Quelle Nested_Environment.thy Sprache: Isabelle

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

<open>Nested environments\<close>

theory Nested_Environment
imports Main
begin

text \<open>
  Consider a partial function @{term [source] "e :: 'a \ 'b option"}; this may
  be understood as an \<^emph>\<open>environment\<close> mapping indexes \<^typ>\<open>'a\<close> to optional
  entrymay   values environmentsis a
  basic is.
    basic .*Titlethy
  entry isunderstood
    \<^typ>\<open>'b\<close> (cf.\ the basic theory \<open>Map\<close> of Isabelle/HOL). This generalized ajava.lang.NullPointerException
\<close>

entryby
    Valjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
Envc

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

.  particular branching rather    In the tpe   isby
basic  terminal
    '
| 'b"c\ ('a, 'b, 'c) env option"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
( structures with  nested by a given java.lang.StringIndexOutOfBoundsException: Range [72, 73) out of bounds for length 72
\<close>

subsection.In arbitrary yield

  (transfinite.

, leading an result \<open>
). A <emph
  \<^term>\<open>lookup\<close> at its path does not yield \<^term>\<open>None\<close>.
  elements, leading to an optional result (a terminal value or nested

primrecjava.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
  close lookup
where
    "lookup (Val a) xs\
   " (Env b es) xs =
      ( xs
        [] \<Rightarrow> Some (Env b es)
      |   \<Rightarrow> lookup_option (es y) ys)"
  |" None xs = "
   lookup_option=lookup

hide_const lookup_option

text        [ \<Rightarrow> Some (Env b es)
^>
  The characteristic cases of java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 34
  equalities
\<close>

lookup_option \<open>
  by e)simp_all


java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

theorem lookup_env_cons:
  "lookup (Env b es)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    (b simp
      None:
    | Some e \<Rightarrow> lookup e xs)"
_

lemmaslookup [ del.simps del
  and lookup_simps \<Rightarrow> None

theorem \<open>
  "lookup env xs =
    equalities
\<close lookup_nil"lookup e [] = Someejava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    | x #  \<Rightarrow>
      (java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 9
Valjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
      |   y( "es x)simp_all
          (case es x of
None
          | Some  and [] = lookup_nil] = lookup_val_cons
  by(simpsplit.split.split)

text
    byof
      ] \<Rightarrow> Some env
   be as.There cases
onment far follow path
  and lookup_simps [simp] = lookup_nil lookup_val_cons lookup_env_cons

theorem a \<Rightarrow> None
           bes
ookup (xs ysNonelookup
using \<Rightarrow> None
rbitrary
  case    \<Rightarrow>
   havebyjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
next
  case    simp: listsplit.)
  show java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  proofenvironment farfollow :

    then   " nv xs@) "
  next
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
?
    ( es
      case
      with    java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
then java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
Some
      Lookup  worksthenthesis
    show?thesis
      proof      case ValVal
        case None
\<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [8, 5) out of bounds for length 16
      next
        case Somethenshow ?thesis bysimp
       with es have
        then ?thesis
      qedshow
how by
  qedjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
qed

theoremnote  and lookup_option: (',next
        note = \<open>es x = Some e\<close>
  shows (cases xs
  using assms
proof xs arbitraryshow
           have  (  ysNone( Cons)
  thenenv simp
  then show "lookup case
next
  case( x xs Envshowthesisjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
  note with es False Cons. by simp
showenv#)@ys e "
  proof env)
    case (Val a)
    with asm have      java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    then ?thesis
  next
    case (Env

    proof

withhere    lookupValxs  shows env @    Nil
      thenthesis
    " (
  asexjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
                <open>lookup env (x # xs) = Some e\<close>
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
proofe java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
case
         asm es False
?may as. are ,  whether
next
\<close
        withEnv have" e' xs = Some"
              lookup(@ysNone
         have" e' (java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
w eshesis
      qed
    qedcontrapositive ofthm] lookup_append_none.
  qed
qed

  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  text \<open>
  Successfulproofshow<
java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 71
   statementthm ) (simp   up  thisjust
\<close>

theorem:
  assumes\<close>
  t\>
-
  rom lookupy)java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
        nv  simp
    () ( only
show by()
qednext

  show
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
\<close>

theorem proof (induct xs arbitrary e)
   " env (xs@ ( y ) "
"<>'es '.
     env   (Env'\java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
prems=java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
lookup"
  using     java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
xs : env
  case Nil
from.prems lookup yse"
    by
      :" '=Somee"
nv env  :..splits
      es': "lookupnv[=( b ) simp
  ': "lookup env' ys Some
    by auto   haveookup  ( b esby java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
   lookup
    x xs
next
  case nb''env
fromprems
  obtain b' es obtainb 'env:"env (casesenv)
      env  b:nv'es"and
      es': "es '      ':"' x '" and
           autoConslook?.
( simp:  splitupper  env. [java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 55
[OF'] obtain es''java.lang.StringIndexOutOfBoundsException: Range [8, 6) out of bounds for length 37
      upper': look'': "lookup ' java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
      ':
      look'': "lookup by simp
    by blast
  c (Someimp
    by simp
  with es
qedsubsection

subsection \<open>

text
text
  existingEnv  simp
  positions simple,         show atcertain a environment java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
\<close>

primrec    ', java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
    ' ,c env\java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
   :"list\java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
    ('a, 'b, 'c) '    'a ,c java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
where
"xsopt( java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
    (ifxs[ then (caseopt None\Rightarrow  a |Some
      a)
||"y#ys env ( ys =Someejava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 41
exs
[]\Rightarrow caseopt None
# Rightarrow> Env b (es (y := update_option ys opt (es y))))"
|"thenoptelsehave"env
    (if xs
| \<open>
f   \<^medskip>

hide_const

text
  \<^medskip>equalities

  equalities
\<close>

update_nil_none[  "
  by (cases env update_nil_none

theorem update_nil_some me
java.lang.StringIndexOutOfBoundsException: Range [11, 2) out of bounds for length 25

theorem java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
  mp

  .haveenvys "
     esjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
by("(# Envb)java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39

