former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
(* Author: Florian Haftmann, TU Muenchen *)
section \<open>Big sum and product over function bodies\<close>
theory Groups_Big_Fun
imports
Main
begin
subsection \<open>Abstract product\<close>
no_notation times (infixl "*" 70)
no_notation Groups.one ("1")
locale comm_monoid_fun = comm_monoid
begin
definition G :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a"
where
expand_set: "G g = comm_monoid_set.F f 1 g {a. g a \<noteq> 1}"
interpretation F: comm_monoid_set f 1
..
lemma expand_superset:
assumes "finite A" and "{a. g a \<noteq> 1} \<subseteq> A"
shows "G g = F.F g A"
apply (simp add: expand_set)
apply (rule F.same_carrierI [of A])
apply (simp_all add: assms)
done
lemma conditionalize:
assumes "finite A"
shows "F.F g A = G (\<lambda>a. if a \<in> A then g a else 1)"
using assms
apply (simp add: expand_set)
apply (rule F.same_carrierI [of A])
apply auto
done
lemma neutral [simp]:
"G (\<lambda>a. 1) = 1"
by (simp add: expand_set)
lemma update [simp]:
assumes "finite {a. g a \<noteq> 1}"
assumes "g a = 1"
shows "G (g(a := b)) = b * G g"
proof (cases "b = 1")
case True with \<open>g a = 1\<close> show ?thesis
by (simp add: expand_set) (rule F.cong, auto)
next
case False
moreover have "{a'. a' \<noteq> a \<longrightarrow> g a' \<noteq> 1} = insert a {a. g a \<noteq> 1}"
by auto
moreover from \<open>g a = 1\<close> have "a \<notin> {a. g a \<noteq> 1}"
by simp
moreover have "F.F (\<lambda>a'. if a' = a then b else g a') {a. g a \<noteq> 1} = F.F g {a. g a \<noteq> 1}"
by (rule F.cong) (auto simp add: \<open>g a = 1\<close>)
ultimately show ?thesis using \<open>finite {a. g a \<noteq> 1}\<close> by (simp add: expand_set)
qed
lemma infinite [simp]:
"\<not> finite {a. g a \<noteq> 1} \<Longrightarrow> G g = 1"
by (simp add: expand_set)
lemma cong:
assumes "\<And>a. g a = h a"
shows "G g = G h"
using assms by (simp add: expand_set)
lemma strong_cong [cong]:
assumes "\<And>a. g a = h a"
shows "G (\<lambda>a. g a) = G (\<lambda>a. h a)"
using assms by (fact cong)
lemma not_neutral_obtains_not_neutral:
assumes "G g \<noteq> 1"
obtains a where "g a \<noteq> 1"
using assms by (auto elim: F.not_neutral_contains_not_neutral simp add: expand_set)
lemma reindex_cong:
assumes "bij l"
assumes "g \<circ> l = h"
shows "G g = G h"
proof -
from assms have unfold: "h = g \<circ> l" by simp
from \<open>bij l\<close> have "inj l" by (rule bij_is_inj)
then have "inj_on l {a. h a \<noteq> 1}" by (rule subset_inj_on) simp
moreover from \<open>bij l\<close> have "{a. g a \<noteq> 1} = l ` {a. h a \<noteq> 1}"
by (auto simp add: image_Collect unfold elim: bij_pointE)
moreover have "\<And>x. x \<in> {a. h a \<noteq> 1} \<Longrightarrow> g (l x) = h x"
by (simp add: unfold)
