src/HOL/Library/Euclidean_Space.thy
author huffman
Sat Feb 21 11:18:50 2009 -0800 (2009-02-21)
changeset 30041 9becd197a40e
parent 30040 e2cd1acda1ab
child 30045 b8ddd7667eed
permissions -rw-r--r--
remove duplicated lemmas about norm
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(* Title:      Library/Euclidean_Space
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   ID:         $Id: 
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   Author:     Amine Chaieb, University of Cambridge
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*)
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header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
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theory Euclidean_Space
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  imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Complex_Main 
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  Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type
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  uses ("normarith.ML")
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begin
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text{* Some common special cases.*}
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lemma forall_1: "(\<forall>(i::'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)$i = of_nat n"
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  apply (induct n)
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  apply vector
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  apply vector
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  done
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lemma zero_index[simp]: 
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  "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (0 :: 'a::zero ^'n)$i = 0" by vector
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lemma one_index[simp]: 
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  "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (1 :: 'a::one ^'n)$i = 1" by vector
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lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \<noteq> 0"
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proof-
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  have "(1::'a) + of_nat n = 0 \<longleftrightarrow> of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp
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  also have "\<dots> \<longleftrightarrow> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff) 
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  finally show ?thesis by simp 
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qed
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instance "^" :: (semiring_char_0,type) semiring_char_0 
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proof (intro_classes) 
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  fix m n ::nat
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  show "(of_nat m :: 'a^'b) = of_nat n \<longleftrightarrow> m = n"
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  proof(induct m arbitrary: n)
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    case 0 thus ?case apply vector 
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      apply (induct n,auto simp add: ring_simps)
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      using dimindex_ge_1 apply auto
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      apply vector
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      by (auto simp add: of_nat_index one_plus_of_nat_neq_0)
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  next
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    case (Suc n m)
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    thus ?case  apply vector
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      apply (induct m, auto simp add: ring_simps of_nat_index zero_index)
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      using dimindex_ge_1 apply simp apply blast
chaieb@29842
   310
      apply (simp add: one_plus_of_nat_neq_0)
chaieb@29842
   311
      using dimindex_ge_1 apply simp apply blast
chaieb@29842
   312
      apply (simp add: vector_component one_index of_nat_index)
chaieb@29842
   313
      apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)
chaieb@29842
   314
      using  dimindex_ge_1 apply simp apply blast
chaieb@29842
   315
      apply (simp add: vector_component one_index of_nat_index)
chaieb@29842
   316
      apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)
chaieb@29842
   317
      using dimindex_ge_1 apply simp apply blast
chaieb@29842
   318
      apply (simp add: vector_component one_index of_nat_index)
chaieb@29842
   319
      done
chaieb@29842
   320
  qed
chaieb@29842
   321
qed
chaieb@29842
   322
chaieb@29842
   323
instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes
huffman@30039
   324
instance "^" :: (ring_char_0,type) ring_char_0 by intro_classes
chaieb@29842
   325
chaieb@29842
   326
lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"  
chaieb@29842
   327
  by (vector mult_assoc)
chaieb@29842
   328
lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" 
chaieb@29842
   329
  by (vector ring_simps)
chaieb@29842
   330
lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" 
chaieb@29842
   331
  by (vector ring_simps)
chaieb@29842
   332
lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
chaieb@29842
   333
lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
chaieb@29842
   334
lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y" 
chaieb@29842
   335
  by (vector ring_simps)
chaieb@29842
   336
lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
chaieb@29842
   337
lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
chaieb@29842
   338
lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
chaieb@29842
   339
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
chaieb@29842
   340
lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x" 
chaieb@29842
   341
  by (vector ring_simps)
chaieb@29842
   342
chaieb@29842
   343
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)" 
chaieb@29842
   344
  apply (auto simp add: vec_def Cart_eq vec_component Cart_lambda_beta )
chaieb@29842
   345
  using dimindex_ge_1 apply auto done
chaieb@29842
   346
huffman@30040
   347
subsection {* Square root of sum of squares *}
huffman@30040
   348
huffman@30040
   349
definition
huffman@30040
   350
  "setL2 f A = sqrt (\<Sum>i\<in>A. (f i)\<twosuperior>)"
huffman@30040
   351
huffman@30040
   352
lemma setL2_cong:
huffman@30040
   353
  "\<lbrakk>A = B; \<And>x. x \<in> B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
huffman@30040
   354
  unfolding setL2_def by simp
huffman@30040
   355
huffman@30040
   356
lemma strong_setL2_cong:
huffman@30040
   357
  "\<lbrakk>A = B; \<And>x. x \<in> B =simp=> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
huffman@30040
   358
  unfolding setL2_def simp_implies_def by simp
huffman@30040
   359
huffman@30040
   360
lemma setL2_infinite [simp]: "\<not> finite A \<Longrightarrow> setL2 f A = 0"
huffman@30040
   361
  unfolding setL2_def by simp
huffman@30040
   362
huffman@30040
   363
lemma setL2_empty [simp]: "setL2 f {} = 0"
huffman@30040
   364
  unfolding setL2_def by simp
huffman@30040
   365
huffman@30040
   366
lemma setL2_insert [simp]:
huffman@30040
   367
  "\<lbrakk>finite F; a \<notin> F\<rbrakk> \<Longrightarrow>
huffman@30040
   368
    setL2 f (insert a F) = sqrt ((f a)\<twosuperior> + (setL2 f F)\<twosuperior>)"
huffman@30040
   369
  unfolding setL2_def by (simp add: setsum_nonneg)
huffman@30040
   370
huffman@30040
   371
lemma setL2_nonneg [simp]: "0 \<le> setL2 f A"
huffman@30040
   372
  unfolding setL2_def by (simp add: setsum_nonneg)
huffman@30040
   373
huffman@30040
   374
lemma setL2_0': "\<forall>a\<in>A. f a = 0 \<Longrightarrow> setL2 f A = 0"
huffman@30040
   375
  unfolding setL2_def by simp
huffman@30040
   376
huffman@30040
   377
lemma setL2_mono:
huffman@30040
   378
  assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i"
huffman@30040
   379
  assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"
huffman@30040
   380
  shows "setL2 f K \<le> setL2 g K"
huffman@30040
   381
  unfolding setL2_def
huffman@30040
   382
  by (simp add: setsum_nonneg setsum_mono power_mono prems)
huffman@30040
   383
huffman@30040
   384
lemma setL2_right_distrib:
huffman@30040
   385
  "0 \<le> r \<Longrightarrow> r * setL2 f A = setL2 (\<lambda>x. r * f x) A"
huffman@30040
   386
  unfolding setL2_def
huffman@30040
   387
  apply (simp add: power_mult_distrib)
huffman@30040
   388
  apply (simp add: setsum_right_distrib [symmetric])
huffman@30040
   389
  apply (simp add: real_sqrt_mult setsum_nonneg)
huffman@30040
   390
  done
huffman@30040
   391
huffman@30040
   392
lemma setL2_left_distrib:
huffman@30040
   393
  "0 \<le> r \<Longrightarrow> setL2 f A * r = setL2 (\<lambda>x. f x * r) A"
huffman@30040
   394
  unfolding setL2_def
huffman@30040
   395
  apply (simp add: power_mult_distrib)
huffman@30040
   396
  apply (simp add: setsum_left_distrib [symmetric])
huffman@30040
   397
  apply (simp add: real_sqrt_mult setsum_nonneg)
huffman@30040
   398
  done
huffman@30040
   399
huffman@30040
   400
lemma setsum_nonneg_eq_0_iff:
huffman@30040
   401
  fixes f :: "'a \<Rightarrow> 'b::pordered_ab_group_add"
huffman@30040
   402
  shows "\<lbrakk>finite A; \<forall>x\<in>A. 0 \<le> f x\<rbrakk> \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
huffman@30040
   403
  apply (induct set: finite, simp)
huffman@30040
   404
  apply (simp add: add_nonneg_eq_0_iff setsum_nonneg)
huffman@30040
   405
  done
huffman@30040
   406
huffman@30040
   407
lemma setL2_eq_0_iff: "finite A \<Longrightarrow> setL2 f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
huffman@30040
   408
  unfolding setL2_def
huffman@30040
   409
  by (simp add: setsum_nonneg setsum_nonneg_eq_0_iff)
huffman@30040
   410
huffman@30040
   411
lemma setL2_triangle_ineq:
huffman@30040
   412
  shows "setL2 (\<lambda>i. f i + g i) A \<le> setL2 f A + setL2 g A"
huffman@30040
   413
proof (cases "finite A")
huffman@30040
   414
  case False
huffman@30040
   415
  thus ?thesis by simp
huffman@30040
   416
next
huffman@30040
   417
  case True
huffman@30040
   418
  thus ?thesis
huffman@30040
   419
  proof (induct set: finite)
huffman@30040
   420
    case empty
huffman@30040
   421
    show ?case by simp
huffman@30040
   422
  next
huffman@30040
   423
    case (insert x F)
huffman@30040
   424
    hence "sqrt ((f x + g x)\<twosuperior> + (setL2 (\<lambda>i. f i + g i) F)\<twosuperior>) \<le>
huffman@30040
   425
           sqrt ((f x + g x)\<twosuperior> + (setL2 f F + setL2 g F)\<twosuperior>)"
huffman@30040
   426
      by (intro real_sqrt_le_mono add_left_mono power_mono insert
huffman@30040
   427
                setL2_nonneg add_increasing zero_le_power2)
huffman@30040
   428
    also have
huffman@30040
   429
      "\<dots> \<le> sqrt ((f x)\<twosuperior> + (setL2 f F)\<twosuperior>) + sqrt ((g x)\<twosuperior> + (setL2 g F)\<twosuperior>)"
huffman@30040
   430
      by (rule real_sqrt_sum_squares_triangle_ineq)
huffman@30040
   431
    finally show ?case
huffman@30040
   432
      using insert by simp
huffman@30040
   433
  qed
huffman@30040
   434
qed
huffman@30040
   435
huffman@30040
   436
lemma sqrt_sum_squares_le_sum:
huffman@30040
   437
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) \<le> x + y"
huffman@30040
   438
  apply (rule power2_le_imp_le)
huffman@30040
   439
  apply (simp add: power2_sum)
huffman@30040
   440
  apply (simp add: mult_nonneg_nonneg)
huffman@30040
   441
  apply (simp add: add_nonneg_nonneg)
huffman@30040
   442
  done
huffman@30040
   443
huffman@30040
   444
lemma setL2_le_setsum [rule_format]:
huffman@30040
   445
  "(\<forall>i\<in>A. 0 \<le> f i) \<longrightarrow> setL2 f A \<le> setsum f A"
huffman@30040
   446
  apply (cases "finite A")
huffman@30040
   447
  apply (induct set: finite)
huffman@30040
   448
  apply simp
huffman@30040
   449
  apply clarsimp
huffman@30040
   450
  apply (erule order_trans [OF sqrt_sum_squares_le_sum])
huffman@30040
   451
  apply simp
huffman@30040
   452
  apply simp
huffman@30040
   453
  apply simp
huffman@30040
   454
  done
huffman@30040
   455
huffman@30040
   456
lemma sqrt_sum_squares_le_sum_abs: "sqrt (x\<twosuperior> + y\<twosuperior>) \<le> \<bar>x\<bar> + \<bar>y\<bar>"
huffman@30040
   457
  apply (rule power2_le_imp_le)
huffman@30040
   458
  apply (simp add: power2_sum)
huffman@30040
   459
  apply (simp add: mult_nonneg_nonneg)
huffman@30040
   460
  apply (simp add: add_nonneg_nonneg)
huffman@30040
   461
  done
huffman@30040
   462
huffman@30040
   463
lemma setL2_le_setsum_abs: "setL2 f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
huffman@30040
   464
  apply (cases "finite A")
huffman@30040
   465
  apply (induct set: finite)
huffman@30040
   466
  apply simp
huffman@30040
   467
  apply simp
huffman@30040
   468
  apply (rule order_trans [OF sqrt_sum_squares_le_sum_abs])
huffman@30040
   469
  apply simp
huffman@30040
   470
  apply simp
huffman@30040
   471
  done
huffman@30040
   472
huffman@30040
   473
lemma setL2_mult_ineq_lemma:
huffman@30040
   474
  fixes a b c d :: real
huffman@30040
   475
  shows "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
huffman@30040
   476
proof -
huffman@30040
   477
  have "0 \<le> (a * d - b * c)\<twosuperior>" by simp
huffman@30040
   478
  also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * d) * (b * c)"
huffman@30040
   479
    by (simp only: power2_diff power_mult_distrib)
huffman@30040
   480
  also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * c) * (b * d)"
huffman@30040
   481
    by simp
huffman@30040
   482
  finally show "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
huffman@30040
   483
    by simp
huffman@30040
   484
qed
huffman@30040
   485
huffman@30040
   486
lemma setL2_mult_ineq: "(\<Sum>i\<in>A. \<bar>f i\<bar> * \<bar>g i\<bar>) \<le> setL2 f A * setL2 g A"
huffman@30040
   487
  apply (cases "finite A")
huffman@30040
   488
  apply (induct set: finite)
huffman@30040
   489
  apply simp
huffman@30040
   490
  apply (rule power2_le_imp_le, simp)
huffman@30040
   491
  apply (rule order_trans)
huffman@30040
   492
  apply (rule power_mono)
huffman@30040
   493
  apply (erule add_left_mono)
huffman@30040
   494
  apply (simp add: add_nonneg_nonneg mult_nonneg_nonneg setsum_nonneg)
huffman@30040
   495
  apply (simp add: power2_sum)
huffman@30040
   496
  apply (simp add: power_mult_distrib)
huffman@30040
   497
  apply (simp add: right_distrib left_distrib)
huffman@30040
   498
  apply (rule ord_le_eq_trans)
huffman@30040
   499
  apply (rule setL2_mult_ineq_lemma)
huffman@30040
   500
  apply simp
huffman@30040
   501
  apply (intro mult_nonneg_nonneg setL2_nonneg)
huffman@30040
   502
  apply simp
huffman@30040
   503
  done
huffman@30040
   504
huffman@30040
   505
lemma member_le_setL2: "\<lbrakk>finite A; i \<in> A\<rbrakk> \<Longrightarrow> f i \<le> setL2 f A"
huffman@30040
   506
  apply (rule_tac s="insert i (A - {i})" and t="A" in subst)
huffman@30040
   507
  apply fast
huffman@30040
   508
  apply (subst setL2_insert)
huffman@30040
   509
  apply simp
huffman@30040
   510
  apply simp
huffman@30040
   511
  apply simp
huffman@30040
   512
  done
huffman@30040
   513
huffman@30040
   514
subsection {* Norms *}
huffman@30040
   515
huffman@30040
   516
instantiation "^" :: (real_normed_vector, type) real_normed_vector
huffman@30040
   517
begin
huffman@30040
   518
huffman@30040
   519
definition vector_norm_def:
huffman@30040
   520
  "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) {1 .. dimindex (UNIV:: 'b set)}"
huffman@30040
   521
huffman@30040
   522
definition vector_sgn_def:
huffman@30040
   523
  "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
huffman@30040
   524
huffman@30040
   525
instance proof
huffman@30040
   526
  fix a :: real and x y :: "'a ^ 'b"
huffman@30040
   527
  show "0 \<le> norm x"
huffman@30040
   528
    unfolding vector_norm_def
huffman@30040
   529
    by (rule setL2_nonneg)
huffman@30040
   530
  show "norm x = 0 \<longleftrightarrow> x = 0"
huffman@30040
   531
    unfolding vector_norm_def
huffman@30040
   532
    by (simp add: setL2_eq_0_iff Cart_eq)
huffman@30040
   533
  show "norm (x + y) \<le> norm x + norm y"
huffman@30040
   534
    unfolding vector_norm_def
huffman@30040
   535
    apply (rule order_trans [OF _ setL2_triangle_ineq])
huffman@30040
   536
    apply (rule setL2_mono)
huffman@30040
   537
    apply (simp add: vector_component norm_triangle_ineq)
huffman@30040
   538
    apply simp
huffman@30040
   539
    done
huffman@30040
   540
  show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
huffman@30040
   541
    unfolding vector_norm_def
huffman@30040
   542
    by (simp add: vector_component norm_scaleR setL2_right_distrib
huffman@30040
   543
             cong: strong_setL2_cong)
huffman@30040
   544
  show "sgn x = scaleR (inverse (norm x)) x"
huffman@30040
   545
    by (rule vector_sgn_def)
huffman@30040
   546
qed
huffman@30040
   547
huffman@30040
   548
end
huffman@30040
   549
chaieb@29842
   550
subsection{* Properties of the dot product.  *}
chaieb@29842
   551
chaieb@29842
   552
lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x" 
chaieb@29842
   553
  by (vector mult_commute)
chaieb@29842
   554
lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \<bullet> z = (x \<bullet> z) + (y \<bullet> z)"
chaieb@29842
   555
  by (vector ring_simps)
chaieb@29842
   556
lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)" 
chaieb@29842
   557
  by (vector ring_simps)
chaieb@29842
   558
lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)" 
chaieb@29842
   559
  by (vector ring_simps)
chaieb@29842
   560
lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)" 
chaieb@29842
   561
  by (vector ring_simps)
chaieb@29842
   562
lemma dot_lmult: "(c *s x) \<bullet> y = (c::'a::ring) * (x \<bullet> y)" by (vector ring_simps)
chaieb@29842
   563
