(* Author: Tobias Nipkow, TU Muenchen *)
section \<open>Sum and product over lists\<close>
theory Groups_List
imports List
begin
locale monoid_list = monoid
begin
definition F :: "'a list \<Rightarrow> 'a"
where
eq_foldr [code]: "F xs = foldr f xs \<^bold>1"
lemma Nil [simp]:
"F [] = \<^bold>1"
by (simp add: eq_foldr)
lemma Cons [simp]:
"F (x # xs) = x \<^bold>* F xs"
by (simp add: eq_foldr)
lemma append [simp]:
"F (xs @ ys) = F xs \<^bold>* F ys"
by (induct xs) (simp_all add: assoc)
end
locale comm_monoid_list = comm_monoid + monoid_list
begin
lemma rev [simp]:
"F (rev xs) = F xs"
by (simp add: eq_foldr foldr_fold fold_rev fun_eq_iff assoc left_commute)
end
locale comm_monoid_list_set = list: comm_monoid_list + set: comm_monoid_set
begin
lemma distinct_set_conv_list:
"distinct xs \<Longrightarrow> set.F g (set xs) = list.F (map g xs)"
by (induct xs) simp_all
lemma set_conv_list [code]:
"set.F g (set xs) = list.F (map g (remdups xs))"
by (simp add: distinct_set_conv_list [symmetric])
end
subsection \<open>List summation\<close>
context monoid_add
begin
sublocale sum_list: monoid_list plus 0
defines
sum_list = sum_list.F ..
end
context comm_monoid_add
begin
sublocale sum_list: comm_monoid_list plus 0
rewrites
"monoid_list.F plus 0 = sum_list"
proof -
show "comm_monoid_list plus 0" ..
then interpret sum_list: comm_monoid_list plus 0 .
from sum_list_def show "monoid_list.F plus 0 = sum_list" by simp
qed
sublocale sum: comm_monoid_list_set plus 0
rewrites
"monoid_list.F plus 0 = sum_list"
and "comm_monoid_set.F plus 0 = sum"
proof -
show "comm_monoid_list_set plus 0" ..
then interpret sum: comm_monoid_list_set plus 0 .
from sum_list_def show "monoid_list.F plus 0 = sum_list" by simp
from sum_def show "comm_monoid_set.F plus 0 = sum" by (auto intro: sym)
qed
end
text \<open>Some syntactic sugar for summing a function over a list:\<close>
syntax (ASCII)
"_sum_list" :: "pttrn => 'a list => 'b => 'b" ("(3SUM _<-_. _)" [0, 51, 10] 10)
syntax
"_sum_list" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
translations \<comment> \<open>Beware of argument permutation!\<close>
"\<Sum>x\<leftarrow>xs. b" == "CONST sum_list (CONST map (\<lambda>x. b) xs)"
context
includes lifting_syntax
begin
lemma sum_list_transfer [transfer_rule]:
"(list_all2 A ===> A) sum_list sum_list"
if [transfer_rule]: "A 0 0" "(A ===> A ===> A) (+) (+)"
unfolding sum_list.eq_foldr [abs_def]
by transfer_prover
end
text \<open>TODO duplicates\<close>
lemmas sum_list_simps = sum_list.Nil sum_list.Cons
lemmas sum_list_append = sum_list.append
lemmas sum_list_rev = sum_list.rev
lemma (in monoid_add) fold_plus_sum_list_rev:
"fold plus xs = plus (sum_list (rev xs))"
proof
fix x
have "fold plus xs x = sum_list (rev xs @ [x])"
by (simp add: foldr_conv_fold sum_list.eq_foldr)
also have "\<dots> = sum_list (rev xs) + x"
by simp
finally show "fold plus xs x = sum_list (rev xs) + x"
.
