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Datei: Weak_Morphisms.thy Sprache: Isabelle

Original von: Isabelle^©

(* 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>

locale comm_monoid_fun = comm_monoid
begin

definition G :: "('b \ 'a) \ 'a"
where
  expand_set: "G g = comm_monoid_set.F f \<^bold>1 g {a. g a \ \<^bold>1}"

interpretation F: comm_monoid_set f "\<^bold>1"
  ..

lemma expand_superset:
  assumes "finite A" and "{a. g a \ \<^bold>1} \ 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 (\a. if a \ A then g a else \<^bold>1)"
  using assms
  apply (simp add: expand_set)
  apply (rule F.same_carrierI [of A])
  apply auto
  done

lemma neutral [simp]:
  "G (\a. \<^bold>1) = \<^bold>1"
  by (simp add: expand_set)

lemma update [simp]:
  assumes "finite {a. g a \ \<^bold>1}"
  assumes "g a = \<^bold>1"
  shows "G (g(a := b)) = b \<^bold>* G g"
proof (cases "b = \<^bold>1")
  case True with \<open>g a = \<^bold>1\<close> show ?thesis
    by (simp add: expand_set) (rule F.cong, auto)
next
  case False
  moreover have "{a'. a' \ a \ g a' \ \<^bold>1} = insert a {a. g a \ \<^bold>1}"
    by auto
  moreover from \<open>g a = \<^bold>1\<close> have "a \<notin> {a. g a \<noteq> \<^bold>1}"
    by simp
  moreover have "F.F (\a'. if a' = a then b else g a') {a. g a \ \<^bold>1} = F.F g {a. g a \ \<^bold>1}"
    by (rule F.cong) (auto simp add: \<open>g a = \<^bold>1\<close>)
  ultimately show ?thesis using \<open>finite {a. g a \<noteq> \<^bold>1}\<close> by (simp add: expand_set)
qed

lemma infinite [simp]:
  "\ finite {a. g a \ \<^bold>1} \ G g = \<^bold>1"
  by (simp add: expand_set)

lemma cong [cong]:
  assumes "\a. g a = h a"
  shows "G g = G h"
  using assms by (simp add: expand_set)

lemma not_neutral_obtains_not_neutral:
  assumes "G g \ \<^bold>1"
  obtains a where "g a \ \<^bold>1"
  using assms by (auto elim: F.not_neutral_contains_not_neutral simp add: expand_set)

lemma reindex_cong:
  assumes "bij l"
  assumes "g \ l = h"
  shows "G g = G h"
proof -
  from assms have unfold: "h = g \ l" by simp
  from \<open>bij l\<close> have "inj l" by (rule bij_is_inj)
  then have "inj_on l {a. h a \ \<^bold>1}" by (rule subset_inj_on) simp
  moreover from \<open>bij l\<close> have "{a. g a \<noteq> \<^bold>1} = l ` {a. h a \<noteq> \<^bold>1}"
    by (auto simp add: image_Collect unfold elim: bij_pointE)
  moreover have "\x. x \ {a. h a \ \<^bold>1} \ g (l x) = h x"
    by (simp add: unfold)
  ultimately have "F.F g {a. g a \ \<^bold>1} = F.F h {a. h a \ \<^bold>1}"
    by (rule F.reindex_cong)
  then show ?thesis by (simp add: expand_set)
qed

lemma distrib:
  assumes "finite {a. g a \ \<^bold>1}" and "finite {a. h a \ \<^bold>1}"
  shows "G (\a. g a \<^bold>* h a) = G g \<^bold>* G h"
proof -
  from assms have "finite ({a. g a \ \<^bold>1} \ {a. h a \ \<^bold>1})" by simp
  moreover have "{a. g a \<^bold>* h a \ \<^bold>1} \ {a. g a \ \<^bold>1} \ {a. h a \ \<^bold>1}"
    by auto (drule sym, simp)
  ultimately show ?thesis
    using assms
    by (simp add: expand_superset [of "{a. g a \ \<^bold>1} \ {a. h a \ \<^bold>1}"] F.distrib)
qed