lemma dot_rmult: "x \<bullet> (c *s y) = (c::'a::comm_ring) * (x \<bullet> y)" by (vector ring_simps)
chaieb@29842
   564
lemma dot_lneg: "(-x) \<bullet> (y::'a::ring ^ 'n) = -(x \<bullet> y)" by vector
chaieb@29842
   565
lemma dot_rneg: "(x::'a::ring ^ 'n) \<bullet> (-y) = -(x \<bullet> y)" by vector
chaieb@29842
   566
lemma dot_lzero[simp]: "0 \<bullet> x = (0::'a::{comm_monoid_add, mult_zero})" by vector
chaieb@29842
   567
lemma dot_rzero[simp]: "x \<bullet> 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector
chaieb@29842
   568
lemma dot_pos_le[simp]: "(0::'a\<Colon>ordered_ring_strict) <= x \<bullet> x"
chaieb@29842
   569
  by (simp add: dot_def setsum_nonneg)
chaieb@29842
   570
chaieb@29842
   571
lemma setsum_squares_eq_0_iff: assumes fS: "finite F" and fp: "\<forall>x \<in> F. f x \<ge> (0 ::'a::pordered_ab_group_add)" shows "setsum f F = 0 \<longleftrightarrow> (ALL x:F. f x = 0)"
chaieb@29842
   572
using fS fp setsum_nonneg[OF fp]
chaieb@29842
   573
proof (induct set: finite)
chaieb@29842
   574
  case empty thus ?case by simp
chaieb@29842
   575
next
chaieb@29842
   576
  case (insert x F)
chaieb@29842
   577
  from insert.prems have Fx: "f x \<ge> 0" and Fp: "\<forall> a \<in> F. f a \<ge> 0" by simp_all
chaieb@29842
   578
  from insert.hyps Fp setsum_nonneg[OF Fp]
chaieb@29842
   579
  have h: "setsum f F = 0 \<longleftrightarrow> (\<forall>a \<in>F. f a = 0)" by metis
chaieb@29842
   580
  from sum_nonneg_eq_zero_iff[OF Fx  setsum_nonneg[OF Fp]] insert.hyps(1,2)
chaieb@29842
   581
  show ?case by (simp add: h)
chaieb@29842
   582
qed
chaieb@29842
   583
chaieb@29842
   584
lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) = 0"
chaieb@29842
   585
proof-
chaieb@29842
   586
  {assume f: "finite (UNIV :: 'n set)"
chaieb@29842
   587
    let ?S = "{Suc 0 .. card (UNIV :: 'n set)}"
chaieb@29842
   588
    have fS: "finite ?S" using f by simp
chaieb@29842
   589
    have fp: "\<forall> i\<in> ?S. x$i * x$i>= 0" by simp
chaieb@29842
   590
    have ?thesis by (vector dimindex_def f setsum_squares_eq_0_iff[OF fS fp])}
chaieb@29842
   591
  moreover
chaieb@29842
   592
  {assume "\<not> finite (UNIV :: 'n set)" then have ?thesis by (vector dimindex_def)}
chaieb@29842
   593
  ultimately show ?thesis by metis
chaieb@29842
   594
qed
chaieb@29842
   595
chaieb@29842
   596
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] 
chaieb@29842
   597
  by (auto simp add: le_less) 
chaieb@29842
   598
huffman@30040
   599
subsection{* The collapse of the general concepts to dimension one. *}
chaieb@29842
   600
chaieb@29842
   601
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
chaieb@29842
   602
  by (vector dimindex_def)
chaieb@29842
   603
chaieb@29842
   604
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
chaieb@29842
   605
  apply auto
chaieb@29842
   606
  apply (erule_tac x= "x$1" in allE)
chaieb@29842
   607
  apply (simp only: vector_one[symmetric])
chaieb@29842
   608
  done
chaieb@29842
   609
huffman@30040
   610
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
huffman@30040
   611
  by (simp add: vector_norm_def dimindex_def)
huffman@30040
   612
chaieb@29842
   613
lemma norm_real: "norm(x::real ^ 1) = abs(x$1)" 
huffman@30040
   614
  by (simp add: norm_vector_1)
chaieb@29842
   615
chaieb@29842
   616
text{* Metric *}
chaieb@29842
   617
huffman@30040
   618
text {* FIXME: generalize to arbitrary @{text real_normed_vector} types *}
chaieb@29842
   619
definition dist:: "real ^ 'n \<Rightarrow> real ^ 'n \<Rightarrow> real" where 
chaieb@29842
   620
  "dist x y = norm (x - y)"
chaieb@29842
   621
chaieb@29842
   622
lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
chaieb@29842
   623
  using dimindex_ge_1[of "UNIV :: 1 set"]
chaieb@29842
   624
  by (auto simp add: norm_real dist_def vector_component Cart_lambda_beta[where ?'a = "1"] )
chaieb@29842
   625
chaieb@29842
   626
subsection {* A connectedness or intermediate value lemma with several applications. *}
chaieb@29842
   627
chaieb@29842
   628
lemma connected_real_lemma:
chaieb@29842
   629
  fixes f :: "real \<Rightarrow> real ^ 'n"
chaieb@29842
   630
  assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
chaieb@29842
   631
  and dst: "\<And>e x. a <= x \<Longrightarrow> x <= b \<Longrightarrow> 0 < e ==> \<exists>d > 0. \<forall>y. abs(y - x) < d \<longrightarrow> dist(f y) (f x) < e"
chaieb@29842
   632
  and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
chaieb@29842
   633
  and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
chaieb@29842
   634
  and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
chaieb@29842
   635
  shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
chaieb@29842
   636
proof-
chaieb@29842
   637
  let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
chaieb@29842
   638
  have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa) 
chaieb@29842
   639
  have Sub: "\<exists>y. isUb UNIV ?S y" 
chaieb@29842
   640
    apply (rule exI[where x= b])
chaieb@29842
   641
    using ab fb e12 by (auto simp add: isUb_def setle_def)  
chaieb@29842
   642
  from reals_complete[OF Se Sub] obtain l where 
chaieb@29842
   643
    l: "isLub UNIV ?S l"by blast
chaieb@29842
   644
  have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
chaieb@29842
   645
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)    
chaieb@29842
   646
    by (metis linorder_linear)
chaieb@29842
   647
  have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
chaieb@29842
   648
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
chaieb@29842
   649
    by (metis linorder_linear not_le)
chaieb@29842
   650
    have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
chaieb@29842
   651
    have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
chaieb@29842
   652
    have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo
chaieb@29842
   653
    {assume le2: "f l \<in> e2"
chaieb@29842
   654
      from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
chaieb@29842
   655
      hence lap: "l - a > 0" using alb by arith
chaieb@29842
   656
      from e2[rule_format, OF le2] obtain e where 
chaieb@29842
   657
	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
chaieb@29842
   658
      from dst[OF alb e(1)] obtain d where 
chaieb@29842
   659
	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
chaieb@29842
   660
      have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1) 
chaieb@29842
   661
	apply ferrack by arith
chaieb@29842
   662
      then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
chaieb@29842
   663
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
chaieb@29842
   664
      from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
chaieb@29842
   665
      moreover
chaieb@29842
   666
      have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
chaieb@29842
   667
      ultimately have False using e12 alb d' by auto}
chaieb@29842
   668
    moreover
chaieb@29842
   669
    {assume le1: "f l \<in> e1"
chaieb@29842
   670
    from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
chaieb@29842
   671
      hence blp: "b - l > 0" using alb by arith
chaieb@29842
   672
      from e1[rule_format, OF le1] obtain e where 
chaieb@29842
   673
	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
chaieb@29842
   674
      from dst[OF alb e(1)] obtain d where 
chaieb@29842
   675
	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
chaieb@29842
   676
      have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo 
chaieb@29842
   677
      then obtain d' where d': "d' > 0" "d' < d" by metis
chaieb@29842
   678
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
chaieb@29842
   679
      hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
chaieb@29842
   680
      with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
chaieb@29842
   681
      with l d' have False 
chaieb@29842
   682
	by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
chaieb@29842
   683
    ultimately show ?thesis using alb by metis
chaieb@29842
   684
qed
chaieb@29842
   685
huffman@29881
   686
text{* One immediately useful corollary is the existence of square roots! --- Should help to get rid of all the development of square-root for reals as a special case @{typ "real^1"} *}
chaieb@29842
   687
chaieb@29842
   688
lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
chaieb@29842
   689
proof-
chaieb@29842
   690
  have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith 
chaieb@29842
   691
  thus ?thesis by (simp add: ring_simps power2_eq_square)
chaieb@29842
   692
qed
chaieb@29842
   693
chaieb@29842
   694
lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
chaieb@29842
   695
  using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_def, rule_format, of e x] apply (auto simp add: power2_eq_square)
chaieb@29842
   696
  apply (rule_tac x="s" in exI)
chaieb@29842
   697
  apply auto
chaieb@29842
   698
  apply (erule_tac x=y in allE)
chaieb@29842
   699
  apply auto
chaieb@29842
   700
  done
chaieb@29842
   701
chaieb@29842
   702
lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
chaieb@29842
   703
  using real_sqrt_le_iff[of x "y^2"] by simp
chaieb@29842
   704
chaieb@29842
   705
lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
chaieb@29842
   706
  using real_sqrt_le_mono[of "x^2" y] by simp
chaieb@29842
   707
chaieb@29842
   708
lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
chaieb@29842
   709
  using real_sqrt_less_mono[of "x^2" y] by simp
chaieb@29842
   710
chaieb@29842
   711
lemma sqrt_even_pow2: assumes n: "even n" 
chaieb@29842
   712
  shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
chaieb@29842
   713
proof-
chaieb@29842
   714
  from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2 
chaieb@29842
   715
    by (auto simp add: nat_number) 
chaieb@29842
   716
  from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
chaieb@29842
   717
    by (simp only: power_mult[symmetric] mult_commute)
chaieb@29842
   718
  then show ?thesis  using m by simp 
chaieb@29842
   719
qed
chaieb@29842
   720
chaieb@29842
   721
lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
chaieb@29842
   722
  apply (cases "x = 0", simp_all)
chaieb@29842
   723
  using sqrt_divide_self_eq[of x]
chaieb@29842
   724
  apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps)
chaieb@29842
   725
  done
chaieb@29842
   726
chaieb@29842
   727
text{* Hence derive more interesting properties of the norm. *}
chaieb@29842
   728
chaieb@29842
   729
lemma norm_0: "norm (0::real ^ 'n) = 0"
huffman@30040
   730
  by (rule norm_zero)
huffman@30040
   731
chaieb@29842
   732
lemma norm_mul: "norm(a *s x) = abs(a) * norm x"
huffman@30040
   733
  by (simp add: vector_norm_def vector_component setL2_right_distrib
huffman@30040
   734
           abs_mult cong: strong_setL2_cong)
chaieb@29842
   735
lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
huffman@30040
   736
  by (simp add: vector_norm_def dot_def setL2_def power2_eq_square)
huffman@30040
   737
lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"
huffman@30040
   738
  by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
chaieb@29842
   739
lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
huffman@30040
   740
  by (simp add: real_vector_norm_def)
huffman@30041
   741
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
chaieb@29842
   742
lemma vector_mul_eq_0: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
chaieb@29842
   743
  by vector
chaieb@29842
   744
lemma vector_mul_lcancel: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
chaieb@29842
   745
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
chaieb@29842
   746
lemma vector_mul_rcancel: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
chaieb@29842
   747
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
chaieb@29842
   748
lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
chaieb@29842
   749
  by (metis vector_mul_lcancel)
chaieb@29842
   750
lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
chaieb@29842
   751
  by (metis vector_mul_rcancel)
chaieb@29842
   752
lemma norm_cauchy_schwarz: "x \<bullet> y <= norm x * norm y"
chaieb@29842
   753
proof-
chaieb@29842
   754
  {assume "norm x = 0"
huffman@30041
   755
    hence ?thesis by (simp add: dot_lzero dot_rzero)}
chaieb@29842
   756
  moreover
chaieb@29842
   757
  {assume "norm y = 0" 
huffman@30041
   758
    hence ?thesis by (simp add: dot_lzero dot_rzero)}
chaieb@29842
   759
  moreover
chaieb@29842
   760
  {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
chaieb@29842
   761
    let ?z = "norm y *s x - norm x *s y"
huffman@30041
   762
    from h have p: "norm x * norm y > 0" by (metis norm_ge_zero le_less zero_compare_simps)
chaieb@29842
   763
    from dot_pos_le[of ?z]
chaieb@29842
   764
    have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"
chaieb@29842
   765
      apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)
chaieb@29842
   766
      by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym)
chaieb@29842
   767
    hence "x\<bullet>y \<le> (norm x ^2 * norm y ^2) / (norm x * norm y)" using p
chaieb@29842
   768
      by (simp add: field_simps)
chaieb@29842
   769
    hence ?thesis using h by (simp add: power2_eq_square)}
chaieb@29842
   770
  ultimately show ?thesis by metis
chaieb@29842
   771
qed
chaieb@29842
   772
chaieb@29842
   773
lemma norm_cauchy_schwarz_abs: "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
chaieb@29842
   774
  using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
huffman@30041
   775
  by (simp add: real_abs_def dot_rneg)
chaieb@29842
   776
chaieb@29842
   777
lemma norm_triangle_sub: "norm (x::real ^'n) <= norm(y) + norm(x - y)"
huffman@30041
   778
  using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
chaieb@29842
   779
lemma norm_triangle_le: "norm(x::real ^'n) + norm y <= e ==> norm(x + y) <= e"
huffman@30041
   780
  by (metis order_trans norm_triangle_ineq)
chaieb@29842
   781
lemma norm_triangle_lt: "norm(x::real ^'n) + norm(y) < e ==> norm(x + y) < e"
huffman@30041
   782
  by (metis basic_trans_rules(21) norm_triangle_ineq)
chaieb@29842
   783
chaieb@29842
   784
lemma setsum_delta: 
chaieb@29842
   785
  assumes fS: "finite S"
chaieb@29842
   786
  shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
chaieb@29842
   787
proof-
chaieb@29842
   788
  let ?f = "(\<lambda>k. if k=a then b k else 0)"
chaieb@29842
   789
  {assume a: "a \<notin> S"
chaieb@29842
   790
    hence "\<forall> k\<in> S. ?f k = 0" by simp
chaieb@29842
   791
    hence ?thesis  using a by simp}
chaieb@29842
   792
  moreover 
chaieb@29842
   793
  {assume a: "a \<in> S"
chaieb@29842
   794
    let ?A = "S - {a}"
chaieb@29842
   795
    let ?B = "{a}"
chaieb@29842
   796
    have eq: "S = ?A \<union> ?B" using a by blast 
chaieb@29842
   797
    have dj: "?A \<inter> ?B = {}" by simp
chaieb@29842
   798
    from fS have fAB: "finite ?A" "finite ?B" by auto  
chaieb@29842
   799
    have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
chaieb@29842
   800
      using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
chaieb@29842
   801
      by simp
chaieb@29842
   802
    then have ?thesis  using a by simp}
chaieb@29842
   803
  ultimately show ?thesis by blast
chaieb@29842
   804
qed
chaieb@29842
   805
  
chaieb@29842
   806
lemma component_le_norm: "i \<in> {1 .. dimindex(UNIV :: 'n set)} ==> \<bar>x$i\<bar> <= norm (x::real ^ 'n)"
huffman@30040
   807
  apply (simp add: vector_norm_def)
huffman@30040
   808
  apply (rule member_le_setL2, simp_all)
huffman@30040
   809
  done
huffman@30040
   810
chaieb@29842
   811
lemma norm_bound_component_le: "norm(x::real ^ 'n) <= e
chaieb@29842
   812
                ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> <= e"
chaieb@29842
   813
  by (metis component_le_norm order_trans)
chaieb@29842
   814
chaieb@29842
   815
lemma norm_bound_component_lt: "norm(x::real ^ 'n) < e
chaieb@29842
   816
                ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> < e"
chaieb@29842
   817
  by (metis component_le_norm basic_trans_rules(21))
chaieb@29842
   818
chaieb@29842
   819
lemma norm_le_l1: "norm (x:: real ^'n) <= setsum(\<lambda>i. \<bar>x$i\<bar>) {1..dimindex(UNIV::'n set)}"
huffman@30040
   820
  by (simp add: vector_norm_def setL2_le_setsum)
chaieb@29842
   821
chaieb@29842
   822
lemma real_abs_norm: "\<bar> norm x\<bar> = norm (x :: real ^'n)" 
huffman@30040
   823
  by (rule abs_norm_cancel)
chaieb@29842
   824
lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n) - norm y\<bar> <= norm(x - y)"
huffman@30040
   825
  by (rule norm_triangle_ineq3)
chaieb@29842
   826
lemma norm_le: "norm(x::real ^ 'n) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
chaieb@29842
   827
  by (simp add: real_vector_norm_def)
chaieb@29842
   828
lemma norm_lt: "norm(x::real ^'n) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
chaieb@29842
   829
  by (simp add: real_vector_norm_def)
chaieb@29842
   830
lemma norm_eq: "norm (x::real ^'n) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
chaieb@29842
   831
  by (simp add: order_eq_iff norm_le)
chaieb@29842
   832
lemma norm_eq_1: "norm(x::real ^ 'n) = 1 \<longleftrightarrow> x \<bullet> x = 1"
chaieb@29842
   833
  by (simp add: real_vector_norm_def)
chaieb@29842
   834
chaieb@29842
   835
text{* Squaring equations and inequalities involving norms.  *}
chaieb@29842
   836
chaieb@29842
   837
lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
chaieb@29842
   838
  by (simp add: real_vector_norm_def  dot_pos_le )
chaieb@29842
   839
chaieb@29842
   840
lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
huffman@30040
   841
  by (auto simp add: real_vector_norm_def)
chaieb@29842
   842
chaieb@29842
   843
lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
chaieb@29842
   844
proof-
chaieb@29842
   845
  have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: ring_simps power2_eq_square)
chaieb@29842
   846
  also have "\<dots> \<longleftrightarrow> \<bar>x\<bar> \<le> \<bar>y\<bar>" apply (simp add: zero_compare_simps real_abs_def not_less) by arith
chaieb@29842
   847
finally show ?thesis ..
