Quelle  Groups_List.thy

(* Author: Tobias Nipkow, TU Muenchen *)

section ‹Sum and product over lists›

theory Groups_List
imports List
begin

locale monoid_list = monoid
begin

definition F :: "'a list ==> 'a"
where
  eq_foldr [code]: "F xs = foldr f xs 🪙1"

lemma Nil [simp]:
  "F [] = 🪙1"
  by (simp add: eq_foldr)

lemma Cons [simp]:
  "F (x # xs) = x 🪙* F xs"
  by (simp add: eq_foldr)

lemma append [simp]:
  "F (xs @ ys) = F xs 🪙* 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 ==> 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])

lemma list_conv_set_nth:
  "list.F xs = set.F (λi. xs ! i) {0..
proof -
  have "xs = map (λi. xs ! i) [0..
    by (simp add: map_nth)
  also have "list.F … = set.F (λi. xs ! i) {0..
    by (subst distinct_set_conv_list [symmetric]) auto
  finally show ?thesis .
qed

lemma atLeastAtMost_conv_list [code_unfold]:
  ‹set.F g {a..b} = list.F (map g (List.interval a b))›
  by (simp flip: List.set_interval_eq add: distinct_set_conv_list)

lemma atLeastLessThan_conv_list [code_unfold]:
  ‹set.F g {a..🚫 = (let d = b - 1 in if d 🚫
  then list.F (map g (List.interval a d))
  else 🪙1)›
  using List.atLeastLessThan_eq_interval [of a b]
  by (simp flip: List.set_interval_eq add: distinct_set_conv_list Let_def)

lemma greaterThanAtMost_conv_list [code_unfold]:
  ‹set.F g {a🚫b} = (let c = a + 1 in if a 🚫
  then list.F (map g (List.interval c b))
  else 🪙1)›
  using List.greaterThanAtMost_eq_interval [of a b]
  by (simp flip: List.set_interval_eq add: distinct_set_conv_list Let_def)

lemma greaterThanLessThan_conv_list [code_unfold]:
  ‹set.F g {a🚫🚫 = (let c = a + 1; d = b - 1 in if a 🚫 ∧ d 🚫
  then list.F (map g (List.interval (a + 1) (b - 1)))
  else 🪙1)›
  using List.greaterThanLessThan_eq_interval [of a b]
  by (simp flip: List.set_interval_eq add: distinct_set_conv_list Let_def)

end


subsection ‹List summation›

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 ‹Some syntactic sugar for summing a function over a list:›

open_bundle sum_list_syntax
begin

syntax (ASCII)
  "_sum_list" :: "pttrn => 'a list => 'b => 'b"    (‹(‹indent=3 notation=‹binder SUM›\›SUM _<-_. _)› [0, 51, 10] 10)
syntax
  "_sum_list" :: "pttrn => 'a list => 'b => 'b"    (‹(‹indent=3 notation=‹binder ∑›\›\←_. _)› [0, 51, 10] 10)
syntax_consts
  "_sum_list" == sum_list
translations 🍋 ‹Beware of argument permutation!›
  "∑x←xs. b" == "CONST sum_list (CONST map (λx. b) xs)"

end

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 ‹TODO duplicates›
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 "… = sum_list (rev xs) + x"
    by simp
  finally show "fold plus xs x = sum_list (rev xs) + x"
    .
qed

lemma sum_list_of_nat: "sum_list (map of_nat xs) = of_nat (sum_list xs)"
  by (induction xs) auto

lemma sum_list_of_int: "sum_list (map of_int xs) = of_int (sum_list xs)"
  by (induction xs) auto

lemma count_list_concat: "count_list (concat xss) x = sum_list (map (λxs. count_list xs x) xss)"
by(induction xss) auto

lemma (in comm_monoid_add) sum_list_map_remove1:
  "x ∈ set xs ==> 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 "∧x. x ∈ set xs ==> ¬ P x ==> 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 ==> nat"
  shows "sum_list (map f (filter P xs)) ≤ sum_list (map f xs)"
by(induction xs; simp)