qed
lemma (in comm_monoid_add) sum_list_map_remove1:
"x \<in> set xs \<Longrightarrow> sum_list (map f xs) = f x + sum_list (map f (remove1 x xs))"
by (induct xs) (auto simp add: ac_simps)
lemma (in monoid_add) size_list_conv_sum_list:
"size_list f xs = sum_list (map f xs) + size xs"
by (induct xs) auto
lemma (in monoid_add) length_concat:
"length (concat xss) = sum_list (map length xss)"
by (induct xss) simp_all
lemma (in monoid_add) length_product_lists:
"length (product_lists xss) = foldr (*) (map length xss) 1"
proof (induct xss)
case (Cons xs xss) then show ?case by (induct xs) (auto simp: length_concat o_def)
qed simp
lemma (in monoid_add) sum_list_map_filter:
assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
shows "sum_list (map f (filter P xs)) = sum_list (map f xs)"
using assms by (induct xs) auto
lemma sum_list_filter_le_nat:
fixes f :: "'a \<Rightarrow> nat"
shows "sum_list (map f (filter P xs)) \<le> sum_list (map f xs)"
by(induction xs; simp)
lemma (in comm_monoid_add) distinct_sum_list_conv_Sum:
"distinct xs \<Longrightarrow> sum_list xs = Sum (set xs)"
by (induct xs) simp_all
lemma sum_list_upt[simp]:
"m \<le> n \<Longrightarrow> sum_list [m..<n] = \<Sum> {m..<n}"
by(simp add: distinct_sum_list_conv_Sum)
context ordered_comm_monoid_add
begin
lemma sum_list_nonneg: "(\<And>x. x \<in> set xs \<Longrightarrow> 0 \<le> x) \<Longrightarrow> 0 \<le> sum_list xs"
by (induction xs) auto
lemma sum_list_nonpos: "(\<And>x. x \<in> set xs \<Longrightarrow> x \<le> 0) \<Longrightarrow> sum_list xs \<le> 0"
by (induction xs) (auto simp: add_nonpos_nonpos)
lemma sum_list_nonneg_eq_0_iff:
"(\<And>x. x \<in> set xs \<Longrightarrow> 0 \<le> x) \<Longrightarrow> sum_list xs = 0 \<longleftrightarrow> (\<forall>x\<in> set xs. x = 0)"
by (induction xs) (simp_all add: add_nonneg_eq_0_iff sum_list_nonneg)
end
context canonically_ordered_monoid_add
begin
lemma sum_list_eq_0_iff [simp]:
"sum_list ns = 0 \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
by (simp add: sum_list_nonneg_eq_0_iff)
lemma member_le_sum_list:
"x \<in> set xs \<Longrightarrow> x \<le> sum_list xs"
by (induction xs) (auto simp: add_increasing add_increasing2)
lemma elem_le_sum_list:
"k < size ns \<Longrightarrow> ns ! k \<le> sum_list (ns)"
by (rule member_le_sum_list) simp
end
lemma (in ordered_cancel_comm_monoid_diff) sum_list_update:
"k < size xs \<Longrightarrow> sum_list (xs[k := x]) = sum_list xs + x - xs ! k"
apply(induction xs arbitrary:k)
apply (auto simp: add_ac split: nat.split)
apply(drule elem_le_sum_list)
by (simp add: local.add_diff_assoc local.add_increasing)
lemma (in monoid_add) sum_list_triv:
"(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
by (induct xs) (simp_all add: distrib_right)
lemma (in monoid_add) sum_list_0 [simp]:
"(\<Sum>x\<leftarrow>xs. 0) = 0"
by (induct xs) (simp_all add: distrib_right)
text\<open>For non-Abelian groups \<open>xs\<close> needs to be reversed on one side:\<close>
lemma (in ab_group_add) uminus_sum_list_map:
"- sum_list (map f xs) = sum_list (map (uminus \<circ> f) xs)"
by (induct xs) simp_all
lemma (in comm_monoid_add) sum_list_addf:
"(\<Sum>x\<leftarrow>xs. f x + g x) = sum_list (map f xs) + sum_list (map g xs)"
by (induct xs) (simp_all add: algebra_simps)
lemma (in ab_group_add) sum_list_subtractf:
"(\<Sum>x\<leftarrow>xs. f x - g x) = sum_list (map f xs) - sum_list (map g xs)"
by (induct xs) (simp_all add: algebra_simps)
lemma (in semiring_0) sum_list_const_mult:
"(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
by (induct xs) (simp_all add: algebra_simps)
lemma (in semiring_0) sum_list_mult_const:
"(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
