply (rule someI_ex)
chaieb@29842
  2212
apply (rule reals_complete)
chaieb@29842
  2213
by (auto simp add: isUb_def setle_def)
chaieb@29842
  2214
chaieb@29842
  2215
lemma rsup_le: assumes Se: "S \<noteq> {}" and Sb: "S *<= b" shows "rsup S \<le> b"
chaieb@29842
  2216
proof-
chaieb@29842
  2217
  from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def)
huffman@30489
  2218
  from rsup[OF Se] Sb have "isLub UNIV S (rsup S)"  by blast
chaieb@29842
  2219
  then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def)
chaieb@29842
  2220
qed
chaieb@29842
  2221
chaieb@29842
  2222
lemma rsup_finite_Max: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2223
  shows "rsup S = Max S"
chaieb@29842
  2224
using fS Se
chaieb@29842
  2225
proof-
chaieb@29842
  2226
  let ?m = "Max S"
chaieb@29842
  2227
  from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis
chaieb@29842
  2228
  with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def)
huffman@30489
  2229
  from Max_in[OF fS Se] lub have mrS: "?m \<le> rsup S"
chaieb@29842
  2230
    by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)
huffman@30489
  2231
  moreover
chaieb@29842
  2232
  have "rsup S \<le> ?m" using Sm lub
chaieb@29842
  2233
    by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
huffman@30489
  2234
  ultimately  show ?thesis by arith
chaieb@29842
  2235
qed
chaieb@29842
  2236
chaieb@29842
  2237
lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2238
  shows "rsup S \<in> S"
chaieb@29842
  2239
  using rsup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis
chaieb@29842
  2240
chaieb@29842
  2241
lemma rsup_finite_Ub: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2242
  shows "isUb S S (rsup S)"
chaieb@29842
  2243
  using rsup_finite_Max[OF fS Se] rsup_finite_in[OF fS Se] Max_ge[OF fS]
chaieb@29842
  2244
  unfolding isUb_def setle_def by metis
chaieb@29842
  2245
chaieb@29842
  2246
lemma rsup_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2247
  shows "a \<le> rsup S \<longleftrightarrow> (\<exists> x \<in> S. a \<le> x)"
chaieb@29842
  2248
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
chaieb@29842
  2249
chaieb@29842
  2250
lemma rsup_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2251
  shows "a \<ge> rsup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)"
chaieb@29842
  2252
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
chaieb@29842
  2253
chaieb@29842
  2254
lemma rsup_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2255
  shows "a < rsup S \<longleftrightarrow> (\<exists> x \<in> S. a < x)"
chaieb@29842
  2256
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
chaieb@29842
  2257
chaieb@29842
  2258
lemma rsup_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2259
  shows "a > rsup S \<longleftrightarrow> (\<forall> x \<in> S. a > x)"
chaieb@29842
  2260
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
chaieb@29842
  2261
chaieb@29842
  2262
lemma rsup_unique: assumes b: "S *<= b" and S: "\<forall>b' < b. \<exists>x \<in> S. b' < x"
chaieb@29842
  2263
  shows "rsup S = b"
huffman@30489
  2264
using b S
chaieb@29842
  2265
unfolding setle_def rsup_alt
chaieb@29842
  2266
apply -
chaieb@29842
  2267
apply (rule some_equality)
chaieb@29842
  2268
apply (metis  linorder_not_le order_eq_iff[symmetric])+
chaieb@29842
  2269
done
chaieb@29842
  2270
chaieb@29842
  2271
lemma rsup_le_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. T *<= b) \<Longrightarrow> rsup S \<le> rsup T"
chaieb@29842
  2272
  apply (rule rsup_le)
chaieb@29842
  2273
  apply simp
chaieb@29842
  2274
  using rsup[of T] by (auto simp add: isLub_def leastP_def setge_def setle_def isUb_def)
chaieb@29842
  2275
chaieb@29842
  2276
lemma isUb_def': "isUb R S = (\<lambda>x. S *<= x \<and> x \<in> R)"
chaieb@29842
  2277
  apply (rule ext)
chaieb@29842
  2278
  by (metis isUb_def)
chaieb@29842
  2279
chaieb@29842
  2280
lemma UNIV_trivial: "UNIV x" using UNIV_I[of x] by (metis mem_def)
chaieb@29842
  2281
lemma rsup_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
chaieb@29842
  2282
  shows "a \<le> rsup S \<and> rsup S \<le> b"