chaieb@29842
   848
qed
chaieb@29842
   849
chaieb@29842
   850
lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
huffman@30040
   851
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
huffman@30041
   852
  using norm_ge_zero[of x]
chaieb@29842
   853
  apply arith
chaieb@29842
   854
  done
chaieb@29842
   855
chaieb@29842
   856
lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2" 
huffman@30040
   857
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
huffman@30041
   858
  using norm_ge_zero[of x]
chaieb@29842
   859
  apply arith
chaieb@29842
   860
  done
chaieb@29842
   861
chaieb@29842
   862
lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
chaieb@29842
   863
  by (metis not_le norm_ge_square)
chaieb@29842
   864
lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
chaieb@29842
   865
  by (metis norm_le_square not_less)
chaieb@29842
   866
chaieb@29842
   867
text{* Dot product in terms of the norm rather than conversely. *}
chaieb@29842
   868
chaieb@29842
   869
lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
chaieb@29842
   870
  by (simp add: norm_pow_2 dot_ladd dot_radd dot_sym)
chaieb@29842
   871
chaieb@29842
   872
lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
chaieb@29842
   873
  by (simp add: norm_pow_2 dot_ladd dot_radd dot_lsub dot_rsub dot_sym)
chaieb@29842
   874
chaieb@29842
   875
chaieb@29842
   876
text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
chaieb@29842
   877
chaieb@29842
   878
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")
chaieb@29842
   879
proof
chaieb@29842
   880
  assume "?lhs" then show ?rhs by simp
chaieb@29842
   881
next
chaieb@29842
   882
  assume ?rhs
chaieb@29842
   883
  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y\<bullet> y = 0" by simp
chaieb@29842
   884
  hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" 
chaieb@29842
   885
    by (simp add: dot_rsub dot_lsub dot_sym)
chaieb@29842
   886
  then have "(x - y) \<bullet> (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub)
chaieb@29842
   887
  then show "x = y" by (simp add: dot_eq_0)
chaieb@29842
   888
qed
chaieb@29842
   889
chaieb@29842
   890
chaieb@29842
   891
subsection{* General linear decision procedure for normed spaces. *}
chaieb@29842
   892
chaieb@29842
   893
lemma norm_cmul_rule_thm: "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(c *s x)"
chaieb@29842
   894
  apply (clarsimp simp add: norm_mul)
chaieb@29842
   895
  apply (rule mult_mono1)
chaieb@29842
   896
  apply simp_all
chaieb@29842
   897
  done
chaieb@29842
   898
chaieb@29842
   899
lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n) \<Longrightarrow> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)"
chaieb@29842
   900
  apply (rule norm_triangle_le) by simp
chaieb@29842
   901
chaieb@29842
   902
lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"
chaieb@29842
   903
  by (simp add: ring_simps)
chaieb@29842
   904
chaieb@29842
   905
lemma pth_1: "(x::real^'n) == 1 *s x" by (simp only: vector_smult_lid)
chaieb@29842
   906
lemma pth_2: "x - (y::real^'n) == x + -y" by (atomize (full)) simp
chaieb@29842
   907
lemma pth_3: "(-x::real^'n) == -1 *s x" by vector
chaieb@29842
   908
lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+
chaieb@29842
   909
lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector
chaieb@29842
   910
lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps)
chaieb@29842
   911
lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all 
chaieb@29842
   912
lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps) 
chaieb@29842
   913
lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z"
chaieb@29842
   914
  "c *s x + (d *s x + z) == (c + d) *s x + z"
chaieb@29842
   915
  "(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+
chaieb@29842
   916
lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector
chaieb@29842
   917
lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y" 
chaieb@29842
   918
  "(c *s x + z) + d *s y == c *s x + (z + d *s y)"
chaieb@29842
   919
  "c *s x + (d *s y + z) == c *s x + (d *s y + z)"
chaieb@29842
   920
  "(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))"
chaieb@29842
   921
  by ((atomize (full)), vector)+
chaieb@29842
   922
lemma pth_c: "(c::real) *s x + d *s y == d *s y + c *s x"
chaieb@29842
   923
  "(c *s x + z) + d *s y == d *s y + (c *s x + z)"
chaieb@29842
   924
  "c *s x + (d *s y + z) == d *s y + (c *s x + z)"
chaieb@29842
   925
  "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+
chaieb@29842
   926
lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector
chaieb@29842
   927
chaieb@29842
   928
lemma norm_imp_pos_and_ge: "norm (x::real ^ 'n) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
huffman@30041
   929
  by (atomize) (auto simp add: norm_ge_zero)
chaieb@29842
   930
chaieb@29842
   931
lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
chaieb@29842
   932
chaieb@29842
   933
lemma norm_pths: 
chaieb@29842
   934
  "(x::real ^'n) = y \<longleftrightarrow> norm (x - y) \<le> 0"
chaieb@29842
   935
  "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
huffman@30041
   936
  using norm_ge_zero[of "x - y"] by auto
chaieb@29842
   937
chaieb@29842
   938
use "normarith.ML"
chaieb@29842
   939
chaieb@29842
   940
method_setup norm = {* Method.ctxt_args (Method.SIMPLE_METHOD' o NormArith.norm_arith_tac)
chaieb@29842
   941
*} "Proves simple linear statements about vector norms"
chaieb@29842
   942
chaieb@29842
   943
chaieb@29842
   944
chaieb@29842
   945
text{* Hence more metric properties. *}
chaieb@29842
   946
chaieb@29842
   947
lemma dist_refl: "dist x x = 0" by norm
chaieb@29842
   948
chaieb@29842
   949
lemma dist_sym: "dist x y = dist y x"by norm
chaieb@29842
   950
chaieb@29842
   951
lemma dist_pos_le: "0 <= dist x y" by norm
chaieb@29842
   952
chaieb@29842
   953
lemma dist_triangle: "dist x z <= dist x y + dist y z" by norm
chaieb@29842
   954
chaieb@29842
   955
lemma dist_triangle_alt: "dist y z <= dist x y + dist x z" by norm
chaieb@29842
   956
chaieb@29842
   957
lemma dist_eq_0: "dist x y = 0 \<longleftrightarrow> x = y" by norm
chaieb@29842
   958
chaieb@29842
   959
lemma dist_pos_lt: "x \<noteq> y ==> 0 < dist x y" by norm 
chaieb@29842
   960
lemma dist_nz:  "x \<noteq> y \<longleftrightarrow> 0 < dist x y" by norm 
chaieb@29842
   961
chaieb@29842
   962
lemma dist_triangle_le: "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e" by norm 
chaieb@29842
   963
chaieb@29842
   964
lemma dist_triangle_lt: "dist x z + dist y z < e ==> dist x y < e" by norm 
chaieb@29842
   965
chaieb@29842
   966
lemma dist_triangle_half_l: "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 ==> dist x1 x2 < e" by norm 
chaieb@29842
   967
chaieb@29842
   968
lemma dist_triangle_half_r: "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 ==> dist x1 x2 < e" by norm 
chaieb@29842
   969
chaieb@29842
   970
lemma dist_triangle_add: "dist (x + y) (x' + y') <= dist x x' + dist y y'"
chaieb@29842
   971
  by norm 
chaieb@29842
   972
chaieb@29842
   973
lemma dist_mul: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y" 
chaieb@29842
   974
  unfolding dist_def vector_ssub_ldistrib[symmetric] norm_mul .. 
chaieb@29842
   975
chaieb@29842
   976
lemma dist_triangle_add_half: " dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 ==> dist(x + y) (x' + y') < e" by norm 
chaieb@29842
   977
chaieb@29842
   978
lemma dist_le_0: "dist x y <= 0 \<longleftrightarrow> x = y" by norm 
chaieb@29842
   979
chaieb@29842
   980
instantiation "^" :: (monoid_add,type) monoid_add
chaieb@29842
   981
begin
chaieb@29842
   982
  instance by (intro_classes)
chaieb@29842
   983
end
chaieb@29842
   984
chaieb@29842
   985
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
chaieb@29842
   986
  apply vector
chaieb@29842
   987
  apply auto
chaieb@29842
   988
  apply (cases "finite S")
chaieb@29842
   989
  apply (rule finite_induct[of S])
chaieb@29842
   990
  apply (auto simp add: vector_component zero_index)
chaieb@29842
   991
  done
chaieb@29842
   992
chaieb@29842
   993
lemma setsum_clauses: 
chaieb@29842
   994
  shows "setsum f {} = 0"
chaieb@29842
   995
  and "finite S \<Longrightarrow> setsum f (insert x S) =
chaieb@29842
   996
                 (if x \<in> S then setsum f S else f x + setsum f S)"
chaieb@29842
   997
  by (auto simp add: insert_absorb)
chaieb@29842
   998
chaieb@29842
   999
lemma setsum_cmul: 
chaieb@29842
  1000
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
chaieb@29842
  1001
  shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
chaieb@29842
  1002
  by (simp add: setsum_eq Cart_eq Cart_lambda_beta vector_component setsum_right_distrib)
chaieb@29842
  1003
chaieb@29842
  1004
lemma setsum_component: 
chaieb@29842
  1005
  fixes f:: " 'a \<Rightarrow> ('b::semiring_1) ^'n"
chaieb@29842
  1006
  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
chaieb@29842
  1007
  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
chaieb@29842
  1008
  using i by (simp add: setsum_eq Cart_lambda_beta)
chaieb@29842
  1009
chaieb@29842
  1010
  (* This needs finiteness assumption due to the definition of fold!!! *)
chaieb@29842
  1011
chaieb@29842
  1012
lemma setsum_superset:
chaieb@29842
  1013
  assumes fb: "finite B" and ab: "A \<subseteq> B" 
chaieb@29842
  1014
  and f0: "\<forall>x \<in> B - A. f x = 0"
chaieb@29842
  1015
  shows "setsum f B = setsum f A"
chaieb@29842
  1016
proof-
chaieb@29842
  1017
  from ab fb have fa: "finite A" by (metis finite_subset)
chaieb@29842
  1018
  from fb have fba: "finite (B - A)" by (metis finite_Diff)
chaieb@29842
  1019
  have d: "A \<inter> (B - A) = {}" by blast
chaieb@29842
  1020
  from ab have b: "B = A \<union> (B - A)" by blast
chaieb@29842
  1021
  from setsum_Un_disjoint[OF fa fba d, of f] b
chaieb@29842
  1022
    setsum_0'[OF f0]
chaieb@29842
  1023
  show "setsum f B = setsum f A" by simp
chaieb@29842
  1024
qed
chaieb@29842
  1025
chaieb@29842
  1026
lemma setsum_restrict_set:
chaieb@29842
  1027
  assumes fA: "finite A"
chaieb@29842
  1028
  shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
chaieb@29842
  1029
proof-
chaieb@29842
  1030
  from fA have fab: "finite (A \<inter> B)" by auto
chaieb@29842
  1031
  have aba: "A \<inter> B \<subseteq> A" by blast
chaieb@29842
  1032
  let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
chaieb@29842
  1033
  from setsum_superset[OF fA aba, of ?g]
chaieb@29842
  1034
  show ?thesis by simp
chaieb@29842
  1035
qed
chaieb@29842
  1036
chaieb@29842
  1037
lemma setsum_cases:
chaieb@29842
  1038
  assumes fA: "finite A"
chaieb@29842
  1039
  shows "setsum (\<lambda>x. if x \<in> B then f x else g x) A =
chaieb@29842
  1040
         setsum f (A \<inter> B) + setsum g (A \<inter> - B)"
chaieb@29842
  1041
proof-
chaieb@29842
  1042
  have a: "A = A \<inter> B \<union> A \<inter> -B" "(A \<inter> B) \<inter> (A \<inter> -B) = {}" 
chaieb@29842
  1043
    by blast+
chaieb@29842
  1044
  from fA 
chaieb@29842
  1045
  have f: "finite (A \<inter> B)" "finite (A \<inter> -B)" by auto
chaieb@29842
  1046
  let ?g = "\<lambda>x. if x \<in> B then f x else g x"
chaieb@29842
  1047
  from setsum_Un_disjoint[OF f a(2), of ?g] a(1)
chaieb@29842
  1048
  show ?thesis by simp
chaieb@29842
  1049
qed
chaieb@29842
  1050
chaieb@29842
  1051
lemma setsum_norm: 
chaieb@29842
  1052
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
chaieb@29842
  1053
  assumes fS: "finite S"
chaieb@29842
  1054
  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
chaieb@29842
  1055
proof(induct rule: finite_induct[OF fS])
huffman@30041
  1056
  case 1 thus ?case by simp
chaieb@29842
  1057
next
chaieb@29842
  1058
  case (2 x S)
chaieb@29842
  1059
  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
chaieb@29842
  1060
  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
chaieb@29842
  1061
    using "2.hyps" by simp
chaieb@29842
  1062
  finally  show ?case  using "2.hyps" by simp
chaieb@29842
  1063
qed
chaieb@29842
  1064
chaieb@29842
  1065
lemma real_setsum_norm: 
chaieb@29842
  1066
  fixes f :: "'a \<Rightarrow> real ^'n"
chaieb@29842
  1067
  assumes fS: "finite S"
chaieb@29842
  1068
  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
chaieb@29842
  1069
proof(induct rule: finite_induct[OF fS])
huffman@30040
  1070
  case 1 thus ?case by simp
chaieb@29842
  1071
next
chaieb@29842
  1072
  case (2 x S)
huffman@30040
  1073
  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
chaieb@29842
  1074
  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
chaieb@29842
  1075
    using "2.hyps" by simp
chaieb@29842
  1076
  finally  show ?case  using "2.hyps" by simp
chaieb@29842
  1077
qed
chaieb@29842
  1078
chaieb@29842
  1079
lemma setsum_norm_le: 
chaieb@29842
  1080
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
chaieb@29842
  1081
  assumes fS: "finite S"
chaieb@29842
  1082
  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
chaieb@29842
  1083
  shows "norm (setsum f S) \<le> setsum g S"
chaieb@29842
  1084
proof-
chaieb@29842
  1085
  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S" 
chaieb@29842
  1086
    by - (rule setsum_mono, simp)
chaieb@29842
  1087
  then show ?thesis using setsum_norm[OF fS, of f] fg
chaieb@29842
  1088
    by arith
chaieb@29842
  1089
qed
chaieb@29842
  1090
chaieb@29842
  1091
lemma real_setsum_norm_le: 
chaieb@29842
  1092
  fixes f :: "'a \<Rightarrow> real ^ 'n"
chaieb@29842
  1093
  assumes fS: "finite S"
chaieb@29842
  1094
  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
chaieb@29842
  1095
  shows "norm (setsum f S) \<le> setsum g S"
chaieb@29842
  1096
proof-
chaieb@29842
  1097
  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S" 
chaieb@29842
  1098
    by - (rule setsum_mono, simp)
chaieb@29842
  1099
  then show ?thesis using real_setsum_norm[OF fS, of f] fg
chaieb@29842
  1100
    by arith
chaieb@29842
  1101
qed
chaieb@29842
  1102
chaieb@29842
  1103
lemma setsum_norm_bound:
chaieb@29842
  1104
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
chaieb@29842
  1105
  assumes fS: "finite S"
chaieb@29842
  1106
  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
chaieb@29842
  1107
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
chaieb@29842
  1108
  using setsum_norm_le[OF fS K] setsum_constant[symmetric]
chaieb@29842
  1109
  by simp
chaieb@29842
  1110
chaieb@29842
  1111
lemma real_setsum_norm_bound:
chaieb@29842
  1112
  fixes f :: "'a \<Rightarrow> real ^ 'n"
chaieb@29842
  1113
  assumes fS: "finite S"
chaieb@29842
  1114
  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
chaieb@29842
  1115
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
chaieb@29842
  1116
  using real_setsum_norm_le[OF fS K] setsum_constant[symmetric]
chaieb@29842
  1117
  by simp
chaieb@29842
  1118
chaieb@29842
  1119
lemma setsum_vmul:
chaieb@29842
  1120
  fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector,semiring, mult_zero}"
chaieb@29842
  1121
  assumes fS: "finite S"
chaieb@29842
  1122
  shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
chaieb@29842
  1123
proof(induct rule: finite_induct[OF fS])
chaieb@29842
  1124
  case 1 then show ?case by (simp add: vector_smult_lzero)
chaieb@29842
  1125
next
chaieb@29842
  1126
  case (2 x F)
chaieb@29842
  1127
  from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v" 
chaieb@29842
  1128
    by simp
chaieb@29842
  1129
  also have "\<dots> = f x *s v + setsum f F *s v" 
chaieb@29842
  1130
    by (simp add: vector_sadd_rdistrib)
chaieb@29842
  1131
  also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp
chaieb@29842
  1132
  finally show ?case .