lemma swap:
  assumes "finite C"
  assumes subset: "{a. \b. g a b \ \<^bold>1} \ {b. \a. g a b \ \<^bold>1} \ C" (is "?A \ ?B \ C")
  shows "G (\a. G (g a)) = G (\b. G (\a. g a b))"
proof -
  from \<open>finite C\<close> subset
    have "finite ({a. \b. g a b \ \<^bold>1} \ {b. \a. g a b \ \<^bold>1})"
    by (rule rev_finite_subset)
  then have fins:
    "finite {b. \a. g a b \ \<^bold>1}" "finite {a. \b. g a b \ \<^bold>1}"
    by (auto simp add: finite_cartesian_product_iff)
  have subsets: "\a. {b. g a b \ \<^bold>1} \ {b. \a. g a b \ \<^bold>1}"
    "\b. {a. g a b \ \<^bold>1} \ {a. \b. g a b \ \<^bold>1}"
    "{a. F.F (g a) {b. \a. g a b \ \<^bold>1} \ \<^bold>1} \ {a. \b. g a b \ \<^bold>1}"
    "{a. F.F (\aa. g aa a) {a. \b. g a b \ \<^bold>1} \ \<^bold>1} \ {b. \a. g a b \ \<^bold>1}"
    by (auto elim: F.not_neutral_contains_not_neutral)
  from F.swap have
    "F.F (\a. F.F (g a) {b. \a. g a b \ \<^bold>1}) {a. \b. g a b \ \<^bold>1} =
      F.F (\<lambda>b. F.F (\<lambda>a. g a b) {a. \<exists>b. g a b \<noteq> \<^bold>1}) {b. \<exists>a. g a b \<noteq> \<^bold>1}" .
  with subsets fins have "G (\a. F.F (g a) {b. \a. g a b \ \<^bold>1}) =
    G (\<lambda>b. F.F (\<lambda>a. g a b) {a. \<exists>b. g a b \<noteq> \<^bold>1})"
    by (auto simp add: expand_superset [of "{b. \a. g a b \ \<^bold>1}"]
      expand_superset [of "{a. \b. g a b \ \<^bold>1}"])
  with subsets fins show ?thesis
    by (auto simp add: expand_superset [of "{b. \a. g a b \ \<^bold>1}"]
      expand_superset [of "{a. \b. g a b \ \<^bold>1}"])
qed

lemma cartesian_product:
  assumes "finite C"
  assumes subset: "{a. \b. g a b \ \<^bold>1} \ {b. \a. g a b \ \<^bold>1} \ C" (is "?A \ ?B \ C")
  shows "G (\a. G (g a)) = G (\(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 (\a. G (g a)) =
    (F.F (\<lambda>a. F.F (g a) {b. \<exists>a. g a b \<noteq> \<^bold>1}) {a. \<exists>b. g a b \<noteq> \<^bold>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) \ g a b) \ \<^bold>1} \ ?A \ ?B"
    by auto
  with subset have **: "{p. (case p of (a, b) \ g a b) \ \<^bold>1} \ 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). \c. g a b c \ \<^bold>1} \ {c. \a b. g a b c \ \<^bold>1} \ D" (is "?AB \ ?C \ D")
  shows "G (\(a, b). G (g a b)) = G (\(a, b, c). g a b c)"
proof -
  have bij: "bij (\(a, b, c). ((a, b), c))"
    by (auto intro!: bijI injI simp add: image_def)
  have "{p. \c. g (fst p) (snd p) c \ \<^bold>1} \ {c. \p. g (fst p) (snd p) c \ \<^bold>1} \ D"
    by auto (insert subset, blast)
  with fin have "G (\p. G (g (fst p) (snd p))) = G (\(p, c). g (fst p) (snd p) c)"
    by (rule cartesian_product)
  then have "G (\(a, b). G (g a b)) = G (\((a, b), c). g a b c)"
    by (auto simp add: split_def)
  also have "G (\((a, b), c). g a b c) = G (\(a, b, c). g a b c)"
    using bij by (rule reindex_cong [of "\(a, b, c). ((a, b), c)"]) (simp add: fun_eq_iff)
  finally show ?thesis .
qed

lemma delta [simp]:
  "G (\b. if b = a then g b else \<^bold>1) = g a"
proof -
  have "{b. (if b = a then g b else \<^bold>1) \ \<^bold>1} \ {a}" by auto
  then show ?thesis by (simp add: expand_superset [of "{a}"])
qed

lemma delta' [simp]:
  "G (\b. if a = b then g b else \<^bold>1) = g a"
proof -
  have "(\b. if a = b then g b else \<^bold>1) = (\b. if b = a then g b else \<^bold>1)"
    by (simp add: fun_eq_iff)
  then have "G (\b. if a = b then g b else \<^bold>1) = G (\b. if b = a then g b else \<^bold>1)"
    by (simp cong del: cong)
  then show ?thesis by simp
qed

