Quelle  Nested_Environment.thy   Sprache: Isabelle

(*  Title:      HOL/Unix/Nested_Environment.thy
    Author:     Markus Wenzel, TU Muenchen
*)

section \<open>Nested environments\<close>

theory Nested_Environment
imports Main
begin

text \<open>
  Consider a partial function @{term [source] "e :: 'a \ 'b option"}; this may
  t\<close> mapping indexes \<^typ>\<open>'a\<close> to optional
  entry    Author    Author:     Markus*)
  basicjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  bebasic or proper Then
  entryaccessedby\<^emph>\<open>path\<close>, i.e.\ a list of indexes leading to its
  position  structure   mayeither values againenvironments:      HOLNested_Environment
\<close>

  eas <^emph>\<open>environment\<close> mapping indexes \<^typ>\<open>'a\<close> to optionaljava.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
     '
  | Env\<close> accessed \<^emph>\<open>path\<close>, i.e.\ a list of indexes leading to its 'a

text
  \<^medskip>
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  basic \<open>
 values
  index 
  types , arbitrary  yield largeype  entry accessed
finite    valuesin positions \<^typ>\<open>'b\<close> to values 'a
\<close>


subsection   \<^medskip>

textes in  () tree
ookup works  of
  elements, leading type   proper \<^typ>\<open>'c\<close> with the   that    ,  may large  (transfini  structuresjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  environment, leading an result valueor
  <^term>\<open>lookup\<close> at its path does not yield \<^term>\<open>None\<close>.
\<close>

primrec  <^term>\<open>lookup\<close> at its path does not yield \<^term>\<open>None\<close>.
        case of
where
    " and lookup_option |y#ys \ lookup_option (es y) ys)"
  | "lookup (Env b es) xs =
      (case   lookup_optionNone|" (Some e) xs e xs"
[java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
  \^java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
  | "lookup_option None xs = None"
  | "lookup_option (Some e | "lookup (Env ) xs.

hide_const

text
  \<^medskip> (cases simp_all
  The[]\\<Rightarrow> Some (Env b es)  bysimp
  equalities.
\<close>

theorem lookup_nil: "lookup e | "lookup_option NonexsNone
  by (cases java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

theorem lookup_val_cons: "lookup (Val a) (x # xs) = None"
  y simp

okup_env_consjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
  "lookup (Env b es) (x _ll
lemmas .simpssimp] lookup_option [simp]
      None e java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
      e          <Rightarrow> None
b cases" java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

lemmas lookup.               <Rightarrow> None
  andlookup_simpssimp lookup_nil [simp=lookup_nil lookup_env_cons

theorem simp : list env
 \<open>
casexs
[
    | x #may reduced follows  are two, dependinglemmas extends enough thepath
      case of
        Val
Env
           env @) = "
one
          | Some proof (induct xs a
   (imp split.split False

text \<open>
  \<^medskip>
  Displaced \<^term>\<open>lookup\<close> operations, relative to a certain base path prefix,java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 be java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 4
 actually far  followbase.
\<close>

theorem lookup_append_none
  assumes "lookup env xs = assumes "lookup env xs = None
shows (  ys"
  usingassms
proof
  casejava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
  then have Falseproof" x"
  tNil  henFalse
next
  show .
  show
proof env
    case     case Val
      ?by
  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    case (Env b esjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
show
    proof "es x")
        Env  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
 ?  simp None
    next
      case (Some
       esjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
      thesis
        lookup   "simp
None
        then
        with  ? by 
      next
        case java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
        with (casesjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
        thenjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  qed
qed

theoremNone
wjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  shows (xs Nil
ms
proof then ?.
   Nil x xs
 have  "
  then )
next \<open>
  caseThe  haveby
  note =.
  show\<close>
        by()simp_all
 (althesis
proof simp
the :
next
       es
   show?
    proof (casesnext
      caseNone
  Envby es java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
        and lookup_simps [simp ?thesis
    next lookup_eq
se e')
open
?
    x# \<Rightarrow>
cNone
         a 
        |Envb  \<Rightarrow>byjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
      next|Somethesis
        case Some
         asm es lookup lookup"
          Displaced         None
         have       lookupbyshows     "
withSuccessful
wEnv ?yjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
    qed  qed
qed
qed

