[isabelledev] Product on lists
Florian Haftmann
florian.haftmann at informatik.tumuenchen.de
Sun Jun 29 18:08:12 CEST 2014
The attached patches provide product on lists, where as much of the
development is shared with sums on lists (similar to products and sums
on sets).
I am reluctant to push such a massive extension so short before a
release without having it exposed to critical eyes. For my part I do
not see much pressure to pursue it by now.
Florian

PGP available:
http://home.informatik.tumuenchen.de/haftmann/pgp/florian_haftmann_at_informatik_tu_muenchen_de
 next part 
# HG changeset patch
# Parent d8a8a6b10f617f99e8041a2e2377a4f11015f1ac
separated listsum material
diff r d8a8a6b10f61 src/HOL/Groups_List.thy
 /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Groups_List.thy Sat Jun 28 21:30:01 2014 +0200
@@ 0,0 +1,212 @@
+
+(* Author: Tobias Nipkow, TU Muenchen *)
+
+header {* Summation over lists *}
+
+theory Groups_List
+imports List
+begin
+
+definition (in monoid_add) listsum :: "'a list \<Rightarrow> 'a" where
+"listsum xs = foldr plus xs 0"
+
+subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
+
+lemma (in monoid_add) listsum_simps [simp]:
+ "listsum [] = 0"
+ "listsum (x # xs) = x + listsum xs"
+ by (simp_all add: listsum_def)
+
+lemma (in monoid_add) listsum_append [simp]:
+ "listsum (xs @ ys) = listsum xs + listsum ys"
+ by (induct xs) (simp_all add: add.assoc)
+
+lemma (in comm_monoid_add) listsum_rev [simp]:
+ "listsum (rev xs) = listsum xs"
+ by (simp add: listsum_def foldr_fold fold_rev fun_eq_iff add_ac)
+
+lemma (in monoid_add) fold_plus_listsum_rev:
+ "fold plus xs = plus (listsum (rev xs))"
+proof
+ fix x
+ have "fold plus xs x = fold plus xs (x + 0)" by simp
+ also have "\<dots> = fold plus (x # xs) 0" by simp
+ also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_conv_fold)
+ also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
+ also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
+ finally show "fold plus xs x = listsum (rev xs) + x" by simp
+qed
+
+text{* Some syntactic sugar for summing a function over a list: *}
+
+syntax
+ "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3SUM _<_. _)" [0, 51, 10] 10)
+syntax (xsymbols)
+ "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
+syntax (HTML output)
+ "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
+
+translations  {* Beware of argument permutation! *}
+ "SUM x<xs. b" == "CONST listsum (CONST map (%x. b) xs)"
+ "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
+
+lemma (in comm_monoid_add) listsum_map_remove1:
+ "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
+ by (induct xs) (auto simp add: ac_simps)
+
+lemma (in monoid_add) size_list_conv_listsum:
+ "size_list f xs = listsum (map f xs) + size xs"
+ by (induct xs) auto
+
+lemma (in monoid_add) length_concat:
+ "length (concat xss) = listsum (map length xss)"
+ by (induct xss) simp_all
+
+lemma (in monoid_add) length_product_lists:
+ "length (product_lists xss) = foldr op * (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) listsum_map_filter:
+ assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
+ shows "listsum (map f (filter P xs)) = listsum (map f xs)"
+ using assms by (induct xs) auto
+
+lemma (in comm_monoid_add) distinct_listsum_conv_Setsum:
+ "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
+ by (induct xs) simp_all
+
+lemma listsum_eq_0_nat_iff_nat [simp]:
+ "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
+ by (induct ns) simp_all
+
+lemma member_le_listsum_nat:
+ "(n :: nat) \<in> set ns \<Longrightarrow> n \<le> listsum ns"
+ by (induct ns) auto
+
+lemma elem_le_listsum_nat:
+ "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
+ by (rule member_le_listsum_nat) simp
+
+lemma listsum_update_nat:
+ "k < size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n  ns ! k"
+apply(induct ns arbitrary:k)
+ apply (auto split:nat.split)
+apply(drule elem_le_listsum_nat)
+apply arith
+done
+
+lemma (in monoid_add) listsum_triv:
+ "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
+ by (induct xs) (simp_all add: distrib_right)
+
+lemma (in monoid_add) listsum_0 [simp]:
+ "(\<Sum>x\<leftarrow>xs. 0) = 0"
+ by (induct xs) (simp_all add: distrib_right)
+
+text{* For nonAbelian groups @{text xs} needs to be reversed on one side: *}
+lemma (in ab_group_add) uminus_listsum_map:
+ " listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
+ by (induct xs) simp_all
+
+lemma (in comm_monoid_add) listsum_addf:
+ "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
+ by (induct xs) (simp_all add: algebra_simps)
+
+lemma (in ab_group_add) listsum_subtractf:
+ "(\<Sum>x\<leftarrow>xs. f x  g x) = listsum (map f xs)  listsum (map g xs)"
+ by (induct xs) (simp_all add: algebra_simps)
+
+lemma (in semiring_0) listsum_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) listsum_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) listsum_abs:
+ "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
+ by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
+
+lemma listsum_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 (in monoid_add) listsum_distinct_conv_setsum_set:
+ "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
+ by (induct xs) simp_all
+
+lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
+ "listsum (map f [m..<n]) = setsum f (set [m..<n])"
+ by (simp add: listsum_distinct_conv_setsum_set)
+
+lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
+ "listsum (map f [k..l]) = setsum f (set [k..l])"
+ by (simp add: listsum_distinct_conv_setsum_set)
+
+text {* General equivalence between @{const listsum} and @{const setsum} *}
+lemma (in monoid_add) listsum_setsum_nth:
+ "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
+ using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
+
+
+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 listsum_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 {* Tools setup *}
+
+lemma setsum_set_upto_conv_listsum_int [code_unfold]:
+ "setsum f (set [i..j::int]) = listsum (map f [i..j])"
+ by (simp add: interv_listsum_conv_setsum_set_int)
+
+lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
+ "setsum f (set [m..<n]) = listsum (map f [m..<n])"
+ by (simp add: interv_listsum_conv_setsum_set_nat)
+
+lemma setsum_code [code]:
+ "setsum f (set xs) = listsum (map f (remdups xs))"
+ by (simp add: listsum_distinct_conv_setsum_set)
+
+context
+begin
+
+interpretation lifting_syntax .
+
+lemma listsum_transfer[transfer_rule]:
+ assumes [transfer_rule]: "A 0 0"
+ assumes [transfer_rule]: "(A ===> A ===> A) op + op +"
+ shows "(list_all2 A ===> A) listsum listsum"
+ unfolding listsum_def[abs_def]
+ by transfer_prover
+
+end
+
+end
\ No newline at end of file
diff r d8a8a6b10f61 src/HOL/List.thy
 a/src/HOL/List.thy Sat Jun 28 21:10:59 2014 +0200
+++ b/src/HOL/List.thy Sat Jun 28 21:30:01 2014 +0200
@@ 110,9 +110,6 @@
"concat [] = []" 
"concat (x # xs) = x @ concat xs"
definition (in monoid_add) listsum :: "'a list \<Rightarrow> 'a" where
"listsum xs = foldr plus xs 0"