chaieb@29842
  1133
qed
chaieb@29842
  1134
chaieb@29842
  1135
(* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \<Rightarrow> real ^'n"]  ---
chaieb@29842
  1136
 Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *)
chaieb@29842
  1137
chaieb@29842
  1138
lemma setsum_add_split: assumes mn: "(m::nat) \<le> n + 1"
chaieb@29842
  1139
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
chaieb@29842
  1140
proof-
chaieb@29842
  1141
  let ?A = "{m .. n}"
chaieb@29842
  1142
  let ?B = "{n + 1 .. n + p}"
chaieb@29842
  1143
  have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto 
chaieb@29842
  1144
  have d: "?A \<inter> ?B = {}" by auto
chaieb@29842
  1145
  from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto
chaieb@29842
  1146
qed
chaieb@29842
  1147
chaieb@29842
  1148
lemma setsum_reindex_nonzero: 
chaieb@29842
  1149
  assumes fS: "finite S"
chaieb@29842
  1150
  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"
chaieb@29842
  1151
  shows "setsum h (f ` S) = setsum (h o f) S"
chaieb@29842
  1152
using nz
chaieb@29842
  1153
proof(induct rule: finite_induct[OF fS])
chaieb@29842
  1154
  case 1 thus ?case by simp
chaieb@29842
  1155
next
chaieb@29842
  1156
  case (2 x F) 
chaieb@29842
  1157
  {assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
chaieb@29842
  1158
    then obtain y where y: "y \<in> F" "f x = f y" by auto 
chaieb@29842
  1159
    from "2.hyps" y have xy: "x \<noteq> y" by auto
chaieb@29842
  1160
    
chaieb@29842
  1161
    from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
chaieb@29842
  1162
    have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
chaieb@29842
  1163
    also have "\<dots> = setsum (h o f) (insert x F)" 
chaieb@29842
  1164
      using "2.hyps" "2.prems" h0  by auto 
chaieb@29842
  1165
    finally have ?case .}
chaieb@29842
  1166
  moreover
chaieb@29842
  1167
  {assume fxF: "f x \<notin> f ` F"
chaieb@29842
  1168
    have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" 
chaieb@29842
  1169
      using fxF "2.hyps" by simp 
chaieb@29842
  1170
    also have "\<dots> = setsum (h o f) (insert x F)"  
chaieb@29842
  1171
      using "2.hyps" "2.prems" fxF
chaieb@29842
  1172
      apply auto apply metis done
chaieb@29842
  1173
    finally have ?case .}
chaieb@29842
  1174
  ultimately show ?case by blast
chaieb@29842
  1175
qed
chaieb@29842
  1176
chaieb@29842
  1177
lemma setsum_Un_nonzero:
chaieb@29842
  1178
  assumes fS: "finite S" and fF: "finite F"
chaieb@29842
  1179
  and f: "\<forall> x\<in> S \<inter> F . f x = (0::'a::ab_group_add)"
chaieb@29842
  1180
  shows "setsum f (S \<union> F) = setsum f S + setsum f F"
chaieb@29842
  1181
  using setsum_Un[OF fS fF, of f] setsum_0'[OF f] by simp
chaieb@29842
  1182
chaieb@29842
  1183
lemma setsum_natinterval_left:
chaieb@29842
  1184
  assumes mn: "(m::nat) <= n" 
chaieb@29842
  1185
  shows "setsum f {m..n} = f m + setsum f {m + 1..n}"
chaieb@29842
  1186
proof-
chaieb@29842
  1187
  from mn have "{m .. n} = insert m {m+1 .. n}" by auto
chaieb@29842
  1188
  then show ?thesis by auto
chaieb@29842
  1189
qed
chaieb@29842
  1190
chaieb@29842
  1191
lemma setsum_natinterval_difff: 
chaieb@29842
  1192
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
chaieb@29842
  1193
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
chaieb@29842
  1194
          (if m <= n then f m - f(n + 1) else 0)"
chaieb@29842
  1195
by (induct n, auto simp add: ring_simps not_le le_Suc_eq)
chaieb@29842
  1196
chaieb@29842
  1197
lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
chaieb@29842
  1198
chaieb@29842
  1199
lemma setsum_setsum_restrict:
chaieb@29842
  1200
  "finite S \<Longrightarrow> finite T \<Longrightarrow> setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y\<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"
chaieb@29842
  1201
  apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
chaieb@29842
  1202
  by (rule setsum_commute)
chaieb@29842
  1203
chaieb@29842
  1204
lemma setsum_image_gen: assumes fS: "finite S"
chaieb@29842
  1205
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
chaieb@29842
  1206
proof-
chaieb@29842
  1207
  {fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto}
chaieb@29842
  1208
  note th0 = this
chaieb@29842
  1209
  have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S" 
chaieb@29842
  1210
    apply (rule setsum_cong2) 
chaieb@29842
  1211
    by (simp add: th0)
chaieb@29842
  1212
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
chaieb@29842
  1213
    apply (rule setsum_setsum_restrict[OF fS])
chaieb@29842
  1214
    by (rule finite_imageI[OF fS])
chaieb@29842
  1215
  finally show ?thesis .
chaieb@29842
  1216
qed
chaieb@29842
  1217
chaieb@29842
  1218
    (* FIXME: Here too need stupid finiteness assumption on T!!! *)
chaieb@29842
  1219
lemma setsum_group:
chaieb@29842
  1220
  assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
chaieb@29842
  1221
  shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
chaieb@29842
  1222
  
chaieb@29842
  1223
apply (subst setsum_image_gen[OF fS, of g f])
chaieb@29842
  1224
apply (rule setsum_superset[OF fT fST])
chaieb@29842
  1225
by (auto intro: setsum_0')
chaieb@29842
  1226
chaieb@29842
  1227
(* FIXME: Change the name to fold_image\<dots> *)
chaieb@29842
  1228
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"
chaieb@29842
  1229
  apply (induct set: finite)
chaieb@29842
  1230
  apply simp by (auto simp add: fold_image_insert)
chaieb@29842
  1231
chaieb@29842
  1232
lemma (in comm_monoid_mult) fold_union_nonzero:
chaieb@29842
  1233
  assumes fS: "finite S" and fT: "finite T"
chaieb@29842
  1234
  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
chaieb@29842
  1235
  shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
chaieb@29842
  1236
proof-
chaieb@29842
  1237
  have "fold_image op * f 1 (S \<inter> T) = 1" 
chaieb@29842
  1238
    apply (rule fold_1')
chaieb@29842
  1239
    using fS fT I0 by auto 
chaieb@29842
  1240
  with fold_image_Un_Int[OF fS fT] show ?thesis by simp
chaieb@29842
  1241
qed
chaieb@29842
  1242
chaieb@29842
  1243
lemma setsum_union_nonzero:  
chaieb@29842
  1244
  assumes fS: "finite S" and fT: "finite T"
chaieb@29842
  1245
  and I0: "\<forall>x \<in> S\<inter>T. f x = 0"
chaieb@29842
  1246
  shows "setsum f (S \<union> T) = setsum f S  + setsum f T"
chaieb@29842
  1247
  using fS fT
chaieb@29842
  1248
  apply (simp add: setsum_def)
chaieb@29842
  1249
  apply (rule comm_monoid_add.fold_union_nonzero)
chaieb@29842
  1250
  using I0 by auto
chaieb@29842
  1251
chaieb@29842
  1252
lemma setprod_union_nonzero:  
chaieb@29842
  1253
  assumes fS: "finite S" and fT: "finite T"
chaieb@29842
  1254
  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
chaieb@29842
  1255
  shows "setprod f (S \<union> T) = setprod f S  * setprod f T"
chaieb@29842
  1256
  using fS fT
chaieb@29842
  1257
  apply (simp add: setprod_def)
chaieb@29842
  1258
  apply (rule fold_union_nonzero)
chaieb@29842
  1259
  using I0 by auto
chaieb@29842
  1260
chaieb@29842
  1261
lemma setsum_unions_nonzero: 
chaieb@29842
  1262
  assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
chaieb@29842
  1263
  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"
chaieb@29842
  1264
  shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
chaieb@29842
  1265
  using fSS f0
chaieb@29842
  1266
proof(induct rule: finite_induct[OF fS])
chaieb@29842
  1267
  case 1 thus ?case by simp
chaieb@29842
  1268
next
chaieb@29842
  1269
  case (2 T F)
chaieb@29842
  1270
  then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F" 
chaieb@29842
  1271
    and H: "setsum f (\<Union> F) = setsum (setsum f) F" by (auto simp add: finite_insert)
chaieb@29842
  1272
  from fTF have fUF: "finite (\<Union>F)" by (auto intro: finite_Union)
chaieb@29842
  1273
  from "2.prems" TF fTF
chaieb@29842
  1274
  show ?case 
chaieb@29842
  1275
    by (auto simp add: H[symmetric] intro: setsum_union_nonzero[OF fTF(1) fUF, of f])
chaieb@29842
  1276
qed
chaieb@29842
  1277
chaieb@29842
  1278
  (* FIXME : Copied from Pocklington --- should be moved to Finite_Set!!!!!!!! *)
chaieb@29842
  1279
chaieb@29842
  1280
chaieb@29842
  1281
lemma (in comm_monoid_mult) fold_related: 
chaieb@29842
  1282
  assumes Re: "R e e" 
chaieb@29842
  1283
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
chaieb@29842
  1284
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
chaieb@29842
  1285
  shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
chaieb@29842
  1286
  using fS by (rule finite_subset_induct) (insert assms, auto)
chaieb@29842
  1287
chaieb@29842
  1288
  (* FIXME: I think we can get rid of the finite assumption!! *)	
chaieb@29842
  1289
lemma (in comm_monoid_mult) 
chaieb@29842
  1290
  fold_eq_general:
chaieb@29842
  1291
  assumes fS: "finite S"
chaieb@29842
  1292
  and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
chaieb@29842
  1293
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
chaieb@29842
  1294
  shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
chaieb@29842
  1295
proof-
chaieb@29842
  1296
  from h f12 have hS: "h ` S = S'" by auto
chaieb@29842
  1297
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
chaieb@29842
  1298
    from f12 h H  have "x = y" by auto }
chaieb@29842
  1299
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
chaieb@29842
  1300
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
chaieb@29842
  1301
  from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
chaieb@29842
  1302
  also have "\<dots> = fold_image (op *) (f2 o h) e S" 
chaieb@29842
  1303
    using fold_image_reindex[OF fS hinj, of f2 e] .
chaieb@29842
  1304
  also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
chaieb@29842
  1305
    by blast
chaieb@29842
  1306
  finally show ?thesis ..
chaieb@29842
  1307
qed
chaieb@29842
  1308
chaieb@29842
  1309
lemma (in comm_monoid_mult) fold_eq_general_inverses:
chaieb@29842
  1310
  assumes fS: "finite S" 
chaieb@29842
  1311
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
chaieb@29842
  1312
  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
chaieb@29842
  1313
  shows "fold_image (op *) f e S = fold_image (op *) g e T"
chaieb@29842
  1314
  using fold_eq_general[OF fS, of T h g f e] kh hk by metis
chaieb@29842
  1315
chaieb@29842
  1316
lemma setsum_eq_general_reverses:
chaieb@29842
  1317
  assumes fS: "finite S" and fT: "finite T"
chaieb@29842
  1318
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
chaieb@29842
  1319
  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
chaieb@29842
  1320
  shows "setsum f S = setsum g T"
chaieb@29842
  1321
  apply (simp add: setsum_def fS fT)
chaieb@29842
  1322
  apply (rule comm_monoid_add.fold_eq_general_inverses[OF fS])
chaieb@29842
  1323
  apply (erule kh)
chaieb@29842
  1324
  apply (erule hk)
chaieb@29842
  1325
  done
chaieb@29842
  1326
chaieb@29842
  1327
lemma vsum_norm_allsubsets_bound:
chaieb@29842
  1328
  fixes f:: "'a \<Rightarrow> real ^'n"
chaieb@29842
  1329
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e" 
chaieb@29842
  1330
  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real (dimindex(UNIV :: 'n set)) *  e"
chaieb@29842
  1331
proof-
chaieb@29842
  1332
  let ?d = "real (dimindex (UNIV ::'n set))"
chaieb@29842
  1333
  let ?nf = "\<lambda>x. norm (f x)"
chaieb@29842
  1334
  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
chaieb@29842
  1335
  have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x $ i\<bar>) P) ?U"
chaieb@29842
  1336
    by (rule setsum_commute)
chaieb@29842
  1337
  have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
chaieb@29842
  1338
  have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
chaieb@29842
  1339
    apply (rule setsum_mono)
chaieb@29842
  1340
    by (rule norm_le_l1)
chaieb@29842
  1341
  also have "\<dots> \<le> 2 * ?d * e"
chaieb@29842
  1342
    unfolding th0 th1
chaieb@29842
  1343
  proof(rule setsum_bounded)
chaieb@29842
  1344
    fix i assume i: "i \<in> ?U"
chaieb@29842
  1345
    let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
chaieb@29842
  1346
    let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
chaieb@29842
  1347
    have thp: "P = ?Pp \<union> ?Pn" by auto
chaieb@29842
  1348
    have thp0: "?Pp \<inter> ?Pn ={}" by auto
chaieb@29842
  1349
    have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
chaieb@29842
  1350
    have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
chaieb@29842
  1351
      using i component_le_norm[OF i, of "setsum (\<lambda>x. f x) ?Pp"]  fPs[OF PpP]
chaieb@29842
  1352
      by (auto simp add: setsum_component intro: abs_le_D1)
chaieb@29842
  1353
    have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
chaieb@29842
  1354
      using i component_le_norm[OF i, of "setsum (\<lambda>x. - f x) ?Pn"]  fPs[OF PnP]
huffman@30041
  1355
      by (auto simp add: setsum_negf setsum_component vector_component intro: abs_le_D1)
chaieb@29842
  1356
    have "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn" 
chaieb@29842
  1357
      apply (subst thp)
chaieb@29842
  1358
      apply (rule setsum_Un_nonzero) 
chaieb@29842
  1359
      using fP thp0 by auto
chaieb@29842
  1360
    also have "\<dots> \<le> 2*e" using Pne Ppe by arith
chaieb@29842
  1361
    finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
chaieb@29842
  1362
  qed
chaieb@29842
  1363
  finally show ?thesis .
