SSL Standardization.thy Sprache: Isabelle

      /.thy
    :     Stefan
       200 TU
*java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

section \<open>Standardization\<close>

theory
imports
begin

text : by( listrelp) (utointro.)
onlecturenotes Matthes <cite
originalproof due Ralph \<^cite>\<open>Loader1998\<close>.
\<close>

subsection \<open>Standard reduction relation\<close>

declare [mono_setshows" R xs ys Longrightarrow R(s xs') (ys @ ys')" using xsys

inductive
  sred :: "dB \ dB \ bool" (infixl \\\<^sub>s\ 50)
  and sredlist :: "dB list \ dB list \ bool" (infixl \[\\<^sub>s]\ 50)
where
  "s [\\<^sub>s] t \ listrelp (\\<^sub>s) s t"
| Var: "rs [\\<^sub>s] rs' \ Var x \\ rs \\<^sub>s Var x \\ rs'"
| Abs: "r \\<^sub>s r' \ ss [\\<^sub>s] ss' \ Abs r \\ ss \\<^sub>s Abs r' \\ ss'"
| Beta: "r[s/0] \\ ss \\<^sub>s t \ Abs r \ s \\ ss \\<^sub>s t"

lemma refl_listrelp: "\x\set xs. R x x \ listrelp R xs xs"
  by (induct (induct arbitrary'ys' auto: listrelpintros

lemma refl_sred: "t \\<^sub>s t"
  by (induct shows

lemma refl_sreds
  ca ( rs'xjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

lemmalistrelp_conj1listrelp\<lambda>x y. R x y \<and> S x y) x y \<Longrightarrow> listrelp R x y"
java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 58

java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
   erule.induct introlistrelp)

lemma listrelp_app:
  assumes from(3)havess
shows Rxs <Longrightarrow> listrelp R (xs @ xs') (ys @ ys')" using xsys
  by (induct arbitrary: xs' ys') (auto intro: listrelp.intros)

lemma lemma1
  assumes r: "r
  shows "r \ s \\<^sub>s r' \ s'" using r
proof induct
  case (Var rs rs' x)
  then have "rs [\\<^sub>s] rs'" by (rule listrelp_conj1)
  moreoverby( sred)
  ltimately have" @ []\\<^sub>s] rs' @ [s']" by (rule listrelp_app)
  hence"Var x \\ (rs @ [s]) \\<^sub>s Var x \\ (rs' @ [s'])" by (rule sred.Var)
  thus ?case by (simp only: app_last)
next
  case (Abs r r' ss ss')
  from Abs3) havehave "ss [[s] ss'" by (rule listrelp_conj1)
  moreover have "[s] [\\<^sub>s] [s']" by (iprover intro: s listrelp.intros)
  ultimately have "ss @ [s] [\\<^sub>s] ss' @ [s']" by (rule listrelp_app)
  with \<open>r \<rightarrow>\<^sub>s r'\<close> have "Abs r \<degree>\<degree> (ss @ [s]) \<rightarrow>\<^sub>s Abs r' \<degree>\<degree> (ss' @ [s'])" ts: " [\\<^sub>s] ts'"
    by (rule
  thus?case simp: app_last
next
  case beta"t\\<^sub>\ u"
  henceshows \<rightarrow>\<^sub>s u" using beta
  henceAbs <degree \<degree>\<degree> [] \<rightarrow>\<^sub>s s[t/0] \<degree>\<degree> []" by (iprover intro: sred.Beta refl_sred)
   case by (simpsimp: app_last
qed

lemma
  case( st u
? by ( intro refl_sred
  by (induct java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
thus?case by simp
lemma
  assumes
  showst
proof java.lang.StringIndexOutOfBoundsException: Index 175 out of bounds for length 175
  caseassumes"
java.lang.StringIndexOutOfBoundsException: Index 139 out of bounds for length 139
    (auto:listrelp_conj2
next
  caseintro:listrelp_betas  converse_rtranclp_into_rtranclp
  thuscase by( intro refl_sred
next
  case appRt java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
  thus by intro refl_sred
next map
    auto)
   case cases auto:sred)
  thus ?case by simp
qed

lemma listrelp_betas:
  assumes ts: "listrelp (\\<^sub>\\<^sup>*) ts ts'"
  shows "\t t'. t \\<^sub>\\<^sup>* t' \ t \\ ts \\<^sub>\\<^sup>* t' \\ ts'" using ts
  by induct auto

lemma lemma2_2 (autointro:.intros
  assumesthus caseby ( intro.Abs)
   "\\<^sub>\\<^sup>* u" using t
  byinduct dest
    intro: listrelp_betas apps_preserves_beta converse_rtranclp_into_rtranclp)