ultimately have "F.F g {a. g a \<noteq> 1} = F.F h {a. h a \<noteq> 1}"
by (rule F.reindex_cong)
then show ?thesis by (simp add: expand_set)
qed
lemma distrib:
assumes "finite {a. g a \<noteq> 1}" and "finite {a. h a \<noteq> 1}"
shows "G (\<lambda>a. g a * h a) = G g * G h"
proof -
from assms have "finite ({a. g a \<noteq> 1} \<union> {a. h a \<noteq> 1})" by simp
moreover have "{a. g a * h a \<noteq> 1} \<subseteq> {a. g a \<noteq> 1} \<union> {a. h a \<noteq> 1}"
by auto (drule sym, simp)
ultimately show ?thesis
using assms
by (simp add: expand_superset [of "{a. g a \<noteq> 1} \<union> {a. h a \<noteq> 1}"] F.distrib)
qed
lemma commute:
assumes "finite C"
assumes subset: "{a. \<exists>b. g a b \<noteq> 1} \<times> {b. \<exists>a. g a b \<noteq> 1} \<subseteq> C" (is "?A \<times> ?B \<subseteq> C")
shows "G (\<lambda>a. G (g a)) = G (\<lambda>b. G (\<lambda>a. g a b))"
proof -
from \<open>finite C\<close> subset
have "finite ({a. \<exists>b. g a b \<noteq> 1} \<times> {b. \<exists>a. g a b \<noteq> 1})"
by (rule rev_finite_subset)
then have fins:
"finite {b. \<exists>a. g a b \<noteq> 1}" "finite {a. \<exists>b. g a b \<noteq> 1}"
by (auto simp add: finite_cartesian_product_iff)
have subsets: "\<And>a. {b. g a b \<noteq> 1} \<subseteq> {b. \<exists>a. g a b \<noteq> 1}"
"\<And>b. {a. g a b \<noteq> 1} \<subseteq> {a. \<exists>b. g a b \<noteq> 1}"
"{a. F.F (g a) {b. \<exists>a. g a b \<noteq> 1} \<noteq> 1} \<subseteq> {a. \<exists>b. g a b \<noteq> 1}"
"{a. F.F (\<lambda>aa. g aa a) {a. \<exists>b. g a b \<noteq> 1} \<noteq> 1} \<subseteq> {b. \<exists>a. g a b \<noteq> 1}"
by (auto elim: F.not_neutral_contains_not_neutral)
from F.commute have
"F.F (\<lambda>a. F.F (g a) {b. \<exists>a. g a b \<noteq> 1}) {a. \<exists>b. g a b \<noteq> 1} =
F.F (\<lambda>b. F.F (\<lambda>a. g a b) {a. \<exists>b. g a b \<noteq> 1}) {b. \<exists>a. g a b \<noteq> 1}" .
with subsets fins have "G (\<lambda>a. F.F (g a) {b. \<exists>a. g a b \<noteq> 1}) =
G (\<lambda>b. F.F (\<lambda>a. g a b) {a. \<exists>b. g a b \<noteq> 1})"
by (auto simp add: expand_superset [of "{b. \<exists>a. g a b \<noteq> 1}"]
expand_superset [of "{a. \<exists>b. g a b \<noteq> 1}"])
with subsets fins show ?thesis
by (auto simp add: expand_superset [of "{b. \<exists>a. g a b \<noteq> 1}"]
expand_superset [of "{a. \<exists>b. g a b \<noteq> 1}"])
qed
lemma cartesian_product:
assumes "finite C"
assumes subset: "{a. \<exists>b. g a b \<noteq> 1} \<times> {b. \<exists>a. g a b \<noteq> 1} \<subseteq> C" (is "?A \<times> ?B \<subseteq> C")
shows "G (\<lambda>a. G (g a)) = G (\<lambda>(a, b). g a b)"
proof -
from subset \<open>finite C\<close> have fin_prod: "finite (?A \<times> ?B)"
by (rule finite_subset)
from fin_prod have "finite ?A" and "finite ?B"
by (auto simp add: finite_cartesian_product_iff)