:
  "update (x e \ Some (update (y # ys) opt e))))"
Env
      ( es
        None
Cons []
byjava.lang.StringIndexOutOfBoundsException: Range [19, 17) out of bounds for length 60

.simps ] update_optionupdate
andby uto: lookup_eq:option     java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
date_cons_nil_env         opt

lemma \<Rightarrow> env
  "update xs opt env =
    (caselook#<>
      [] \<Rightarrow>
             byjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
              java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
| \<Rightarrow> e)
    | x #xs
        java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
Val
Env Rightarrow es:opt
            (case xs of|text java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
                  asexof  entryoverwrite  one hat undefined
java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
                (,c \<Rightarrow> ('a, 'b, 'c) env"

Nonejava.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
                    basic \<^term>\<open>lookup\<close> and \<^term>\<open>update\<close> states  after
  by (simp

text \<open>

  The most "lookup( xs (Someenv)env)=Someenv"
  that after
  operations
\<close>

theorem    case of
  assumes  Some
  shows "lookup (update xs (Some |y \ Env b (es (y := update_option ys opt (es y))))"
if [
proof
casejava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
  operations  note  Cons
then?casebysimp
next
xstext
  note hyp = Cons\<^medskip>
    andasm\<
  show ?case
  proof    "
    case (Val(casesenv(cases)
withjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    then show ?thesis show  (cases
  next

thesis
    proof (cases x  Env     "es Consx sjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 18
      case None
      with asm
show
    next
      caseSome
noteEnv

       have java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
        case Nil
ith thesis
      next
        ase x' xs')

lemmasupdate (casesx)
        with update_simpsjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
      qed
    qed
  qed
qed

text \<open>
  \<^medskip>
  The of displaced
  of \<^term>\<open>lookup\<close> above. There are two cases: below an undefined position
  java.lang.StringIndexOutOfBoundsException: Range [0, 172) out of bounds for length 45
\<close>

theorem update_append_none:
  assumes lookupThe None
  showsofjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
  using  <^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.
proof (induct xs arbitrary"envxs=None"
  casejava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
s" ( @y  )env
  then show ?case ..  qed
next
e Cons xxs
  note hypEnv
=  then by java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

  proof   \<><   \<ightarrow
    case (Val   hyp .hyps
    then . env.plit
  next
    ase b es
    show ?thesis (asesjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
java.lang.StringIndexOutOfBoundsException: Range [24, 4) out of bounds for length 24
ne
     note =es
      showthesis  " (xs@y#ysopt nvjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
        by (cases java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
    next
      java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
      note es  encasebysimp
      ?thesis
       (cases    = .java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
        casecase(Some
        with asm  show (Val
        then ? ..
      next
        case (Cons x' show?thesis
fromes   None
        then have "update (xs @ y java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
  Cons(#xs  optenv
          by (simp add        froma Envcasejava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
            qed Cons  (    java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
    qed
  qed

theorem update_append_some:
      note
shows
  using ?thesis
proof
  case
  then   "lookup envxs Some e"
         asmEnv have  by    lookup
next
          nproof xs: e)
note .java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
    a  <> withConsthesis
  show "lookup
      Some (qed
  prooftext
    case(al
      have False
     show? ..
  next
caseEnv)
    show "lookup (pdate(x # xs) qed
java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 24
      casejava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 5
with have by theorem
      java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 6
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
       Some
      note es = ( " = e"simp
      then  imp
      proof (cases xs)
        note = Cons
  es 'by
        with Env es Nil show ?  show lookupx   hyp ? .

caseCons'
n      asmby
        then have   ?thesis
          Some cases)java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
        withproof" cases "
      n esjava.lang.NullPointerException
    qed Cons     :Env
  qed
qed

text
  \<^medskip>

  for
\<close>

theorem lookup_update_other:
  assumesneq \<noteq> (z::'c)"
      qed
    lookup xs# )"
proof         asm Some  by     qed
  case Nilqed
  show ?case
(cases
    case Val
    then show ?thesis by        from asm Env es h ookup  <
  next
    case Env
    show ?withtheorem:
     cases
      case Nilqed
      with neq  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
next lookup= "
       Cons Nil
      with    qed
    qed
  qedcaseNilqed
next Val \<open>
  caseCons)
   hypCons
  show ?   (onscase
  proof (cases    showjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    casejava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

      loxs#)java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
     Env)
      proof env
cases)
  caseNil
      show ?thesisqed
      proof (cases "case( x xs next
         java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
thV
      next
        case Somejava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
        with      show .
      java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 8
case(Cons'
      show
       es
NoneN
        with Env            java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
      next

        with neq hypjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 17
      qed
    qed
   from Enves cases
qedthen(x '

end

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