primrec drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
drop_Nil: "drop n [] = []" 
drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs  Suc m \<Rightarrow> drop m xs)"
@@ 318,8 +315,7 @@
@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp}\\
@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate_def eval_nat_numeral)}\\
@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
@{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
@{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)}
+@{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}
\end{tabular}}
\caption{Characteristic examples}
\label{fig:Characteristic}
@@ 3458,149 +3454,6 @@
auto simp add: injD[OF assms])
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}

lemma (in monoid_add) listsum_simps [simp]:
 "listsum [] = 0"
 "listsum (x # xs) = x + listsum xs"
 by (simp_all add: listsum_def)

lemma (in monoid_add) listsum_append [simp]:
 "listsum (xs @ ys) = listsum xs + listsum ys"
 by (induct xs) (simp_all add: add.assoc)

lemma (in comm_monoid_add) listsum_rev [simp]:
 "listsum (rev xs) = listsum xs"
 by (simp add: listsum_def foldr_fold fold_rev fun_eq_iff add_ac)

lemma (in monoid_add) fold_plus_listsum_rev:
 "fold plus xs = plus (listsum (rev xs))"
proof
 fix x
 have "fold plus xs x = fold plus xs (x + 0)" by simp
 also have "\<dots> = fold plus (x # xs) 0" by simp
 also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_conv_fold)
 also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
 also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
 finally show "fold plus xs x = listsum (rev xs) + x" by simp
qed

text{* Some syntactic sugar for summing a function over a list: *}

syntax
 "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3SUM _<_. _)" [0, 51, 10] 10)
syntax (xsymbols)
 "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
syntax (HTML output)
 "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)

