diff --git a/src/HOL/Groups_List.thy b/src/HOL/Groups_List.thy
--- a/src/HOL/Groups_List.thy
+++ b/src/HOL/Groups_List.thy
@@ -1,576 +1,582 @@
 (* 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_replicate: "sum_list (replicate n c) = of_nat n * c"
 by(induction n)(auto simp add: distrib_right)
 
 
 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>Horner sums\<close>
 
 context comm_semiring_0
 begin
 
 definition horner_sum :: \<open>('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'a\<close>
   where horner_sum_foldr: \<open>horner_sum f a xs = foldr (\<lambda>x b. f x + a * b) xs 0\<close>
 
 lemma horner_sum_simps [simp]:
   \<open>horner_sum f a [] = 0\<close>
   \<open>horner_sum f a (x # xs) = f x + a * horner_sum f a xs\<close>
   by (simp_all add: horner_sum_foldr)
 
 lemma horner_sum_eq_sum_funpow:
   \<open>horner_sum f a xs = (\<Sum>n = 0..<length xs. ((*) a ^^ n) (f (xs ! n)))\<close>
 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]:
   \<open>((B ===> A) ===> A ===> list_all2 B ===> A) horner_sum horner_sum\<close>
   if [transfer_rule]: \<open>A 0 0\<close>
     and [transfer_rule]: \<open>(A ===> A ===> A) (+) (+)\<close>
     and [transfer_rule]: \<open>(A ===> A ===> A) (*) (*)\<close>
   by (unfold horner_sum_foldr) transfer_prover
 
 end
 
 context comm_semiring_1
 begin
 
 lemma horner_sum_eq_sum:
   \<open>horner_sum f a xs = (\<Sum>n = 0..<length xs. f (xs ! n) * a ^ n)\<close>
 proof -
   have \<open>(*) a ^^ n = (*) (a ^ n)\<close> 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:
+  \<open>horner_sum f a (xs @ ys) = horner_sum f a xs + a ^ length xs * horner_sum f a ys\<close>
+  using sum.atLeastLessThan_shift_bounds [of _ 0 \<open>length xs\<close> \<open>length ys\<close>]
+    atLeastLessThan_add_Un [of 0 \<open>length xs\<close> \<open>length ys\<close>]
+  by (simp add: horner_sum_eq_sum sum_distrib_left sum.union_disjoint ac_simps nth_append power_add)
+
 end
 
 context semiring_bit_shifts
 begin
 
 lemma horner_sum_bit_eq_take_bit:
   \<open>horner_sum of_bool 2 (map (bit a) [0..<n]) = take_bit n a\<close>
 proof (induction a arbitrary: n rule: bits_induct)
   case (stable a)
   moreover have \<open>bit a = (\<lambda>_. odd a)\<close>
     using stable by (simp add: stable_imp_bit_iff_odd fun_eq_iff)
   moreover have \<open>{q. q < n} = {0..<n}\<close>
     by auto
   ultimately show ?case
     by (simp add: stable_imp_take_bit_eq horner_sum_eq_sum mask_eq_sum_exp)
 next
   case (rec a b)
   show ?case
   proof (cases n)
     case 0
     then show ?thesis
       by simp
   next
     case (Suc m)
     have \<open>map (bit (of_bool b + 2 * a)) [0..<Suc m] = b # map (bit (of_bool b + 2 * a)) [Suc 0..<Suc m]\<close>
       by (simp only: upt_conv_Cons) simp
     also have \<open>\<dots> = b # map (bit a) [0..<m]\<close>
       by (simp only: flip: map_Suc_upt) (simp add: bit_Suc rec.hyps)
     finally show ?thesis
       using Suc rec.IH [of m] by (simp add: take_bit_Suc rec.hyps, simp add: ac_simps mod_2_eq_odd)
   qed
 qed
 
 end
 
 context unique_euclidean_semiring_with_bit_shifts
 begin
 
-lemma bit_horner_sum_bit_iff:
+lemma bit_horner_sum_bit_iff [bit_simps]:
   \<open>bit (horner_sum of_bool 2 bs) n \<longleftrightarrow> n < length bs \<and> bs ! n\<close>
 proof (induction bs arbitrary: n)
   case Nil
   then show ?case
     by simp
 next
   case (Cons b bs)
   show ?case
   proof (cases n)
     case 0
     then show ?thesis
       by simp
   next
     case (Suc m)
     with bit_rec [of _ n] Cons.prems Cons.IH [of m]
     show ?thesis by simp
   qed
 qed
 
 lemma take_bit_horner_sum_bit_eq:
   \<open>take_bit n (horner_sum of_bool 2 bs) = horner_sum of_bool 2 (take n bs)\<close>
   by (auto simp add: bit_eq_iff bit_take_bit_iff bit_horner_sum_bit_iff)
 
 end
 
 lemma horner_sum_of_bool_2_less:
   \<open>(horner_sum of_bool 2 bs :: int) < 2 ^ length bs\<close>
 proof -
   have \<open>(\<Sum>n = 0..<length bs. of_bool (bs ! n) * (2::int) ^ n) \<le> (\<Sum>n = 0..<length bs. 2 ^ n)\<close>
     by (rule sum_mono) simp
   also have \<open>\<dots> = 2 ^ length bs - 1\<close>
     by (induction bs) simp_all
   finally show ?thesis
     by (simp add: horner_sum_eq_sum)
 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