SSL Quotient_List.thy Sprache: Isabelle

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

section \<open>Quotient infrastructure for the list type\<close>

theory Quotient_List
imports Quotient_Set Quotient_Product Quotient_Option
begin

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]:
  "list_all2 (=) = (=)"
proof (rule ext)+
  fix xs ys
  show "list_all2 (=) xs ys \ xs = ys"
    by (induct xs ys rule: list_induct2') simp_all
qed Quotient_Set  Quotient_Option

reflp_list_all2
  assumes R
  shows "java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
  " = =()java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
   assms have java.lang.NullPointerException

  show   java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
b induct)simp_all: )
qed

lemma list_symp:
  assumes "symp R"
  shows "symp (list_all2 R "reflp"
proof (rule "reflp (list_all2R"
  from have : "<> . R R ys xs" by (rule sympE)
  fix xs ys
  assume "list_all2 R xs ys"
then " R ys xs"
    by ( xs "list_all2 Rxsxsjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
qed

lemma list_transp list_symp
  ssumes
showslist_all2 (rule )
proof( transpIfrom have :"xs ys. R xs ys \ R ys xs" by (rule sympE)
  have \<And>xs ys zs. R xs ys \<Longrightarrow> R ys zs \<Longrightarrow> R xs zs" by (rule transpE)
   xszs
  assume "list_all2 Rjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
showRxs
    by (induct have * "<> ys zs.R xsys\ R ys zs \ R xs zs" by (rule transpE)
qed

lemma list_equivp [quot_equiv]:
  "equivp R then show "list_all2 R xs zs"
  by (blastby induct: zs) (auto simp intro)

list_quotient3]:
  assumes "Quotient3 R equivp R\
  shows list_all2 )map
proof (rule Quotient3I)
  from assms have "\x. Abs (Rep x) = x" by (rule Quotient3_abs_rep)
thenjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
next
   "Quotient3( R) (map Abs (mapRep)"
then "\xs. list_all2 R (map Rep xs) (map Rep xs)"
    by (simpfrom have "\x. Abs (Rep x) = x" by (rule Quotient3_abs_rep)
next
  fix xs ys
  from assms have "\x y. R x x \ R y y \ Abs x = Abs y \ R x y" by (rule Quotient3_rel)
  then shownext
    by(induct ys rulelist_induct2
qed

declare [[mapQ3 list = (list_all2, list_quotient3

lemma   fromassms"<>xy.Rx \ R y y \ Abs x = Abs y \ R x y" by (rule Quotient3_rel)
  assumes q: "Quotient3 R Abs Rep"
  shows "(Rep ---> (mapthenshow" R xs \<longleftrightarrow> list_all2 R xs xs \<and> list_all2 R ys ys \<and> map Abs xs = map Abs ys"
   (autosimp: comp_def [OF]

lemma cons_rsp [quot_respect]:
  assumes " R Abs Repjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
  showsR=>R== )
  by auto

lemmalemma  [quot_preserve
: "Quotient3 R Abs Repjava.lang.StringIndexOutOfBoundsException: Range [34, 35) out of bounds for length 34
  showsAbs"
  by simp

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
assumes:Quotient3 "
  shows "list_all2 R [] []"
  by simp

lemma map_prs_aux:
  assumes a: "Quotient3 R1 abs1 rep1"
  and     b: "Quotient3 R2 assumes q " R AbsRep
shows abs2 ( ---> rep2) f)(ap l)) = map f ljava.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
  by (nduct)
(simp_all: Quotient3_abs_repOF Quotient3_abs_rep b])

lemma map_prs "map Abs [ =[]"
  assumes a: "Quotient3 R1 abs1 rep1"
  and     b: "Quotient3 R2 abs2 rep2"
  by simp
  and   "((abs1 ---> id) ---> map rep1 ---> id) map = map"
  by
   ( add: [OF a] Quotient3_abs_rep b])

lemma map_rspassumesq: "Quotient3 R Abs Rep"
  assumesq1: "Quotient3 Abs1 Rep1"
  and     q2 "Quotient3R2 Abs2 Rep2"
  shows(R1=  =>( R1=>list_all2 "
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
  unfolding [symmetric byrule.map_transfer

lemma foldr_prs_aux:
  assumes a: "Quotient3 R1 abs1 rep1"
  and     b: "Quotient3 R2 abs2 rep2"
  shows    (inductl)
  by (induct l) (simp_all add: Quotient3_abs_rep     ( add: Quotient3_abs_rep[OFa] Quotient3_abs_rep b])

lemma foldr_prs [quot_preserve]:
  assumes a: "Quotient3 R1 abs1 rep1"
  and: " R2 abs2 rep2"
  shows "((abs1 b: "Quotient3 abs2"
  apply(imp: fun_eq_iff
  and(( --  ->maprep1> )map"
     (simp)

lemma foldl_prs_aux:
  assumesby( only: fun_eq_iff[OFab] )
  andb:" R2 abs2 rep2"
  shows "abs1 (foldl ((abs1 -- abs2 -->rep1)f)(rep1 e) (map rep2 l) =foldl "
  by (induct l java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemmaassumes : "Quotient3 R1 Abs1 Rep1"
  assumes a "Quotient3R1abs1rep1java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
  and" R2 abs2 rep2java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
showsabs1>abs2 --->rep1) --> rep1 -->(map) --->) foldl foldl
  by (simp add list_all2_eq symmetric by (rule.map_transfer)+

lemma foldl_rsp
  assumes foldr_prs_aux
  and     q2  assumes a: "Quotient3R1abs1 rep1"
   "((R1 ===>R2 == R1)=== R1=== list_all2 ===> R1)foldlfoldl"
  by (rule foldl_transfer)

lemma foldr_rsp[quot_respect]:
  assumes q1: "Quotient3 R1 Abs1 Rep1"
  and     q2   "abs2 (foldr ((abs1 >abs2-->rep2)f (map rep1 l) (rep2 e)) = foldrfle"
  shows  by(induct (simp_all: Quotient3_abs_rep[ a] Quotient3_abs_rep b])
  by (rule)

lemma list_all2_rspassumes "Quotient3 R1 abs1 rep1"
  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  shows "((abs1 ---> abs2 --> rep2 -->(map rep1 -- rep2 ---> abs2) foldr= foldr"
  by (induct arbitrary: a b rule: list_all2_induct,
    auto simp  apply ( add: fun_eq_iff)

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

lemmaassumes Quotient3abs1
   a: "Quotient3R abs1 rep1"
  shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_all2 = list_all2"
  apply (simpadd)
  apply
  apply induct_tac xb rule: list_induct2
  apply (simp_all   ( l arbitrary:e)(simp_all add Quotient3_abs_rep a] Quotient3_abs_repOFjava.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
  done

lemma [quot_preserve  and "uotient3R2 abs2 rep2"
  assumes a: Quotient3rep1
  shows "(list_all2 (simp add: fun_eq_iff foldl_prs_aux OF ab])
  by (inductmrule') (simp_all add:java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77

lemma list_all2_find_element:
  assumes a: "x \ set a"
  and b: "list_all2 R a b"
  shows \existsy y\<in> set b \<and> R x y)"
  usingabyinduct

lemma list_all2_refl)
   a: "\x y. R x y = (R x = R y)"
  showsRxx"
   (induct ( simp add a)

end

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