src/HOL/Groups_List.thy
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(* Author: Tobias Nipkow, TU Muenchen *)
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section \<open>Sum and product over lists\<close>
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theory Groups_List
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imports List
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begin
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no_notation times (infixl "*" 70)
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no_notation Groups.one ("1")
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locale monoid_list = monoid
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begin
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definition F :: "'a list \<Rightarrow> 'a"
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where
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  eq_foldr [code]: "F xs = foldr f xs 1"
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lemma Nil [simp]:
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  "F [] = 1"
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  by (simp add: eq_foldr)
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lemma Cons [simp]:
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  "F (x # xs) = x * F xs"
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  by (simp add: eq_foldr)
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lemma append [simp]:
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  "F (xs @ ys) = F xs * F ys"
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  by (induct xs) (simp_all add: assoc)
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end
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locale comm_monoid_list = comm_monoid + monoid_list
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begin
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lemma rev [simp]:
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  "F (rev xs) = F xs"
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  by (simp add: eq_foldr foldr_fold  fold_rev fun_eq_iff assoc left_commute)
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end
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locale comm_monoid_list_set = list: comm_monoid_list + set: comm_monoid_set
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begin
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lemma distinct_set_conv_list:
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  "distinct xs \<Longrightarrow> set.F g (set xs) = list.F (map g xs)"
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  by (induct xs) simp_all
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lemma set_conv_list [code]:
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  "set.F g (set xs) = list.F (map g (remdups xs))"
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  by (simp add: distinct_set_conv_list [symmetric])
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end
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notation times (infixl "*" 70)
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notation Groups.one ("1")
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subsection \<open>List summation\<close>
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context monoid_add
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begin
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sublocale listsum: monoid_list plus 0
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defines
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  listsum = listsum.F ..
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end
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context comm_monoid_add
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begin
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sublocale listsum: comm_monoid_list plus 0
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rewrites
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  "monoid_list.F plus 0 = listsum"
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proof -
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  show "comm_monoid_list plus 0" ..
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  then interpret listsum: comm_monoid_list plus 0 .
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  from listsum_def show "monoid_list.F plus 0 = listsum" by simp
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qed
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sublocale setsum: comm_monoid_list_set plus 0
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rewrites
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  "monoid_list.F plus 0 = listsum"
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  and "comm_monoid_set.F plus 0 = setsum"
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proof -
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  show "comm_monoid_list_set plus 0" ..
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  then interpret setsum: comm_monoid_list_set plus 0 .
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  from listsum_def show "monoid_list.F plus 0 = listsum" by simp
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  from setsum_def show "comm_monoid_set.F plus 0 = setsum" by (auto intro: sym)
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qed
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end
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text \<open>Some syntactic sugar for summing a function over a list:\<close>
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syntax
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  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
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syntax (xsymbols)
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  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
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translations -- \<open>Beware of argument permutation!\<close>
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  "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
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  "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
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text \<open>TODO duplicates\<close>
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lemmas listsum_simps = listsum.Nil listsum.Cons
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lemmas listsum_append = listsum.append
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lemmas listsum_rev = listsum.rev
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lemma (in monoid_add) fold_plus_listsum_rev:
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  "fold plus xs = plus (listsum (rev xs))"
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proof
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  fix x
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  have "fold plus xs x = listsum (rev xs @ [x])"
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    by (simp add: foldr_conv_fold listsum.eq_foldr)
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  also have "\<dots> = listsum (rev xs) + x"
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    by simp
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  finally show "fold plus xs x = listsum (rev xs) + x"
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    .
