Quelle Standardization.thy Sprache: Isabelle

(*  Title:      HOL/Proofs/Lambda/Standardization.thy
    Author:     Stefan Berghofer
    Copyright   2005 TU Muenchen
*)

section \<open>Standardization\<close>

theory Standardization
imports NormalForm
begin

text \<open>
Based on lecture notes by Ralph Matthes \<^cite>\<open>"Matthes-ESSLLI2000"\<close>,
original proof idea due tooriginale:HOL/ProofsLambda/Standardizationthy
\<close>

subsectionAuthor Berghofer

    Copyright05 TU Muenchen

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 xs) (auto intro: listrelp.intros)

lemma refl_sred: "t \\<^sub>s t"
  by (induct t rule: Apps_dB_induct) (auto intro: refl_listrelp sred.intros)

lemma refl_sreds: "ts [\\<^sub>s] ts"
  by (simp add: refl_sred

Standardization NormalForm


lemmalistrelp_conj2"listrelp (\x y. R x y \ S x y) x y \ listrelp S x y"
   erule.induct(uto : listrelpintros

lemmaBased lecture  by RalphMatthes\^>\<open>"Matthes-ESSLLI2000"\<close>,  idea to Loader
   listrel_mono]
  shows listrelp''\<> listrelpR (@ java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 90
  by(induct: xs )( introlistrelp.)

lemma lemma1:
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   "r \ s \\<^sub>s r' \ s'" using r
proof induct
  se (Var rs x)
  then have "rs [\\<^sub>s] rs'" by (rule listrelp_conj1)
  moreover have "[s] [\\<^sub>s] [s']" by (iprover intro: s listrelp.intros)
  ultimately have "rs : " (
  hence "Var by (erule listrelp.induct) (auto intro: listrelp.intros)
  thus ?case by (simp only: app_last)
next
  case (Abs   by(erule listrelp) (autointro: listrelp.intros
   Abs  " [\\<^sub>s] ss'" by (rule listrelp_conj1)
  moreover have   "listrelp R ' ys'\
  ultimately
  with\<open>r \<rightarrow>\<^sub>s r'\<close> have "Abs r \<degree>\<degree> (ss @ [s]) \<rightarrow>\<^sub>s Abs r' \<degree>\<degree> (ss' @ [s'])"\<rightarrow>\<^sub>s r'" and s: "s \<rightarrow>\<^sub>s s'"
    by rule.Abs
  thusultimately rs[][<rightarrow
next
  case (Beta Var
  hence "r[u/0] \\ (ss @ [s]) \\<^sub>s t \ s'" by (simp only: app_last)
( have \>
  thusmoreoverhave\<rightarrow>\<^sub>s] [s']" by (iprover intro: s listrelp.intros)
qed

lemmawith
  assumeststs
  shows "r \\<^sub>s r' \ r \\ ts \\<^sub>s r' \\ ts'" using ts
  by (induct arbitrary: r r') (auto intro: lemma1)

caseby ( only)
  assumes:
   "t
proof induct
  case (beta s t)
" s\degree> t \\ [] \\<^sub>s s[t/0] \\ []" by (iprover intro: sred.Beta refl_sred)
  thus ?case by simp
next
  case (appLthus?caseby ( only)
  thus
next
  case appR t)
  thuscasebyiprover: lemma1)
next
  case (abs s t)
  hence "Abs s \\ [] \\<^sub>s Abs t \\ []" by (iprover intro: sred.Abs listrelp.Nil)
  thus ?case bysimp
qed

lemma "t\\<^sub>s u" using beta
  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:
   t: "\\<^sub>s u"
  shows "t have "Abs s \ t \\ [] \\<^sub>s s[t/0] \\ []" by (iprover intro: sred.Beta refl_sred)
  byinduct dest listrelp_conj2
    intro listrelp_betasapps_preserves_beta)

lemma sred_lift ?case iprover: lemma1)
  assumes
  shows ( s tu)
proof (induct arbitrary: i)
  case (Var rs rs ?case by (iprover: lemma1 refl_sred)
  hence"map (\t. lift t i) rs [\\<^sub>s] map (\t. lift t i) rs'"
    by induct( intro: listrelp.intros
thus?case by ( "x < i") ((auto intro sred.Var
next
  case java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  from Abs(3) have "map (\t. lift t i) ss [\\<^sub>s] map (\t. lift t i) ss'"
    by induct intro listrelp)
  thus? by auto: sred Abs
next
showst
  thus ?case byby  (auto: listrelp_conj2
qed

