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) => 'a list => 'a list" (infix "*\<^sub>s" 70)
where "x *\<^sub>s xs ≡ 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 => '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"
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 => '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 *\<^sub>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