by (induct xs) (simp_all add: algebra_simps)
lemma (in ordered_ab_group_add_abs) sum_list_abs:
"\<bar>sum_list xs\<bar> \<le> sum_list (map abs xs)"
by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
lemma sum_list_mono:
fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
shows "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sum>x\<leftarrow>xs. f x) \<le> (\<Sum>x\<leftarrow>xs. g x)"
by (induct xs) (simp, simp add: add_mono)
lemma sum_list_strict_mono:
fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, strict_ordered_ab_semigroup_add}"
shows "\<lbrakk> xs \<noteq> []; \<And>x. x \<in> set xs \<Longrightarrow> f x < g x \<rbrakk>
\<Longrightarrow> sum_list (map f xs) < sum_list (map g xs)"
proof (induction xs)
case Nil thus ?case by simp
next
case C: (Cons _ xs)
show ?case
proof (cases xs)
case Nil thus ?thesis using C.prems by simp
next
case Cons thus ?thesis using C by(simp add: add_strict_mono)
qed
qed
lemma (in monoid_add) sum_list_distinct_conv_sum_set:
"distinct xs \<Longrightarrow> sum_list (map f xs) = sum f (set xs)"
by (induct xs) simp_all
lemma (in monoid_add) interv_sum_list_conv_sum_set_nat:
"sum_list (map f [m..<n]) = sum f (set [m..<n])"
by (simp add: sum_list_distinct_conv_sum_set)
lemma (in monoid_add) interv_sum_list_conv_sum_set_int:
"sum_list (map f [k..l]) = sum f (set [k..l])"
by (simp add: sum_list_distinct_conv_sum_set)
text \<open>General equivalence between \<^const>\<open>sum_list\<close> and \<^const>\<open>sum\<close>\<close>
lemma (in monoid_add) sum_list_sum_nth:
"sum_list xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
using interv_sum_list_conv_sum_set_nat [of "(!) xs" 0 "length xs"] by (simp add: map_nth)
lemma sum_list_map_eq_sum_count:
"sum_list (map f xs) = sum (\<lambda>x. count_list xs x * f x) (set xs)"
proof(induction xs)
case (Cons x xs)
show ?case (is "?l = ?r")
proof cases
assume "x \<in> set xs"
have "?l = f x + (\<Sum>x\<in>set xs. count_list xs x * f x)" by (simp add: Cons.IH)
also have "set xs = insert x (set xs - {x})" using \<open>x \<in> set xs\<close>by blast
also have "f x + (\<Sum>x\<in>insert x (set xs - {x}). count_list xs x * f x) = ?r"
by (simp add: sum.insert_remove eq_commute)
finally show ?thesis .
next
assume "x \<notin> set xs"
hence "\<And>xa. xa \<in> set xs \<Longrightarrow> x \<noteq> xa" by blast
thus ?thesis by (simp add: Cons.IH \<open>x \<notin> set xs\<close>)
qed
qed simp
lemma sum_list_map_eq_sum_count2:
assumes "set xs \<subseteq> X" "finite X"
shows "sum_list (map f xs) = sum (\<lambda>x. count_list xs x * f x) X"
proof-
let ?F = "\<lambda>x. count_list xs x * f x"
have "sum ?F X = sum ?F (set xs \<union> (X - set xs))"
using Un_absorb1[OF assms(1)] by(simp)
also have "\<dots> = sum ?F (set xs)"
using assms(2)
by(simp add: sum.union_disjoint[OF _ _ Diff_disjoint] del: Un_Diff_cancel)
finally show ?thesis by(simp add:sum_list_map_eq_sum_count)
qed
lemma sum_list_nonneg:
"(\<And>x. x \<in> set xs \<Longrightarrow> (x :: 'a :: ordered_comm_monoid_add) \<ge> 0) \<Longrightarrow> sum_list xs \<ge> 0"
by (induction xs) simp_all
lemma sum_list_Suc:
"sum_list (map (\<lambda>x. Suc(f x)) xs) = sum_list (map f xs) + length xs"
by(induction xs; simp)
lemma (in monoid_add) sum_list_map_filter':
"sum_list (map f (filter P xs)) = sum_list (map (\<lambda>x. if P x then f x else 0) xs)"
by (induction xs) simp_all
text \<open>Summation of a strictly ascending sequence with length \<open>n\<close>
can be upper-bounded by summation over \<open>{0..<n}\<close>.\<close>
lemma sorted_wrt_less_sum_mono_lowerbound:
fixes f :: "nat \<Rightarrow> ('b::ordered_comm_monoid_add)"
assumes mono: "\<And>x y. x\<le>y \<Longrightarrow> f x \<le> f y"