have *: "G (\<lambda>a. G (g a)) =
(F.F (\<lambda>a. F.F (g a) {b. \<exists>a. g a b \<noteq> 1}) {a. \<exists>b. g a b \<noteq> 1})"
apply (subst expand_superset [of "?B"])
apply (rule \<open>finite ?B\<close>)
apply auto
apply (subst expand_superset [of "?A"])
apply (rule \<open>finite ?A\<close>)
apply auto
apply (erule F.not_neutral_contains_not_neutral)
apply auto
done
have "{p. (case p of (a, b) \<Rightarrow> g a b) \<noteq> 1} \<subseteq> ?A \<times> ?B"
by auto
with subset have **: "{p. (case p of (a, b) \<Rightarrow> g a b) \<noteq> 1} \<subseteq> C"
by blast
show ?thesis
apply (simp add: *)
apply (simp add: F.cartesian_product)
apply (subst expand_superset [of C])
apply (rule \<open>finite C\<close>)
apply (simp_all add: **)
apply (rule F.same_carrierI [of C])
apply (rule \<open>finite C\<close>)
apply (simp_all add: subset)
apply auto
done
qed
lemma cartesian_product2:
assumes fin: "finite D"
assumes subset: "{(a, b). \<exists>c. g a b c \<noteq> 1} \<times> {c. \<exists>a b. g a b c \<noteq> 1} \<subseteq> D" (is "?AB \<times> ?C \<subseteq> D")
shows "G (\<lambda>(a, b). G (g a b)) = G (\<lambda>(a, b, c). g a b c)"
proof -
have bij: "bij (\<lambda>(a, b, c). ((a, b), c))"
by (auto intro!: bijI injI simp add: image_def)
have "{p. \<exists>c. g (fst p) (snd p) c \<noteq> 1} \<times> {c. \<exists>p. g (fst p) (snd p) c \<noteq> 1} \<subseteq> D"
by auto (insert subset, blast)
with fin have "G (\<lambda>p. G (g (fst p) (snd p))) = G (\<lambda>(p, c). g (fst p) (snd p) c)"
by (rule cartesian_product)
then have "G (\<lambda>(a, b). G (g a b)) = G (\<lambda>((a, b), c). g a b c)"
by (auto simp add: split_def)
also have "G (\<lambda>((a, b), c). g a b c) = G (\<lambda>(a, b, c). g a b c)"
using bij by (rule reindex_cong [of "\<lambda>(a, b, c). ((a, b), c)"]) (simp add: fun_eq_iff)
finally show ?thesis .
qed
lemma delta [simp]:
"G (\<lambda>b. if b = a then g b else 1) = g a"
proof -
have "{b. (if b = a then g b else 1) \<noteq> 1} \<subseteq> {a}" by auto
then show ?thesis by (simp add: expand_superset [of "{a}"])
qed
lemma delta' [simp]:
"G (\<lambda>b. if a = b then g b else 1) = g a"
proof -
have "(\<lambda>b. if a = b then g b else 1) = (\<lambda>b. if b = a then g b else 1)"
by (simp add: fun_eq_iff)
then have "G (\<lambda>b. if a = b then g b else 1) = G (\<lambda>b. if b = a then g b else 1)"
by (simp cong del: strong_cong)
then show ?thesis by simp
qed
end
notation times (infixl "*" 70)
notation Groups.one ("1")
subsection \<open>Concrete sum\<close>
context comm_monoid_add
begin
sublocale Sum_any: comm_monoid_fun plus 0
defines
Sum_any = Sum_any.G
rewrites
"comm_monoid_set.F plus 0 = setsum"
proof -
show "comm_monoid_fun plus 0" ..
then interpret Sum_any: comm_monoid_fun plus 0 .
