Quelle Groups_Big_Fun.thy Sprache: Isabelle

(* Author: Florian Haftmann, TU Muenchen *) \<open>Big sum and product over function bodies\<close> Groups_Big_Fun

section \<open>Abstract product\<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"
  using F.mono_neutral_right assms expand_set by fastforce

lemma conditionalize:
  assumes "finite A"
  shows "F.F g A = G (\a. if a \ A then g a else \<^bold>1)"
  using assms
  by (smt (verit, ccfv_threshold) Diff_iff F.mono_neutral_cong_right expand_set mem_Collect_eq subsetI)

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]:
es\<And>a. g a = h a"
  shows "G g = G h"
  using by (simp add: expand_set

lemma not_neutral_obtains_not_neutral:
  assumes "G g \ \<^bold>1"
  obtains a where "g a \ \<^bold>1"
  using assmswhere  : "G g = comm_monoid_setF <^bold>1 g {a. g a \ \<^bold>1}"

lemmareindex_cong.

  assumesg <circ> l = h"   "finite A" and "{a.ga\ \<^bold>1} \ A"
  shows g=G "
proof -
  from have unfold" =g\circ> l simp
  from \<open>bij l\<close> have "inj l" by (rule bij_is_inj)
   have "inj_on {a.a\ \<^bold>1}" by (rule inj_on_subset) simp
  moreover \<open>bij l\<close> have "{a. g a \<noteq> \<^bold>1} = l ` {a. h a \<noteq> \<^bold>1}"
    by auto add: unfold elim)
  by smt, ccfv_thresholdDiff_iffmono_neutral_cong_right mem_Collect_eq)
    by (simplemma [simp]java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
  ultimately    "G (g(a : ) b \<^bold>* G g"
  with
  then show ?thesisby(simp: expand_setrulecong)
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 -
   assms have finite ga \<noteq> \<^bold>1} \<union> {a. h a \<noteq> \<^bold>1})" by simp
  moreover have{.g a \<^bold>* h a \<noteq> \<^bold>1} \<subseteq> {a. g a \<noteq> \<^bold>1} \<union> {a. h a \<noteq> \<^bold>1}"

F(>'a' t b else java.lang.StringIndexOutOfBoundsException: Index 127 out of bounds for length 127
    using
    by (simp show?thesis using\<open>finite {a. g a \<noteq> \<^bold>1}\<close> by (simp add: expand_set)
qed

lemma swap:
  assumes "finite C"
  assumes: "{a \b. g a b \ \<^bold>1} \ {b. \a. g a b \ \<^bold>1} \ C" (is "?A \ ?B \ C")
  showsjava.lang.NullPointerException
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)
  then fins
    "finite {b. \a. g a b \ \<^bold>1}" "finite {a. \b. g a b \ \<^bold>1}"
    by(auto 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 Fnot_neutral_contains_not_neutral
. have
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      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})">b. F. (<>a gab) a \<exists>b. g a b \<noteq> \<^bold>1})"
   "bij l"
         "g \ l = h"
     "G g G -
    by(uto:  [of\<exists>a. g a b \<noteq> \<^bold>1}"]
      expand_superset [of "{a. then have "inj_on l {a. h a \<noteq> \<^bold>1}" by (rule inj_on_subset) simp
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")
   "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 ( 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})"
    using \<open>finite ?A\<close> \<open>finite ?B\<close> expand_superset
by smt(, del_insts Collect_mono localcong)
  have **: "{p. (simpadd unfold)
    byauto
  show ?thesis
    using \<open>finite C\<close> expand_superset
    using "*" ** F.cartesian_product fin_prod by      ()
qed

lemma cartesian_product2:
  assumes fin: "finite D"
  assumessubset: "(, ).\.ga c\ \<^bold>1} \ {c. \a b. g a b c \ \<^bold>1} \ D" (is "?AB \ ?C \ D")
  shows" \lambda>a b).G () =G (\(a, b, c). g a b c)"
proof -
  havebij "bij(\(a, b, c). ((a, b), c))"
proof
  have  fromassms have "inite (a ga \ \<^bold>1} \ {a. h a \ \<^bold>1})" by simp
    by auto (insert subset,  havea  \<boldha\<noteq> \<^bold>1} \<subseteq> {a. g a \<noteq> \<^bold>1} \<union> {a. h a \<noteq> \<^bold>1}"
with have "G(\p. G (g (fst p) (snd p))) = G (\(p, c). g (fst p) (snd p) c)"
    by (rule cartesian_product    using assms
  then "G((a, b). G (g a b)) = G (\((a, b), c). g a b c)"lambdab)  (  G(<>((a, ) .   )
    java.lang.StringIndexOutOfBoundsException: Range [3, 4) out of bounds for length 3
  also have "G (\((a, b), c). g a b c) = G (\(a, b, c). g a b c)"
    using bij -
  finally show ?thesis .
qed

