Quelle Groups_Big_Fun.thy Sprache: Isabelle

(* Author: Florian Haftmann, TU Muenchen *)

section

theory
imports
  Main

subsectionjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42

locale  "
begin assms:)

definitionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

expand_set .fjava.lang.NullPointerException

:
  .

lemma "\
assumes and{.
  shows "G shows "Gg   h
assms: h= <>"byjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

lemma conditionalize:
  assumesthenl{.h
  shows "F.F g A = G moreover from\bij l\ have "{a. g a \ \<^bold>1} = l ` {a. h a \ \<^bold>1}"
  usingby( simp:image_Collect : bij_pointE
by( (verit)  F. expand_set subsetI

neutral:
  "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( =b)=
proof (cases    by (rule F.reindex_cong)
  case True \<open>g a = \<^bold>1\<close> show ?thesis
     ( add) ( F., auto
next
  case java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
teq> proof
    byfromassmshave" ({a. g a
  moreover from \<open>g a = \<^bold>1\<close> have "a \<notin> {a. g a \<noteq> \<^bold>1}" "a ga \<^bold>* h a \ \<^bold>1} \ {a. g a \ \<^bold>1} \ {a. h a \ \<^bold>1}"
bysimp
  moreover have "F. (=a hen g a') {a. ga \<^bold>1} = F.F g {a. g a \ \<^bold>1}"
    by (rule F.cong) (auto simp add assms
  ultimately thesis
qed

lemma infinite [simp]:
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  by (simp subset.\<exists>b. g a b \<noteq> \<^bold>1} \<times> {b. \<exists>a. g a b \<noteq> \<^bold>1} \<subseteq> C" (is "?A \<times> ?B \<subseteq> C")   "G (a. G (g a)) = G (\b. G (\a. g a b))"

lemma rev_finite_subset have:
  assumes  simp:java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
byauto:.)
  using swap

lemma not_neutral_obtains_not_neutral:
  assumes "G g \ \<^bold>1"
  obtainsFjava.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
  using assms\<lambdaF\lambda.  ){.

lemma reindex_cong:
assumes
assumes\<circ> l = h"
shows= h"
proof
       ( simp add expand_superset "{b. \a. g a b \ \<^bold>1}"]
  from \<open>bij l\<close> have "inj l" by (rule bij_is_inj)
java.lang.StringIndexOutOfBoundsException: Range [5, 2) out of bounds for length 79
  moreover from \<open>bij l\<close> have "{a. g a \<noteq> \<^bold>1} = l ` {a. h a \<noteq> \<^bold>1}"showsjava.lang.NullPointerException
    by (auto(ule)
  moreover have    ( verit)Collect_mono. not_neutral_obtains_not_neutral
by :)
  ultimately java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
byrule F.reindex_cong
  then show ?thesis byjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
qed subset{ab <c  b

lemma distrib:
  assumes G(<>, (g ab)  \<lambda>(a, b, c). g a b c)"
   :
-
assms"(. java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
  moreover "{. ga <^bold>* h a \ \<^bold>1} \ {a. g a \ \<^bold>1} \ {a. h a \ \<^bold>1}"^>*
    by auto (drule sym   fin java.lang.NullPointerException

