src/HOL/Library/Euclidean_Space.thy
author huffman
Sat, 21 Feb 2009 15:39:59 -0800
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real_inner class instance for vectors
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4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
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(* Title:      Library/Euclidean_Space
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   ID:         $Id: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
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   Author:     Amine Chaieb, University of Cambridge
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*)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
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4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
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header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
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theory Euclidean_Space
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  imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Complex_Main 
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  Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type
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  Inner_Product
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  uses ("normarith.ML")
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begin
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4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
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text{* Some common special cases.*}
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lemma forall_1: "(\<forall>(i::'a::{order,one}). 1 <= i \<and> i <= 1 --> P i) \<longleftrightarrow> P 1"
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  by (metis order_eq_iff)
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lemma forall_dimindex_1: "(\<forall>i \<in> {1..dimindex(UNIV:: 1 set)}. P i) \<longleftrightarrow> P 1"
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  by (simp add: dimindex_def)
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lemma forall_2: "(\<forall>(i::nat). 1 <= i \<and> i <= 2 --> P i) \<longleftrightarrow> P 1 \<and> P 2"
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proof-
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  have "\<And>i::nat. 1 <= i \<and> i <= 2 \<longleftrightarrow> i = 1 \<or> i = 2" by arith
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  thus ?thesis by metis
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qed
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lemma forall_3: "(\<forall>(i::nat). 1 <= i \<and> i <= 3 --> P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
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proof-
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  have "\<And>i::nat. 1 <= i \<and> i <= 3 \<longleftrightarrow> i = 1 \<or> i = 2 \<or> i = 3" by arith
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  thus ?thesis by metis
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qed
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lemma setsum_singleton[simp]: "setsum f {x} = f x" by simp
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lemma setsum_1: "setsum f {(1::'a::{order,one})..1} = f 1" 
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  by (simp add: atLeastAtMost_singleton)
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lemma setsum_2: "setsum f {1::nat..2} = f 1 + f 2" 
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  by (simp add: nat_number  atLeastAtMostSuc_conv add_commute)
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lemma setsum_3: "setsum f {1::nat..3} = f 1 + f 2 + f 3" 
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  by (simp add: nat_number  atLeastAtMostSuc_conv add_commute)
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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 : {1 .. dimindex (UNIV :: 'b set)}.
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  x$i <= y$i)"
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definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i : {1 ..
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  dimindex (UNIV :: 'b set)}. x$i < y$i)"
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instance by (intro_classes)
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end
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instantiation "^" :: (scaleR, type) scaleR
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begin
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definition vector_scaleR_def: "scaleR = (\<lambda> r x.  (\<chi> i. scaleR r (x$i)))" 
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instance ..
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end
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text{* Also the scalar-vector multiplication. *}
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definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'n" (infixr "*s" 75)
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  where "c *s x = (\<chi> i. c * (x$i))"
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text{* Constant Vectors *}
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definition "vec x = (\<chi> i. x)"
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text{* Dot products. *}
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definition dot :: "'a::{comm_monoid_add, times} ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" (infix "\<bullet>" 70) where
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  "x \<bullet> y = setsum (\<lambda>i. x$i * y$i) {1 .. dimindex (UNIV:: 'n set)}"
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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 dimindex_def)
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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 dimindex_def nat_number)
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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 dimindex_def nat_number)
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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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lemmas Cart_lambda_beta' = Cart_lambda_beta[rule_format]
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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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  Method.thms_args (Method.SIMPLE_METHOD' o vector_arith_tac)
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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: " i \<in> {1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (vec x :: 'a ^ 'n)$i = x" 
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  by (vector vec_def) 
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lemma vector_add_component: 
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  fixes x y :: "'a::{plus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
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  shows "(x + y)$i = x$i + y$i"
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  using i by vector
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lemma vector_minus_component: 
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  fixes x y :: "'a::{minus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
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  shows "(x - y)$i = x$i - y$i"
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  using i  by vector
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lemma vector_mult_component: 
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  fixes x y :: "'a::{times} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
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  shows "(x * y)$i = x$i * y$i"
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  using i by vector
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lemma vector_smult_component: 
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  fixes y :: "'a::{times} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
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  shows "(c *s y)$i = c * (y$i)"
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  using i by vector
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lemma vector_uminus_component: 
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  fixes x :: "'a::{uminus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
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  shows "(- x)$i = - (x$i)"
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  using i by vector
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lemma vector_scaleR_component:
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  fixes x :: "'a::scaleR ^ 'n"
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  assumes i: "i \<in> {1 .. dimindex(UNIV :: 'n set)}"
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  shows "(scaleR r x)$i = scaleR r (x$i)"
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  using i by vector
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lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
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lemmas vector_component =
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  vec_component vector_add_component vector_mult_component
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  vector_smult_component vector_minus_component vector_uminus_component
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  vector_scaleR_component cond_component
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subsection {* Some frequently useful arithmetic lemmas over vectors. *}
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instance "^" :: (semigroup_add,type) semigroup_add 
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  apply (intro_classes) by (vector add_assoc)
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instance "^" :: (monoid_add,type) monoid_add 
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  apply (intro_classes) by vector+ 
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instance "^" :: (group_add,type) group_add 
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  apply (intro_classes) by (vector algebra_simps)+ 
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instance "^" :: (ab_semigroup_add,type) ab_semigroup_add 
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  apply (intro_classes) by (vector add_commute)
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instance "^" :: (comm_monoid_add,type) comm_monoid_add
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  apply (intro_classes) by vector
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instance "^" :: (ab_group_add,type) ab_group_add 
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  apply (intro_classes) by vector+
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instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add 
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  apply (intro_classes)
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  by (vector Cart_eq)+
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instance "^" :: (cancel_ab_semigroup_add,type) cancel_ab_semigroup_add
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  apply (intro_classes)
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  by (vector Cart_eq)
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instance "^" :: (real_vector, type) real_vector
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  by default (vector scaleR_left_distrib scaleR_right_distrib)+
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instance "^" :: (semigroup_mult,type) semigroup_mult 
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  apply (intro_classes) by (vector mult_assoc)
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instance "^" :: (monoid_mult,type) monoid_mult 
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  apply (intro_classes) by vector+
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instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult 
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  apply (intro_classes) by (vector mult_commute)
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instance "^" :: (ab_semigroup_idem_mult,type) ab_semigroup_idem_mult 
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  apply (intro_classes) by (vector mult_idem)
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instance "^" :: (comm_monoid_mult,type) comm_monoid_mult 
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  apply (intro_classes) by vector
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fun vector_power :: "('a::{one,times} ^'n) \<Rightarrow> nat \<Rightarrow> 'a^'n" where
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  "vector_power x 0 = 1"
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  | "vector_power x (Suc n) = x * vector_power x n"
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instantiation "^" :: (recpower,type) recpower 