chaieb@29842
  2283
proof-
chaieb@29842
  2284
  from rsup[OF Se] u have lub: "isLub UNIV S (rsup S)" by blast
chaieb@29842
  2285
  hence b: "rsup S \<le> b" using u by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
chaieb@29842
  2286
  from Se obtain y where y: "y \<in> S" by blast
chaieb@29842
  2287
  from lub l have "a \<le> rsup S" apply (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
chaieb@29842
  2288
    apply (erule ballE[where x=y])
chaieb@29842
  2289
    apply (erule ballE[where x=y])
chaieb@29842
  2290
    apply arith
chaieb@29842
  2291
    using y apply auto
chaieb@29842
  2292
    done
chaieb@29842
  2293
  with b show ?thesis by blast
chaieb@29842
  2294
qed
chaieb@29842
  2295
chaieb@29842
  2296
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
  2297
  unfolding abs_le_interval_iff  using rsup_bounds[of S "-a" a]
chaieb@29842
  2298
  by (auto simp add: setge_def setle_def)
chaieb@29842
  2299
chaieb@29842
  2300
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
  2301
proof-
chaieb@29842
  2302
  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
  2303
  show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th
chaieb@29842
  2304
    by  (auto simp add: setge_def setle_def)
chaieb@29842
  2305
qed
chaieb@29842
  2306
chaieb@29842
  2307
definition rinf:: "real set \<Rightarrow> real" where
chaieb@29842
  2308
  "rinf S = (SOME a. isGlb UNIV S a)"
chaieb@29842
  2309
chaieb@29842
  2310
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
  2311
chaieb@29842
  2312
lemma reals_complete_Glb: assumes Se: "\<exists>x. x \<in> S" and lb: "\<exists> y. isLb UNIV S y"
chaieb@29842
  2313
  shows "\<exists>(t::real). isGlb UNIV S t"
chaieb@29842
  2314
proof-
chaieb@29842
  2315
  let ?M = "uminus ` S"
chaieb@29842
  2316
  from lb have th: "\<exists>y. isUb UNIV ?M y" apply (auto simp add: isUb_def isLb_def setle_def setge_def)
chaieb@29842
  2317
    by (rule_tac x="-y" in exI, auto)
chaieb@29842
  2318
  from Se have Me: "\<exists>x. x \<in> ?M" by blast
chaieb@29842
  2319
  from reals_complete[OF Me th] obtain t where t: "isLub UNIV ?M t" by blast
chaieb@29842
  2320
  have "isGlb UNIV S (- t)" using t
chaieb@29842
  2321
    apply (auto simp add: isLub_def isGlb_def leastP_def greatestP_def setle_def setge_def isUb_def isLb_def)
chaieb@29842
  2322
    apply (erule_tac x="-y" in allE)
chaieb@29842
  2323
    apply auto
chaieb@29842
  2324
    done
chaieb@29842
  2325
  then show ?thesis by metis
chaieb@29842
  2326
qed
chaieb@29842
  2327
chaieb@29842
  2328
lemma rinf: assumes Se: "S \<noteq> {}" and b: "\<exists>b. b <=* S"
chaieb@29842
  2329
  shows "isGlb UNIV S (rinf S)"
chaieb@29842
  2330
using Se b
chaieb@29842
  2331
unfolding rinf_def
chaieb@29842
  2332
apply clarify
chaieb@29842
  2333
apply (rule someI_ex)
chaieb@29842
  2334
apply (rule reals_complete_Glb)
chaieb@29842
  2335
apply (auto simp add: isLb_def setle_def setge_def)
chaieb@29842
  2336
done
chaieb@29842
  2337
chaieb@29842
  2338
lemma rinf_ge: assumes Se: "S \<noteq> {}" and Sb: "b <=* S" shows "rinf S \<ge> b"
chaieb@29842
  2339
proof-
chaieb@29842
  2340
  from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def)
huffman@30489
  2341
  from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)"  by blast
chaieb@29842
  2342
  then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def)
chaieb@29842
  2343
qed
chaieb@29842
  2344
chaieb@29842
  2345
lemma rinf_finite_Min: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2346
  shows "rinf S = Min S"
chaieb@29842
  2347
using fS Se
chaieb@29842
  2348
proof-
chaieb@29842
  2349
  let ?m = "Min S"
chaieb@29842
  2350
  from Min_le[OF fS] have Sm: "\<forall> x\<in> S. x \<ge> ?m" by metis
chaieb@29842
  2351
  with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def)
huffman@30489
  2352
  from Min_in[OF fS Se] glb have mrS: "?m \<ge> rinf S"
chaieb@29842
  2353
    by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def)
huffman@30489
  2354
  moreover
chaieb@29842
  2355
  have "rinf S \<ge> ?m" using Sm glb
chaieb@29842
  2356
    by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def)