lemma (in comm_monoid_add) distinct_sum_list_conv_Sum:
  "distinct xs ==> sum_list xs = Sum (set xs)"
  by (metis local.sum.set_conv_list local.sum_list_def map_ident remdups_id_iff_distinct)

lemma sum_list_upt[simp]:
  "m ≤ n ==> sum_list [m..∑ {m..
by(simp add: distinct_sum_list_conv_Sum)

context ordered_comm_monoid_add
begin

lemma sum_list_nonneg: "(∧x. x ∈ set xs ==> 0 ≤ x) ==> 0 ≤ sum_list xs"
  by (induction xs) auto

lemma sum_list_nonpos: "(∧x. x ∈ set xs ==> x ≤ 0) ==> sum_list xs ≤ 0"
  by (induction xs) (auto simp: add_nonpos_nonpos)

lemma sum_list_nonneg_eq_0_iff:
  "(∧x. x ∈ set xs ==> 0 ≤ x) ==> sum_list xs = 0 ⟷ (∀x∈ 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 ⟷ (∀n ∈ set ns. n = 0)"
  by (simp add: sum_list_nonneg_eq_0_iff)

lemma member_le_sum_list:
  "x ∈ set xs ==> x ≤ sum_list xs"
  by (induction xs) (auto simp: add_increasing add_increasing2)

lemma elem_le_sum_list:
  "k < size ns ==> ns ! k ≤ sum_list (ns)"
  by (simp add: member_le_sum_list)

end

lemma (in ordered_cancel_comm_monoid_diff) sum_list_update:
  "k < size xs ==> sum_list (xs[k := x]) = sum_list xs + x - xs ! k"
proof (induction xs arbitrary:k)
  case Nil
  then show ?case by auto
next
  case (Cons a xs)
  then show ?case
    apply (simp add: add_ac split: nat.split)
    using add_increasing diff_add_assoc elem_le_sum_list zero_le by force
qed

lemma (in monoid_add) sum_list_triv:
  "(∑x←xs. r) = of_nat (length xs) * r"
  by (induct xs) (simp_all add: distrib_right)

lemma (in monoid_add) sum_list_0 [simp]:
  "(∑x←xs. 0) = 0"
  by (induct xs) (simp_all add: distrib_right)

text‹For non-Abelian groups ‹xs› needs to be reversed on one side:›
lemma (in ab_group_add) uminus_sum_list_map:
  "- sum_list (map f xs) = sum_list (map (uminus ∘ f) xs)"
  by (induct xs) simp_all

lemma (in comm_monoid_add) sum_list_addf:
  "(∑x←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:
  "(∑x←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:
  "(∑x←xs. c * f x) = c * (∑x←xs. f x)"
  by (induct xs) (simp_all add: algebra_simps)

lemma (in semiring_0) sum_list_mult_const:
  "(∑x←xs. f x * c) = (∑x←xs. f x) * c"
  by (induct xs) (simp_all add: algebra_simps)

lemma (in ordered_ab_group_add_abs) sum_list_abs:
  "∣sum_list xs∣ ≤ 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 ==> 'b::{monoid_add, ordered_ab_semigroup_add}"
  shows "(∧x. x ∈ set xs ==> f x ≤ g x) ==> (∑x←xs. f x) ≤ (∑x←xs. g x)"
by (induct xs) (simp, simp add: add_mono)

lemma sum_list_strict_mono:
  fixes f g :: "'a ==> 'b::{monoid_add, strict_ordered_ab_semigroup_add}"
  shows "[ xs ≠ []; ∧x. x ∈ set xs ==> f x < g x ]
    ==> 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

text ‹A much more general version of this monotonicity lemma
 can be formulated with multisets and the multiset order›

lemma sum_list_mono2: fixes xs :: "'a ::ordered_comm_monoid_add list"
shows "[ length xs = length ys; ∧i. i < length xs ⟶ xs!i ≤ ys!i ]
  ==> sum_list xs ≤ sum_list ys"
  by (induction xs ys rule: list_induct2) (auto simp: nth_Cons' less_Suc_eq_0_disj imp_ex add_mono)