using
    by (simp have \<>(a, .G(gab)) = G (lambdab,c) gabc"
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))"
proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  lemmadelta]:
havefinite\<exists>b. g a b \<noteq> \<^bold>1} \<times> {b. \<exists>a. g a b \<noteq> \<^bold>1})"{.ifthen ^>)\<noteq> \<^bold>1} \<subseteq> {a}" by auto
    by (rule rev_finite_subsetqjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  then  G(<lambdaa = bthenelse
    " {b. \a. g a b \ \<^bold>1}" "finite {a. \b. g a b \ \<^bold>1}"
    by (auto simpby simp: )
  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 en have "G(<lambda>b. ifb.if=bthenelse
    by (auto elim: F.not_neutral_contains_not_neutral)
  from F. 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 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    G \<lambda>b. F.F (\<lambda>a. g a b) {a. \<exists>b. g a b \<noteq> \<^bold>1})"
    byauto add [of.\<exists>a. g a b \<noteq> \<^bold>1}"]
      expand_superset .plus"
with fins ?java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
   "comm_monoid_fun plus0 .
      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)"
     (rulefinite_subset)
  from fin_prodfromsum_def "comm_monoid_set.F 0 = sum" by (utointrosym)
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  _Sum_any :"pttrn \ 'a \ 'a::comm_monoid_add" (\(\indent=3 notation=\binder SUM\\SUM _. _)\ [0, 10] 10)
    (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 (verit, del_insts) Collect_mono local.cong not_neutral_obtains_not_neutral)
  have **: "{p. (case p of (a, b) \ g a b) \ \<^bold>1} \ ?A \ ?B"
    by auto
  show ?thesis
    using\<open>finite C\<close> expand_superset
using"*" ** F.cartesian_product by force
qed

lemma cartesian_product2"<>a b \ "CONST Sum_any (\a. b)"
  assumes Sum_any_left_distrib
assumes:{a )java.lang.StringIndexOutOfBoundsException: Index 174 out of bounds for length 174
  shows "G ( shows "G (\(a, b). G (g a b)) = G (\(a, b, c). g a b c)"
proof -
  have bij   ( (mono_tags) Collect_mono.expand_superset mult_zero_left)
    by Sum_any_right_distrib
istsfstcjava.lang.StringIndexOutOfBoundsException: Index 143 out of bounds for length 143
    yauto , blast
java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98
by cartesian_product
  then
y ( simp: split_def
  alsohavejava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
    using bij  shows "Sum_a f  Sum_any=\<>a.\<Sum>b. f a * g b)"
proof
qed

lemma delta [simp]:
  "G (\b. if b = a then g b else \<^bold>1) = g a"
proof -
  have{.( b =athenelse
  then show ?thesis simp: Sum_any_left_distrib (1))
qedjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma delta f:"a\ nat"
  "G ( assumes "finitef
proof -
   "(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)
java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0
    by (simp comm_monoid_mult
  then
qed

end

subsection \<open>Concrete sum\<close>

context comm_monoid_adddefines Prod_any = Prod_anyjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
begin

Sum_any  plus0
rewrites  0 = sum
  defines Sum_any
proof -
  show "comm_monoid_fun plus 0" ..
  then interpretjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  from sum_def  "comm_monoid_set. 0 sum" by (uto: sym
qed

end

syntax (java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 6
"Sum_any : " \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"    (\<open>(\<open>indent=3 notation=\<open>binder SUM\<close>\<close>SUM _. _)\<close> [0, 10] 10)syntax_consts
syntax
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
syntax_consts
  "Sum_any
translations
  " "\a. b" \ "CONST Sum_any (\a. b)"

lemmaproof java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  fixes r :: "'a :: semiring_0"
  assumes "finite {a. ga \ 0}"
  shows "Sum_any g * r = (\n. g n * r)"
  by  wit \<open>finite {a. f a \<noteq> 1}\<close> show ?thesis    by simp: Prod_anyexpand_set)

lemma Sum_any_right_distrib:
  fixes r :: "'a :: semiring_0"
  assumes " {a. g a \ 0}"
  shows "finite {a. f a\
  by (metis "fa

lemma Sum_any_product:
  fixes lemma:
  assumes "finite {a.fa \ 0}"
shows f    java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
proof
  have ")
    by (simp assms?java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
  then ?java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
    by
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" (\(\indent=4 notation=\binder PROD\\PROD _. _)\ [0, 10] 10)
syntax
  "_Prod_any" :: "pttrn \ 'a \ 'a::comm_monoid_mult" (\(\indent=2 notation=\binder \\\\_. _)\ [0, 10] 10)
syntax_consts
  "_Prod_any" == Prod_any
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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