Theory HOL.Groups_List
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..<length xs}"
proof -
have "xs = map (λi. xs ! i) [0..<length xs]"
by (simp add: map_nth)
also have "list.F … = set.F (λi. xs ! i) {0..<length xs}"
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..<b} = (let d = b - 1 in if d < b
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 < c
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<..<b} = (let c = a + 1; d = b - 1 in if a < c ∧ d < b
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
"∑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..<n] = ∑ {m..<n}"
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..<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 ‹General equivalence between \<^const>‹sum_list› and \<^const>‹sum››
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..<n}›.›
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..<length ns}. f i) ≤ (∑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..<length (ns @ [n])}
= sum f {0..<length ns} + f (length ns)"
by simp
also have "sum f {0..<length ns} ≤ 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..<length (ns @ [n])} ≤ sum_list (map f ns) + f n"
using mono add_mono by blast
thus ?case by simp
qed
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..<length 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..<length 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 < 2 ^ 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 < a›
using Cons * by simp
moreover have ‹1 ≤ a›
using ‹0 < a› by (simp add: less_eq_iff_succ_less)
ultimately have ‹0 + 1 < a + a›
by (rule add_less_le_mono)
then have ‹1 < a * 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) < 2 ^ 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 < horner_sum 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) < horner_sum 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 \<^const>‹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..<n]) = sum_list (map f [m..<n])"
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
"∏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