lemma (in monoid_add) sum_list_distinct_conv_sum_set:
  "distinct xs ==> 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..
  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 ‹General equivalence between 🍋‹sum_list› and 🍋‹sum›\<close>
lemma (in monoid_add) sum_list_sum_nth:
  "sum_list xs = (∑ 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 (λx. count_list xs x * f x) (set xs)"
proof(induction xs)
  case (Cons x xs)
  show ?case (is "?l = ?r")
  proof cases
    assume "x ∈ set xs"
    have "?l = f x + (∑x∈set xs. count_list xs x * f x)" by (simp add: Cons.IH)
    also have "set xs = insert x (set xs - {x})" using ‹x ∈ set xs›by blast
    also have "f x + (∑x∈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 ∉ set xs"
    hence "∧xa. xa ∈ set xs ==> x ≠ xa" by blast
    thus ?thesis by (simp add: Cons.IH ‹x ∉ set xs›)
  qed
qed simp

lemma sum_list_map_eq_sum_count2:
assumes "set xs ⊆ X" "finite X"
shows "sum_list (map f xs) = sum (λx. count_list xs x * f x) X"
proof-
  let ?F = "λx. count_list xs x * f x"
  have "sum ?F X = sum ?F (set xs ∪ (X - set xs))"
    using Un_absorb1[OF assms(1)] by(simp)
  also have "… = 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_replicate: "sum_list (replicate n c) = of_nat n * c"
by(induction n)(auto simp add: distrib_right)


lemma sum_list_nonneg:
    "(∧x. x ∈ set xs ==> (x :: 'a :: ordered_comm_monoid_add) ≥ 0) ==> sum_list xs ≥ 0"
  by (induction xs) simp_all

lemma sum_list_Suc:
  "sum_list (map (λx. 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 (λx. if P x then f x else 0) xs)"
  by (induction xs) simp_all

text ‹Summation of a strictly ascending sequence with length ‹n›
  can be upper-bounded by summation over ‹{0..🚫›.›

lemma sorted_wrt_less_sum_mono_lowerbound:
  fixes f :: "nat ==> ('b::ordered_comm_monoid_add)"
  assumes mono: "∧x y. x≤y ==> f x ≤ f y"
  shows "sorted_wrt (<) ns ==>
    (∑i∈{0..≤ (∑i←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..
      = sum f {0..
    by simp
  also have "sum f {0..≤ sum_list (map f ns)"
    using snoc by (auto simp: sorted_wrt_append)
  also have "length ns ≤ n"
    using sorted_wrt_less_idx[OF snoc.prems(1), of "length ns"] by auto
  finally have "sum f {0..≤ sum_list (map f ns) + f n"
    using mono add_mono by blast
  thus ?case by simp
qed

(*Note that we also have this for class canonically_ordered_monoid_add*)
lemma member_le_sum_list:
  fixes x :: "'a :: ordered_comm_monoid_add"
  assumes "x ∈ set xs" "∧x. x ∈ set xs ==> x ≥ 0"
  shows   "x ≤ sum_list xs"
  using assms
proof (induction xs)
  case (Cons y xs)
  show ?case
  proof (cases "y = x")
    case True
    have "x + 0 ≤ x + sum_list xs"
      by (intro add_mono order.refl sum_list_nonneg) (use Cons in auto)
    thus ?thesis
      using True by auto
  next
    case False
    have "0 + x ≤ y + sum_list xs"
      by (intro add_mono Cons.IH Cons.prems) (use Cons.prems False in auto)
    thus ?thesis
      by auto
  qed
qed auto

subsection ‹Horner sums›

context comm_semiring_0
begin

definition horner_sum :: ‹('b ==> 'a) ==> 'a ==> 'b list ==> 'a›
  where horner_sum_foldr: ‹horner_sum f a xs = foldr (λx b. f x + a * b) xs 0›

