(*  Author: Tobias Nipkow, 2007 *)

header {* Lists as vectors *}

theory ListVector

imports List Main

begin

text{* \noindent

A vector-space like structure of lists and arithmetic operations on them.

Is only a vector space if restricted to lists of the same length. *}

text{* Multiplication with a scalar: *}

abbreviation scale :: "('a::times) \<Rightarrow> 'a list \<Rightarrow> 'a list" (infix "*\<^sub>s" 70)

where "x *\<^sub>s xs \<equiv> map (op * x) xs"

lemma scale1[simp]: "(1::'a::monoid_mult) *\<^sub>s xs = xs"

by (induct xs) simp_all

subsection {* @{text"+"} and @{text"-"} *}

fun zipwith0 :: "('a::zero \<Rightarrow> 'b::zero \<Rightarrow> 'c) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'c list"

where

"zipwith0 f [] [] = []" |

"zipwith0 f (x#xs) (y#ys) = f x y # zipwith0 f xs ys" |

"zipwith0 f (x#xs) [] = f x 0 # zipwith0 f xs []" |

"zipwith0 f [] (y#ys) = f 0 y # zipwith0 f [] ys"

instantiation list :: ("{zero, plus}") plus

begin

definition

  list_add_def: "op + = zipwith0 (op +)"

instance ..

end

instantiation list :: ("{zero, uminus}") uminus

begin

definition

  list_uminus_def: "uminus = map uminus"

instance ..

end

instantiation list :: ("{zero,minus}") minus

begin

definition

  list_diff_def: "op - = zipwith0 (op -)"

instance ..

end

lemma zipwith0_Nil[simp]: "zipwith0 f [] ys = map (f 0) ys"

by(induct ys) simp_all

lemma list_add_Nil[simp]: "[] + xs = (xs::'a::monoid_add list)"

by (induct xs) (auto simp:list_add_def)

lemma list_add_Nil2[simp]: "xs + [] = (xs::'a::monoid_add list)"

by (induct xs) (auto simp:list_add_def)

lemma list_add_Cons[simp]: "(x#xs) + (y#ys) = (x+y)#(xs+ys)"

by(auto simp:list_add_def)

lemma list_diff_Nil[simp]: "[] - xs = -(xs::'a::group_add list)"

by (induct xs) (auto simp:list_diff_def list_uminus_def)

lemma list_diff_Nil2[simp]: "xs - [] = (xs::'a::group_add list)"

by (induct xs) (auto simp:list_diff_def)

lemma list_diff_Cons_Cons[simp]: "(x#xs) - (y#ys) = (x-y)#(xs-ys)"

by (induct xs) (auto simp:list_diff_def)

lemma list_uminus_Cons[simp]: "-(x#xs) = (-x)#(-xs)"

by (induct xs) (auto simp:list_uminus_def)

lemma self_list_diff:

  "xs - xs = replicate (length(xs::'a::group_add list)) 0"

by(induct xs) simp_all

lemma list_add_assoc: fixes xs :: "'a::monoid_add list"

shows "(xs+ys)+zs = xs+(ys+zs)"

apply(induct xs arbitrary: ys zs)

 apply simp

apply(case_tac ys)

 apply(simp)

apply(simp)

apply(case_tac zs)

 apply(simp)

apply(simp add:add_assoc)

done

subsection "Inner product"

definition iprod :: "'a::ring list \<Rightarrow> 'a list \<Rightarrow> 'a" ("\<langle>_,_\<rangle>") where

"\<langle>xs,ys\<rangle> = (\<Sum>(x,y) \<leftarrow> zip xs ys. x*y)"

lemma iprod_Nil[simp]: "\<langle>[],ys\<rangle> = 0"

by(simp add:iprod_def)

lemma iprod_Nil2[simp]: "\<langle>xs,[]\<rangle> = 0"

by(simp add:iprod_def)

lemma iprod_Cons[simp]: "\<langle>x#xs,y#ys\<rangle> = x*y + \<langle>xs,ys\<rangle>"

by(simp add:iprod_def)

lemma iprod0_if_coeffs0: "\<forall>c\<in>set cs. c = 0 \<Longrightarrow> \<langle>cs,xs\<rangle> = 0"

apply(induct cs arbitrary:xs)

 apply simp

apply(case_tac xs) apply simp

apply auto

done

lemma iprod_uminus[simp]: "\<langle>-xs,ys\<rangle> = -\<langle>xs,ys\<rangle>"

by(simp add: iprod_def uminus_listsum_map o_def split_def map_zip_map list_uminus_def)

lemma iprod_left_add_distrib: "\<langle>xs + ys,zs\<rangle> = \<langle>xs,zs\<rangle> + \<langle>ys,zs\<rangle>"

apply(induct xs arbitrary: ys zs)

apply (simp add: o_def split_def)

apply(case_tac ys)

apply simp

apply(case_tac zs)

apply (simp)

apply(simp add:left_distrib)

done

lemma iprod_left_diff_distrib: "\<langle>xs - ys, zs\<rangle> = \<langle>xs,zs\<rangle> - \<langle>ys,zs\<rangle>"

apply(induct xs arbitrary: ys zs)

apply (simp add: o_def split_def)

apply(case_tac ys)

apply simp

apply(case_tac zs)

apply (simp)

apply(simp add:left_diff_distrib)

done

lemma iprod_assoc: "\<langle>x *\<^sub>s xs, ys\<rangle> = x * \<langle>xs,ys\<rangle>"

apply(induct xs arbitrary: ys)

apply simp

apply(case_tac ys)

apply (simp)

apply (simp add:right_distrib mult_assoc)

done

end