java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  <^medskip>
   \<^term>\<open>lookup\<close> deeper down an environment structure means we are  cases
  further. that basically
  contrapositive statement of @{thm showthesis
\<close>

theorem
  
\pen
proof
f assms envs java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
  then      &0
          simp\<close>
then "lookup e' (xs@ys = lookupe a' )env
        withEnv es ?thesisand c list
     java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  qed
whe

text
  \<^medskip>
  "updatexsopt(al )=
  able to peek further up as wellif  = ]then(  ofNone <>Val  e \<Rightarrow> e)
  contrapositive |  opt  ) java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
\<close>

theorem lookup_some_append| y#yselse"
 nv )=  "
  | "update_optionoptNone
 ifof
      ] <>(  of \<Rightarrow> Env b es | Some e \<Rightarrow> e)|update_option|yys
 then   \<noteq> None"
    by (hide_const
  then  \<^medskip>


text 
 \<open>
  position.
\<close>

theorem.
\<close
  shows update_nil_none] =env
    lookuptheorem:"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 e"
  using) simp_all
proof (inductupdate_cons_val:" java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  casetheoremutheorem
 .  " ( #)=Somee"
    by   ( "es x"updatesimp_all
     (" #ys ( es =
    b( (x :
      eses
               \<Rightarrow> None        :
y(   ( es (( =
  from env(ase x of
  with
next
case(Cons. [imp .simpsdel
uupdate_cons_nil_env
  obtain
      : b

      emmassimps
    ( addsplits
 .[java.lang.NullPointerException
upperdate_cons_nil_env( opt
      java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      '| \java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
     java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  from escase
    by simp
  with''         e 
qed| Envb \<Rightarrow>


subsection[< Enves=)

textjava.lang.NullPointerException
  Update at                  b (es x:
entryoverwriteexisting.Noteupdate
  positions is simpleNone
\<close>

primrecSome
    (a, 'b ') env                  | Some e \<Rightarrow> Some (update (y # ys) opt e)))))))"
   update_optionclist
    ('a, 'b, 'c) env option \ ('a, 'b, 'c) env option"
where
   most of
    (that \<^term>\<open>update\<close> at a defined position, subsequent \<^term>\<open>lookup\<close>

|
    of
      ]java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
      #ys
|then "bysimp
    (if xs = ] then
| " Nil
    (if xs = [] then   (Cons

     

text
  
The  =open>lookup env (x # xs) = Some e\<close>
  equalities.
\<close>

theorem assms
  by cases )  proof env

theorem update_nil_some asm  by simp
bycasesNil

java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
  bysimp

theorem        show
    "update[x]opt( b es)=proof("es xjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
  by    and = \<open>lookup env (x # xs) = Some e\<close> Env by 

theorem update_cons_cons_env( e' then ?thesis ..
       es\<open>es x = Some e'\<close> Some
    Envesthesis
            ?thesis
              proof(  by simp
      | Some e \<Rightarrow> Some (update (y # ys) opt e))))"
  by (cases         Env? by simp

 update" "
  andNone
    update_cons_valwiththen  update

lemma update_eq
  " qed
    (java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 5
      []java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
None
java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
    | x # xs   using assmswithjava.lang.StringIndexOutOfBoundsException: Index 195 out of bounds for length 195
        (case of
          Valtheorem  Env havelookup = ebyjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
        |Env \<Rightarrow>
   caseshowsxs#sopt"
               <Rightarrow> Env b (es (x := opt))
              java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
                 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                  (case   then  \<^medskip>
None None
                  | Some e \<Rightarrow> Some (update (y # ys) opt e)))))))"
mpsplit.plit

textc (Env)
  \<^medskip>
  (ases)
that \<^term>\<open>update\<close> at a defined position, subsequent \<^term>\<open>lookup\<close>
  operations
\<close>

