(* ID: $Id: ListVector.thy,v 1.1 2008/02/27 17:01:10 nipkow Exp $ Author: Tobias Nipkow, 2007 *) header "Lists as vectors" theory ListVector imports 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) => 'a list => 'a list" (infix "*s" 70) where "x *s xs ≡ map (op * x) xs" lemma scale1[simp]: "(1::'a::monoid_mult) *s xs = xs" by (induct xs) simp_all subsection {* @{text"+"} and @{text"-"} *} fun zipwith0 :: "('a::zero => 'b::zero => 'c) => 'a list => 'b list => '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" instance list :: ("{zero,plus}")plus list_add_def : "op + ≡ zipwith0 (op +)" .. instance list :: ("{zero,uminus}")uminus list_uminus_def: "uminus ≡ map uminus" .. instance list :: ("{zero,minus}")minus list_diff_def: "op - ≡ zipwith0 (op -)" .. 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 => 'a list => 'a" ("〈_,_〉") where "〈xs,ys〉 = (∑(x,y) \<leftarrow> zip xs ys. x*y)" lemma iprod_Nil[simp]: "〈[],ys〉 = 0" by(simp add:iprod_def) lemma iprod_Nil2[simp]: "〈xs,[]〉 = 0" by(simp add:iprod_def) lemma iprod_Cons[simp]: "〈x#xs,y#ys〉 = x*y + 〈xs,ys〉" by(simp add:iprod_def) lemma iprod0_if_coeffs0: "∀c∈set cs. c = 0 ==> 〈cs,xs〉 = 0" apply(induct cs arbitrary:xs) apply simp apply(case_tac xs) apply simp apply auto done lemma iprod_uminus[simp]: "〈-xs,ys〉 = -〈xs,ys〉" by(simp add: iprod_def uminus_listsum_map o_def split_def map_zip_map list_uminus_def) lemma iprod_left_add_distrib: "〈xs + ys,zs〉 = 〈xs,zs〉 + 〈ys,zs〉" 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: "〈xs - ys, zs〉 = 〈xs,zs〉 - 〈ys,zs〉" 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: "〈x *s xs, ys〉 = x * 〈xs,ys〉" apply(induct xs arbitrary: ys) apply simp apply(case_tac ys) apply (simp) apply (simp add:right_distrib mult_assoc) done end
lemma scale1:
(1::'a) *s xs = xs
lemma zipwith0_Nil:
zipwith0 f [] ys = map (f (0::'b)) ys
lemma list_add_Nil:
[] + xs = xs
lemma list_add_Nil2:
xs + [] = xs
lemma list_add_Cons:
x # xs + (y # ys) = (x + y) # xs + ys
lemma list_diff_Nil:
[] - xs = - xs
lemma list_diff_Nil2:
xs - [] = xs
lemma list_diff_Cons_Cons:
x # xs - (y # ys) = (x - y) # xs - ys
lemma list_uminus_Cons:
- (x # xs) = - x # - xs
lemma self_list_diff:
xs - xs = replicate (length xs) (0::'a)
lemma list_add_assoc:
xs + ys + zs = xs + (ys + zs)
lemma iprod_Nil:
〈[],ys〉 = (0::'a)
lemma iprod_Nil2:
〈xs,[]〉 = (0::'a)
lemma iprod_Cons:
〈x # xs,y # ys〉 = x * y + 〈xs,ys〉
lemma iprod0_if_coeffs0:
∀c∈set cs. c = (0::'a) ==> 〈cs,xs〉 = (0::'a)
lemma iprod_uminus:
〈- xs,ys〉 = - 〈xs,ys〉
lemma iprod_left_add_distrib:
〈xs + ys,zs〉 = 〈xs,zs〉 + 〈ys,zs〉
lemma iprod_left_diff_distrib:
〈xs - ys,zs〉 = 〈xs,zs〉 - 〈ys,zs〉
lemma iprod_assoc:
〈x *s xs,ys〉 = x * 〈xs,ys〉