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begin
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  definition vec_power_def: "op ^ \<equiv> vector_power"
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  instance 
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  apply (intro_classes) by (simp_all add: vec_power_def) 
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end
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instance "^" :: (semiring,type) semiring
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  apply (intro_classes) by (vector ring_simps)+
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instance "^" :: (semiring_0,type) semiring_0
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  apply (intro_classes) by (vector ring_simps)+
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instance "^" :: (semiring_1,type) semiring_1
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  apply (intro_classes) apply vector using dimindex_ge_1 by auto 
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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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  "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
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  apply (induct n)
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  apply vector
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  apply vector
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  done
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lemma zero_index[simp]: 
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   284
  "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (0 :: 'a::zero ^'n)$i = 0" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   285
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   286
lemma one_index[simp]: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   287
  "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (1 :: 'a::one ^'n)$i = 1" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   288
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   289
lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \<noteq> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   290
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   291
  have "(1::'a) + of_nat n = 0 \<longleftrightarrow> of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   292
  also have "\<dots> \<longleftrightarrow> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff) 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   293
  finally show ?thesis by simp 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   294
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   295
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   296
instance "^" :: (semiring_char_0,type) semiring_char_0 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   297
proof (intro_classes) 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   298
  fix m n ::nat
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   299
  show "(of_nat m :: 'a^'b) = of_nat n \<longleftrightarrow> m = n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   300
  proof(induct m arbitrary: n)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   301
    case 0 thus ?case apply vector 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   302
      apply (induct n,auto simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   303
      using dimindex_ge_1 apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   304
      apply vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   305
      by (auto simp add: of_nat_index one_plus_of_nat_neq_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   306
  next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   307
    case (Suc n m)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   308
    thus ?case  apply vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   309
      apply (induct m, auto simp add: ring_simps of_nat_index zero_index)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   310
      using dimindex_ge_1 apply simp apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   311
      apply (simp add: one_plus_of_nat_neq_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   312
      using dimindex_ge_1 apply simp apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   313
      apply (simp add: vector_component one_index of_nat_index)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   314
      apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   315
      using  dimindex_ge_1 apply simp apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   316
      apply (simp add: vector_component one_index of_nat_index)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   317
      apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   318
      using dimindex_ge_1 apply simp apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   319
      apply (simp add: vector_component one_index of_nat_index)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   320
      done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   321
  qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   322
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   323
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   324
instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes
30039
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   325
instance "^" :: (ring_char_0,type) ring_char_0 by intro_classes
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   326
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   327
lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"  
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   328
  by (vector mult_assoc)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   329
lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   330
  by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   331
lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   332
  by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   333
lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   334
lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   335
lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   336
  by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   337
lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   338
lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   339
lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   340
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   341
lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   342
  by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   343
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   344
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   345
  apply (auto simp add: vec_def Cart_eq vec_component Cart_lambda_beta )
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   346
  using dimindex_ge_1 apply auto done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   347
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   348
subsection {* Square root of sum of squares *}
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   349
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   350
definition
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   351
  "setL2 f A = sqrt (\<Sum>i\<in>A. (f i)\<twosuperior>)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   352
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   353
lemma setL2_cong:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   354
  "\<lbrakk>A = B; \<And>x. x \<in> B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   355
  unfolding setL2_def by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   356
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   357
lemma strong_setL2_cong:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   358
  "\<lbrakk>A = B; \<And>x. x \<in> B =simp=> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   359
  unfolding setL2_def simp_implies_def by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   360
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   361
lemma setL2_infinite [simp]: "\<not> finite A \<Longrightarrow> setL2 f A = 0"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   362
  unfolding setL2_def by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   363
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   364
lemma setL2_empty [simp]: "setL2 f {} = 0"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   365
  unfolding setL2_def by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   366
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   367
lemma setL2_insert [simp]:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   368
  "\<lbrakk>finite F; a \<notin> F\<rbrakk> \<Longrightarrow>
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   369
    setL2 f (insert a F) = sqrt ((f a)\<twosuperior> + (setL2 f F)\<twosuperior>)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   370
  unfolding setL2_def by (simp add: setsum_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   371
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   372
lemma setL2_nonneg [simp]: "0 \<le> setL2 f A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   373
  unfolding setL2_def by (simp add: setsum_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   374
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   375
lemma setL2_0': "\<forall>a\<in>A. f a = 0 \<Longrightarrow> setL2 f A = 0"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   376
  unfolding setL2_def by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   377
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   378
lemma setL2_mono:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   379
  assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   380
  assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   381
  shows "setL2 f K \<le> setL2 g K"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   382
  unfolding setL2_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   383
  by (simp add: setsum_nonneg setsum_mono power_mono prems)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   384
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   385
lemma setL2_right_distrib:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   386
  "0 \<le> r \<Longrightarrow> r * setL2 f A = setL2 (\<lambda>x. r * f x) A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   387
  unfolding setL2_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   388
  apply (simp add: power_mult_distrib)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   389
  apply (simp add: setsum_right_distrib [symmetric])
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   390
  apply (simp add: real_sqrt_mult setsum_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   391
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   392
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   393
lemma setL2_left_distrib:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   394
  "0 \<le> r \<Longrightarrow> setL2 f A * r = setL2 (\<lambda>x. f x * r) A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   395
  unfolding setL2_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   396
  apply (simp add: power_mult_distrib)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   397
  apply (simp add: setsum_left_distrib [symmetric])
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   398
  apply (simp add: real_sqrt_mult setsum_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   399
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   400
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   401
lemma setsum_nonneg_eq_0_iff:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   402
  fixes f :: "'a \<Rightarrow> 'b::pordered_ab_group_add"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   403
  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)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   404
  apply (induct set: finite, simp)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   405
  apply (simp add: add_nonneg_eq_0_iff setsum_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   406
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   407
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   408
lemma setL2_eq_0_iff: "finite A \<Longrightarrow> setL2 f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   409
  unfolding setL2_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   410
  by (simp add: setsum_nonneg setsum_nonneg_eq_0_iff)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   411
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   412
lemma setL2_triangle_ineq:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   413
  shows "setL2 (\<lambda>i. f i + g i) A \<le> setL2 f A + setL2 g A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   414
proof (cases "finite A")
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   415
  case False
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   416
  thus ?thesis by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   417