chaieb@29842
  1364
qed
chaieb@29842
  1365
chaieb@29842
  1366
lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> (y::'a::{comm_ring}^'n) = setsum (\<lambda>x. f x \<bullet> y) S "
chaieb@29842
  1367
  by (induct rule: finite_induct, auto simp add: dot_lzero dot_ladd)
chaieb@29842
  1368
chaieb@29842
  1369
lemma dot_rsum: "finite S \<Longrightarrow> (y::'a::{comm_ring}^'n) \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
chaieb@29842
  1370
  by (induct rule: finite_induct, auto simp add: dot_rzero dot_radd)
chaieb@29842
  1371
chaieb@29842
  1372
subsection{* Basis vectors in coordinate directions. *}
chaieb@29842
  1373
chaieb@29842
  1374
chaieb@29842
  1375
definition "basis k = (\<chi> i. if i = k then 1 else 0)"
chaieb@29842
  1376
chaieb@29842
  1377
lemma delta_mult_idempotent: 
chaieb@29842
  1378
  "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" by (cases "k=a", auto)
chaieb@29842
  1379
chaieb@29842
  1380
lemma norm_basis:
chaieb@29842
  1381
  assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
chaieb@29842
  1382
  shows "norm (basis k :: real ^'n) = 1"
chaieb@29842
  1383
  using k 
chaieb@29842
  1384
  apply (simp add: basis_def real_vector_norm_def dot_def)
chaieb@29842
  1385
  apply (vector delta_mult_idempotent)
chaieb@29842
  1386
  using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "k" "\<lambda>k. 1::real"]
chaieb@29842
  1387
  apply auto
chaieb@29842
  1388
  done
chaieb@29842
  1389
chaieb@29842
  1390
lemma norm_basis_1: "norm(basis 1 :: real ^'n) = 1"
chaieb@29842
  1391
  apply (simp add: basis_def real_vector_norm_def dot_def)
chaieb@29842
  1392
  apply (vector delta_mult_idempotent)
chaieb@29842
  1393
  using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "1" "\<lambda>k. 1::real"] dimindex_nonzero[of "UNIV :: 'n set"]
chaieb@29842
  1394
  apply auto
chaieb@29842
  1395
  done
chaieb@29842
  1396
chaieb@29842
  1397
lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c"
chaieb@29842
  1398
  apply (rule exI[where x="c *s basis 1"])
chaieb@29842
  1399
  by (simp only: norm_mul norm_basis_1)
chaieb@29842
  1400
chaieb@29842
  1401
lemma vector_choose_dist: assumes e: "0 <= e" 
chaieb@29842
  1402
  shows "\<exists>(y::real^'n). dist x y = e"
chaieb@29842
  1403
proof-
chaieb@29842
  1404
  from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"
chaieb@29842
  1405
    by blast
chaieb@29842
  1406
  then have "dist x (x - c) = e" by (simp add: dist_def)
chaieb@29842
  1407
  then show ?thesis by blast
chaieb@29842
  1408
qed
chaieb@29842
  1409
chaieb@29842
  1410
lemma basis_inj: "inj_on (basis :: nat \<Rightarrow> real ^'n) {1 .. dimindex (UNIV :: 'n set)}"
chaieb@29842
  1411
  by (auto simp add: inj_on_def basis_def Cart_eq Cart_lambda_beta)
chaieb@29842
  1412
chaieb@29842
  1413
lemma basis_component: "i \<in> {1 .. dimindex(UNIV:: 'n set)} ==> (basis k ::('a::semiring_1)^'n)$i = (if k=i then 1 else 0)"
chaieb@29842
  1414
  by (simp add: basis_def Cart_lambda_beta)
chaieb@29842
  1415
chaieb@29842
  1416
lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
chaieb@29842
  1417
  by auto
chaieb@29842
  1418
chaieb@29842
  1419
lemma basis_expansion:
chaieb@29842
  1420
  "setsum (\<lambda>i. (x$i) *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
chaieb@29842
  1421
  by (auto simp add: Cart_eq basis_component[where ?'n = "'n"] setsum_component vector_component cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
chaieb@29842
  1422
chaieb@29842
  1423
lemma basis_expansion_unique: 
chaieb@29842
  1424
  "setsum (\<lambda>i. f i *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i\<in>{1 .. dimindex(UNIV:: 'n set)}. f i = x$i)"
chaieb@29842
  1425
  by (simp add: Cart_eq setsum_component vector_component basis_component setsum_delta cond_value_iff cong del: if_weak_cong)
chaieb@29842
  1426
chaieb@29842
  1427
lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
chaieb@29842
  1428
  by auto
chaieb@29842
  1429
chaieb@29842
  1430
lemma dot_basis:
chaieb@29842
  1431
  assumes i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
chaieb@29842
  1432
  shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i :: 'a^'n) = (x$i :: 'a::semiring_1)"
chaieb@29842
  1433
  using i
chaieb@29842
  1434
  by (auto simp add: dot_def basis_def Cart_lambda_beta cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)
chaieb@29842
  1435
chaieb@29842
  1436
lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> i \<notin> {1..dimindex(UNIV ::'n set)}"
chaieb@29842
  1437
  by (auto simp add: Cart_eq basis_component zero_index)
chaieb@29842
  1438
chaieb@29842
  1439
lemma basis_nonzero: 
chaieb@29842
  1440
  assumes k: "k \<in> {1 .. dimindex(UNIV ::'n set)}"
chaieb@29842
  1441
  shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
chaieb@29842
  1442
  using k by (simp add: basis_eq_0)
chaieb@29842
  1443
chaieb@29842
  1444
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = (z::'a::semiring_1^'n)"
chaieb@29842
  1445
  apply (auto simp add: Cart_eq dot_basis)
chaieb@29842
  1446
  apply (erule_tac x="basis i" in allE)
chaieb@29842
  1447
  apply (simp add: dot_basis)
chaieb@29842
  1448
  apply (subgoal_tac "y = z")
chaieb@29842
  1449
  apply simp
chaieb@29842
  1450
  apply vector
chaieb@29842
  1451
  done
chaieb@29842
  1452
chaieb@29842
  1453
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = (y::'a::semiring_1^'n)"
chaieb@29842
  1454
  apply (auto simp add: Cart_eq dot_basis)
chaieb@29842
  1455
  apply (erule_tac x="basis i" in allE)
chaieb@29842
  1456
  apply (simp add: dot_basis)
chaieb@29842
  1457
  apply (subgoal_tac "x = y")
chaieb@29842
  1458
  apply simp
chaieb@29842
  1459
  apply vector
chaieb@29842
  1460
  done
chaieb@29842
  1461
chaieb@29842
  1462
subsection{* Orthogonality. *}
chaieb@29842
  1463
chaieb@29842
  1464
definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
chaieb@29842
  1465
chaieb@29842
  1466
lemma orthogonal_basis:
chaieb@29842
  1467
  assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}" 
chaieb@29842
  1468
  shows "orthogonal (basis i :: 'a^'n) x \<longleftrightarrow> x$i = (0::'a::ring_1)"
chaieb@29842
  1469
  using i
chaieb@29842
  1470
  by (auto simp add: orthogonal_def dot_def basis_def Cart_lambda_beta cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
chaieb@29842
  1471
chaieb@29842
  1472
lemma orthogonal_basis_basis:
chaieb@29842
  1473
  assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}" 
chaieb@29842
  1474
  and j: "j \<in> {1 .. dimindex(UNIV ::'n set)}" 
chaieb@29842
  1475
  shows "orthogonal (basis i :: 'a::ring_1^'n) (basis j) \<longleftrightarrow> i \<noteq> j" 
chaieb@29842
  1476
  unfolding orthogonal_basis[OF i] basis_component[OF i] by simp
chaieb@29842
  1477
chaieb@29842
  1478
  (* FIXME : Maybe some of these require less than comm_ring, but not all*)
chaieb@29842
  1479
lemma orthogonal_clauses:
chaieb@29842
  1480
  "orthogonal a (0::'a::comm_ring ^'n)"
chaieb@29842
  1481
  "orthogonal a x ==> orthogonal a (c *s x)"
chaieb@29842
  1482
  "orthogonal a x ==> orthogonal a (-x)"
chaieb@29842
  1483
  "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x + y)"
chaieb@29842
  1484
  "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x - y)"
chaieb@29842
  1485
  "orthogonal 0 a"
chaieb@29842
  1486
  "orthogonal x a ==> orthogonal (c *s x) a"
chaieb@29842
  1487
  "orthogonal x a ==> orthogonal (-x) a"
chaieb@29842
  1488
  "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x + y) a"
chaieb@29842
  1489
  "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x - y) a"
chaieb@29842
  1490
  unfolding orthogonal_def dot_rneg dot_rmult dot_radd dot_rsub
chaieb@29842
  1491
  dot_lzero dot_rzero dot_lneg dot_lmult dot_ladd dot_lsub
chaieb@29842
  1492
  by simp_all
chaieb@29842
  1493
chaieb@29842
  1494
lemma orthogonal_commute: "orthogonal (x::'a::{ab_semigroup_mult,comm_monoid_add} ^'n)y \<longleftrightarrow> orthogonal y x"
chaieb@29842
  1495
  by (simp add: orthogonal_def dot_sym)
chaieb@29842
  1496
chaieb@29842
  1497
subsection{* Explicit vector construction from lists. *}
chaieb@29842
  1498
chaieb@29842
  1499
lemma Cart_lambda_beta_1[simp]: "(Cart_lambda g)$1 = g 1"
chaieb@29842
  1500
  apply (rule Cart_lambda_beta[rule_format])
chaieb@29842
  1501
  using dimindex_ge_1 apply auto done
chaieb@29842
  1502
chaieb@29842
  1503
lemma Cart_lambda_beta_1'[simp]: "(Cart_lambda g)$(Suc 0) = g 1"
chaieb@29842
  1504
  by (simp only: One_nat_def[symmetric] Cart_lambda_beta_1)
chaieb@29842
  1505
chaieb@29842
  1506
definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"
chaieb@29842
  1507
chaieb@29842
  1508
lemma vector_1: "(vector[x]) $1 = x"
chaieb@29842
  1509
  using dimindex_ge_1
chaieb@29842
  1510
  by (auto simp add: vector_def Cart_lambda_beta[rule_format])
chaieb@29842
  1511
lemma dimindex_2[simp]: "2 \<in> {1 .. dimindex (UNIV :: 2 set)}"
chaieb@29842
  1512
  by (auto simp add: dimindex_def)
chaieb@29842
  1513
lemma dimindex_2'[simp]: "2 \<in> {Suc 0 .. dimindex (UNIV :: 2 set)}"
chaieb@29842
  1514
  by (auto simp add: dimindex_def)
chaieb@29842
  1515
lemma dimindex_3[simp]: "2 \<in> {1 .. dimindex (UNIV :: 3 set)}" "3 \<in> {1 .. dimindex (UNIV :: 3 set)}"
chaieb@29842
  1516
  by (auto simp add: dimindex_def)
chaieb@29842
  1517
chaieb@29842
  1518
lemma dimindex_3'[simp]: "2 \<in> {Suc 0 .. dimindex (UNIV :: 3 set)}" "3 \<in> {Suc 0 .. dimindex (UNIV :: 3 set)}"
chaieb@29842
  1519
  by (auto simp add: dimindex_def)
chaieb@29842
  1520
chaieb@29842
  1521
lemma vector_2:
chaieb@29842
  1522
 "(vector[x,y]) $1 = x"
chaieb@29842
  1523
 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
chaieb@29842
  1524
  apply (simp add: vector_def)
chaieb@29842
  1525
  using Cart_lambda_beta[rule_format, OF dimindex_2, of "\<lambda>i. if i \<le> length [x,y] then [x,y] ! (i - 1) else (0::'a)"]
chaieb@29842
  1526
  apply (simp only: vector_def )
chaieb@29842
  1527
  apply auto
chaieb@29842
  1528
  done
chaieb@29842
  1529
chaieb@29842
  1530
lemma vector_3:
chaieb@29842
  1531
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
chaieb@29842
  1532
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
chaieb@29842
  1533
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
chaieb@29842
  1534
apply (simp_all add: vector_def Cart_lambda_beta dimindex_3)
chaieb@29842
  1535
  using Cart_lambda_beta[rule_format, OF dimindex_3(1), of "\<lambda>i. if i \<le> length [x,y,z] then [x,y,z] ! (i - 1) else (0::'a)"]   using Cart_lambda_beta[rule_format, OF dimindex_3(2), of "\<lambda>i. if i \<le> length [x,y,z] then [x,y,z] ! (i - 1) else (0::'a)"]
chaieb@29842
  1536
  by simp_all
chaieb@29842
  1537
chaieb@29842
  1538
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
chaieb@29842
  1539
  apply auto
chaieb@29842
  1540
  apply (erule_tac x="v$1" in allE)
chaieb@29842
  1541
  apply (subgoal_tac "vector [v$1] = v")
chaieb@29842
  1542
  apply simp
chaieb@29842
  1543
  by (vector vector_def dimindex_def)
chaieb@29842
  1544
chaieb@29842
  1545
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
chaieb@29842
  1546
  apply auto
chaieb@29842
  1547
  apply (erule_tac x="v$1" in allE)
chaieb@29842
  1548
  apply (erule_tac x="v$2" in allE)
chaieb@29842
  1549
  apply (subgoal_tac "vector [v$1, v$2] = v")
chaieb@29842
  1550
  apply simp
chaieb@29842
  1551
  apply (vector vector_def dimindex_def)
chaieb@29842
  1552
  apply auto
chaieb@29842
  1553
  apply (subgoal_tac "i = 1 \<or> i =2", auto)
chaieb@29842
  1554
  done
chaieb@29842
  1555
chaieb@29842
  1556
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
chaieb@29842
  1557
  apply auto
chaieb@29842
  1558
  apply (erule_tac x="v$1" in allE)
chaieb@29842
  1559
  apply (erule_tac x="v$2" in allE)
chaieb@29842
  1560
  apply (erule_tac x="v$3" in allE)
chaieb@29842
  1561
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
chaieb@29842
  1562
  apply simp
chaieb@29842
  1563
  apply (vector vector_def dimindex_def)
chaieb@29842
  1564
  apply auto
chaieb@29842
  1565
  apply (subgoal_tac "i = 1 \<or> i =2 \<or> i = 3", auto)
chaieb@29842
  1566
  done
chaieb@29842
  1567
chaieb@29842
  1568
subsection{* Linear functions. *}
chaieb@29842
  1569
chaieb@29842
  1570
definition "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *s x) = c *s f x)"
chaieb@29842
  1571
chaieb@29842
  1572
lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. (c::'a::comm_semiring) *s f x)"
chaieb@29842
  1573
  by (vector linear_def Cart_eq Cart_lambda_beta[rule_format] ring_simps)
chaieb@29842
  1574
chaieb@29842
  1575
lemma linear_compose_neg: "linear (f :: 'a ^'n \<Rightarrow> 'a::comm_ring ^'m) ==> linear (\<lambda>x. -(f(x)))" by (vector linear_def Cart_eq)
chaieb@29842
  1576
chaieb@29842
  1577
lemma linear_compose_add: "linear (f :: 'a ^'n \<Rightarrow> 'a::semiring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
chaieb@29842
  1578
  by (vector linear_def Cart_eq ring_simps)
chaieb@29842
  1579
chaieb@29842
  1580
lemma linear_compose_sub: "linear (f :: 'a ^'n \<Rightarrow> 'a::ring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
chaieb@29842
  1581
  by (vector linear_def Cart_eq ring_simps)
chaieb@29842
  1582
chaieb@29842
  1583
lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
chaieb@29842
  1584
  by (simp add: linear_def)
chaieb@29842
  1585
chaieb@29842
  1586
lemma linear_id: "linear id" by (simp add: linear_def id_def)
chaieb@29842
  1587
chaieb@29842
  1588
lemma linear_zero: "linear (\<lambda>x. 0::'a::semiring_1 ^ 'n)" by (simp add: linear_def)
chaieb@29842
  1589
chaieb@29842
  1590
lemma linear_compose_setsum:
chaieb@29842
  1591
  assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a :: 'a::semiring_1 ^ 'n \<Rightarrow> 'a ^ 'm)"
chaieb@29842
  1592
  shows "linear(\<lambda>x. setsum (\<lambda>a. f a x :: 'a::semiring_1 ^'m) S)"
chaieb@29842
  1593
  using lS
chaieb@29842
  1594
  apply (induct rule: finite_induct[OF fS])
chaieb@29842
  1595
  by (auto simp add: linear_zero intro: linear_compose_add)
chaieb@29842
  1596
chaieb@29842
  1597
lemma linear_vmul_component:
chaieb@29842
  1598
  fixes f:: "'a::semiring_1^'m \<Rightarrow> 'a^'n"
chaieb@29842
  1599
  assumes lf: "linear f" and k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
chaieb@29842
  1600
  shows "linear (\<lambda>x. f x $ k *s v)"
chaieb@29842
  1601
  using lf k
chaieb@29842
  1602
  apply (auto simp add: linear_def )
chaieb@29842
  1603
  by (vector ring_simps)+
chaieb@29842
  1604
chaieb@29842
  1605
lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)"
chaieb@29842
  1606
  unfolding linear_def
chaieb@29842
  1607
  apply clarsimp
chaieb@29842
  1608
  apply (erule allE[where x="0::'a"])
chaieb@29842
  1609
  apply simp
chaieb@29842
  1610
  done
chaieb@29842
  1611
chaieb@29842
  1612
lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def)
chaieb@29842
  1613
chaieb@29842
  1614
lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \<Rightarrow> _) ==> f (-x) = - f x"
chaieb@29842
  1615
  unfolding vector_sneg_minus1
chaieb@29842
  1616
  using linear_cmul[of f] by auto 
chaieb@29842
  1617
chaieb@29842
  1618
lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def) 
chaieb@29842
  1619
chaieb@29842
  1620
lemma linear_sub: "linear (f::'a::ring_1 ^'n \<Rightarrow> _) ==> f(x - y) = f x - f y"
chaieb@29842
  1621
  by (simp add: diff_def linear_add linear_neg)
chaieb@29842
  1622
chaieb@29842
  1623
lemma linear_setsum: 
chaieb@29842
  1624
  fixes f:: "'a::semiring_1^'n \<Rightarrow> _"
chaieb@29842
  1625
  assumes lf: "linear f" and fS: "finite S"
chaieb@29842
  1626
  shows "f (setsum g S) = setsum (f o g) S"
chaieb@29842
  1627
proof (induct rule: finite_induct[OF fS])
chaieb@29842
  1628
  case 1 thus ?case by (simp add: linear_0[OF lf])
chaieb@29842
  1629
next
chaieb@29842
  1630
  case (2 x F)
chaieb@29842
  1631
  have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
chaieb@29842
  1632
    by simp
chaieb@29842
  1633
  also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
chaieb@29842
  1634
  also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
chaieb@29842
  1635
  finally show ?case .