translations  {* Beware of argument permutation! *}
 "SUM x<xs. b" == "CONST listsum (CONST map (%x. b) xs)"
 "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"

lemma (in comm_monoid_add) listsum_map_remove1:
 "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
 by (induct xs) (auto simp add: ac_simps)

lemma (in monoid_add) size_list_conv_listsum:
 "size_list f xs = listsum (map f xs) + size xs"
 by (induct xs) auto

lemma (in monoid_add) length_concat:
 "length (concat xss) = listsum (map length xss)"
 by (induct xss) simp_all

lemma (in monoid_add) length_product_lists:
 "length (product_lists xss) = foldr op * (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) listsum_map_filter:
 assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
 shows "listsum (map f (filter P xs)) = listsum (map f xs)"
 using assms by (induct xs) auto

lemma (in comm_monoid_add) distinct_listsum_conv_Setsum:
 "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
 by (induct xs) simp_all

lemma listsum_eq_0_nat_iff_nat [simp]:
 "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
 by (induct ns) simp_all

lemma member_le_listsum_nat:
 "(n :: nat) \<in> set ns \<Longrightarrow> n \<le> listsum ns"
 by (induct ns) auto

lemma elem_le_listsum_nat:
 "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
 by (rule member_le_listsum_nat) simp

lemma listsum_update_nat:
 "k < size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n  ns ! k"
apply(induct ns arbitrary:k)
 apply (auto split:nat.split)
apply(drule elem_le_listsum_nat)
apply arith
done

lemma (in monoid_add) listsum_triv:
 "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
 by (induct xs) (simp_all add: distrib_right)

lemma (in monoid_add) listsum_0 [simp]:
 "(\<Sum>x\<leftarrow>xs. 0) = 0"
 by (induct xs) (simp_all add: distrib_right)

text{* For nonAbelian groups @{text xs} needs to be reversed on one side: *}
lemma (in ab_group_add) uminus_listsum_map:
 " listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
 by (induct xs) simp_all

lemma (in comm_monoid_add) listsum_addf:
 "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
 by (induct xs) (simp_all add: algebra_simps)

lemma (in ab_group_add) listsum_subtractf:
 "(\<Sum>x\<leftarrow>xs. f x  g x) = listsum (map f xs)  listsum (map g xs)"
 by (induct xs) (simp_all add: algebra_simps)

lemma (in semiring_0) listsum_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) listsum_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) listsum_abs:
 "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
 by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])

lemma listsum_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 (in monoid_add) listsum_distinct_conv_setsum_set:
 "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
 by (induct xs) simp_all

lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
 "listsum (map f [m..<n]) = setsum f (set [m..<n])"
 by (simp add: listsum_distinct_conv_setsum_set)

lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
 "listsum (map f [k..l]) = setsum f (set [k..l])"
 by (simp add: listsum_distinct_conv_setsum_set)

text {* General equivalence between @{const listsum} and @{const setsum} *}
lemma (in monoid_add) listsum_setsum_nth:
 "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
 using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)


subsubsection {* @{const insert} *}
lemma in_set_insert [simp]:
@@ 4282,9 +4135,6 @@
lemma n_lists_Nil [simp]: "List.n_lists n [] = (if n = 0 then [[]] else [])"
by (induct n) simp_all
lemma length_n_lists: "length (List.n_lists n xs) = length xs ^ n"
 by (induct n) (auto simp add: length_concat o_def listsum_triv)

lemma length_n_lists_elem: "ys \<in> set (List.n_lists n xs) \<Longrightarrow> length ys = n"
by (induct n arbitrary: ys) auto
@@ 4303,28 +4153,6 @@
qed
qed
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

subsubsection {* @{const splice} *}
@@ 6232,10 +6060,6 @@
"(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
by auto
lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
 "setsum f (set [m..<n]) = listsum (map f [m..<n])"
 by (simp add: interv_listsum_conv_setsum_set_nat)

text{* Bounded @{text LEAST} operator: *}
definition "Bleast S P = (LEAST x. x \<in> S \<and> P x)"
@@ 6283,10 +6107,6 @@
lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]
lemma setsum_set_upto_conv_listsum_int [code_unfold]:
 "setsum f (set [i..j::int]) = listsum (map f [i..j])"
 by (simp add: interv_listsum_conv_setsum_set_int)