next
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   418
  case True
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   419
  thus ?thesis
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   420
  proof (induct set: finite)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   421
    case empty
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   422
    show ?case by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   423
  next
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   424
    case (insert x F)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   425
    hence "sqrt ((f x + g x)\<twosuperior> + (setL2 (\<lambda>i. f i + g i) F)\<twosuperior>) \<le>
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   426
           sqrt ((f x + g x)\<twosuperior> + (setL2 f F + setL2 g F)\<twosuperior>)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   427
      by (intro real_sqrt_le_mono add_left_mono power_mono insert
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   428
                setL2_nonneg add_increasing zero_le_power2)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   429
    also have
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   430
      "\<dots> \<le> sqrt ((f x)\<twosuperior> + (setL2 f F)\<twosuperior>) + sqrt ((g x)\<twosuperior> + (setL2 g F)\<twosuperior>)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   431
      by (rule real_sqrt_sum_squares_triangle_ineq)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   432
    finally show ?case
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   433
      using insert by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   434
  qed
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   435
qed
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   436
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   437
lemma sqrt_sum_squares_le_sum:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   438
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) \<le> x + y"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   439
  apply (rule power2_le_imp_le)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   440
  apply (simp add: power2_sum)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   441
  apply (simp add: mult_nonneg_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   442
  apply (simp add: add_nonneg_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   443
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   444
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   445
lemma setL2_le_setsum [rule_format]:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   446
  "(\<forall>i\<in>A. 0 \<le> f i) \<longrightarrow> setL2 f A \<le> setsum f A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   447
  apply (cases "finite A")
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   448
  apply (induct set: finite)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   449
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   450
  apply clarsimp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   451
  apply (erule order_trans [OF sqrt_sum_squares_le_sum])
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   452
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   453
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   454
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   455
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   456
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   457
lemma sqrt_sum_squares_le_sum_abs: "sqrt (x\<twosuperior> + y\<twosuperior>) \<le> \<bar>x\<bar> + \<bar>y\<bar>"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   458
  apply (rule power2_le_imp_le)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   459
  apply (simp add: power2_sum)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   460
  apply (simp add: mult_nonneg_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   461
  apply (simp add: add_nonneg_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   462
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   463
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   464
lemma setL2_le_setsum_abs: "setL2 f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   465
  apply (cases "finite A")
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   466
  apply (induct set: finite)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   467
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   468
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   469
  apply (rule order_trans [OF sqrt_sum_squares_le_sum_abs])
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   470
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   471
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   472
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   473
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   474
lemma setL2_mult_ineq_lemma:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   475
  fixes a b c d :: real
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   476
  shows "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   477
proof -
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   478
  have "0 \<le> (a * d - b * c)\<twosuperior>" by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   479
  also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * d) * (b * c)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   480
    by (simp only: power2_diff power_mult_distrib)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   481
  also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * c) * (b * d)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   482
    by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   483
  finally show "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   484
    by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   485
qed
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   486
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   487
lemma setL2_mult_ineq: "(\<Sum>i\<in>A. \<bar>f i\<bar> * \<bar>g i\<bar>) \<le> setL2 f A * setL2 g A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   488
  apply (cases "finite A")
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   489
  apply (induct set: finite)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   490
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   491
  apply (rule power2_le_imp_le, simp)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   492
  apply (rule order_trans)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   493
  apply (rule power_mono)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   494
  apply (erule add_left_mono)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   495
  apply (simp add: add_nonneg_nonneg mult_nonneg_nonneg setsum_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   496
  apply (simp add: power2_sum)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   497
  apply (simp add: power_mult_distrib)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   498
  apply (simp add: right_distrib left_distrib)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   499
  apply (rule ord_le_eq_trans)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   500
  apply (rule setL2_mult_ineq_lemma)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   501
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   502
  apply (intro mult_nonneg_nonneg setL2_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   503
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   504
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   505
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   506
lemma member_le_setL2: "\<lbrakk>finite A; i \<in> A\<rbrakk> \<Longrightarrow> f i \<le> setL2 f A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   507
  apply (rule_tac s="insert i (A - {i})" and t="A" in subst)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   508
  apply fast
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   509
  apply (subst setL2_insert)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   510
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   511
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   512
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   513
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   514
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   515
subsection {* Norms *}
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   516
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   517
instantiation "^" :: (real_normed_vector, type) real_normed_vector
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   518
begin
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   519
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   520
definition vector_norm_def:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   521
  "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) {1 .. dimindex (UNIV:: 'b set)}"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   522
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   523
definition vector_sgn_def:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   524
  "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   525
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   526
instance proof
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   527
  fix a :: real and x y :: "'a ^ 'b"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   528
  show "0 \<le> norm x"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   529
    unfolding vector_norm_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   530
    by (rule setL2_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   531
  show "norm x = 0 \<longleftrightarrow> x = 0"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   532
    unfolding vector_norm_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   533
    by (simp add: setL2_eq_0_iff Cart_eq)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   534
  show "norm (x + y) \<le> norm x + norm y"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   535
    unfolding vector_norm_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   536
    apply (rule order_trans [OF _ setL2_triangle_ineq])
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   537
    apply (rule setL2_mono)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   538
    apply (simp add: vector_component norm_triangle_ineq)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   539
    apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   540
    done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   541
  show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   542
    unfolding vector_norm_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   543
    by (simp add: vector_component norm_scaleR setL2_right_distrib
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   544
             cong: strong_setL2_cong)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   545
  show "sgn x = scaleR (inverse (norm x)) x"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   546
    by (rule vector_sgn_def)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   547
qed
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   548
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   549
end
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   550
30045
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   551
subsection {* Inner products *}
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   552
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   553
instantiation "^" :: (real_inner, type) real_inner
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   554
begin
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   555
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   556
definition vector_inner_def:
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   557
  "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) {1 .. dimindex(UNIV::'b set)}"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   558
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   559
instance proof
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   560
  fix r :: real and x y z :: "'a ^ 'b"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   561
  show "inner x y = inner y x"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   562
    unfolding vector_inner_def
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   563
    by (simp add: inner_commute)
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   564
  show "inner (x + y) z = inner x z + inner y z"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   565
    unfolding vector_inner_def
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   566
    by (vector inner_left_distrib)
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   567
  show "inner (scaleR r x) y = r * inner x y"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   568
    unfolding vector_inner_def
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   569