chaieb@29842
  1636
qed
chaieb@29842
  1637
chaieb@29842
  1638
lemma linear_setsum_mul:
chaieb@29842
  1639
  fixes f:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m"
chaieb@29842
  1640
  assumes lf: "linear f" and fS: "finite S"
chaieb@29842
  1641
  shows "f (setsum (\<lambda>i. c i *s v i) S) = setsum (\<lambda>i. c i *s f (v i)) S"
chaieb@29842
  1642
  using linear_setsum[OF lf fS, of "\<lambda>i. c i *s v i" , unfolded o_def]
chaieb@29842
  1643
  linear_cmul[OF lf] by simp 
chaieb@29842
  1644
chaieb@29842
  1645
lemma linear_injective_0:
chaieb@29842
  1646
  assumes lf: "linear (f:: 'a::ring_1 ^ 'n \<Rightarrow> _)"
chaieb@29842
  1647
  shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
chaieb@29842
  1648
proof-
chaieb@29842
  1649
  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
chaieb@29842
  1650
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
chaieb@29842
  1651
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)" 
chaieb@29842
  1652
    by (simp add: linear_sub[OF lf])
chaieb@29842
  1653
  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
chaieb@29842
  1654
  finally show ?thesis .
chaieb@29842
  1655
qed
chaieb@29842
  1656
chaieb@29842
  1657
lemma linear_bounded:
chaieb@29842
  1658
  fixes f:: "real ^'m \<Rightarrow> real ^'n"
chaieb@29842
  1659
  assumes lf: "linear f"
chaieb@29842
  1660
  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
chaieb@29842
  1661
proof-
chaieb@29842
  1662
  let ?S = "{1..dimindex(UNIV:: 'm set)}"
chaieb@29842
  1663
  let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
chaieb@29842
  1664
  have fS: "finite ?S" by simp
chaieb@29842
  1665
  {fix x:: "real ^ 'm"
chaieb@29842
  1666
    let ?g = "(\<lambda>i::nat. (x$i) *s (basis i) :: real ^ 'm)"
chaieb@29842
  1667
    have "norm (f x) = norm (f (setsum (\<lambda>i. (x$i) *s (basis i)) ?S))"
chaieb@29842
  1668
      by (simp only:  basis_expansion)
chaieb@29842
  1669
    also have "\<dots> = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)"
chaieb@29842
  1670
      using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf]
chaieb@29842
  1671
      by auto
chaieb@29842
  1672
    finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)" .
chaieb@29842
  1673
    {fix i assume i: "i \<in> ?S"
chaieb@29842
  1674
      from component_le_norm[OF i, of x]
chaieb@29842
  1675
      have "norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x"
chaieb@29842
  1676
      unfolding norm_mul
chaieb@29842
  1677
      apply (simp only: mult_commute)
chaieb@29842
  1678
      apply (rule mult_mono)
huffman@30041
  1679
      by (auto simp add: ring_simps norm_ge_zero) }
chaieb@29842
  1680
    then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
chaieb@29842
  1681
    from real_setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (f (basis i))", OF th]
chaieb@29842
  1682
    have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
chaieb@29842
  1683
  then show ?thesis by blast
chaieb@29842
  1684
qed
chaieb@29842
  1685
chaieb@29842
  1686
lemma linear_bounded_pos:
chaieb@29842
  1687
  fixes f:: "real ^'n \<Rightarrow> real ^ 'm"
chaieb@29842
  1688
  assumes lf: "linear f"
chaieb@29842
  1689
  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
chaieb@29842
  1690
proof-
chaieb@29842
  1691
  from linear_bounded[OF lf] obtain B where 
chaieb@29842
  1692
    B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
chaieb@29842
  1693
  let ?K = "\<bar>B\<bar> + 1"
chaieb@29842
  1694
  have Kp: "?K > 0" by arith
chaieb@29842
  1695
    {assume C: "B < 0"
huffman@30041
  1696
      have "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff)
chaieb@29842
  1697
      with C have "B * norm (1:: real ^ 'n) < 0"
chaieb@29842
  1698
	by (simp add: zero_compare_simps)
huffman@30041
  1699
      with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp
chaieb@29842
  1700
    }
chaieb@29842
  1701
    then have Bp: "B \<ge> 0" by ferrack
chaieb@29842
  1702
    {fix x::"real ^ 'n"
chaieb@29842
  1703
      have "norm (f x) \<le> ?K *  norm x"
huffman@30041
  1704
      using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
huffman@30040
  1705
      apply (auto simp add: ring_simps split add: abs_split)
huffman@30040
  1706
      apply (erule order_trans, simp)
huffman@30040
  1707
      done
chaieb@29842
  1708
  }
chaieb@29842
  1709
  then show ?thesis using Kp by blast
chaieb@29842
  1710
qed
chaieb@29842
  1711
chaieb@29842
  1712
subsection{* Bilinear functions. *}
chaieb@29842
  1713
chaieb@29842
  1714
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
chaieb@29842
  1715
chaieb@29842
  1716
lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
chaieb@29842
  1717
  by (simp add: bilinear_def linear_def)
chaieb@29842
  1718
lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
chaieb@29842
  1719
  by (simp add: bilinear_def linear_def)
chaieb@29842
  1720
chaieb@29842
  1721
lemma bilinear_lmul: "bilinear h ==> h (c *s x) y = c *s (h x y)"
chaieb@29842
  1722
  by (simp add: bilinear_def linear_def)
chaieb@29842
  1723
chaieb@29842
  1724
lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)"
chaieb@29842
  1725
  by (simp add: bilinear_def linear_def)
chaieb@29842
  1726
chaieb@29842
  1727
lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)"
chaieb@29842
  1728
  by (simp only: vector_sneg_minus1 bilinear_lmul)
chaieb@29842
  1729
chaieb@29842
  1730
lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y"
chaieb@29842
  1731
  by (simp only: vector_sneg_minus1 bilinear_rmul)
chaieb@29842
  1732
chaieb@29842
  1733
lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
chaieb@29842
  1734
  using add_imp_eq[of x y 0] by auto
chaieb@29842
  1735
    
chaieb@29842
  1736
lemma bilinear_lzero: 
chaieb@29842
  1737
  fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h 0 x = 0"
chaieb@29842
  1738
  using bilinear_ladd[OF bh, of 0 0 x] 
chaieb@29842
  1739
    by (simp add: eq_add_iff ring_simps)
chaieb@29842
  1740
chaieb@29842
  1741
lemma bilinear_rzero: 
chaieb@29842
  1742
  fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h x 0 = 0"
chaieb@29842
  1743
  using bilinear_radd[OF bh, of x 0 0 ] 
chaieb@29842
  1744
    by (simp add: eq_add_iff ring_simps)
chaieb@29842
  1745
chaieb@29842
  1746
lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ 'n)) z = h x z - h y z"
chaieb@29842
  1747
  by (simp  add: diff_def bilinear_ladd bilinear_lneg)
chaieb@29842
  1748
chaieb@29842
  1749
lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ 'n)) = h z x - h z y"
chaieb@29842
  1750
  by (simp  add: diff_def bilinear_radd bilinear_rneg)
chaieb@29842
  1751
chaieb@29842
  1752
lemma bilinear_setsum:
chaieb@29842
  1753
  fixes h:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m \<Rightarrow> 'a ^ 'k"
chaieb@29842
  1754
  assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
chaieb@29842
  1755
  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
chaieb@29842
  1756
proof- 
chaieb@29842
  1757
  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
chaieb@29842
  1758
    apply (rule linear_setsum[unfolded o_def])
chaieb@29842
  1759
    using bh fS by (auto simp add: bilinear_def)
chaieb@29842
  1760
  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
chaieb@29842
  1761
    apply (rule setsum_cong, simp)
chaieb@29842
  1762
    apply (rule linear_setsum[unfolded o_def])
chaieb@29842
  1763
    using bh fT by (auto simp add: bilinear_def)
chaieb@29842
  1764
  finally show ?thesis unfolding setsum_cartesian_product .
chaieb@29842
  1765
qed
chaieb@29842
  1766
chaieb@29842
  1767
lemma bilinear_bounded:
chaieb@29842
  1768
  fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^ 'k"
chaieb@29842
  1769
  assumes bh: "bilinear h"
chaieb@29842
  1770
  shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
chaieb@29842
  1771
proof- 
chaieb@29842
  1772
  let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
chaieb@29842
  1773
  let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
chaieb@29842
  1774
  let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
chaieb@29842
  1775
  have fM: "finite ?M" and fN: "finite ?N" by simp_all
chaieb@29842
  1776
  {fix x:: "real ^ 'm" and  y :: "real^'n"
chaieb@29842
  1777
    have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$i) *s basis i) ?M) (setsum (\<lambda>i. (y$i) *s basis i) ?N))" unfolding basis_expansion ..
chaieb@29842
  1778
    also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$i) *s basis i) ((y$j) *s basis j)) (?M \<times> ?N))"  unfolding bilinear_setsum[OF bh fM fN] ..
chaieb@29842
  1779
    finally have th: "norm (h x y) = \<dots>" .