lemmaassumes:"
    shows "s
  shows "lift s i \\<^sub>s lift t i" using s
proof (induct arbitrary: i)
   Var rs
  hence "map (\t. lift t i) rs [\\<^sub>s] map (\t. lift t i) rs'"case Var rs
    by induct (autoinduct ( intro.intros)
?  cases i) (auto: sred)
next
  case (Abs r r' ss ss')
f Abs(
    by True? by ( refl_sred
  thuscaseFalse
next
  by(cases" x)( simpa: Var intro: )
  thus
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma lemma3[s/x\rightarrow
assumes"\\<^sub>s r'"
showss \<rightarrow>\<^sub>s s' \<Longrightarrow> r[s/x] \<rightarrow>\<^sub>s r'[s'/x]" using rshow ? simp sred)
proofinductjava.lang.StringIndexOutOfBoundsException: Range [23, 15) out of bounds for length 32
  caseVar')
  hence "map (\t. t[s/x]) rs [\\<^sub>s] map (\t. t[s'/x]) rs'"
    by lemma4_aux:
   rs listrelp\<lambda>t u. t \<rightarrow>\<^sub>s u \<and> (\<forall>r. u \<rightarrow>\<^sub>\<beta> r \<longrightarrow> t \<rightarrow>\<^sub>s r)) rs rs'"
  proofcase
case thus   ( refl_sred
  next
     False
    thus ?thesis
      by (cases "y = x") (auto simp add: Var intro: refl_sred)
  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
show?asesimprule')
next
  case (Abs rnext
  from Abs(4) havehencess:y'#java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 37
  hence " hence"x#xs\<rightarrow>\<^sub>s] y' # ys'"
  moreoverassumeH y\rightarrow
byinduct (auto intro listrelp.intros Abs)
  ultimately show? by  (rulesred)
next
  case        from' have" [\\<^sub>s] ys" by (iprover dest: listrelp_conj1)
  thus ?case by (auto simp addwith Hs ?thesis java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
qed

lemma:
  assumes
rs
proof (induct
   java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
  thus ?case by cases (auto intro: listrelp.Nil)
next
  case (Cons x y xs ys)
  note Cons' = Cons
  show ?case
proof( ss)
    case Nil withshowsr
  next
    casecase(Var rs' x)
    hence: "ss= '# ys"bysimp
    from Cons Cons' have "y \\<^sub>\ y' \ ys' = ys \ y' = y \ ys => ys'" by simp
    hence "x # xs [\\<^sub>s] y' # ys'"
proof
      assume    Var \<degree>\<degree> rs \<rightarrow>\<^sub>s Var x \<degree>\<degree> ss" by (rule sred.Var) r''  casebysimp
      with Cons' have "x \\<^sub>s y'" by blast
      fromCons xs
      ultimately have "x # xs [\\<^sub>s] y' # ys" by (rule listrelp.Cons)
      with H show ?thesis obtain''  s: s Abs'"andr' r' \\<^sub>\ r'''" by cases auto
    next
      assume H: "y moreover from Abs have "[<rightarrow>\<^sub>s] ss'" by (iprover dest: listrelp_conj1)
      with Cons' have "x \\<^sub>s y'" by blast
      moreoverfromHhavexs
      ultimately show '
    qed
    with showthesis
  qed
  assume"r'=r java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

lemma lemma4:
assumesr:r\rightarrow
  shows "r' \\<^sub>\ r'' \ r \\<^sub>s r''" using r
proof (induct arbitrary    with have"[/]\<
  case (Var rs rs' x "Abs r \ u \\ us \\<^sub>s r'[u'/0] \\ us'" by (rule sred.Beta)
  then obtain ss where rsmoreover  " r =Abs"  "r' =['/]\<>java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
    byjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  from Var(by( sred) ( Beta
  hencejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  with' ? by simp
next "r\\<^sub>s r'" using r
  case (Abs r r' by induct (iprover intro lemma4)+
  from subsection \<open>Leftmost reduction and weakly normalizing terms\<close>
  proof
    fix s
    assumer' "r' = s \<degree>\<degree> ss'"
      and lredlist dB
hen obtain r''where s  '" ' r'\<rightarrow>\<^sub>\<beta> r'''" by cases auto
Var" [
    moreover fromAbshave" \java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
    ultimately have "| Abs r _{r'\ Abs r \\<^sub>l Abs r'"}
    with r'' s show "Abs r \\ ss \\<^sub>s r''" by simp
  assumes: "s \\<^sub>l t"
     rs'
    assume "ss' induct
     Abs) have "ss [\\<^sub>s] rs'" by (rule lemma4_aux)
    with java.lang.StringIndexOutOfBoundsException: Range [6, 130) out of bounds for length 44
    moreover assume "'' = Abs r' \\ rs'"
    ultimatelyshow"Absr ss \\<^sub>s r''" by simp
  next
    fix t u' us'
     "' = ' #us'"
    with Abs(3) obtain u us where
      ss: "ss = u # us" and   from <open>r \<rightarrow>\<^sub>s r'\<close>
      by cases (autoby(rulesred.Absjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
    have java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
    with haveru0 \<degree>\<degree> us \<rightarrow>\<^sub>s r'[u'/0] \<degree>\<degree> us'" by (rule lemma1')
    hence "Abs r \ u \\ us \\<^sub>s r'[u'/0] \\ us'" by (rule sred.Beta)
    moreover assume "Abs r' = Abs t" java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 3
u  " r \\ ss \\<^sub>s r''" using ss by simp
  qed
next
  casewjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  show ?case
    by(ule sred) ( Beta+
qed