    by (vector inner_scaleR_left)
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   570
  show "0 \<le> inner x x"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   571
    unfolding vector_inner_def
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   572
    by (simp add: setsum_nonneg)
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   573
  show "inner x x = 0 \<longleftrightarrow> x = 0"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   574
    unfolding vector_inner_def
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   575
    by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   576
  show "norm x = sqrt (inner x x)"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   577
    unfolding vector_inner_def vector_norm_def setL2_def
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   578
    by (simp add: power2_norm_eq_inner)
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   579
qed
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   580
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   581
end
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   582
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   583
subsection{* Properties of the dot product.  *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   584
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   585
lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   586
  by (vector mult_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   587
lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \<bullet> z = (x \<bullet> z) + (y \<bullet> z)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   588
  by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   589
lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   590
  by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   591
lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   592
  by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   593
lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   594
  by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   595
lemma dot_lmult: "(c *s x) \<bullet> y = (c::'a::ring) * (x \<bullet> y)" by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   596
lemma dot_rmult: "x \<bullet> (c *s y) = (c::'a::comm_ring) * (x \<bullet> y)" by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   597
lemma dot_lneg: "(-x) \<bullet> (y::'a::ring ^ 'n) = -(x \<bullet> y)" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   598
lemma dot_rneg: "(x::'a::ring ^ 'n) \<bullet> (-y) = -(x \<bullet> y)" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   599
lemma dot_lzero[simp]: "0 \<bullet> x = (0::'a::{comm_monoid_add, mult_zero})" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   600
lemma dot_rzero[simp]: "x \<bullet> 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   601
lemma dot_pos_le[simp]: "(0::'a\<Colon>ordered_ring_strict) <= x \<bullet> x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   602
  by (simp add: dot_def setsum_nonneg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   603
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   604
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)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   605
using fS fp setsum_nonneg[OF fp]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   606
proof (induct set: finite)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   607
  case empty thus ?case by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   608
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   609
  case (insert x F)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   610
  from insert.prems have Fx: "f x \<ge> 0" and Fp: "\<forall> a \<in> F. f a \<ge> 0" by simp_all
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   611
  from insert.hyps Fp setsum_nonneg[OF Fp]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   612
  have h: "setsum f F = 0 \<longleftrightarrow> (\<forall>a \<in>F. f a = 0)" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   613
  from sum_nonneg_eq_zero_iff[OF Fx  setsum_nonneg[OF Fp]] insert.hyps(1,2)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   614
  show ?case by (simp add: h)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   615
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   616
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   617
lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   618
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   619
  {assume f: "finite (UNIV :: 'n set)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   620
    let ?S = "{Suc 0 .. card (UNIV :: 'n set)}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   621
    have fS: "finite ?S" using f by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   622
    have fp: "\<forall> i\<in> ?S. x$i * x$i>= 0" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   623
    have ?thesis by (vector dimindex_def f setsum_squares_eq_0_iff[OF fS fp])}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   624
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   625
  {assume "\<not> finite (UNIV :: 'n set)" then have ?thesis by (vector dimindex_def)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   626
  ultimately show ?thesis by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   627
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   628
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   629
lemma dot_pos_lt: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x] 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   630
  by (auto simp add: le_less) 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   631
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   632
subsection{* The collapse of the general concepts to dimension one. *}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   633
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   634
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   635
  by (vector dimindex_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   636
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   637
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   638
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   639
  apply (erule_tac x= "x$1" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   640
  apply (simp only: vector_one[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   641
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   642
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   643
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   644
  by (simp add: vector_norm_def dimindex_def)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   645
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   646
lemma norm_real: "norm(x::real ^ 1) = abs(x$1)" 
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   647
  by (simp add: norm_vector_1)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   648
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   649
text{* Metric *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   650
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   651
text {* FIXME: generalize to arbitrary @{text real_normed_vector} types *}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   652
definition dist:: "real ^ 'n \<Rightarrow> real ^ 'n \<Rightarrow> real" where 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   653
  "dist x y = norm (x - y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   654
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   655
lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   656
  using dimindex_ge_1[of "UNIV :: 1 set"]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   657
  by (auto simp add: norm_real dist_def vector_component Cart_lambda_beta[where ?'a = "1"] )
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   658
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   659
subsection {* A connectedness or intermediate value lemma with several applications. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   660
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   661
lemma connected_real_lemma:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   662
  fixes f :: "real \<Rightarrow> real ^ 'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   663
  assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   664
  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"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   665
  and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   666
  and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   667
  and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   668
  shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   669
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   670
  let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   671
  have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa) 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   672
  have Sub: "\<exists>y. isUb UNIV ?S y" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   673
    apply (rule exI[where x= b])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   674
    using ab fb e12 by (auto simp add: isUb_def setle_def)  
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   675
  from reals_complete[OF Se Sub] obtain l where 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   676
    l: "isLub UNIV ?S l"by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   677
  have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   678
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)    
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   679
    by (metis linorder_linear)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   680
  have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   681
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   682
    by (metis linorder_linear not_le)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   683
    have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   684
    have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   685
    have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   686
    {assume le2: "f l \<in> e2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   687
      from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   688
      hence lap: "l - a > 0" using alb by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   689
      from e2[rule_format, OF le2] obtain e where 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   690
	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   691
      from dst[OF alb e(1)] obtain d where 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   692
	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   693
      have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1) 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   694
	apply ferrack by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   695
      then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   696
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   697
      from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   698
      moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   699
      have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   700
      ultimately have False using e12 alb d' by auto}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   701
    moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   702
    {assume le1: "f l \<in> e1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   703
    from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   704
      hence blp: "b - l > 0" using alb by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   705
      from e1[rule_format, OF le1] obtain e where 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   706
	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   707
      from dst[OF alb e(1)] obtain d where 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   708
	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   709
      have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   710
      then obtain d' where d': "d' > 0" "d' < d" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   711
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   712
      hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   713
      with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   714
      with l d' have False 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   715
	by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   716
    ultimately show ?thesis using alb by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   717
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   718
29881
58f3c48dbbb7 fix document generation
huffman
parents: 29844
diff changeset
   719
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"} *}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   720
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   721
lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   722
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   723