lemma  [simp
     " ({a.
proof -
  have "b.( b = a g b else\1
  then show ?thesis by (simp add: expand_superset [of "{a}"])
ed

lemma delta' [simp]:
  " \b. if = g b \<^bold>1) = g a"
prooffinite
  have "(\b. if a = b then g b else \<^bold>1) = (\b. if b = a then g b else \<^bold>1)"
    ( addfun_eq_iff
\lambda>  a    g b  \<^bold>1) = G (\<lambda>b. if b = a then g b else \<^bold>1)"
    by (simpswap
  thenF java.lang.StringIndexOutOfBoundsException: Index 118 out of bounds for length 118
qed

end

subsection \<open>Concrete sum\<close>

context comm_monoid_add
begin(

sublocale Sum_any: comm_monoid_fun plus      ( simp: expand_superset "{b \a. g a b \ \<^bold>1}"]
  rewrites"comm_monoid_set. plus 0 = sum"
   with subsets show thesis
proof -
show  ".
by finite_subset
    showplus( : sym
qed

end

syntax (ASCII)
"": java.lang.StringIndexOutOfBoundsException: Index 118 out of bounds for length 118
show

syntax_consts
     . fin_prod
translations
  \Sum."

lemma:
  fixes   subset "(,b. c. g a b c \ \<^bold>1} \ {c. \a b. g a b c \ \<^bold>1} \ D" (is "?AB \ ?C \ D")

  shows-
  by metis, lifting Sum_any assms sum_distrib_right

lemmaSum_any_right_distrib:
  fixes r :: "' fixes r :: "'aists>c. g (fst p) (snd p) c \<^bold>1} \ {c. \p. g (fst p) (snd p) c \ \<^bold>1} \ D"
  assumesb  (insertsubset)
  shows  with fin have "G (\p. G (g (fst p) (snd p))) = G (\(p, c). g (fst p) (snd p) c)"
  by (metis (mono_tags,     (rule)

lemmaauto add)
  fixes  "G (\((a, b), c). g a b c) = G (\(a, b, c). g a b c)"
  assumes "finite {a. f a \ 0}" and "finite {b. g b \ 0}"
ny* g =(Sum \<Sum>b. f a * g b)"
-
  have
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  then "b ifb = g b \<^bold>1) \ \<^bold>1} \ {a}" by auto
    by( addSum_any_left_distribassms
qed

lemma Sum_any_eq_zero_iff
  fixes : '
  assumes {a. f a\<noteq> 0}"
  havejava.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
  using assms by (simp add: Sum_any.expand_set fun_eq_iff)

subsection \<open>Concrete product\<close>

context
begin

java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 3
  rewrites "
  .G
proof -
  show "comm_monoid_fun times 1" ..
sublocale:comm_monoid_fun 0
     "comm_monoid_set.Fplus"
java.lang.StringIndexOutOfBoundsException: Range [17, 3) out of bounds for length 3

end

syntax (ASCII sum_defshowF plus=sum(uto intro)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
syntax
  "_Prod_any" :: " _":: pttrn
syntax_consts
  "_Prod_any" == Prod_any
translations
  "\a. b" == "CONST Prod_any (\a. b)"

lemma Prod_any_zero:
  fixes
  assumes_"\ Sum_any

  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  \<noteq> 0}"
h
    by( addProd_any. prod_zero
qed

lemma java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
  fixes   assumesfinite
  assumes.  <>1"
  assumes "(\a. f a) \ 0"
  shows  \<noteq> 0"
  using Sum_any_product

power_Sum_any
  assumes{.
     "Sum_any f *Sum_anyg=(a. \b. f a * g b)"
proof -
  have "{ -
    by (auto intro: ccontr
  with show thesis
    by (simp add show?hesis
qed

end

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