from setsum_def show "comm_monoid_set.F plus 0 = setsum" by (auto intro: sym)
qed
end
syntax (ASCII)
"_Sum_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add" ("(3SUM _. _)" [0, 10] 10)
syntax
"_Sum_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add" ("(3\<Sum>_. _)" [0, 10] 10)
translations
"\<Sum>a. b" \<rightleftharpoons> "CONST Sum_any (\<lambda>a. b)"
lemma Sum_any_left_distrib:
fixes r :: "'a :: semiring_0"
assumes "finite {a. g a \<noteq> 0}"
shows "Sum_any g * r = (\<Sum>n. g n * r)"
proof -
note assms
moreover have "{a. g a * r \<noteq> 0} \<subseteq> {a. g a \<noteq> 0}" by auto
ultimately show ?thesis
by (simp add: setsum_left_distrib Sum_any.expand_superset [of "{a. g a \<noteq> 0}"])
qed
lemma Sum_any_right_distrib:
fixes r :: "'a :: semiring_0"
assumes "finite {a. g a \<noteq> 0}"
shows "r * Sum_any g = (\<Sum>n. r * g n)"
proof -
note assms
moreover have "{a. r * g a \<noteq> 0} \<subseteq> {a. g a \<noteq> 0}" by auto
ultimately show ?thesis
by (simp add: setsum_right_distrib Sum_any.expand_superset [of "{a. g a \<noteq> 0}"])
qed
lemma Sum_any_product:
fixes f g :: "'b \<Rightarrow> 'a::semiring_0"
assumes "finite {a. f a \<noteq> 0}" and "finite {b. g b \<noteq> 0}"
shows "Sum_any f * Sum_any g = (\<Sum>a. \<Sum>b. f a * g b)"
proof -
have subset_f: "{a. (\<Sum>b. f a * g b) \<noteq> 0} \<subseteq> {a. f a \<noteq> 0}"
by rule (simp, rule, auto)
moreover have subset_g: "\<And>a. {b. f a * g b \<noteq> 0} \<subseteq> {b. g b \<noteq> 0}"
by rule (simp, rule, auto)
ultimately show ?thesis using assms
by (auto simp add: Sum_any.expand_set [of f] Sum_any.expand_set [of g]
Sum_any.expand_superset [of "{a. f a \<noteq> 0}"] Sum_any.expand_superset [of "{b. g b \<noteq> 0}"]
setsum_product)
qed
lemma Sum_any_eq_zero_iff [simp]:
fixes f :: "'a \<Rightarrow> nat"
assumes "finite {a. f a \<noteq> 0}"
shows "Sum_any f = 0 \<longleftrightarrow> f = (\<lambda>_. 0)"
using assms by (simp add: Sum_any.expand_set fun_eq_iff)
subsection \<open>Concrete product\<close>
context comm_monoid_mult
begin
sublocale Prod_any: comm_monoid_fun times 1
defines
Prod_any = Prod_any.G
rewrites
"comm_monoid_set.F times 1 = setprod"
proof -
show "comm_monoid_fun times 1" ..
then interpret Prod_any: comm_monoid_fun times 1 .
from setprod_def show "comm_monoid_set.F times 1 = setprod" by (auto intro: sym)
qed
end
syntax (ASCII)
"_Prod_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult" ("(3PROD _. _)" [0, 10] 10)
syntax
"_Prod_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult" ("(3\<Prod>_. _)" [0, 10] 10)
translations
"\<Prod>a. b" == "CONST Prod_any (\<lambda>a. b)"
lemma Prod_any_zero:
fixes f :: "'b \<Rightarrow> 'a :: comm_semiring_1"
assumes "finite {a. f a \<noteq> 1}"
assumes "f a = 0"
shows "(\<Prod>a. f a) = 0"
proof -
from \<open>f a = 0\<close> have "f a \<noteq> 1" by simp
with \<open>f a = 0\<close> have "\<exists>a. f a \<noteq> 1 \<and> f a = 0" by blast
with \<open>finite {a. f a \<noteq> 1}\<close> show ?thesis
by (simp add: Prod_any.expand_set setprod_zero)
qed
lemma Prod_any_not_zero:
fixes f :: "'b \<Rightarrow> 'a :: comm_semiring_1"
assumes "finite {a. f a \<noteq> 1}"
assumes "(\<Prod>a. f a) \<noteq> 0"
shows "f a \<noteq> 0"
using assms Prod_any_zero [of f] by blast
lemma power_Sum_any:
assumes "finite {a. f a \<noteq> 0}"
shows "c ^ (\<Sum>a. f a) = (\<Prod>a. c ^ f a)"
proof -
have "{a. c ^ f a \<noteq> 1} \<subseteq> {a. f a \<noteq> 0}"
by (auto intro: ccontr)
with assms show ?thesis
by (simp add: Sum_any.expand_set Prod_any.expand_superset power_setsum)
qed
end