  have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   724
  thus ?thesis by (simp add: ring_simps power2_eq_square)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   725
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   726
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   727
lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   728
  using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_def, rule_format, of e x] apply (auto simp add: power2_eq_square)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   729
  apply (rule_tac x="s" in exI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   730
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   731
  apply (erule_tac x=y in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   732
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   733
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   734
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   735
lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   736
  using real_sqrt_le_iff[of x "y^2"] by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   737
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   738
lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   739
  using real_sqrt_le_mono[of "x^2" y] by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   740
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   741
lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   742
  using real_sqrt_less_mono[of "x^2" y] by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   743
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   744
lemma sqrt_even_pow2: assumes n: "even n" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   745
  shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   746
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   747
  from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   748
    by (auto simp add: nat_number) 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   749
  from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   750
    by (simp only: power_mult[symmetric] mult_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   751
  then show ?thesis  using m by simp 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   752
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   753
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   754
lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   755
  apply (cases "x = 0", simp_all)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   756
  using sqrt_divide_self_eq[of x]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   757
  apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   758
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   759
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   760
text{* Hence derive more interesting properties of the norm. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   761
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   762
lemma norm_0: "norm (0::real ^ 'n) = 0"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   763
  by (rule norm_zero)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   764
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   765
lemma norm_mul: "norm(a *s x) = abs(a) * norm x"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   766
  by (simp add: vector_norm_def vector_component setL2_right_distrib
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   767
           abs_mult cong: strong_setL2_cong)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   768
lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   769
  by (simp add: vector_norm_def dot_def setL2_def power2_eq_square)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   770
lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   771
  by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   772
lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   773
  by (simp add: real_vector_norm_def)
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   774
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   775
lemma vector_mul_eq_0: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   776
  by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   777
lemma vector_mul_lcancel: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   778
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   779
lemma vector_mul_rcancel: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   780
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   781
lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   782
  by (metis vector_mul_lcancel)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   783
lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   784
  by (metis vector_mul_rcancel)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   785
lemma norm_cauchy_schwarz: "x \<bullet> y <= norm x * norm y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   786
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   787
  {assume "norm x = 0"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   788
    hence ?thesis by (simp add: dot_lzero dot_rzero)}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   789
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   790
  {assume "norm y = 0" 
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   791
    hence ?thesis by (simp add: dot_lzero dot_rzero)}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   792
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   793
  {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   794
    let ?z = "norm y *s x - norm x *s y"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   795
    from h have p: "norm x * norm y > 0" by (metis norm_ge_zero le_less zero_compare_simps)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   796
    from dot_pos_le[of ?z]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   797
    have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   798
      apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   799
      by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   800
    hence "x\<bullet>y \<le> (norm x ^2 * norm y ^2) / (norm x * norm y)" using p
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   801
      by (simp add: field_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   802
    hence ?thesis using h by (simp add: power2_eq_square)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   803
  ultimately show ?thesis by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   804
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   805
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   806
lemma norm_cauchy_schwarz_abs: "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   807
  using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   808
  by (simp add: real_abs_def dot_rneg)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   809
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   810
lemma norm_triangle_sub: "norm (x::real ^'n) <= norm(y) + norm(x - y)"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   811
  using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   812
lemma norm_triangle_le: "norm(x::real ^'n) + norm y <= e ==> norm(x + y) <= e"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   813
  by (metis order_trans norm_triangle_ineq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   814
lemma norm_triangle_lt: "norm(x::real ^'n) + norm(y) < e ==> norm(x + y) < e"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   815
  by (metis basic_trans_rules(21) norm_triangle_ineq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   816
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   817
lemma setsum_delta: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   818
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   819
  shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   820
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   821
  let ?f = "(\<lambda>k. if k=a then b k else 0)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   822
  {assume a: "a \<notin> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   823
    hence "\<forall> k\<in> S. ?f k = 0" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   824
    hence ?thesis  using a by simp}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   825
  moreover 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   826
  {assume a: "a \<in> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   827
    let ?A = "S - {a}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   828
    let ?B = "{a}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   829
    have eq: "S = ?A \<union> ?B" using a by blast 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   830
    have dj: "?A \<inter> ?B = {}" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   831
    from fS have fAB: "finite ?A" "finite ?B" by auto  
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   832
    have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   833
      using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   834
      by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   835
    then have ?thesis  using a by simp}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   836
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   837
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   838
  
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   839
lemma component_le_norm: "i \<in> {1 .. dimindex(UNIV :: 'n set)} ==> \<bar>x$i\<bar> <= norm (x::real ^ 'n)"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   840
  apply (simp add: vector_norm_def)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   841
  apply (rule member_le_setL2, simp_all)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   842
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   843
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   844
lemma norm_bound_component_le: "norm(x::real ^ 'n) <= e
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   845
                ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> <= e"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   846
  by (metis component_le_norm order_trans)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   847
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   848
lemma norm_bound_component_lt: "norm(x::real ^ 'n) < e
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   849
                ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> < e"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   850
  by (metis component_le_norm basic_trans_rules(21))
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   851
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   852
lemma norm_le_l1: "norm (x:: real ^'n) <= setsum(\<lambda>i. \<bar>x$i\<bar>) {1..dimindex(UNIV::'n set)}"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   853
  by (simp add: vector_norm_def setL2_le_setsum)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   854
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   855
lemma real_abs_norm: "\<bar> norm x\<bar> = norm (x :: real ^'n)" 
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   856
  by (rule abs_norm_cancel)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   857
lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n) - norm y\<bar> <= norm(x - y)"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   858
  by (rule norm_triangle_ineq3)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   859
lemma norm_le: "norm(x::real ^ 'n) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   860
  by (simp add: real_vector_norm_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   861
lemma norm_lt: "norm(x::real ^'n) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   862
  by (simp add: real_vector_norm_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   863
lemma norm_eq: "norm (x::real ^'n) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   864
  by (simp add: order_eq_iff norm_le)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   865
lemma norm_eq_1: "norm(x::real ^ 'n) = 1 \<longleftrightarrow> x \<bullet> x = 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   866
  by (simp add: real_vector_norm_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   867
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   868
text{* Squaring equations and inequalities involving norms.  *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   869
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   870
lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   871
  by (simp add: real_vector_norm_def  dot_pos_le )
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   872
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   873
lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   874
  by (auto simp add: real_vector_norm_def)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   875
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   876
lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   877
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   878
