theory about sum and product on function bodies

--- /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

--- a/src/HOL/Library/Library.thy	Sat Sep 06 20:12:34 2014 +0200
+++ b/src/HOL/Library/Library.thy	Sat Sep 06 20:12:36 2014 +0200
@@ -27,6 +27,7 @@
   Function_Growth
   Fundamental_Theorem_Algebra
   Fun_Lexorder
+  Groups_Big_Fun
   Indicator_Function
   Infinite_Set
   Inner_Product