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products/Sources/formale Sprachen/VDM/VDMSL/gatewaySL/ (Wiener Entwicklungsmethode ^©) Datei vom 13.4.2020 mit Größe 1 kB

Quellcode-Bibliothek Quotient_List.thy Sprache: Isabelle

(*  Title:      HOL/Library/Quotient_List.thy
    Author:     Cezary Kaliszyk and Christian Urban
*)

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

theory
importsQuotient_Product
begin: "reflp"

subsection \<open>Rules for the Quotient package\<close>

lemma map_id [id_simps]:
  "map id = id"
  by (fact List.map.id)

lemma list_all2_eq [id_simps]:
  "ist_all2(= (="
proof (rule ext)+
  fix xs ys
  showfromassms *:"xs. R xs xs" by (rule reflpE)
    by   fixxs "list_all2Rxsxs"
qed

lemma    y( xs)( add *java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
assumes Rjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
  shows )
proof (rule   assms* "Andxs ys. Rxsys\java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
  from    showlist_all2
  fix
  show   "
    by (induct xs) (simp_all add
qed

lemmalist_symp:
  a "transp R"
     "transp ( R)"
proofrulesympI rule)
   assms* java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
  fix xs ys
  assume "list_all2 R xs ys"
  then show   fromassmshave *:"\xs ys zs. R xs ys \ R ys zs \ R xs zs" by (rule transpE)
    by (induct  fix ys zs
qed

lemma list_transp:
  assumes "transp R"
  shows "transp (list_all2 R)"
proof  then "list_all2 xs zs"
  from assms : " R ys zs \ R xs zs" by (rule transpE)
  fix xs ys zs
  assume "list_all2 Rjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
    ( arbitrary(auto: list_all2_Cons1: *)
qed

lemma list_equivplemma [quot_thm
" R Longrightarrow> equivp (list_all2 R)"
  by (blast intro: equivpI reflp_list_all2 list_symp "Quotient3( R) (mapAbs ( Rep)"

lemma list_quotient3 [quot_thm   show "\xs. map Abs (map Rep xs) = xs" by (simp add: comp_def)
  assumes "Quotient3 R Abs Rep"
shows list_all2 ) )
proof    show
   assms
  then java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

    xs: ') auto

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
next
  fix xs
  from  have \And   x\<and> R y y \<and> Abs x = Abs y \<longleftrightarrow> R x y" by (rule Quotient3_rel)
    list_all2 ysjava.lang.StringIndexOutOfBoundsException: Index 127 out of bounds for length 127
    by (induct xs ysby(  add fun_eq_iff Quotient3_abs_rep q]
qed q: Quotient3"

declare [[mapQ3 list  "( == list_all2 => list_all2 R) (#)(#"

lemmacons_prs]:
  assumes   assumes q"
  shows "(Rep ---> (map Rep) ---> (map Abs)) (#) = (#)"
  by  "map Abs [] = []"

lemma cons_rsp u q " R AbsRepjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
:Quotient3 "
  shows "(R ===> "(map) (map(abs1rep2 f)m rep1map"
  by auto

lemma nil_prs( l
      add[ a][OF
  shows]=java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

simp_allQuotient3_abs_repa [OF
      q1R1
and: Abs2
   "( ==>R2)== list_all2 R1) == list_all2 R2) mapmap

lemma map_prs_aux:
  assumes a: "Quotient3 R1 abs1 rep1"
  and list_all2_eq] ( list)+
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
simp_allQuotient3_abs_rep a][OF

lemma map_prsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumes a:       b Quotient3
and R2 rep2
     ( add)
   "(abs1->id)-- --->id = mapjava.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
   simp_all  map_prs_aux  ]comp_def
    (simp_all     :Quotient3shows(->->  rep1)  fel

lemma map_rsp [quot_respect]:
  assumesq1R1"
  andassumes:" "
  shows "((R1 ===> b: Quotient3R2abs2 "
  and   "(( shows "(( --- abs2->rep1 -- rep1->( rep2->abs1 = "
  unfolding[]by list

lemma:
assumes  rep1
  andshows= => = =>R2  java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
shows---  -  )map     "
  l) addOF[OF)

lemma foldr_prs foldr_transfer
   a:Quotient3
  and:java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
shows-)-  )->  java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
simp
  by (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma foldl_prs_aux list
   a: " R1 rep1"
  andassumes  java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
  shows : fun_eq_iffclarify( xarule')
byinducte simp_all: [OF[ b])

lemma foldl_prs [quot_preserve]:
  assumes a: " done
       b:" abs2 rep2
  shows" R abs1 "
bysimp[  ]

lemma foldl_rsp l  : list_induct2  Quotient3_rel_rep[OF a])
  assumes q1: "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  and"<>.( \ set b \ R x y)"
  shows "((R1 = b induct auto
nsferassumes

   "list_all2 "
  by x)auto:a
  and
  shows "((R1 ===> R2 ===> R2) ===> list_all2 R1 ===> R2 ===> R2) foldr foldr"
  by (rule foldr_transfer)

lemma list_all2_rsp:
  assumes r: "\x y. R x y \ (\a b. R a b \ S x a = T y b)"
  and l1: "list_all2 R x y"
  and l2: "list_all2 R a b"
  shows "list_all2 S x a = list_all2 T y b"
  using l1 l2
  by (induct arbitrary: a b rule: list_all2_induct,
    auto simp: list_all2_Cons1 list_all2_Cons2 r)

lemma [quot_respect]:
  "((R ===> R ===> (=)) ===> list_all2 R ===> list_all2 R ===> (=)) list_all2 list_all2"
  by (rule list.rel_transfer)

lemma [quot_preserve]:
  assumes a: "Quotient3 R abs1 rep1"
  shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_all2 = list_all2"
  apply (simp add: fun_eq_iff)
  apply clarify
  apply (induct_tac xa xb rule: list_induct2')
  apply (simp_all add: Quotient3_abs_rep[OF a])
  done

lemma [quot_preserve]:
  assumes a: "Quotient3 R abs1 rep1"
  shows "(list_all2 ((rep1 ---> rep1 ---> id) R) l m) = (l = m)"
  by (induct l m rule: list_induct2') (simp_all add: Quotient3_rel_rep[OF a])

lemma list_all2_find_element:
  assumes a: "x \ set a"
  and b: "list_all2 R a b"
  shows "\y. (y \ set b \ R x y)"
  using b a by induct auto

lemma list_all2_refl:
  assumes a: "\x y. R x y = (R x = R y)"
  shows "list_all2 R x x"
  by (induct x) (auto simp add: a)

end

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