  have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: ring_simps power2_eq_square)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   879
  also have "\<dots> \<longleftrightarrow> \<bar>x\<bar> \<le> \<bar>y\<bar>" apply (simp add: zero_compare_simps real_abs_def not_less) by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   880
finally show ?thesis ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   881
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   882
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   883
lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   884
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   885
  using norm_ge_zero[of x]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   886
  apply arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   887
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   888
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   889
lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2" 
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   890
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   891
  using norm_ge_zero[of x]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   892
  apply arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   893
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   894
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   895
lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   896
  by (metis not_le norm_ge_square)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   897
lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   898
  by (metis norm_le_square not_less)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   899
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   900
text{* Dot product in terms of the norm rather than conversely. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   901
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   902
lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   903
  by (simp add: norm_pow_2 dot_ladd dot_radd dot_sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   904
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   905
lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   906
  by (simp add: norm_pow_2 dot_ladd dot_radd dot_lsub dot_rsub dot_sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   907
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   908
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   909
text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   910
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   911
lemma vector_eq: "(x:: real ^ 'n) = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y\<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   912
proof
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   913
  assume "?lhs" then show ?rhs by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   914
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   915
  assume ?rhs
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   916
  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y\<bullet> y = 0" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   917
  hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   918
    by (simp add: dot_rsub dot_lsub dot_sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   919
  then have "(x - y) \<bullet> (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   920
  then show "x = y" by (simp add: dot_eq_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   921
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   922
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   923
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   924
subsection{* General linear decision procedure for normed spaces. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   925
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   926
lemma norm_cmul_rule_thm: "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(c *s x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   927
  apply (clarsimp simp add: norm_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   928
  apply (rule mult_mono1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   929
  apply simp_all
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   930
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   931
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   932
lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n) \<Longrightarrow> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   933
  apply (rule norm_triangle_le) by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   934
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   935
lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   936
  by (simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   937
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   938
lemma pth_1: "(x::real^'n) == 1 *s x" by (simp only: vector_smult_lid)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   939
lemma pth_2: "x - (y::real^'n) == x + -y" by (atomize (full)) simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   940
lemma pth_3: "(-x::real^'n) == -1 *s x" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   941
lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   942
lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   943
lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   944
lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   945
lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps) 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   946
lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   947
  "c *s x + (d *s x + z) == (c + d) *s x + z"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   948
  "(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   949
lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   950
lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   951
  "(c *s x + z) + d *s y == c *s x + (z + d *s y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   952
  "c *s x + (d *s y + z) == c *s x + (d *s y + z)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   953
  "(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   954
  by ((atomize (full)), vector)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   955
lemma pth_c: "(c::real) *s x + d *s y == d *s y + c *s x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   956
  "(c *s x + z) + d *s y == d *s y + (c *s x + z)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   957
  "c *s x + (d *s y + z) == d *s y + (c *s x + z)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   958
  "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   959
lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   960
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   961
lemma norm_imp_pos_and_ge: "norm (x::real ^ 'n) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   962
  by (atomize) (auto simp add: norm_ge_zero)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   963
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   964
lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   965
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   966
lemma norm_pths: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   967
  "(x::real ^'n) = y \<longleftrightarrow> norm (x - y) \<le> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   968
  "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   969
  using norm_ge_zero[of "x - y"] by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   970
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   971
use "normarith.ML"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   972
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   973
method_setup norm = {* Method.ctxt_args (Method.SIMPLE_METHOD' o NormArith.norm_arith_tac)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   974
*} "Proves simple linear statements about vector norms"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   975
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   976
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   977
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   978
text{* Hence more metric properties. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   979
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   980
lemma dist_refl: "dist x x = 0" by norm
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   981
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   982
lemma dist_sym: "dist x y = dist y x"by norm
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   983
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   984
lemma dist_pos_le: "0 <= dist x y" by norm
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   985
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   986
lemma dist_triangle: "dist x z <= dist x y + dist y z" by norm
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   987
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   988
lemma dist_triangle_alt: "dist y z <= dist x y + dist x z" by norm
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   989
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   990
lemma dist_eq_0: "dist x y = 0 \<longleftrightarrow> x = y" by norm
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   991
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   992
lemma dist_pos_lt: "x \<noteq> y ==> 0 < dist x y" by norm 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   993
lemma dist_nz:  "x \<noteq> y \<longleftrightarrow> 0 < dist x y" by norm 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   994
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   995
lemma dist_triangle_le: "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e" by norm 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   996
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   997
lemma dist_triangle_lt: "dist x z + dist y z < e ==> dist x y < e" by norm 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   998
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   999
lemma dist_triangle_half_l: "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 ==> dist x1 x2 < e" by norm 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1000
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1001
lemma dist_triangle_half_r: "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 ==> dist x1 x2 < e" by norm 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1002
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1003
lemma dist_triangle_add: "dist (x + y) (x' + y') <= dist x x' + dist y y'"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1004
  by norm 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1005
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1006
lemma dist_mul: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1007
  unfolding dist_def vector_ssub_ldistrib[symmetric] norm_mul .. 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1008
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1009
lemma dist_triangle_add_half: " dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 ==> dist(x + y) (x' + y') < e" by norm 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1010
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1011
lemma dist_le_0: "dist x y <= 0 \<longleftrightarrow> x = y" by norm 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1012
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1013
instantiation "^" :: (monoid_add,type) monoid_add
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1014
begin
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1015
  instance by (intro_classes)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1016
end
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1017
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1018
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1019
  apply vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1020
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1021
  apply (cases "finite S")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1022
  apply (rule finite_induct[of S])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1023
  apply (auto simp add: vector_component zero_index)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1024
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1025
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1026
lemma setsum_clauses: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1027
  shows "setsum f {} = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1028
  and "finite S \<Longrightarrow> setsum f (insert x S) =
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1029
                 (if x \<in> S then setsum f S else f x + setsum f S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1030
  by (auto simp add: insert_absorb)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1031
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1032
lemma setsum_cmul: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1033
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1034
  shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1035
  by (simp add: setsum_eq Cart_eq Cart_lambda_beta vector_component setsum_right_distrib)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1036
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1037
lemma setsum_component: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1038
  fixes f:: " 'a \<Rightarrow> ('b::semiring_1) ^'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1039
  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1040
  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1041
  using i by (simp add: setsum_eq Cart_lambda_beta)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1042
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1043
  (* This needs finiteness assumption due to the definition of fold!!! *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1044
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1045
lemma setsum_superset:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1046
  assumes fb: "finite B" and ab: "A \<subseteq> B" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1047
  and f0: "\<forall>x \<in> B - A. f x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1048
  shows "setsum f B = setsum f A"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1049
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1050
  from ab fb have fa: "finite A" by (metis finite_subset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1051
  from fb have fba: "finite (B - A)" by (metis finite_Diff)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1052
  have d: "A \<inter> (B - A) = {}" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1053
  from ab have b: "B = A \<union> (B - A)" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1054
  from setsum_Un_disjoint[OF fa fba d, of f] b
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1055
    setsum_0'[OF f0]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1056
  show "setsum f B = setsum f A" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1057
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1058
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1059
lemma setsum_restrict_set:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1060
  assumes fA: "finite A"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1061
  shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1062
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1063
  from fA have fab: "finite (A \<inter> B)" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1064
  have aba: "A \<inter> B \<subseteq> A" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1065
  let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1066
  from setsum_superset[OF fA aba, of ?g]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1067
  show ?thesis by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1068
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1069
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1070
lemma setsum_cases:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1071
  assumes fA: "finite A"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1072
  shows "setsum (\<lambda>x. if x \<in> B then f x else g x) A =
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1073
         setsum f (A \<inter> B) + setsum g (A \<inter> - B)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1074
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1075
  have a: "A = A \<inter> B \<union> A \<inter> -B" "(A \<inter> B) \<inter> (A \<inter> -B) = {}" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1076
    by blast+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1077
  from fA 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1078
  have f: "finite (A \<inter> B)" "finite (A \<inter> -B)" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1079
  let ?g = "\<lambda>x. if x \<in> B then f x else g x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1080
  from setsum_Un_disjoint[OF f a(2), of ?g] a(1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1081
  show ?thesis by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1082
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1083
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1084
lemma setsum_norm: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1085
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1086
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1087
  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1088
proof(induct rule: finite_induct[OF fS])
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  1089
  case 1 thus ?case by simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1090
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1091
  case (2 x S)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1092
  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1093
  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1094
    using "2.hyps" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1095
  finally  show ?case  using "2.hyps" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1096
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1097
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1098
lemma real_setsum_norm: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1099
  fixes f :: "'a \<Rightarrow> real ^'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1100
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1101
  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1102
proof(induct rule: finite_induct[OF fS])
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
  1103
  case 1 thus ?case by simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1104
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1105
  case (2 x S)
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
  1106
  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1107
  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1108
    using "2.hyps" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1109
  finally  show ?case  using "2.hyps" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1110
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1111
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1112
lemma setsum_norm_le: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1113
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1114
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1115
  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1116
  shows "norm (setsum f S) \<le> setsum g S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1117
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1118
  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1119
    by - (rule setsum_mono, simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1120
  then show ?thesis using setsum_norm[OF fS, of f] fg
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1121
    by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1122
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1123
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1124
lemma real_setsum_norm_le: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1125
  fixes f :: "'a \<Rightarrow> real ^ 'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1126
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1127
  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1128
  shows "norm (setsum f S) \<le> setsum g S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1129
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1130
  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1131
    by - (rule setsum_mono, simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1132
  then show ?thesis using real_setsum_norm[OF fS, of f] fg
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1133
    by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1134
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1135
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1136
lemma setsum_norm_bound:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1137
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1138
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1139
  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1140
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1141
  using setsum_norm_le[OF fS K] setsum_constant[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1142
  by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1143
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1144
lemma real_setsum_norm_bound:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1145
  fixes f :: "'a \<Rightarrow> real ^ 'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1146
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1147
  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1148
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1149
  using real_setsum_norm_le[OF fS K] setsum_constant[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1150
  by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1151
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1152
lemma setsum_vmul:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1153
  fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector,semiring, mult_zero}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1154
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1155
  shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1156
proof(induct rule: finite_induct[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1157
  case 1 then show ?case by (simp add: vector_smult_lzero)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1158
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1159
  case (2 x F)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1160
  from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1161
    by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1162
  also have "\<dots> = f x *s v + setsum f F *s v" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1163
    by (simp add: vector_sadd_rdistrib)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1164
  also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1165
  finally show ?case .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1166
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1167
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1168
(* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \<Rightarrow> real ^'n"]  ---
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1169
 Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1170
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1171
lemma setsum_add_split: assumes mn: "(m::nat) \<le> n + 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1172
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1173
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1174
  let ?A = "{m .. n}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1175
  let ?B = "{n + 1 .. n + p}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1176
  have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1177
  have d: "?A \<inter> ?B = {}" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1178
  from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1179
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1180
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1181
lemma setsum_reindex_nonzero: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1182
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1183
  and nz: "\<And> x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1184
  shows "setsum h (f ` S) = setsum (h o f) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1185
using nz
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1186
proof(induct rule: finite_induct[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1187
  case 1 thus ?case by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1188
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1189
  case (2 x F) 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1190
  {assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1191
    then obtain y where y: "y \<in> F" "f x = f y" by auto 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1192
    from "2.hyps" y have xy: "x \<noteq> y" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1193
    
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1194
    from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1195
    have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1196
    also have "\<dots> = setsum (h o f) (insert x F)" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1197
      using "2.hyps" "2.prems" h0  by auto 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1198
    finally have ?case .}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1199
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1200
  {assume fxF: "f x \<notin> f ` F"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1201
    have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1202
      using fxF "2.hyps" by simp 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1203
    also have "\<dots> = setsum (h o f) (insert x F)"  