lemma rtrancl_beta_sred:
  assumes r:
  showslistrelp_imp_listsp1
  by induct H: " (\x y. P x) xs ys"

subsection

inductive
  lred
  andlredlist:"dB list \ dB list \ bool" (infixl \[\\<^sub>l]\ 50)
where
  "s [\\<^sub>l] t \ listrelp (\\<^sub>l) s t"
| : " [\\<^sub>l] rs' \ Var x \\ rs \\<^sub>l Var x \\ rs'"
| Abs: "r \\<^sub>l r' \ Abs r \\<^sub>l Abs r'"
:/

by auto
  assumes lred: "s \\<^sub>l t"
  shows"s \\<^sub>s t" using lred
proof induct
  case (Var rs "WN r"and"NF r'" using
  then have " (iproverdest: listrelp_conj1 listrelp_conj2
    byinduct intro.intros
  then showNF [simplified])+
next
  case (Abs wnWNjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
  ase rs
Abs \<degree>\<degree> []" using listrelp.Nil
    by (rule sred.Abs)
  thenshow ?caseby
next
   (Beta s sst)
(iproverintro: lredintros
  show
qed

inductive WN :   "NFr
  where
    :   java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
  | Lambda cases java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
  | Beta induct

lemmash ? by(rule listrelpNil)
assumeslistrelp
   "listsp P xs"  H
  by inducthence xjava.lang.NullPointerException

lemma listrelp_imp_listsp2:
  assumes H: b ?case  (rule.Var
  showscase( rr  ss
  by induct auto ':"' []" (rule )

lemma lemma5:
  assumes lred: "r \\<^sub>l r'"
  shows "WN r" and  from ss  " Abs r')"') simp
  by induct
    iprover:listrelp_conj1
     listrelp_imp_listsp1 Abs java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
     . [simplified])+

lemma lemma6:
  assumes wn: "WN r"
  shows "\r'. r \\<^sub>l r'" using wn
proof induct
  case (Var rs n)
  then have "\rs'. rs [\\<^sub>l] rs'"
    by induct (iprover intro: listrelp
  then show ?case bylemmaWN_eq WN  \exists>t.t\<>
qed (iprover "WN t"

lemma lemma7:
  assumesr "r \\<^sub>s r'"
"NFr'\\<^sub>l r'" using r
proof induct
  case (Varthen F:" t by rulelemma5
  from\<
    by cases simp_all
  withthenhave "\\<^sub>\\<^sup>* t'" by (rule lemma2_2)
  duct
    case Nil
    show ?case by (rule listrelp.Nil)
  next
    case (Cons x y xs ys)
    hence " t' havehave t\\<^sub>s t'" by (rule rtrancl_beta_sred)
thus ? by(rulelistrelp)
  qed
  thus
next
  case (Abs r r' ss ss')
  from \<open>NF (Abs r' \<degree>\<degree> ss')\<close>
  have ss': "ss' = []" by (rule Abs_NF)
  from Abs(3) have ss: "ss = []" using ss'
    by cases simp_all
  from ss' Abs have "NF (Abs r')" by simp
  hence "NF r'" by cases simp_all
  with Abs have "r \\<^sub>l r'" by simp
  hence "Abs r \\<^sub>l Abs r'" by (rule lred.Abs)
  with ss ss' show ?case by simp
next
  case (Beta r s ss t)
  hence "r[s/0] \\ ss \\<^sub>l t" by simp
  thus ?case by (rule lred.Beta)
qed

lemma WN_eq: "WN t = (\t'. t \\<^sub>\\<^sup>* t' \ NF t')"
proof
  assume "WN t"
  then have "\t'. t \\<^sub>l t'" by (rule lemma6)
  then obtain t' where t': "t \\<^sub>l t'" ..
  then have NF: "NF t'" by (rule lemma5)
  from t' have "t \\<^sub>s t'" by (rule lred_imp_sred)
  then have "t \\<^sub>\\<^sup>* t'" by (rule lemma2_2)
  with NF show "\t'. t \\<^sub>\\<^sup>* t' \ NF t'" by iprover
next
  assume "\t'. t \\<^sub>\\<^sup>* t' \ NF t'"
  then obtain t' where t': "t \\<^sub>\\<^sup>* t'" and NF: "NF t'"
    by iprover
  from t' have "t \\<^sub>s t'" by (rule rtrancl_beta_sred)
  then have "t \\<^sub>l t'" using NF by (rule lemma7)
  then show "WN t" by (rule lemma5)
qed

end

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