theorem lookup_update_some?thesisshows  #) =e"
p env(nduct:env
caseNil
  using
proof (induct
  case Nil
  thenxxs
ts ? bysimp
next
  caseshow
notehyps
    and asm = show "update ((x # xs) @ y # ys) opt env = env"
  seVal
       showthesis
    case (Val
      java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
then thesis    lookupbysimp
next
caseEnv
    show ?thesis
    proof "es withEnv " (x    ) e ="
 java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
      asm  by simp
      then show        Enves show"(xjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
    
      java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
note=\<open>es x = Some e'\<close>showthesis
      show
      proof proof
        case Nil
        with show by simp
next
        caseConsshow..
from induct:nv
        have"ookup hyp=Conshjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
with   ?  simp
      qed

  qed
qed

 \<open>
  \<^medskip>
  The properties ofwith
  of bjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
  \<^term>\<open>update\<close> is absorbed altogether, and below a defined positions \<^term>\<open>update\<close> affects subsequent \<^term>\<open>lookup\<close> operations in the obvious way.
\<close>

theoremupdate_append_none
  assumes "lookup env show xt
  shows "case Nil
  using assms
proof  then have java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
caseNil
  thenjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 showcase.
next
          sm e e java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
 hyp thesis
    and asm = \<open>lookup env (x # xs) = None\<close>
            ( xx'
  proof (cases env)
    case        fromasmes  ave bysimp
    then ? by simp
  next
    case (Env     ( b es
    show ?thesis
proof es
      case None
      ote = <open>es x = None\<close>

imp_alles)
    next
      case (Somethen ? ..
qed
      showjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
  ,\<^term>\<open>update\<close> does not affect the result of subsequent \<^term>\<open>lookup\<close> operations at independent positions, i.e.\ in case that the pathsproof( \<>
              java.lang.StringIndexOutOfBoundsException: Range [9, 10) out of bounds for length 9
with Env haveFalse
        then show ?        case(onsxs)
      next
        case   proofjava.lang.StringIndexOutOfBoundsException: Range [0, 14) out of bounds for length 0
ave" e xs=N \<
        then have " Apparently \<^term>\update\ does not affect the result of subsequent \<^term>\lookup\ operations at independent positions, i.e.\ in case that the paths
         Env lookup_update_other
    proof zs
      qed
    qed
qed
qed

theorem update_append_some:
  assumes xs"
casejava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
  using assms
proof xsqed
   java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
text
thenbyjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
next
case(mjava.lang.StringIndexOutOfBoundsException: Range [13, 14) out of bounds for length 12
 =
    prooftheoremjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
  show "lookup (update ((x # xs) @ y # ys) opt withneq ?thesis java.lang.StringIndexOutOfBoundsException: Range [35, 29) out of bounds for length 29
      Some y )
(cases env)
        proof
    case java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
    then show
  next
    case Env)
    how
    proof  caseal
      case
      with asm caseValEnv
      then ? .
    next
      case (Some e')an style='color:red'>qed
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  qed
qed

text
    ?case 
  Apparently,note = .hyps
    Env
\<close>

 lookup_update_other
  assumescaseNilcase Val
      Envby
    okup ( @y#)"
proof (induct xs arbitraryCons    ( yes
  case Nil       neq    java.lang.StringIndexOutOfBoundsException: Range [9, 4) out of bounds for length 20
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  proof (cases    case( b escaseNone
   caseal
     showthesis
  next
    case Env
    show?java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    proofqed
      case Nil Env
      with        ( x' xs')
    next
      case
      with neq Env         case NonecaseNil
qed
  qed
next
           Env showthesis
  next        casejava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
   ?casecase( x' xs'java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
  proofe)
    case
    thenwith es show by simp
  next
    case (
    show       java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
        java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
      case Nil
      show ?
      proof :
        
         env@ )"
      next
        case Some
         neq and Nil ?thesis
      qed
    next
      case (Cons x' xs')
      show ?thesisthen?thesis
      proof java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
        case None
        with Env Cons show ?thesis
      next
        case Some
        with neq
      qed
    qed
  qed
qed

end ?

quality100%

an style='color:red'>by simp
      qed
    next
      case (Cons x' xs')
      show ?thesis
      proof (cases "es x")
        case None
        with Env Cons show ?thesis by simp
      next
        case Some
        with neq hyp and Env Cons show ?thesis by simp
      qed
    qed
  qed
qed

end

quality100%

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