shows "sorted_wrt (<) ns \<Longrightarrow>
(\<Sum>i\<in>{0..<length ns}. f i) \<le> (\<Sum>i\<leftarrow>ns. f i)"
proof (induction ns rule: rev_induct)
case Nil
then show ?case by simp
next
case (snoc n ns)
have "sum f {0..<length (ns @ [n])}
= sum f {0..<length ns} + f (length ns)"
by simp
also have "sum f {0..<length ns} \<le> sum_list (map f ns)"
using snoc by (auto simp: sorted_wrt_append)
also have "length ns \<le> n"
using sorted_wrt_less_idx[OF snoc.prems(1), of "length ns"] by auto
finally have "sum f {0..<length (ns @ [n])} \<le> sum_list (map f ns) + f n"
using mono add_mono by blast
thus ?case by simp
qed
subsection \<open>Further facts about \<^const>\<open>List.n_lists\<close>\<close>
lemma length_n_lists: "length (List.n_lists n xs) = length xs ^ n"
by (induct n) (auto simp add: comp_def length_concat sum_list_triv)
lemma distinct_n_lists:
assumes "distinct xs"
shows "distinct (List.n_lists n xs)"
proof (rule card_distinct)
from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
have "card (set (List.n_lists n xs)) = card (set xs) ^ n"
proof (induct n)
case 0 then show ?case by simp
next
case (Suc n)
moreover have "card (\<Union>ys\<in>set (List.n_lists n xs). (\<lambda>y. y # ys) ` set xs)
= (\<Sum>ys\<in>set (List.n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
by (rule card_UN_disjoint) auto
moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
by (rule card_image) (simp add: inj_on_def)
ultimately show ?case by auto
qed
also have "\<dots> = length xs ^ n" by (simp add: card_length)
finally show "card (set (List.n_lists n xs)) = length (List.n_lists n xs)"
by (simp add: length_n_lists)
qed
subsection \<open>Tools setup\<close>
lemmas sum_code = sum.set_conv_list
lemma sum_set_upto_conv_sum_list_int [code_unfold]:
"sum f (set [i..j::int]) = sum_list (map f [i..j])"
by (simp add: interv_sum_list_conv_sum_set_int)
lemma sum_set_upt_conv_sum_list_nat [code_unfold]:
"sum f (set [m..<n]) = sum_list (map f [m..<n])"
by (simp add: interv_sum_list_conv_sum_set_nat)
subsection \<open>List product\<close>
context monoid_mult
begin
sublocale prod_list: monoid_list times 1
defines
prod_list = prod_list.F ..
end
context comm_monoid_mult
begin
sublocale prod_list: comm_monoid_list times 1
rewrites
"monoid_list.F times 1 = prod_list"
proof -
show "comm_monoid_list times 1" ..
then interpret prod_list: comm_monoid_list times 1 .
from prod_list_def show "monoid_list.F times 1 = prod_list" by simp
qed
sublocale prod: comm_monoid_list_set times 1
rewrites
"monoid_list.F times 1 = prod_list"
and "comm_monoid_set.F times 1 = prod"
proof -
show "comm_monoid_list_set times 1" ..
then interpret prod: comm_monoid_list_set times 1 .
from prod_list_def show "monoid_list.F times 1 = prod_list" by simp
from prod_def show "comm_monoid_set.F times 1 = prod" by (auto intro: sym)
qed
end
text \<open>Some syntactic sugar:\<close>
syntax (ASCII)
"_prod_list" :: "pttrn => 'a list => 'b => 'b" ("(3PROD _<-_. _)" [0, 51, 10] 10)
syntax
"_prod_list" :: "pttrn => 'a list => 'b => 'b" ("(3\<Prod>_\<leftarrow>_. _)" [0, 51, 10] 10)
translations \<comment> \<open>Beware of argument permutation!\<close>
"\<Prod>x\<leftarrow>xs. b" \<rightleftharpoons> "CONST prod_list (CONST map (\<lambda>x. b) xs)"
context
includes lifting_syntax
begin
lemma prod_list_transfer [transfer_rule]:
"(list_all2 A ===> A) prod_list prod_list"
if [transfer_rule]: "A 1 1" "(A ===> A ===> A) (*) (*)"
unfolding prod_list.eq_foldr [abs_def]
by transfer_prover
end
lemma prod_list_zero_iff:
"prod_list xs = 0 \<longleftrightarrow> (0 :: 'a :: {semiring_no_zero_divisors, semiring_1}) \<in> set xs"
by (induction xs) simp_all
end