huffman@30489
  2357
  ultimately  show ?thesis by arith
chaieb@29842
  2358
qed
chaieb@29842
  2359
chaieb@29842
  2360
lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2361
  shows "rinf S \<in> S"
chaieb@29842
  2362
  using rinf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis
chaieb@29842
  2363
chaieb@29842
  2364
lemma rinf_finite_Lb: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2365
  shows "isLb S S (rinf S)"
chaieb@29842
  2366
  using rinf_finite_Min[OF fS Se] rinf_finite_in[OF fS Se] Min_le[OF fS]
chaieb@29842
  2367
  unfolding isLb_def setge_def by metis
chaieb@29842
  2368
chaieb@29842
  2369
lemma rinf_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2370
  shows "a \<le> rinf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)"
chaieb@29842
  2371
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
chaieb@29842
  2372
chaieb@29842
  2373
lemma rinf_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2374
  shows "a \<ge> rinf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)"
chaieb@29842
  2375
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
chaieb@29842
  2376
chaieb@29842
  2377
lemma rinf_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2378
  shows "a < rinf S \<longleftrightarrow> (\<forall> x \<in> S. a < x)"
chaieb@29842
  2379
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
chaieb@29842
  2380
chaieb@29842
  2381
lemma rinf_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2382
  shows "a > rinf S \<longleftrightarrow> (\<exists> x \<in> S. a > x)"
chaieb@29842
  2383
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
chaieb@29842
  2384
chaieb@29842
  2385
lemma rinf_unique: assumes b: "b <=* S" and S: "\<forall>b' > b. \<exists>x \<in> S. b' > x"
chaieb@29842
  2386
  shows "rinf S = b"
huffman@30489
  2387
using b S
chaieb@29842
  2388
unfolding setge_def rinf_alt
chaieb@29842
  2389
apply -
chaieb@29842
  2390
apply (rule some_equality)
chaieb@29842
  2391
apply (metis  linorder_not_le order_eq_iff[symmetric])+
chaieb@29842
  2392
done
chaieb@29842
  2393
chaieb@29842
  2394
lemma rinf_ge_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. b <=* T) \<Longrightarrow> rinf S >= rinf T"
chaieb@29842
  2395
  apply (rule rinf_ge)
chaieb@29842
  2396
  apply simp
chaieb@29842
  2397
  using rinf[of T] by (auto simp add: isGlb_def greatestP_def setge_def setle_def isLb_def)
chaieb@29842
  2398
chaieb@29842
  2399
lemma isLb_def': "isLb R S = (\<lambda>x. x <=* S \<and> x \<in> R)"
chaieb@29842
  2400
  apply (rule ext)
chaieb@29842
  2401
  by (metis isLb_def)
chaieb@29842
  2402
chaieb@29842
  2403
lemma rinf_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
chaieb@29842
  2404
  shows "a \<le> rinf S \<and> rinf S \<le> b"
chaieb@29842
  2405
proof-
chaieb@29842
  2406
  from rinf[OF Se] l have lub: "isGlb UNIV S (rinf S)" by blast
chaieb@29842
  2407
  hence b: "a \<le> rinf S" using l by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
chaieb@29842
  2408
  from Se obtain y where y: "y \<in> S" by blast
chaieb@29842
  2409
  from lub u have "b \<ge> rinf S" apply (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
chaieb@29842
  2410
    apply (erule ballE[where x=y])
chaieb@29842
  2411
    apply (erule ballE[where x=y])
chaieb@29842
  2412
    apply arith
chaieb@29842
  2413
    using y apply auto
chaieb@29842
  2414
    done
chaieb@29842
  2415
  with b show ?thesis by blast
chaieb@29842
  2416
qed
chaieb@29842
  2417
chaieb@29842
  2418
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
  2419
  unfolding abs_le_interval_iff  using rinf_bounds[of S "-a" a]
chaieb@29842
  2420
  by (auto simp add: setge_def setle_def)
chaieb@29842
  2421
chaieb@29842
  2422
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
  2423
proof-
chaieb@29842
  2424
  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
  2425
  show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th
chaieb@29842
  2426
    by  (auto simp add: setge_def setle_def)
chaieb@29842
  2427
qed
chaieb@29842
  2428
chaieb@29842
  2429
chaieb@29842
  2430
chaieb@29842
  2431<