--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Groups_Big_Fun.thy Sat Sep 06 20:12:36 2014 +0200
@@ -0,0 +1,340 @@
+(* Author: Florian Haftmann, TU Muenchen *)
+
+header \<open>Big sum and product over function bodies\<close>
+
+theory Groups_Big_Fun
+imports
+ Main
+ "~~/src/Tools/Permanent_Interpretation"
+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 `g a = 1` 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 `g a = 1` 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: `g a = 1`)
+ ultimately show ?thesis using `finite {a. g a \<noteq> 1}` 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 `bij l` 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 `bij l` 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 `finite C` 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 `finite C` 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 `finite ?B`)
+ apply auto
+ apply (subst expand_superset [of "?A"])
+ apply (rule `finite ?A`)
+ 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 `finite C`)
+ apply (simp_all add: **)
+ apply (rule F.same_carrierI [of C])
+ apply (rule `finite C`)
+ 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, auto)
+ 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
+
+definition Sum_any :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a"
+where
+ "Sum_any = comm_monoid_fun.G plus 0"
+
+permanent_interpretation Sum_any!: comm_monoid_fun plus 0
+where
+ "comm_monoid_fun.G plus 0 = Sum_any" and
+ "comm_monoid_set.F plus 0 = setsum"
+proof -
+ show "comm_monoid_fun plus 0" ..
+ then interpret Sum_any!: comm_monoid_fun plus 0 .
+ from Sum_any_def show "comm_monoid_fun.G plus 0 = Sum_any" by rule
+ from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
+qed
+
+end
+
+syntax
+ "_Sum_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add" ("(3SUM _. _)" [0, 10] 10)
+syntax (xsymbols)
+ "_Sum_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add" ("(3\<Sum>_. _)" [0, 10] 10)
+syntax (HTML output)
+ "_Sum_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add" ("(3\<Sum>_. _)" [0, 10] 10)
+
+translations
+ "\<Sum>a. b" == "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
+
+
+subsection \<open>Concrete product\<close>
+
+context comm_monoid_mult
+begin
+
+definition Prod_any :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a"
+where
+ "Prod_any = comm_monoid_fun.G times 1"
+
+permanent_interpretation Prod_any!: comm_monoid_fun times 1
+where
+ "comm_monoid_fun.G times 1 = Prod_any" and
+ "comm_monoid_set.F times 1 = setprod"
+proof -
+ show "comm_monoid_fun times 1" ..
+ then interpret Prod_any!: comm_monoid_fun times 1 .
+ from Prod_any_def show "comm_monoid_fun.G times 1 = Prod_any" by rule
+ from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
+qed
+
+end
+
+syntax
+ "_Prod_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult" ("(3PROD _. _)" [0, 10] 10)
+syntax (xsymbols)
+ "_Prod_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult" ("(3\<Prod>_. _)" [0, 10] 10)
+syntax (HTML output)
+ "_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 `f a = 0` have "f a \<noteq> 1" by simp
+ with `f a = 0` have "\<exists>a. f a \<noteq> 1 \<and> f a = 0" by blast
+ with `finite {a. f a \<noteq> 1}` 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
+
+end