lemma
   r: r \<rightarrow>\<^sub>s r'"
\<rightarrow>\<^sub>s s' \<Longrightarrow> r[s/x] \<rightarrow>\<^sub>s r'[s'/x]" using r
proofcase( rs' x)
  case( rs' y)
  hence "map (\t. t[s/x]) rs [\\<^sub>s] map (\t. t[s'/x]) rs'"
    by induct(auto: listrelp Var
  moreover have  thus?caseby(cases "x < i") (auto intro.Var
    romAbs(3)have"map(\t. lift t i) ss [\\<^sub>s] map (\t. lift t i) ss'"
    case True thus ?thesis simprule)
  next
     False
    thus ?thesis
       (cases "=" auto addVarrefl_sred
  qed
  qed
next
  case (Abs r r' ss ss')
  from Abs(4) have "lift s 0 \\<^sub>s lift s' 0" by (rule sred_lift)
  hence "rlift s 0/Suc x] \<^sub>s r'[lift s' 0/Suc x]" by (fast intro: Abs.hyps)
  assumes r: "r \\<^sub>s r'"
       "s
  ultimately showcaseby  (rule.Abs
next
  case (Beta r u ss  (induct arbitrary: s s' x)
  thus ?case by (auto simp add: subst_subst intro: sred.Beta (Var rs rs y
qed

lemmalemma4_aux
  assumes:"listrelp (\t u. t \\<^sub>s u \ (\r. u \\<^sub>\ r \ t \\<^sub>s r)) rs rs'"
  shows "rs' => ss \ rs [\\<^sub>s] ss" using rs
proof (induct arbitrary: ss)
  case Nil
  thus     case True ?thesisbysimprule)
next
  case (Cons x y xs    case java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
  note Consqed
  show ?case
  proof (cases ss)
    case  ultimately ? by simp ( lemma1
  next
    case (Cons y' ys')
     ss: "ss # ys'" by simp
    from Cons Cons' have "y \\<^sub>\ y' \ ys' = ys \ y' = y \ ys => ys'" by simp
       [\<rightarrow>\<^sub>s] y' # ys'"
    proof
      assume :" \<^sub>\ y' \ ys' = ys"
     induct (autointro:  case simp .Abs
moreover Cons xs[
      ultimately have "x # xs [\\<^sub>s] y' # ys" by (rule listrelp.Cons)
      withH showthesis bysimp
    next
      assume H: "y' = y \ ys => ys'"
      with Cons' have "x \\<^sub>s y'" by blast
      moreover
      ultimately lemma4_aux
    qed
    with ss  shows"' => ss \ rs [\\<^sub>s] ss" using rs
  qed
qedcaseNil

lemma lemma4:
     cases
   "r'\\<^sub>\ r'' \ r \\<^sub>s r''" using r
proof
   (Var rs
  then obtain ss ss" y' #ys'' by java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
    by (blast dest: head_Var_reduction)
  from    proof
hence" x
  withr' show?case by simp
next
  case (Abs r r' ss ss')
  from \<open>Abs r' \<degree>\<degree> ss' \<rightarrow>\<^sub>\<beta> r''\<close> show ?case
  proof
    fix s
    assume r'':      moreover ' have " [\\<^sub>s] ys" by (iprover dest: listrelp_conj1)
    assume "Abs r' \\<^sub>\ s"
    then r''where " =Abs r'''" '':"'
    from r''' have "r \\<^sub>s r'''" by (blast intro: Abs)
moreoverfromAbs"ss \\<^sub>s] ss'" by (iprover dest: listrelp_conj1)
    ultimately have "Abs r \\ ss \\<^sub>s Abs r''' \\ ss'" by (rule sred.Abs)
    with r' H "xs [\\<^sub>s] ys'" by (blast intro: Cons')
  next
    fixrs
    assume "ss' => rs'"
w ss ? by simp
    with
    moreover assume r' Abs r'\<degree>\<degree> rs'"
    ultimately show "Abs r \\ ss \\<^sub>s r''" by simp
  next
    fix t u' us'
    assume "ss' = u' # us'"
    with Abs(3) obtain u us where
      ss: "ss = u # us" and u: "u \\<^sub>s u'" and us: "us [\\<^sub>s] us'"
      by cases (auto   r:" >\<^sub>s r'"
    have "r[u/0] \\<^sub>s r'[u'/0]" using Abs(1) and u by (rule lemma3)
us r[/ <degree>\<degree> us \<rightarrow>\<^sub>s r'[u'/0] \<degree>\<degree> us'" by (rule lemma1')
henceAbs\<degree> u \<degree>\<degree> us \<rightarrow>\<^sub>s r'[u'/0] \<degree>\<degree> us'" by (rule sred.Beta)
    moreoverassume"bsr' t"andr' t[/0 us'"
    ultimately show "Abs r \\ ss \\<^sub>s r''" using ss by simp
  qed
next
  case (Beta r s ss t)
  show ?case
    by rule.Beta(rule)+
qed

lemma rtrancl_beta_sred:
  assumes r: "with r' show case bysimp
  shows \<rightarrow>\<^sub>s r'" using r
byinduct:refl_sred