chaieb@29842
  1780
    have "norm (h x y) \<le> ?B * norm x * norm y"
chaieb@29842
  1781
      apply (simp add: setsum_left_distrib th)
chaieb@29842
  1782
      apply (rule real_setsum_norm_le)
chaieb@29842
  1783
      using fN fM
chaieb@29842
  1784
      apply simp
chaieb@29842
  1785
      apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)
chaieb@29842
  1786
      apply (rule mult_mono)
huffman@30041
  1787
      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
chaieb@29842
  1788
      apply (rule mult_mono)
huffman@30041
  1789
      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
chaieb@29842
  1790
      done}
chaieb@29842
  1791
  then show ?thesis by metis
chaieb@29842
  1792
qed
chaieb@29842
  1793
chaieb@29842
  1794
lemma bilinear_bounded_pos:
chaieb@29842
  1795
  fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^ 'k"
chaieb@29842
  1796
  assumes bh: "bilinear h"
chaieb@29842
  1797
  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
chaieb@29842
  1798
proof-
chaieb@29842
  1799
  from bilinear_bounded[OF bh] obtain B where 
chaieb@29842
  1800
    B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
chaieb@29842
  1801
  let ?K = "\<bar>B\<bar> + 1"
chaieb@29842
  1802
  have Kp: "?K > 0" by arith
chaieb@29842
  1803
  have KB: "B < ?K" by arith
chaieb@29842
  1804
  {fix x::"real ^'m" and y :: "real ^'n"
chaieb@29842
  1805
    from KB Kp
chaieb@29842
  1806
    have "B * norm x * norm y \<le> ?K * norm x * norm y"
chaieb@29842
  1807
      apply - 
chaieb@29842
  1808
      apply (rule mult_right_mono, rule mult_right_mono)
huffman@30041
  1809
      by (auto simp add: norm_ge_zero)
chaieb@29842
  1810
    then have "norm (h x y) \<le> ?K * norm x * norm y"
chaieb@29842
  1811
      using B[rule_format, of x y] by simp} 
chaieb@29842
  1812
  with Kp show ?thesis by blast
chaieb@29842
  1813
qed
chaieb@29842
  1814
chaieb@29842
  1815
subsection{* Adjoints. *}
chaieb@29842
  1816
chaieb@29842
  1817
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
chaieb@29842
  1818
chaieb@29842
  1819
lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
chaieb@29842
  1820
chaieb@29842
  1821
lemma adjoint_works_lemma:
chaieb@29842
  1822
  fixes f:: "'a::ring_1 ^'n \<Rightarrow> 'a ^ 'm"
chaieb@29842
  1823
  assumes lf: "linear f"
chaieb@29842
  1824
  shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
chaieb@29842
  1825
proof-
chaieb@29842
  1826
  let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
chaieb@29842
  1827
  let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
chaieb@29842
  1828
  have fN: "finite ?N" by simp
chaieb@29842
  1829
  have fM: "finite ?M" by simp
chaieb@29842
  1830
  {fix y:: "'a ^ 'm"
chaieb@29842
  1831
    let ?w = "(\<chi> i. (f (basis i) \<bullet> y)) :: 'a ^ 'n"
chaieb@29842
  1832
    {fix x
chaieb@29842
  1833
      have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *s basis i) ?N) \<bullet> y"
chaieb@29842
  1834
	by (simp only: basis_expansion)
chaieb@29842
  1835
      also have "\<dots> = (setsum (\<lambda>i. (x$i) *s f (basis i)) ?N) \<bullet> y"
chaieb@29842
  1836
	unfolding linear_setsum[OF lf fN] 
chaieb@29842
  1837
	by (simp add: linear_cmul[OF lf])
chaieb@29842
  1838
      finally have "f x \<bullet> y = x \<bullet> ?w"
chaieb@29842
  1839
	apply (simp only: )
chaieb@29842
  1840
	apply (simp add: dot_def setsum_component Cart_lambda_beta setsum_left_distrib setsum_right_distrib vector_component setsum_commute[of _ ?M ?N] ring_simps del: One_nat_def)
chaieb@29842
  1841
	done}
chaieb@29842
  1842
  }
chaieb@29842
  1843
  then show ?thesis unfolding adjoint_def 
chaieb@29842
  1844
    some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
chaieb@29842
  1845
    using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
chaieb@29842
  1846
    by metis
chaieb@29842
  1847
qed
chaieb@29842
  1848
chaieb@29842
  1849
lemma adjoint_works:
chaieb@29842
  1850
  fixes f:: "'a::ring_1 ^'n \<Rightarrow> 'a ^ 'm"
chaieb@29842
  1851
  assumes lf: "linear f"
chaieb@29842
  1852
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
chaieb@29842
  1853
  using adjoint_works_lemma[OF lf] by metis
chaieb@29842
  1854
chaieb@29842
  1855
chaieb@29842
  1856
lemma adjoint_linear:
chaieb@29842
  1857
  fixes f :: "'a::comm_ring_1 ^'n \<Rightarrow> 'a ^ 'm"
chaieb@29842
  1858
  assumes lf: "linear f"
chaieb@29842
  1859
  shows "linear (adjoint f)"
chaieb@29842
  1860
  by (simp add: linear_def vector_eq_ldot[symmetric] dot_radd dot_rmult adjoint_works[OF lf])
chaieb@29842
  1861
chaieb@29842
  1862
lemma adjoint_clauses:
chaieb@29842
  1863
  fixes f:: "'a::comm_ring_1 ^'n \<Rightarrow> 'a ^ 'm"
chaieb@29842
  1864
  assumes lf: "linear f"
chaieb@29842
  1865
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
chaieb@29842
  1866
  and "adjoint f y \<bullet> x = y \<bullet> f x"
chaieb@29842
  1867
  by (simp_all add: adjoint_works[OF lf] dot_sym )
chaieb@29842
  1868
chaieb@29842
  1869
lemma adjoint_adjoint:
chaieb@29842
  1870
  fixes f:: "'a::comm_ring_1 ^ 'n \<Rightarrow> _"
chaieb@29842
  1871
  assumes lf: "linear f"
chaieb@29842
  1872
  shows "adjoint (adjoint f) = f"
chaieb@29842
  1873
  apply (rule ext)
chaieb@29842
  1874
  by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf])
chaieb@29842
  1875
chaieb@29842
  1876
lemma adjoint_unique:
chaieb@29842
  1877
  fixes f:: "'a::comm_ring_1 ^ 'n \<Rightarrow> 'a ^ 'm"
chaieb@29842
  1878
  assumes lf: "linear f" and u: "\<forall>x y. f' x \<bullet> y = x \<bullet> f y"
chaieb@29842
  1879
  shows "f' = adjoint f"
chaieb@29842
  1880
  apply (rule ext)
chaieb@29842
  1881
  using u
chaieb@29842
  1882
  by (simp add: vector_eq_rdot[symmetric] adjoint_clauses[OF lf])
chaieb@29842
  1883
huffman@29881
  1884
text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
chaieb@29842
  1885
chaieb@29842
  1886
consts generic_mult :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" (infixr "\<star>" 75)
chaieb@29842
  1887
chaieb@29842
  1888
defs (overloaded) 
chaieb@29842
  1889
matrix_matrix_mult_def: "(m:: ('a::semiring_1) ^'n^'m) \<star> (m' :: 'a ^'p^'n) \<equiv> (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) {1 .. dimindex (UNIV :: 'n set)}) ::'a ^ 'p ^'m"
chaieb@29842
  1890
chaieb@29842
  1891
abbreviation 
chaieb@29842
  1892
  matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)
chaieb@29842
  1893
  where "m ** m' == m\<star> m'"
chaieb@29842
  1894
chaieb@29842
  1895
defs (overloaded) 
chaieb@29842
  1896
  matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) {1..dimindex(UNIV ::'n set)}) :: 'a^'m"
chaieb@29842
  1897
chaieb@29842
  1898
abbreviation 
chaieb@29842
  1899
  matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)
chaieb@29842
  1900
  where 
chaieb@29842
  1901
  "m *v v == m \<star> v"
chaieb@29842
  1902
chaieb@29842
  1903
defs (overloaded) 
chaieb@29842
  1904
  vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (x$i)) {1..dimindex(UNIV :: 'm set)}) :: 'a^'n"
chaieb@29842
  1905
chaieb@29842
  1906
abbreviation 
chaieb@29842
  1907
  vactor_matrix_mult' :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)
chaieb@29842
  1908
  where 
chaieb@29842
  1909
  "v v* m == v \<star> m"
chaieb@29842
  1910
chaieb@29842
  1911
definition "(mat::'a::zero => 'a ^'n^'m) k = (\<chi> i j. if i = j then k else 0)"
chaieb@29842
  1912
definition "(transp::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
chaieb@29842
  1913
definition "(row::nat => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
chaieb@29842
  1914
definition "(column::nat =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
chaieb@29842
  1915
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> {1 .. dimindex(UNIV :: 'm set)}}"
chaieb@29842
  1916
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> {1 .. dimindex(UNIV :: 'n set)}}"
chaieb@29842
  1917
chaieb@29842
  1918
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
chaieb@29842
  1919
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \<star> B) + (A \<star> C)"
chaieb@29842
  1920
  by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps)
chaieb@29842
  1921
chaieb@29842
  1922
lemma setsum_delta': 
chaieb@29842
  1923
  assumes fS: "finite S" shows 
chaieb@29842
  1924
  "setsum (\<lambda>k. if a = k then b k else 0) S = 
chaieb@29842
  1925
     (if a\<in> S then b a else 0)"
chaieb@29842
  1926
  using setsum_delta[OF fS, of a b, symmetric] 
chaieb@29842
  1927
  by (auto intro: setsum_cong)
chaieb@29842
  1928
chaieb@29842
  1929
lemma matrix_mul_lid: "mat 1 ** A = A"
chaieb@29842
  1930
  apply (simp add: matrix_matrix_mult_def mat_def)
chaieb@29842
  1931
  apply vector
chaieb@29842
  1932
  by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite_atLeastAtMost]  mult_1_left mult_zero_left if_True)
chaieb@29842
  1933
chaieb@29842
  1934
chaieb@29842
  1935
lemma matrix_mul_rid: "A ** mat 1 = A"
chaieb@29842
  1936
  apply (simp add: matrix_matrix_mult_def mat_def)
chaieb@29842
  1937
  apply vector
chaieb@29842
  1938
  by (auto simp only: cond_value_iff cond_application_beta setsum_delta[OF finite_atLeastAtMost]  mult_1_right mult_zero_right if_True cong: if_cong)
chaieb@29842
  1939
chaieb@29842
  1940
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
chaieb@29842
  1941
  apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
chaieb@29842
  1942
  apply (subst setsum_commute)
chaieb@29842
  1943
  apply simp
chaieb@29842
  1944
  done
chaieb@29842
  1945
chaieb@29842
  1946
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
chaieb@29842
  1947
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
chaieb@29842
  1948
  apply (subst setsum_commute)
chaieb@29842
  1949
  apply simp
chaieb@29842
  1950
  done
chaieb@29842
  1951
chaieb@29842
  1952
lemma matrix_vector_mul_lid: "mat 1 *v x = x"
chaieb@29842
  1953
  apply (vector matrix_vector_mult_def mat_def)
chaieb@29842
  1954
  by (simp add: cond_value_iff cond_application_beta 
chaieb@29842
  1955
    setsum_delta' cong del: if_weak_cong)
chaieb@29842
  1956
chaieb@29842
  1957
lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)"
chaieb@29842
  1958
  by (simp add: matrix_matrix_mult_def transp_def Cart_eq Cart_lambda_beta mult_commute)
chaieb@29842
  1959
chaieb@29842
  1960
lemma matrix_eq: "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@29842
  1961
  apply auto
chaieb@29842
  1962
  apply (subst Cart_eq)
chaieb@29842
  1963
  apply clarify
chaieb@29842
  1964
  apply (clarsimp simp add: matrix_vector_mult_def basis_def cond_value_iff cond_application_beta Cart_eq Cart_lambda_beta cong del: if_weak_cong)
chaieb@29842
  1965
  apply (erule_tac x="basis ia" in allE)
chaieb@29842
  1966
  apply (erule_tac x="i" in ballE)
chaieb@29842
  1967
  by (auto simp add: basis_def cond_value_iff cond_application_beta Cart_lambda_beta setsum_delta[OF finite_atLeastAtMost] cong del: if_weak_cong)
chaieb@29842
  1968
chaieb@29842
  1969
lemma matrix_vector_mul_component: 
chaieb@29842
  1970
  assumes k: "k \<in> {1.. dimindex (UNIV :: 'm set)}"
chaieb@29842
  1971
  shows "((A::'a::semiring_1^'n'^'m) *v x)$k = (A$k) \<bullet> x"
chaieb@29842
  1972
  using k
chaieb@29842
  1973
  by (simp add: matrix_vector_mult_def Cart_lambda_beta dot_def)
chaieb@29842
  1974
chaieb@29842
  1975
lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) v* A) \<bullet> y = x \<bullet> (A *v y)"
chaieb@29842
  1976
  apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib Cart_lambda_beta mult_ac)
chaieb@29842
  1977
  apply (subst setsum_commute)
chaieb@29842
  1978
  by simp
chaieb@29842
  1979
chaieb@29842
  1980
lemma transp_mat: "transp (mat n) = mat n"
chaieb@29842
  1981
  by (vector transp_def mat_def)
chaieb@29842
  1982
chaieb@29842
  1983
lemma transp_transp: "transp(transp A) = A"
chaieb@29842
  1984
  by (vector transp_def)
chaieb@29842
  1985
chaieb@29842
  1986
lemma row_transp: 
chaieb@29842
  1987
  fixes A:: "'a::semiring_1^'n^'m"
chaieb@29842
  1988
  assumes i: "i \<in> {1.. dimindex (UNIV :: 'n set)}"
chaieb@29842
  1989
  shows "row i (transp A) = column i A"
chaieb@29842
  1990
  using i 
chaieb@29842
  1991
  by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)
chaieb@29842
  1992
chaieb@29842
  1993
lemma column_transp:
chaieb@29842
  1994
  fixes A:: "'a::semiring_1^'n^'m"
chaieb@29842
  1995
  assumes i: "i \<in> {1.. dimindex (UNIV :: 'm set)}"
chaieb@29842
  1996
  shows "column i (transp A) = row i A"
chaieb@29842
  1997
  using i 
chaieb@29842
  1998
  by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)
chaieb@29842
  1999
chaieb@29842
  2000
lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A"
chaieb@29842
  2001
apply (auto simp add: rows_def columns_def row_transp intro: set_ext)
chaieb@29842
  2002
apply (rule_tac x=i in exI)
chaieb@29842
  2003
apply (auto simp add: row_transp)
chaieb@29842
  2004
done
chaieb@29842
  2005
chaieb@29842
  2006
lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp)
chaieb@29842
  2007
chaieb@29842
  2008
text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
chaieb@29842
  2009
chaieb@29842
  2010
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
chaieb@29842
  2011
  by (simp add: matrix_vector_mult_def dot_def)
chaieb@29842
  2012
chaieb@29842
  2013
lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) {1 .. dimindex(UNIV:: 'n set)}"
chaieb@29842
  2014
  by (simp add: matrix_vector_mult_def Cart_eq setsum_component Cart_lambda_beta vector_component column_def mult_commute)
chaieb@29842
  2015
chaieb@29842
  2016
lemma vector_componentwise:
chaieb@29842
  2017
  "(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) {1..dimindex(UNIV :: 'n set)})"
chaieb@29842
  2018
  apply (subst basis_expansion[symmetric])
chaieb@29842
  2019
  by (vector Cart_eq Cart_lambda_beta setsum_component)
chaieb@29842
  2020
chaieb@29842
  2021
lemma linear_componentwise:
chaieb@29842
  2022
  fixes f:: "'a::ring_1 ^ 'm \<Rightarrow> 'a ^ 'n"
chaieb@29842
  2023
  assumes lf: "linear f" and j: "j \<in> {1 .. dimindex (UNIV :: 'n set)}"
chaieb@29842
  2024
  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) {1 .. dimindex (UNIV :: 'm set)}" (is "?lhs = ?rhs")
chaieb@29842
  2025
proof-
chaieb@29842
  2026
  let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
chaieb@29842
  2027
  let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
chaieb@29842
  2028
  have fM: "finite ?M" by simp
chaieb@29842
  2029
  have "?rhs = (setsum (\<lambda>i.(x$i) *s f (basis i) ) ?M)$j"
chaieb@29842
  2030
    unfolding vector_smult_component[OF j, symmetric]
chaieb@29842
  2031
    unfolding setsum_component[OF j, of "(\<lambda>i.(x$i) *s f (basis i :: 'a^'m))" ?M]
chaieb@29842
  2032
    ..
chaieb@29842
  2033
  then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion ..
chaieb@29842
  2034
qed
chaieb@29842
  2035
chaieb@29842
  2036
text{* Inverse matrices  (not necessarily square) *}
chaieb@29842
  2037
chaieb@29842
  2038
definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
chaieb@29842
  2039
chaieb@29842
  2040
definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
chaieb@29842
  2041
        (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
chaieb@29842
  2042
chaieb@29842
  2043
text{* Correspondence between matrices and linear operators. *}
chaieb@29842
  2044
chaieb@29842
  2045
definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
chaieb@29842
  2046
where "matrix f = (\<chi> i j. (f(basis j))$i)"
chaieb@29842
  2047
chaieb@29842
  2048
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::'a::comm_semiring_1 ^ 'n))"
chaieb@29842
  2049
  by (simp add: linear_def matrix_vector_mult_def Cart_eq Cart_lambda_beta vector_component ring_simps setsum_right_distrib setsum_addf)
chaieb@29842
  2050
chaieb@29842
  2051
lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n)"
chaieb@29842
  2052
apply (simp add: matrix_def matrix_vector_mult_def Cart_eq Cart_lambda_beta mult_commute del: One_nat_def)
chaieb@29842
  2053
apply clarify
chaieb@29842
  2054
apply (rule linear_componentwise[OF lf, symmetric])
chaieb@29842
  2055
apply simp
chaieb@29842
  2056
done
chaieb@29842
  2057
chaieb@29842
  2058
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::'a::comm_ring_1 ^ 'n))" by (simp add: ext matrix_works)
chaieb@29842
  2059
chaieb@29842
  2060
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n)) = A"
chaieb@29842
  2061
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
chaieb@29842
  2062
chaieb@29842
  2063
lemma matrix_compose: 
chaieb@29842
  2064
  assumes lf: "linear (f::'a::comm_ring_1^'n \<Rightarrow> _)" and lg: "linear g" 
chaieb@29842
  2065
  shows "matrix (g o f) = matrix g ** matrix f"
chaieb@29842
  2066
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
chaieb@29842
  2067
  by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
chaieb@29842
  2068
chaieb@29842
  2069
lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s ((transp A)$i)) {1..dimindex(UNIV:: 'n set)}"
chaieb@29842
  2070
  by (simp add: matrix_vector_mult_def transp_def Cart_eq Cart_lambda_beta setsum_component vector_component mult_commute)
chaieb@29842
  2071
chaieb@29842
  2072
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::'a::comm_ring_1^'n^'m) *v x) = (\<lambda>x. transp A *v x)"
chaieb@29842
  2073
  apply (rule adjoint_unique[symmetric])
chaieb@29842
  2074
  apply (rule matrix_vector_mul_linear)
chaieb@29842
  2075
  apply (simp add: transp_def dot_def Cart_lambda_beta matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
chaieb@29842
  2076
  apply (subst setsum_commute)
chaieb@29842
  2077
  apply (auto simp add: mult_ac)
chaieb@29842
  2078
  done
chaieb@29842
  2079
chaieb@29842
  2080
lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n \<Rightarrow> 'a ^ 'm)"
chaieb@29842
  2081
  shows "matrix(adjoint f) = transp(matrix f)"
chaieb@29842
  2082
  apply (subst matrix_vector_mul[OF lf])
chaieb@29842
  2083
  unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
chaieb@29842
  2084
chaieb@29842
  2085
subsection{* Interlude: Some properties of real sets *}
chaieb@29842
  2086
chaieb@29842
  2087
lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
chaieb@29842
  2088
  shows "\<forall>n \<ge> m. d n < e m"
chaieb@29842
  2089
  using prems apply auto
chaieb@29842
  2090
  apply (erule_tac x="n" in allE)
chaieb@29842
  2091
  apply (erule_tac x="n" in allE)
chaieb@29842
  2092
  apply auto
chaieb@29842
  2093
  done
chaieb@29842
  2094
chaieb@29842
  2095
chaieb@29842
  2096
lemma real_convex_bound_lt: 
chaieb@29842
  2097
  assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"
chaieb@29842
  2098
  and uv: "u + v = 1" 
chaieb@29842
  2099
  shows "u * x + v * y < a"
chaieb@29842
  2100
proof-
chaieb@29842
  2101
  have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
chaieb@29842
  2102
  have "a = a * (u + v)" unfolding uv  by simp
chaieb@29842
  2103
  hence th: "u * a + v * a = a" by (simp add: ring_simps)
chaieb@29842
  2104
  from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_compare_simps)
chaieb@29842
  2105
  from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_compare_simps)
chaieb@29842
  2106
  from xa ya u v have "u * x + v * y < u * a + v * a"
chaieb@29842
  2107
    apply (cases "u = 0", simp_all add: uv')
chaieb@29842
  2108
    apply(rule mult_strict_left_mono)
chaieb@29842
  2109
    using uv' apply simp_all
chaieb@29842
  2110
    
chaieb@29842
  2111
    apply (rule add_less_le_mono)
chaieb@29842
  2112
    apply(rule mult_strict_left_mono)
chaieb@29842
  2113
    apply simp_all
chaieb@29842
  2114
    apply (rule mult_left_mono)
chaieb@29842
  2115
    apply simp_all
chaieb@29842
  2116
    done
chaieb@29842
  2117
  thus ?thesis unfolding th .