end

subsection \<open>Concrete sum\<close>

context comm_monoid_add
begin

sublocale Sum_any: comm_monoid_fun plus 0
  rewrites "comm_monoid_set.F plus 0 = sum"
  defines Sum_any = Sum_any.G
proof -
  show "comm_monoid_fun plus 0" ..
  then interpret Sum_any: comm_monoid_fun plus 0 .
  from sum_def show "comm_monoid_set.F plus 0 = sum" by (auto intro: sym)
qed

end

syntax (ASCII)
  "_Sum_any" :: "pttrn \ 'a \ 'a::comm_monoid_add" ("(3SUM _. _)" [0, 10] 10)
syntax
  "_Sum_any" :: "pttrn \ 'a \ 'a::comm_monoid_add" ("(3\_. _)" [0, 10] 10)
translations
  "\a. b" \ "CONST Sum_any (\a. b)"

lemma Sum_any_left_distrib:
  fixes r :: "'a :: semiring_0"
  assumes "finite {a. g a \ 0}"
  shows "Sum_any g * r = (\n. g n * r)"
proof -
  note assms
  moreover have "{a. g a * r \ 0} \ {a. g a \ 0}" by auto
  ultimately show ?thesis
    by (simp add: sum_distrib_right Sum_any.expand_superset [of "{a. g a \ 0}"])
qed

lemma Sum_any_right_distrib:
  fixes r :: "'a :: semiring_0"
  assumes "finite {a. g a \ 0}"
  shows "r * Sum_any g = (\n. r * g n)"
proof -
  note assms
  moreover have "{a. r * g a \ 0} \ {a. g a \ 0}" by auto
  ultimately show ?thesis
    by (simp add: sum_distrib_left Sum_any.expand_superset [of "{a. g a \ 0}"])
qed

lemma Sum_any_product:
  fixes f g :: "'b \ 'a::semiring_0"
  assumes "finite {a. f a \ 0}" and "finite {b. g b \ 0}"
  shows "Sum_any f * Sum_any g = (\a. \b. f a * g b)"
proof -
  have subset_f: "{a. (\b. f a * g b) \ 0} \ {a. f a \ 0}"
    by rule (simp, rule, auto)
  moreover have subset_g: "\a. {b. f a * g b \ 0} \ {b. g b \ 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 \ 0}"] Sum_any.expand_superset [of "{b. g b \ 0}"]
      sum_product)
qed

lemma Sum_any_eq_zero_iff [simp]:
  fixes f :: "'a \ nat"
  assumes "finite {a. f a \ 0}"
  shows "Sum_any f = 0 \ f = (\_. 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
  rewrites "comm_monoid_set.F times 1 = prod"
  defines Prod_any = Prod_any.G
proof -
  show "comm_monoid_fun times 1" ..
  then interpret Prod_any: comm_monoid_fun times 1 .
  from prod_def show "comm_monoid_set.F times 1 = prod" by (auto intro: sym)
qed

end

syntax (ASCII)
  "_Prod_any" :: "pttrn \ 'a \ 'a::comm_monoid_mult" ("(3PROD _. _)" [0, 10] 10)
syntax
  "_Prod_any" :: "pttrn \ 'a \ 'a::comm_monoid_mult" ("(3\_. _)" [0, 10] 10)
translations
  "\a. b" == "CONST Prod_any (\a. b)"

lemma Prod_any_zero:
  fixes f :: "'b \ 'a :: comm_semiring_1"
  assumes "finite {a. f a \ 1}"
  assumes "f a = 0"
  shows "(\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 prod_zero)
qed

lemma Prod_any_not_zero:
  fixes f :: "'b \ 'a :: comm_semiring_1"
  assumes "finite {a. f a \ 1}"
  assumes "(\a. f a) \ 0"
  shows "f a \ 0"
  using assms Prod_any_zero [of f] by blast

lemma power_Sum_any:
  assumes "finite {a. f a \ 0}"
  shows "c ^ (\a. f a) = (\a. c ^ f a)"
proof -
  have "{a. c ^ f a \ 1} \ {a. f a \ 0}"
    by (auto intro: ccontr)
  with assms show ?thesis
    by (simp add: Sum_any.expand_set Prod_any.expand_superset power_sum)
qed

end

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