lemma horner_sum_simps [simp]:
  ‹horner_sum f a [] = 0›
  ‹horner_sum f a (x # xs) = f x + a * horner_sum f a xs›
  by (simp_all add: horner_sum_foldr)

lemma horner_sum_eq_sum_funpow:
  ‹horner_sum f a xs = (∑n = 0..🚫xs. ((*) a ^^ n) (f (xs ! n)))›
proof (induction xs)
  case Nil
  then show ?case
    by simp
next
  case (Cons x xs)
  then show ?case
    by (simp add: sum.atLeast0_lessThan_Suc_shift sum_distrib_left del: sum.op_ivl_Suc)
qed

end

context
  includes lifting_syntax
begin

lemma horner_sum_transfer [transfer_rule]:
  ‹((B ===> A) ===> A ===> list_all2 B ===> A) horner_sum horner_sum›
  if [transfer_rule]: ‹A 0 0›
    and [transfer_rule]: ‹(A ===> A ===> A) (+) (+)›
    and [transfer_rule]: ‹(A ===> A ===> A) (*) (*)›
  by (unfold horner_sum_foldr) transfer_prover

end

context comm_semiring_1
begin

lemma horner_sum_eq_sum:
  ‹horner_sum f a xs = (∑n = 0..🚫xs. f (xs ! n) * a ^ n)›
proof -
  have ‹(*) a ^^ n = (*) (a ^ n)\ for n
  by (induction n) (simp_all add: ac_simps)
  then show ?thesis
  by (simp add: horner_sum_eq_sum_funpow ac_simps)
 qed
 
 lemma horner_sum_append:
  ‹horner_sum f a (xs @ ys) = horner_sum f a xs + a ^ length xs * horner_sum f a ys›
  using sum.atLeastLessThan_shift_bounds [of _ 0 ‹length xs› ‹length ys›]
    atLeastLessThan_add_Un [of 0 ‹length xs› ‹length ys›]
  by (simp add: horner_sum_eq_sum sum_distrib_left sum.union_disjoint ac_simps nth_append power_add)

end

context linordered_semidom
begin

lemma horner_sum_nonnegative:
  ‹0 ≤ horner_sum of_bool 2 bs›
  by (induction bs) simp_all

end

context discrete_linordered_semidom
begin

lemma horner_sum_bound:
  ‹horner_sum of_bool 2 bs 🚫 ^ length bs›
proof (induction bs)
  case Nil
  then show ?case
    by simp
next
  case (Cons b bs)
  moreover define a where ‹a = 2 ^ length bs - horner_sum of_bool 2 bs›
  ultimately have *: ‹2 ^ length bs = horner_sum of_bool 2 bs + a›
    by simp
  have ‹0 🚫›
    using Cons * by simp
  moreover have ‹1 ≤ a›
    using ‹0 🚫› by (simp add: less_eq_iff_succ_less)
  ultimately have ‹0 + 1 🚫 + a›
    by (rule add_less_le_mono)
  then have ‹1 🚫 * 2›
    by (simp add: mult_2_right)
  with Cons show ?case
    by (simp add: * algebra_simps)
qed

lemma horner_sum_of_bool_2_less:
  ‹(horner_sum of_bool 2 bs) 🚫 ^ length bs›
  by (fact horner_sum_bound)

end

lemma nat_horner_sum [simp]:
  ‹nat (horner_sum of_bool 2 bs) = horner_sum of_bool 2 bs›
  by (induction bs) (auto simp add: nat_add_distrib horner_sum_nonnegative)

context discrete_linordered_semidom
begin

lemma horner_sum_less_eq_iff_lexordp_eq:
  ‹horner_sum of_bool 2 bs ≤ horner_sum of_bool 2 cs ⟷ lexordp_eq (rev bs) (rev cs)›
  if ‹length bs = length cs›
proof -
  have ‹horner_sum of_bool 2 (rev bs) ≤ horner_sum of_bool 2 (rev cs) ⟷ lexordp_eq bs cs›
    if ‹length bs = length cs› for bs cs
  using that proof (induction bs cs rule: list_induct2)
    case Nil
    then show ?case
      by simp
  next
    case (Cons b bs c cs)
    with horner_sum_nonnegative [of ‹rev bs›] horner_sum_nonnegative [of ‹rev cs›]
      horner_sum_bound [of ‹rev bs›] horner_sum_bound [of ‹rev cs›]
    show ?case
      by (auto simp add: horner_sum_append not_le Cons intro: add_strict_increasing2 add_increasing)
  qed
  from that this [of ‹rev bs› ‹rev cs›] show ?thesis
    by simp
qed