subsubsection {* Optimizing by rewriting *}
@@ 6597,10 +6417,6 @@
"Pow (set (x # xs)) = (let A = Pow (set xs) in A \<union> insert x ` A)"
by (simp_all add: Pow_insert Let_def)
lemma setsum_code [code]:
 "setsum f (set xs) = listsum (map f (remdups xs))"
by (simp add: listsum_distinct_conv_setsum_set)

definition map_project :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a set \<Rightarrow> 'b set" where
"map_project f A = {b. \<exists> a \<in> A. f a = Some b}"
@@ 6851,13 +6667,6 @@
apply (erule_tac xs=x in list_all2_induct, simp, simp add: rel_fun_def)
done
lemma listsum_transfer[transfer_rule]:
 assumes [transfer_rule]: "A 0 0"
 assumes [transfer_rule]: "(A ===> A ===> A) op + op +"
 shows "(list_all2 A ===> A) listsum listsum"
 unfolding listsum_def[abs_def]
 by transfer_prover

end
end
diff r d8a8a6b10f61 src/HOL/Random.thy
 a/src/HOL/Random.thy Sat Jun 28 21:10:59 2014 +0200
+++ b/src/HOL/Random.thy Sat Jun 28 21:30:01 2014 +0200
@@ 4,7 +4,7 @@
header {* A HOL random engine *}
theory Random
imports List
+imports List Groups_List
begin
notation fcomp (infixl "\<circ>>" 60)
 next part 
# HG changeset patch
# Parent 8f486cccad0396b6d1d3f78938e52b4bc11541bf
abstract product over lists
diff r 8f486cccad03 src/HOL/Groups_List.thy
 a/src/HOL/Groups_List.thy Sun Jun 29 17:34:37 2014 +0200
+++ b/src/HOL/Groups_List.thy Sun Jun 29 17:53:40 2014 +0200
@@ 1,43 +1,108 @@
(* Author: Tobias Nipkow, TU Muenchen *)
header {* Summation over lists *}
+header {* Sum and product over lists *}
theory Groups_List
imports List
begin
definition (in monoid_add) listsum :: "'a list \<Rightarrow> 'a" where
"listsum xs = foldr plus xs 0"
+no_notation times (infixl "*" 70)
+no_notation Groups.one ("1")
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
+locale monoid_list = monoid
+begin
lemma (in monoid_add) listsum_simps [simp]:
 "listsum [] = 0"
 "listsum (x # xs) = x + listsum xs"
 by (simp_all add: listsum_def)
+definition F :: "'a list \<Rightarrow> 'a"
+where
+ eq_fold [code]: "F xs = foldr f xs 1"
lemma (in monoid_add) listsum_append [simp]:
 "listsum (xs @ ys) = listsum xs + listsum ys"
 by (induct xs) (simp_all add: add.assoc)
+lemma Nil [simp]:
+ "F [] = 1"
+ by (simp add: eq_fold)
lemma (in comm_monoid_add) listsum_rev [simp]:
 "listsum (rev xs) = listsum xs"
 by (simp add: listsum_def foldr_fold fold_rev fun_eq_iff add_ac)
+lemma Cons [simp]:
+ "F (x # xs) = x * F xs"
+ by (simp add: eq_fold)
lemma (in monoid_add) fold_plus_listsum_rev:
 "fold plus xs = plus (listsum (rev xs))"
proof
 fix x
 have "fold plus xs x = fold plus xs (x + 0)" by simp
 also have "\<dots> = fold plus (x # xs) 0" by simp
 also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_conv_fold)
 also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
 also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
 finally show "fold plus xs x = listsum (rev xs) + x" by simp
+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_fold 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
+
+notation times (infixl "*" 70)
+notation Groups.one ("1")
+
+
+subsection {* List summation *}
+
+context monoid_add
+begin
+
+definition listsum :: "'a list \<Rightarrow> 'a"
+where
+ "listsum = monoid_list.F plus 0"
+
+sublocale listsum!: monoid_list plus 0
+where
+ "monoid_list.F plus 0 = listsum"
+proof 
+ show "monoid_list plus 0" ..
+ then interpret listsum!: monoid_list plus 0 .
+ from listsum_def show "monoid_list.F plus 0 = listsum" by rule
qed
text{* Some syntactic sugar for summing a function over a list: *}
+end
+
+context comm_monoid_add
+begin
+
+sublocale listsum!: comm_monoid_list plus 0
+where
+ "monoid_list.F plus 0 = listsum"
+proof 
+ show "comm_monoid_list plus 0" ..
+ then interpret listsum!: comm_monoid_list plus 0 .
+ from listsum_def show "monoid_list.F plus 0 = listsum" by rule
+qed
+
+sublocale setsum!: comm_monoid_list_set plus 0
+where
+ "monoid_list.F plus 0 = listsum"
+ and "comm_monoid_set.F plus 0 = setsum"
+proof 
+ show "comm_monoid_list_set plus 0" ..
+ then interpret setsum!: comm_monoid_list_set plus 0 .
+ from listsum_def show "monoid_list.F plus 0 = listsum" by rule
+ from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
+qed
+
+end
+
+text {* Some syntactic sugar for summing a function over a list: *}
syntax
"_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3SUM _<_. _)" [0, 51, 10] 10)
@@ 50,6 +115,11 @@
"SUM x<xs. b" == "CONST listsum (CONST map (%x. b) xs)"
"\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
+text {* TODO duplicates *}
+lemmas listsum_simps = listsum.Nil listsum.Cons
+lemmas listsum_append = listsum.append
+lemmas listsum_rev = listsum.rev
+
lemma (in comm_monoid_add) listsum_map_remove1:
"x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
by (induct xs) (auto simp add: ac_simps)
@@ 183,6 +253,8 @@
subsection {* Tools setup *}
+lemmas setsum_code = setsum.set_conv_list
+
lemma setsum_set_upto_conv_listsum_int [code_unfold]:
"setsum f (set [i..j::int]) = listsum (map f [i..j])"
by (simp add: interv_listsum_conv_setsum_set_int)
@@ 191,10 +263,6 @@
"setsum f (set [m..<n]) = listsum (map f [m..<n])"
by (simp add: interv_listsum_conv_setsum_set_nat)
lemma setsum_code [code]:
 "setsum f (set xs) = listsum (map f (remdups xs))"
 by (simp add: listsum_distinct_conv_setsum_set)