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qed
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lemma (in comm_monoid_add) listsum_map_remove1:
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  "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
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  by (induct xs) (auto simp add: ac_simps)
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lemma (in monoid_add) size_list_conv_listsum:
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  "size_list f xs = listsum (map f xs) + size xs"
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  by (induct xs) auto
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lemma (in monoid_add) length_concat:
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  "length (concat xss) = listsum (map length xss)"
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  by (induct xss) simp_all
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lemma (in monoid_add) length_product_lists:
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  "length (product_lists xss) = foldr op * (map length xss) 1"
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proof (induct xss)
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  case (Cons xs xss) then show ?case by (induct xs) (auto simp: length_concat o_def)
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qed simp
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lemma (in monoid_add) listsum_map_filter:
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  assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
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  shows "listsum (map f (filter P xs)) = listsum (map f xs)"
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  using assms by (induct xs) auto
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lemma (in comm_monoid_add) distinct_listsum_conv_Setsum:
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  "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
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  by (induct xs) simp_all
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lemma listsum_upt[simp]:
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  "m \<le> n \<Longrightarrow> listsum [m..<n] = \<Sum> {m..<n}"
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by(simp add: distinct_listsum_conv_Setsum)
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lemma listsum_eq_0_nat_iff_nat [simp]:
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  "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
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  by (induct ns) simp_all
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lemma member_le_listsum_nat:
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  "(n :: nat) \<in> set ns \<Longrightarrow> n \<le> listsum ns"
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  by (induct ns) auto
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lemma elem_le_listsum_nat:
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  "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
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  by (rule member_le_listsum_nat) simp
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lemma listsum_update_nat:
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  "k < size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
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apply(induct ns arbitrary:k)
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 apply (auto split:nat.split)
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apply(drule elem_le_listsum_nat)
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apply arith
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done
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lemma (in monoid_add) listsum_triv:
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  "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
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  by (induct xs) (simp_all add: distrib_right)
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lemma (in monoid_add) listsum_0 [simp]:
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  "(\<Sum>x\<leftarrow>xs. 0) = 0"
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  by (induct xs) (simp_all add: distrib_right)
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text\<open>For non-Abelian groups @{text xs} needs to be reversed on one side:\<close>
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lemma (in ab_group_add) uminus_listsum_map:
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  "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
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  by (induct xs) simp_all
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lemma (in comm_monoid_add) listsum_addf:
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  "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
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  by (induct xs) (simp_all add: algebra_simps)
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lemma (in ab_group_add) listsum_subtractf:
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  "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
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  by (induct xs) (simp_all add: algebra_simps)
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lemma (in semiring_0) listsum_const_mult:
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  "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
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  by (induct xs) (simp_all add: algebra_simps)
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lemma (in semiring_0) listsum_mult_const:
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  "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
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  by (induct xs) (simp_all add: algebra_simps)
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lemma (in ordered_ab_group_add_abs) listsum_abs:
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  "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
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  by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
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lemma listsum_mono:
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  fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
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  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)"
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  by (induct xs) (simp, simp add: add_mono)
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lemma (in monoid_add) listsum_distinct_conv_setsum_set:
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  "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
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  by (induct xs) simp_all
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lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
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  "listsum (map f [m..<n]) = setsum f (set [m..<n])"
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  by (simp add: listsum_distinct_conv_setsum_set)
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lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
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  "listsum (map f [k..l]) = setsum f (set [k..l])"
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  by (simp add: listsum_distinct_conv_setsum_set)
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text \<open>General equivalence between @{const listsum} and @{const setsum}\<close>
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lemma (in monoid_add) listsum_setsum_nth:
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  "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
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  using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
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lemma listsum_map_eq_setsum_count:
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  "listsum (map f xs) = setsum (\<lambda>x. count_list xs x * f x) (set xs)"
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proof(induction xs)
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  case (Cons x xs)
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  show ?case (is "?l = ?r")
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  proof cases
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    assume "x \<in> set xs"
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    have "?l = f x + (\<Sum>x\<in>set xs. count_list xs x * f x)" by (simp add: Cons.IH)
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    also have "set xs = insert x (set xs - {x})" using \<open>x \<in> set xs\<close>by blast
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    also have "f x + (\<Sum>x\<in>insert x (set xs - {x}). count_list xs x * f x) = ?r"
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      by (simp add: setsum.insert_remove eq_commute)
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    finally show ?thesis .