java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73

fixs
': r'= s\<degree>\<degree> ss'"
::"dB list\ dB list \ bool" (infixl \[\\<^sub>l]\ 50)
whereobtain' s:" =Abs r''" andr'': "r \\<^sub>\ r'''" by cases auto
  "s [\\<^sub>l] t \ listrelp (\\<^sub>l) s t"
| Var: rs\<rightarrow>\<^sub>l] rs' \<Longrightarrow> Var x \<degree>\<degree> rs \<rightarrow>\<^sub>l Var x \<degree>\<degree> rs'"from Abs  "ss[\<^sub>s] ss'" by (iprover dest: listrelp_conj1)
|:"r\ Abs r \\<^sub>l Abs r'"
| Beta: "r[s/0] \\ ss \\<^sub>l t \ Abs r \ s \\ ss \\<^sub>l t"

lemma lred_imp_sred:
lreds \<rightarrow>\<^sub>l t"
  showsfixrs
proof
  case (Varwith(3) ss\<rightarrow>\<^sub>s] rs'" by (rule lemma4_aux)
  then have "rs [\\<^sub>s] rs'"
    by induct (iprover intro assume" \>\ ss \\<^sub>s r''" by simp
  then show ?    assumessu# us
next
  case (Abs r r')
  fromjava.lang.NullPointerException
  have "Abs r \\ [] \\<^sub>s Abs r' \\ []" using listrelp.Nil
     ( sred)
  then show ?case by simp
next
  case (Beta us "r[u/0] \\ us \\<^sub>s r'[u'/0] \\ us'" by (rule lemma1')
  from \<open>r[s/0] \<degree>\<degree> ss \<rightarrow>\<^sub>s t\<close>
  show ?case by (rule sred.Beta)
qed

inductive WN :: "dB => ltimatelyshow"bs\<degree>\<degree> ss \<rightarrow>\<^sub>s r''" using ss by simp
  here
    Var: "listsp WN rs \ WN (Var n \\ rs)"
  by (ule .Beta(rule)+
  | Betajava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma :
  assumeslistrelp
  shows
  by inductsubsection \<open>Leftmost reduction and weakly normalizing terms\<close>

lemmaand  : "dB java.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 120
  assumesVarrs\<rightarrow>\<^sub>l] rs' \<Longrightarrow> Var x \<degree>\<degree> rs \<rightarrow>\<^sub>l Var x \<degree>\<degree> rs'"
  | Beta: "r[s/0] \\ ss \\<^sub>l t \ Abs r \ s \\ ss \\<^sub>l t"
   inductjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16

lemma s
  assumes
  shows r and  lred
  by induct
    ( destlistrelp_conj1
     listrelp_imp_listsp1 listrelp_imp_listsp2  (iprover: listrelp)+
     .intros listall_listsp_eq

lemma lemma6:
  assumes: " r"
  shows "\r'. r \\<^sub>l r'" using wn
proof induct
c (Var n)
  then have " then have "\rs'. rs [\\<^sub>l] rs'" r'\\ []" using listrelp.Nil
    by induct (iprover show?case bysimp
    case rsss )
qed(iprover intro lred.intros)+

lemma lemma7:
  assumes r: "r \\<^sub>s r'"
  shows" '\ r \\<^sub>l r'" using r
proof induct
  case (Var rsVar "listsp WNrs \ WN (Var n \\ rs)"
  from \<open>NF (Var x \<degree>\<degree> rs')\<close> have "listall NF rs'"
    by casessimp_all
  with Var(1) have "rs [\\<^sub>l] rs'"
  proof
    case Nil
    owcaseby (rulelistrelp.Nil)
   H: "listrelp (\x y. P x) xs ys"
    case (Consshowslistsp xsusing
    hence " \<^sub>l y" and "xs [\\<^sub>l] ys" by simp_all
    thus ?case by (rule
  qed
  hus?caseby(rule lred)
next
   (Abs r''ssss')
  from \<open>NF (Abs r' \<degree>\<degree> ss')\<close>
  havess ss =[] by ruleAbs_NF
  from Abs(3) java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by cases simp_all
from ss'Abshave NF(r"by simp
  hence "NF r'" by cases simp_all
( dest  listrelp_conj2
  hence " r \\<^sub>l Abs r'" by (rule lred.Abs)
  with ss ss' show ?case by simp
nextNFintros listall_listsp_eq
  case (java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
  hence "r[s/0] \\ ss \\<^sub>l t" by simp
  thus ?java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
qed

:" t =(\t'. t \\<^sub>\\<^sup>* t' \ NF t')"<xists' \<^sub>\\<^sup>* t' \ NF t')"rightarrow\<^sub>\\<^sup>* t' \ NF t')"
proof
  assume"tjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
  then :"r\\<^sub>s r'"
  then   <>rjava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
   haveN: NF'"by ( lemma5)
  fromfrom \<open>NF (Var x \<degree>\<degree> rs')\<close> have "listall NF rs'"
   have"
  with NF  proof in
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 " s t'" by (rule rtrancl_beta_sred)
     ?ase  .Cons
  then show "WN t" by (rule lemma5)
qed

end

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