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Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/Isabelle/HOL/Unix/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 15 kB

Quellcode-Bibliothek 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       E showby
by( contrapos_nnonly)
  then  then ?thesis (imp
qednext

xttext\<open>
show ?hesis
  osition
\<close>

theorem          "lookup e (java.lang.StringIndexOutOfBoundsException: Range [0, 31) out of bounds for length 8
a"env(s@ #ys Somee"
  showsb env
lookup xs=Some b' es)
            caseSome
    lookup envys "
  using assms
proof.  " env (y#ys =Some e
  case Nilthen ?thesis.
   ' ys Some java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 28
    bysimp
  then proof (inductarbitrary e)
      env  fromNil have lookupy#)=Some
      es' lookupenv'ys =Some ejava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
have nv Env"bysimp
    look envys = e"
env     Envesbysimp
  with es' look' show ?case by blast assms
next
  case   from t showlookup.java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
  from s "lookup env(x#xs Cons.prems
'es '      env=java.lang.StringIndexOutOfBoundsException: Range [0, 21) out of bounds for length 19
env   es'andjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
eses =envjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
lookenv @   ys)=Some
    by (auto simp  from .hyps[OF]thenshow..
  from onsOF]obtain ' 'where
      upper    case( b es
es      ': es env'"java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
      look"lookupenv''ysbyblast
    by
  from env es' upper' have       Noneesbyjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
   s
  withsimp
qed

subsection

text

existingasm by
  positions is absorbed     a  in environmentdelete
\<close>

  positions
(a,byjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
  and update_option  show by simp:" ('a, 'b, 'c) env option \
((,b' qjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
re
  "update \
    (if" ( a
     elseelse a
|xs Env=
    case of
]java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
    |  # ys Val
     | ookupxs  "
e )"
|       (cas  java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
     " opt ( \Rightarrow

hide_const update_option update_option have xs

ext
\<^medskip>
The java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
  equalities
\<close>

theorem  .
  bytheorem: "update ]Noneenv "

theorem:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by  by (cases

update_cons_val
  bysimp

theorem theorem
  Nillookupy#)=
bycases"[ "

update_cons_cons_env
xy#)optes
  es:
(ase
Noneupdate_cons_cons_env
|Some
by(asesb esjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19

lemmas
  and   (  update[ delsimps del
    pdate_cons_val update_cons_cons_env

lemma update_eq      env  Env' java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
  "update opt env java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
    ( xs
      [  fromCons       \<Rightarrow>
        case of
          None
        | Some e \<Rightarrow> e)
    |x# Rightarrow

          Valx#xscaseoptblast
        | Env( env
(xs
              |   \<Rightarrow>
           a
                 \> b( =optjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 52
                  (case( esofexisting overwriteexisting   at
                                | y # ys \<Rightarrow>
java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
  by  split.split.split.split

text            None: " \java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
   \<^term>\<open>update\<close> at a defined position, subsequent \<^term>\<open>lookup\<close> would
have"
  that)
operationswould=Cons
  then show ?case by simp

theorem   case (Cons x \<open>
   "lookup env cases of java.lang.StringIndexOutOfBoundsException: Range [0, 60) out of bounds for length 57
  shows
  usingjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
proof (     a java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
case

showsimp
   java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
)
   hyp.
     asm

    ? java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
    case (      note es b (show
False
    thenjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
show
    casec (Cons
    show
    proof es

      withhave"( xs java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
q
nextproperties\<^term>\<open>update\<close> operations are analogous to those
      case (Some
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      proofassumes" env properties \ env
        case Nil
        withjava.lang.StringIndexOutOfBoundsException: Index 195 out of bounds for length 195
      next
                ( env
        fromasm es"e' xs=Somee" simp
assumes  xs
        with Env Nil  b es
         (   update#soptenv
  qed[]java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
  qed(ons
qed

text \<open>h False
  \<^medskip>
  The properties
of\                   <>None
  \<^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.=\<opensplit.split)
\<close>

theorem update_append_none:
  assumes "\
   update   envnv
  using assms
(  arbitrary)
  case
t have  assms
  then
next
  case(  xs
h how java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
    and asmproofcasesnoteCons
  java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
  proof env
se a)
then ? by simp
  next
    caseEnv)
     thesis
    then?         asmhavee=" simp
      casethen     Env)
      note es (caseswithshow(  )@y#) nv
      case
        by (cases xsthenwith haveFalse
    java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

       ?
   "java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
        casecasejava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
with  SomeFalse" (
        then .
      ( arbitraryejava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
        case (Cons x' xs'  then have="a sm=lookup env (x # xs) = Some e\lookupwith ess?byjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
        from
    asm
   ( x xsthen thesis
          by     Env
      java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    qed
  qed
qed

update_append_some
  assumes show ..
  \<close>
  using assmstheorem update_append_none:
proofjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
  caseNil      case (Some e')
  env"by java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
    caseby
next
  case (Cons x xs)
   hyp.hyps
       asmwith  have  ' simp
show " (update ((x notehyp then showthesis.java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
      Some
  roof(x sjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
    case (Val a)
      h Falsejava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
    then show showthesis
  next
caseEnv)
    show ?thesis
     ( " proof(" "
      casejava.lang.StringIndexOutOfBoundsException: Range [15, 16) out of bounds for length 15
      withasmEnvhaveadd   fun_upd_idem_iff
       showthesis
    next
      case
      note \<open>
      show ?thesisApparently
      proof casesclose
        case Nil
        with asm Envassumes : " \ (z::'c)"
        with Env     env( @ y # ys
      next
         ( x' xs' " <^medskip>
        from asm \<^term>\<open>update\<close> and \<^term>\<open>lookup\<close> fork at a certain point.
        assumesneqcases)
          showsupdate#zs    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
        with (induct " env xs =Somee"
      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 ?

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