context
begin
@@ 204,9 +272,68 @@
assumes [transfer_rule]: "A 0 0"
assumes [transfer_rule]: "(A ===> A ===> A) op + op +"
shows "(list_all2 A ===> A) listsum listsum"
 unfolding listsum_def[abs_def]
+ unfolding listsum.eq_fold [abs_def]
by transfer_prover
end
end
\ No newline at end of file
+
+subsection {* List product *}
+
+context monoid_mult
+begin
+
+definition listprod :: "'a list \<Rightarrow> 'a"
+where
+ "listprod = monoid_list.F times 1"
+
+sublocale listprod!: monoid_list times 1
+where
+ "monoid_list.F times 1 = listprod"
+proof 
+ show "monoid_list times 1" ..
+ then interpret listprod!: monoid_list times 1 .
+ from listprod_def show "monoid_list.F times 1 = listprod" by rule
+qed
+
+end
+
+context comm_monoid_mult
+begin
+
+sublocale listprod!: comm_monoid_list times 1
+where
+ "monoid_list.F times 1 = listprod"
+proof 
+ show "comm_monoid_list times 1" ..
+ then interpret listprod!: comm_monoid_list times 1 .
+ from listprod_def show "monoid_list.F times 1 = listprod" by rule
+qed
+
+sublocale setprod!: comm_monoid_list_set times 1
+where
+ "monoid_list.F times 1 = listprod"
+ and "comm_monoid_set.F times 1 = setprod"
+proof 
+ show "comm_monoid_list_set times 1" ..
+ then interpret setprod!: comm_monoid_list_set times 1 .
+ from listprod_def show "monoid_list.F times 1 = listprod" by rule
+ from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
+qed
+
+end
+
+text {* Some syntactic sugar: *}
+
+syntax
+ "_listprod" :: "pttrn => 'a list => 'b => 'b" ("(3PROD _<_. _)" [0, 51, 10] 10)
+syntax (xsymbols)
+ "_listprod" :: "pttrn => 'a list => 'b => 'b" ("(3\<Prod>_\<leftarrow>_. _)" [0, 51, 10] 10)
+syntax (HTML output)
+ "_listprod" :: "pttrn => 'a list => 'b => 'b" ("(3\<Prod>_\<leftarrow>_. _)" [0, 51, 10] 10)
+
+translations  {* Beware of argument permutation! *}
+ "PROD x<xs. b" == "CONST listprod (CONST map (%x. b) xs)"
+ "\<Prod>x\<leftarrow>xs. b" == "CONST listprod (CONST map (%x. b) xs)"
+
+end
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