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1204
      using "2.hyps" "2.prems" fxF
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1205
      apply auto apply metis done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1206
    finally have ?case .}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1207
  ultimately show ?case by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1208
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1209
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1210
lemma setsum_Un_nonzero:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1211
  assumes fS: "finite S" and fF: "finite F"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1212
  and f: "\<forall> x\<in> S \<inter> F . f x = (0::'a::ab_group_add)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1213
  shows "setsum f (S \<union> F) = setsum f S + setsum f F"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1214
  using setsum_Un[OF fS fF, of f] setsum_0'[OF f] by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1215
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1216
lemma setsum_natinterval_left:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1217
  assumes mn: "(m::nat) <= n" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1218
  shows "setsum f {m..n} = f m + setsum f {m + 1..n}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1219
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1220
  from mn have "{m .. n} = insert m {m+1 .. n}" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1221
  then show ?thesis by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1222
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1223
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1224
lemma setsum_natinterval_difff: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1225
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1226
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1227
          (if m <= n then f m - f(n + 1) else 0)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1228
by (induct n, auto simp add: ring_simps not_le le_Suc_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1229
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1230
lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1231
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1232
lemma setsum_setsum_restrict:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1233
  "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"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1234
  apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1235
  by (rule setsum_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1236
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1237
lemma setsum_image_gen: assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1238
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1239
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1240
  {fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1241
  note th0 = this
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1242
  have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1243
    apply (rule setsum_cong2) 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1244
    by (simp add: th0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1245
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1246
    apply (rule setsum_setsum_restrict[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1247
    by (rule finite_imageI[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1248
  finally show ?thesis .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1249
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1250
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1251
    (* FIXME: Here too need stupid finiteness assumption on T!!! *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1252
lemma setsum_group:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1253
  assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1254
  shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1255
  
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1256
apply (subst setsum_image_gen[OF fS, of g f])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1257
apply (rule setsum_superset[OF fT fST])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1258
by (auto intro: setsum_0')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1259
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1260
(* FIXME: Change the name to fold_image\<dots> *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1261
lemma (in comm_monoid_mult) fold_1': "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1262
  apply (induct set: finite)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1263
  apply simp by (auto simp add: fold_image_insert)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1264
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1265
lemma (in comm_monoid_mult) fold_union_nonzero:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1266
  assumes fS: "finite S" and fT: "finite T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1267
  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1268
  shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1269
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1270
  have "fold_image op * f 1 (S \<inter> T) = 1" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1271
    apply (rule fold_1')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1272
    using fS fT I0 by auto 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1273
  with fold_image_Un_Int[OF fS fT] show ?thesis by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1274
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1275
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1276
lemma setsum_union_nonzero:  
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1277
  assumes fS: "finite S" and fT: "finite T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1278
  and I0: "\<forall>x \<in> S\<inter>T. f x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1279
  shows "setsum f (S \<union> T) = setsum f S  + setsum f T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1280
  using fS fT
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1281
  apply (simp add: setsum_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1282
  apply (rule comm_monoid_add.fold_union_nonzero)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1283
  using I0 by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1284
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1285
lemma setprod_union_nonzero:  
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1286
  assumes fS: "finite S" and fT: "finite T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1287
  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1288
  shows "setprod f (S \<union> T) = setprod f S  * setprod f T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1289
  using fS fT
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1290
  apply (simp add: setprod_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1291
  apply (rule fold_union_nonzero)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1292
  using I0 by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1293
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1294
lemma setsum_unions_nonzero: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1295
  assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1296
  and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1297
  shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1298
  using fSS f0
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1299
proof(induct rule: finite_induct[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1300
  case 1 thus ?case by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1301
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1302
  case (2 T F)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1303
  then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1304
    and H: "setsum f (\<Union> F) = setsum (setsum f) F" by (auto simp add: finite_insert)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1305
  from fTF have fUF: "finite (\<Union>F)" by (auto intro: finite_Union)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1306
  from "2.prems" TF fTF
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1307
  show ?case 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1308
    by (auto simp add: H[symmetric] intro: setsum_union_nonzero[OF fTF(1) fUF, of f])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1309
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1310
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1311
  (* FIXME : Copied from Pocklington --- should be moved to Finite_Set!!!!!!!! *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1312
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1313
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1314
lemma (in comm_monoid_mult) fold_related: 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1315
  assumes Re: "R e e" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1316
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1317
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1318
  shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1319
  using fS by (rule finite_subset_induct) (insert assms, auto)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1320
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1321
  (* FIXME: I think we can get rid of the finite assumption!! *)	
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1322
lemma (in comm_monoid_mult) 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1323
  fold_eq_general:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1324
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1325
  and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1326
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1327
  shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1328
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1329
  from h f12 have hS: "h ` S = S'" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1330
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1331
    from f12 h H  have "x = y" by auto }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1332
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1333
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1334
  from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1335
  also have "\<dots> = fold_image (op *) (f2 o h) e S" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1336
    using fold_image_reindex[OF fS hinj, of f2 e] .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1337
  also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1338
    by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1339
  finally show ?thesis ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1340
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1341
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1342
lemma (in comm_monoid_mult) fold_eq_general_inverses:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1343
  assumes fS: "finite S" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1344
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1345
  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1346
  shows "fold_image (op *) f e S = fold_image (op *) g e T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1347
  using fold_eq_general[OF fS, of T h g f e] kh hk by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1348
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1349
lemma setsum_eq_general_reverses:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1350
  assumes fS: "finite S" and fT: "finite T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1351
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1352
  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1353
  shows "setsum f S = setsum g T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1354
  apply (simp add: setsum_def fS fT)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1355
  apply (rule comm_monoid_add.fold_eq_general_inverses[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1356
  apply (erule kh)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1357
  apply (erule hk)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1358
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1359
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1360
lemma vsum_norm_allsubsets_bound:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1361
  fixes f:: "'a \<Rightarrow> real ^'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1362
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e" 
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1363
  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real (dimindex(UNIV :: 'n set)) *  e"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1364
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1365
  let ?d = "real (dimindex (UNIV ::'n set))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1366
  let ?nf = "\<lambda>x. norm (f x)"
4ac60c7d9b78 (Real) Vectors in Eucl