chaieb@29842
  2118
qed
chaieb@29842
  2119
chaieb@29842
  2120
lemma real_convex_bound_le: 
chaieb@29842
  2121
  assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"
chaieb@29842
  2122
  and uv: "u + v = 1" 
chaieb@29842
  2123
  shows "u * x + v * y \<le> a"
chaieb@29842
  2124
proof-
chaieb@29842
  2125
  from xa ya u v have "u * x + v * y \<le> u * a + v * a" by (simp add: add_mono mult_left_mono)
chaieb@29842
  2126
  also have "\<dots> \<le> (u + v) * a" by (simp add: ring_simps)
chaieb@29842
  2127
  finally show ?thesis unfolding uv by simp
chaieb@29842
  2128
qed
chaieb@29842
  2129
chaieb@29842
  2130
lemma infinite_enumerate: assumes fS: "infinite S"
chaieb@29842
  2131
  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
chaieb@29842
  2132
unfolding subseq_def
chaieb@29842
  2133
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
chaieb@29842
  2134
chaieb@29842
  2135
lemma approachable_lt_le: "(\<exists>(d::real)>0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
chaieb@29842
  2136
apply auto
chaieb@29842
  2137
apply (rule_tac x="d/2" in exI)
chaieb@29842
  2138
apply auto
chaieb@29842
  2139
done
chaieb@29842
  2140
chaieb@29842
  2141
chaieb@29842
  2142
lemma triangle_lemma: 
chaieb@29842
  2143
  assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
chaieb@29842
  2144
  shows "x <= y + z"
chaieb@29842
  2145
proof-
chaieb@29842
  2146
  have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y  by (simp add: zero_compare_simps)
chaieb@29842
  2147
  with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square ring_simps)
chaieb@29842
  2148
  from y z have yz: "y + z \<ge> 0" by arith
chaieb@29842
  2149
  from power2_le_imp_le[OF th yz] show ?thesis .
chaieb@29842
  2150
qed
chaieb@29842
  2151
chaieb@29842
  2152
chaieb@29842
  2153
lemma lambda_skolem: "(\<forall>i \<in> {1 .. dimindex(UNIV :: 'n set)}. \<exists>x. P i x) \<longleftrightarrow>
chaieb@29842
  2154
   (\<exists>x::'a ^ 'n. \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@29842
  2155
proof-
chaieb@29842
  2156
  let ?S = "{1 .. dimindex(UNIV :: 'n set)}"
chaieb@29842
  2157
  {assume H: "?rhs"
chaieb@29842
  2158
    then have ?lhs by auto}
chaieb@29842
  2159
  moreover
chaieb@29842
  2160
  {assume H: "?lhs"
chaieb@29842
  2161
    then obtain f where f:"\<forall>i\<in> ?S. P i (f i)" unfolding Ball_def choice_iff by metis
chaieb@29842
  2162
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
chaieb@29842
  2163
    {fix i assume i: "i \<in> ?S"
chaieb@29842
  2164
      with f i have "P i (f i)" by metis
chaieb@29842
  2165
      then have "P i (?x$i)" using Cart_lambda_beta[of f, rule_format, OF i] by auto 
chaieb@29842
  2166
    }
chaieb@29842
  2167
    hence "\<forall>i \<in> ?S. P i (?x$i)" by metis
chaieb@29842
  2168
    hence ?rhs by metis }
chaieb@29842
  2169
  ultimately show ?thesis by metis
chaieb@29842
  2170
qed 
chaieb@29842
  2171
chaieb@29842
  2172
(* Supremum and infimum of real sets *)
chaieb@29842
  2173
chaieb@29842
  2174
chaieb@29842
  2175
definition rsup:: "real set \<Rightarrow> real" where
chaieb@29842
  2176
  "rsup S = (SOME a. isLub UNIV S a)"
chaieb@29842
  2177
chaieb@29842
  2178
lemma rsup_alt: "rsup S = (SOME a. (\<forall>x \<in> S. x \<le> a) \<and> (\<forall>b. (\<forall>x \<in> S. x \<le> b) \<longrightarrow> a \<le> b))"  by (auto simp  add: isLub_def rsup_def leastP_def isUb_def setle_def setge_def)
chaieb@29842
  2179
chaieb@29842
  2180
lemma rsup: assumes Se: "S \<noteq> {}" and b: "\<exists>b. S *<= b"
chaieb@29842
  2181
  shows "isLub UNIV S (rsup S)"
chaieb@29842
  2182
using Se b
chaieb@29842
  2183
unfolding rsup_def
chaieb@29842
  2184
apply clarify
chaieb@29842
  2185
apply (rule someI_ex)
chaieb@29842
  2186
apply (rule reals_complete)
chaieb@29842
  2187
by (auto simp add: isUb_def setle_def)
chaieb@29842
  2188
chaieb@29842
  2189
lemma rsup_le: assumes Se: "S \<noteq> {}" and Sb: "S *<= b" shows "rsup S \<le> b"
chaieb@29842
  2190
proof-
chaieb@29842
  2191
  from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def)
chaieb@29842
  2192
  from rsup[OF Se] Sb have "isLub UNIV S (rsup S)"  by blast 
chaieb@29842
  2193
  then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def)
chaieb@29842
  2194
qed
chaieb@29842
  2195
chaieb@29842
  2196
lemma rsup_finite_Max: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2197
  shows "rsup S = Max S"
chaieb@29842
  2198
using fS Se
chaieb@29842
  2199
proof-
chaieb@29842
  2200
  let ?m = "Max S"
chaieb@29842
  2201
  from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis
chaieb@29842
  2202
  with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def)
chaieb@29842
  2203
  from Max_in[OF fS Se] lub have mrS: "?m \<le> rsup S" 
chaieb@29842
  2204
    by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)
chaieb@29842
  2205
  moreover 
chaieb@29842
  2206
  have "rsup S \<le> ?m" using Sm lub
chaieb@29842
  2207
    by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
chaieb@29842
  2208
  ultimately  show ?thesis by arith 
chaieb@29842
  2209
qed
chaieb@29842
  2210
chaieb@29842
  2211
lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2212
  shows "rsup S \<in> S"
chaieb@29842
  2213
  using rsup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis
chaieb@29842
  2214
chaieb@29842
  2215
lemma rsup_finite_Ub: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2216
  shows "isUb S S (rsup S)"
chaieb@29842
  2217
  using rsup_finite_Max[OF fS Se] rsup_finite_in[OF fS Se] Max_ge[OF fS]
chaieb@29842
  2218
  unfolding isUb_def setle_def by metis
chaieb@29842
  2219
chaieb@29842
  2220
lemma rsup_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2221
  shows "a \<le> rsup S \<longleftrightarrow> (\<exists> x \<in> S. a \<le> x)"
chaieb@29842
  2222
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
chaieb@29842
  2223
chaieb@29842
  2224
lemma rsup_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2225
  shows "a \<ge> rsup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)"
chaieb@29842
  2226
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
chaieb@29842
  2227
chaieb@29842
  2228
lemma rsup_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2229
  shows "a < rsup S \<longleftrightarrow> (\<exists> x \<in> S. a < x)"
chaieb@29842
  2230
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
chaieb@29842
  2231
chaieb@29842
  2232
lemma rsup_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2233
  shows "a > rsup S \<longleftrightarrow> (\<forall> x \<in> S. a > x)"
chaieb@29842
  2234
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
chaieb@29842
  2235
chaieb@29842
  2236
lemma rsup_unique: assumes b: "S *<= b" and S: "\<forall>b' < b. \<exists>x \<in> S. b' < x"
chaieb@29842
  2237
  shows "rsup S = b"
chaieb@29842
  2238
using b S  
chaieb@29842
  2239
unfolding setle_def rsup_alt
chaieb@29842
  2240
apply -
chaieb@29842
  2241
apply (rule some_equality)
chaieb@29842
  2242
apply (metis  linorder_not_le order_eq_iff[symmetric])+
chaieb@29842
  2243
done
chaieb@29842
  2244
chaieb@29842
  2245
lemma rsup_le_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. T *<= b) \<Longrightarrow> rsup S \<le> rsup T"
chaieb@29842
  2246
  apply (rule rsup_le)
chaieb@29842
  2247
  apply simp
chaieb@29842
  2248
  using rsup[of T] by (auto simp add: isLub_def leastP_def setge_def setle_def isUb_def)
chaieb@29842
  2249
chaieb@29842
  2250
lemma isUb_def': "isUb R S = (\<lambda>x. S *<= x \<and> x \<in> R)"
chaieb@29842
  2251
  apply (rule ext)
chaieb@29842
  2252
  by (metis isUb_def)
chaieb@29842
  2253
chaieb@29842
  2254
lemma UNIV_trivial: "UNIV x" using UNIV_I[of x] by (metis mem_def)
chaieb@29842
  2255
lemma rsup_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
chaieb@29842
  2256
  shows "a \<le> rsup S \<and> rsup S \<le> b"
chaieb@29842
  2257
proof-
chaieb@29842
  2258
  from rsup[OF Se] u have lub: "isLub UNIV S (rsup S)" by blast
chaieb@29842
  2259
  hence b: "rsup S \<le> b" using u by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
chaieb@29842
  2260
  from Se obtain y where y: "y \<in> S" by blast
chaieb@29842
  2261
  from lub l have "a \<le> rsup S" apply (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
chaieb@29842
  2262
    apply (erule ballE[where x=y])
chaieb@29842
  2263
    apply (erule ballE[where x=y])
chaieb@29842
  2264
    apply arith
chaieb@29842
  2265
    using y apply auto
chaieb@29842
  2266
    done
chaieb@29842
  2267
  with b show ?thesis by blast
chaieb@29842
  2268
qed
chaieb@29842
  2269
chaieb@29842
  2270
lemma rsup_abs_le: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rsup S\<bar> \<le> a"
chaieb@29842
  2271
  unfolding abs_le_interval_iff  using rsup_bounds[of S "-a" a]
chaieb@29842
  2272
  by (auto simp add: setge_def setle_def)
chaieb@29842
  2273
chaieb@29842
  2274
lemma rsup_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rsup S - l\<bar> \<le> e"
chaieb@29842
  2275
proof-
chaieb@29842
  2276
  have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
chaieb@29842
  2277
  show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th 
chaieb@29842
  2278
    by  (auto simp add: setge_def setle_def)
chaieb@29842
  2279
qed
chaieb@29842
  2280
chaieb@29842
  2281
definition rinf:: "real set \<Rightarrow> real" where
chaieb@29842
  2282
  "rinf S = (SOME a. isGlb UNIV S a)"
chaieb@29842
  2283
chaieb@29842
  2284
lemma rinf_alt: "rinf S = (SOME a. (\<forall>x \<in> S. x \<ge> a) \<and> (\<forall>b. (\<forall>x \<in> S. x \<ge> b) \<longrightarrow> a \<ge> b))"  by (auto simp  add: isGlb_def rinf_def greatestP_def isLb_def setle_def setge_def)
chaieb@29842
  2285
chaieb@29842
  2286
lemma reals_complete_Glb: assumes Se: "\<exists>x. x \<in> S" and lb: "\<exists> y. isLb UNIV S y"
chaieb@29842
  2287
  shows "\<exists>(t::real). isGlb UNIV S t"
chaieb@29842
  2288
proof-
chaieb@29842
  2289
  let ?M = "uminus ` S"
chaieb@29842
  2290
  from lb have th: "\<exists>y. isUb UNIV ?M y" apply (auto simp add: isUb_def isLb_def setle_def setge_def)
chaieb@29842
  2291
    by (rule_tac x="-y" in exI, auto)
chaieb@29842
  2292
  from Se have Me: "\<exists>x. x \<in> ?M" by blast
chaieb@29842
  2293
  from reals_complete[OF Me th] obtain t where t: "isLub UNIV ?M t" by blast
chaieb@29842
  2294
  have "isGlb UNIV S (- t)" using t
chaieb@29842
  2295
    apply (auto simp add: isLub_def isGlb_def leastP_def greatestP_def setle_def setge_def isUb_def isLb_def)
chaieb@29842
  2296
    apply (erule_tac x="-y" in allE)
chaieb@29842
  2297
    apply auto
chaieb@29842
  2298
    done
chaieb@29842
  2299
  then show ?thesis by metis
chaieb@29842
  2300
qed
chaieb@29842
  2301
chaieb@29842
  2302
lemma rinf: assumes Se: "S \<noteq> {}" and b: "\<exists>b. b <=* S"
chaieb@29842
  2303
  shows "isGlb UNIV S (rinf S)"
chaieb@29842
  2304
using Se b
chaieb@29842
  2305
unfolding rinf_def
chaieb@29842
  2306
apply clarify
chaieb@29842
  2307
apply (rule someI_ex)
chaieb@29842
  2308
apply (rule reals_complete_Glb)
chaieb@29842
  2309
apply (auto simp add: isLb_def setle_def setge_def)
chaieb@29842
  2310
done
chaieb@29842
  2311
chaieb@29842
  2312
lemma rinf_ge: assumes Se: "S \<noteq> {}" and Sb: "b <=* S" shows "rinf S \<ge> b"
chaieb@29842
  2313
proof-
chaieb@29842
  2314
  from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def)
chaieb@29842
  2315
  from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)"  by blast 
chaieb@29842
  2316
  then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def)
chaieb@29842
  2317
qed
chaieb@29842
  2318
chaieb@29842
  2319
lemma rinf_finite_Min: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2320
  shows "rinf S = Min S"
chaieb@29842
  2321
using fS Se
chaieb@29842
  2322
proof-
chaieb@29842
  2323
  let ?m = "Min S"
chaieb@29842
  2324
  from Min_le[OF fS] have Sm: "\<forall> x\<in> S. x \<ge> ?m" by metis
chaieb@29842
  2325
  with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def)
chaieb@29842
  2326
  from Min_in[OF fS Se] glb have mrS: "?m \<ge> rinf S" 
chaieb@29842
  2327
    by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def)
chaieb@29842
  2328
  moreover 
chaieb@29842
  2329
  have "rinf S \<ge> ?m" using Sm glb
chaieb@29842
  2330
    by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def)
chaieb@29842
  2331
  ultimately  show ?thesis by arith 
chaieb@29842
  2332
qed
chaieb@29842
  2333
chaieb@29842
  2334
lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2335
  shows "rinf S \<in> S"
chaieb@29842
  2336
  using rinf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis
chaieb@29842
  2337
chaieb@29842
  2338
lemma rinf_finite_Lb: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2339
  shows "isLb S S (rinf S)"
chaieb@29842
  2340
  using rinf_finite_Min[OF fS Se] rinf_finite_in[OF fS Se] Min_le[OF fS]
chaieb@29842
  2341
  unfolding isLb_def setge_def by metis
chaieb@29842
  2342
chaieb@29842
  2343
lemma rinf_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2344
  shows "a \<le> rinf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)"
chaieb@29842
  2345
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
chaieb@29842
  2346
chaieb@29842
  2347
lemma rinf_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2348
  shows "a \<ge> rinf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)"
chaieb@29842
  2349
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
chaieb@29842
  2350
chaieb@29842
  2351
lemma rinf_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
chaieb@29842
  2352
  shows "a < rinf S \<longleftrightarrow> (\<forall> x \<in> S. a < x)"
chaieb@29842
  2353
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
chaieb@29842
  2354