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  next
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    assume "x \<notin> set xs"
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    hence "\<And>xa. xa \<in> set xs \<Longrightarrow> x \<noteq> xa" by blast
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    thus ?thesis by (simp add: Cons.IH \<open>x \<notin> set xs\<close>)
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  qed
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qed simp
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lemma listsum_map_eq_setsum_count2:
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assumes "set xs \<subseteq> X" "finite X"
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shows "listsum (map f xs) = setsum (\<lambda>x. count_list xs x * f x) X"
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proof-
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  let ?F = "\<lambda>x. count_list xs x * f x"
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  have "setsum ?F X = setsum ?F (set xs \<union> (X - set xs))"
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    using Un_absorb1[OF assms(1)] by(simp)
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  also have "\<dots> = setsum ?F (set xs)"
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    using assms(2)
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    by(simp add: setsum.union_disjoint[OF _ _ Diff_disjoint] del: Un_Diff_cancel)
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  finally show ?thesis by(simp add:listsum_map_eq_setsum_count)
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qed
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subsection \<open>Further facts about @{const List.n_lists}\<close>
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lemma length_n_lists: "length (List.n_lists n xs) = length xs ^ n"
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  by (induct n) (auto simp add: comp_def length_concat listsum_triv)
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lemma distinct_n_lists:
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  assumes "distinct xs"
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  shows "distinct (List.n_lists n xs)"
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proof (rule card_distinct)
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  from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
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  have "card (set (List.n_lists n xs)) = card (set xs) ^ n"
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  proof (induct n)
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    case 0 then show ?case by simp
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  next
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    case (Suc n)
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    moreover have "card (\<Union>ys\<in>set (List.n_lists n xs). (\<lambda>y. y # ys) ` set xs)
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      = (\<Sum>ys\<in>set (List.n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
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      by (rule card_UN_disjoint) auto
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    moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
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      by (rule card_image) (simp add: inj_on_def)
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    ultimately show ?case by auto
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  qed
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  also have "\<dots> = length xs ^ n" by (simp add: card_length)
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  finally show "card (set (List.n_lists n xs)) = length (List.n_lists n xs)"
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   286
    by (simp add: length_n_lists)
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qed
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subsection \<open>Tools setup\<close>
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lemmas setsum_code = setsum.set_conv_list
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lemma setsum_set_upto_conv_listsum_int [code_unfold]:
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  "setsum f (set [i..j::int]) = listsum (map f [i..j])"
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  by (simp add: interv_listsum_conv_setsum_set_int)
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   297
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lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
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  "setsum f (set [m..<n]) = listsum (map f [m..<n])"
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  by (simp add: interv_listsum_conv_setsum_set_nat)
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context
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begin
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interpretation lifting_syntax .
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   306
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lemma listsum_transfer[transfer_rule]:
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  assumes [transfer_rule]: "A 0 0"
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  assumes [transfer_rule]: "(A ===> A ===> A) op + op +"
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  shows "(list_all2 A ===> A) listsum listsum"
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  unfolding listsum.eq_foldr [abs_def]
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  by transfer_prover
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end
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subsection \<open>List product\<close>
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context monoid_mult
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begin
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sublocale listprod: monoid_list times 1
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defines
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  listprod = listprod.F ..
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end
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context comm_monoid_mult
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begin
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sublocale listprod: comm_monoid_list times 1
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rewrites
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  "monoid_list.F times 1 = listprod"
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   334
proof -
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   335
  show "comm_monoid_list times 1" ..
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   336
  then interpret listprod: comm_monoid_list times 1 .
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  from listprod_def show "monoid_list.F times 1 = listprod" by simp
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qed
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   339
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sublocale setprod: comm_monoid_list_set times 1
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   341
rewrites
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  "monoid_list.F times 1 = listprod"
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  and "comm_monoid_set.F times 1 = setprod"
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   344
proof -
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  show "comm_monoid_list_set times 1" ..
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  then interpret setprod: comm_monoid_list_set times 1 .
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  from listprod_def show "monoid_list.F times 1 = listprod" by simp
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  from setprod_def show "comm_monoid_set.F times 1 = setprod" by (auto intro: sym)
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qed
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end
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d8d85a8172b5 isabelle update_cartouches;
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text \<open>Some syntactic sugar:\<close>
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syntax
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  "_listprod" :: "pttrn => 'a list => 'b => 'b"    ("(3PROD _<-_. _)" [0, 51, 10] 10)
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syntax (xsymbols)
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  "_listprod" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Prod>_\<leftarrow>_. _)" [0, 51, 10] 10)
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translations -- \<open>Beware of argument permutation!\<close>
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  "PROD x<-xs. b" == "CONST listprod (CONST map (%x. b) xs)"
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  "\<Prod>x\<leftarrow>xs. b" == "CONST listprod (CONST map (%x. b) xs)"
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end