lemma horner_sum_less_iff_lexordp:
  ‹horner_sum of_bool 2 bs 🚫 of_bool 2 cs ⟷ ord_class.lexordp (rev bs) (rev cs)›
  if ‹length bs = length cs›
proof -
  have ‹horner_sum of_bool 2 (rev bs) 🚫 of_bool 2 (rev cs) ⟷ ord_class.lexordp bs cs›
    if ‹length bs = length cs› for bs cs
  using that proof (induction bs cs rule: list_induct2)
    case Nil
    then show ?case
      by simp
  next
    case (Cons b bs c cs)
    with horner_sum_nonnegative [of ‹rev bs›] horner_sum_nonnegative [of ‹rev cs›]
      horner_sum_bound [of ‹rev bs›] horner_sum_bound [of ‹rev cs›]
    show ?case
      by (auto simp add: horner_sum_append not_less Cons intro: add_strict_increasing2 add_increasing)
  qed
  from that this [of ‹rev bs› ‹rev cs›] show ?thesis
    by simp
qed

end


subsection ‹Further facts about 🍋‹List.n_lists›\›

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 (∪ys∈set (List.n_lists n xs). (λy. y # ys) ` set xs)
      = (∑ys∈set (List.n_lists n xs). card ((λy. y # ys) ` set xs))"
      by (rule card_UN_disjoint) auto
    moreover have "∧ys. card ((λy. y # ys) ` set xs) = card (set xs)"
      by (rule card_image) (simp add: inj_on_def)
    ultimately show ?case by auto
  qed
  also have "… = 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 ‹Tools setup›

lemmas sum_code = sum.set_conv_list

lemma sum_set_upto_conv_sum_list_int:
  "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:
  "sum f (set [m..
  by (simp add: interv_sum_list_conv_sum_set_nat)


subsection ‹List product›

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 ‹Some syntactic sugar:›

open_bundle prod_list_syntax
begin

syntax (ASCII)
  "_prod_list" :: "pttrn => 'a list => 'b => 'b"    (‹(‹indent=3 notation=‹binder PROD›\›PROD _🚫. _)› [0, 51, 10] 10)
syntax
  "_prod_list" :: "pttrn => 'a list => 'b => 'b"    (‹(‹indent=3 notation=‹binder ∏›\›\∏_←_. _)› [0, 51, 10] 10)
syntax_consts
  "_prod_list" ⇌ prod_list
translations 🍋 ‹Beware of argument permutation!›
  "∏x←xs. b" ⇌ "CONST prod_list (CONST map (λx. b) xs)"

end

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 ⟷ (0 :: 'a :: {semiring_no_zero_divisors, semiring_1}) ∈ set xs"
  by (induction xs) simp_all

lemma prod_list_nonneg: "(∧ x. (x :: 'a :: ordered_semiring_1) ∈ set xs ==> x ≥ 0) ==> prod_list xs ≥ 0"
  by (induct xs) auto

lemma prod_list_replicate[simp]: "prod_list (replicate n a) = a ^ n"
  by (induct n) auto

lemma prod_list_power: 
  fixes xs :: "'a :: comm_monoid_mult list"
  shows "prod_list xs ^ n = (∏x←xs. x ^ n)"
  by (induct xs, auto simp: power_mult_distrib)

lemma prod_list_dvd: 
  assumes "(x :: 'a :: comm_monoid_mult) ∈ set xs"
  shows "x dvd prod_list xs"
  by (metis assms dvd_mult dvd_triv_left in_set_conv_decomp prod_list.Cons prod_list.append)

end