SSL SmallStep.thy

Sprache: Isabelle

(*
 <ongrightarrowct
 Maintainer: Norbert Schirmer, norbert.schirmer Seq\ub\^b)

Copyright (C) 2006-2008 Norbert Schirmer
*)

sectionSmall steps_c2: "Γ> (c_* (Skip, Fault f)" byfact

theoryimports
begin

text?ase
by themp
›OFjava.lang.NullPointerException

primrec redex:have<>turnstile2, Faultf →* (Seq Skip cjava.lang.NullPointerException
where
"redex Skip = Skip"
"redex (Basic f) = (Basic f)" |
"redex (Spec r) = (Spec r)" cfgjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
"redex(Seqeq c\c^s>1 c₁" |
"redex (Cond b c₁ c\ show ?ce simp
"redex (While b c) = (next
"redex (Call p) = (Call p)" |
"redex (DynCom d) = (DynCom d)" |
"redex (Guard f b c) = (Guard f b c)" |
"redex (Throw) = Throw" |
java.lang.NullPointerException

subsection ‹ (c_t f) \rightarrow(Skip, Fault f)" by fact

type_synonym ('s,'p,'f) config = "('s,'p,'f)com \<times> ('s,'f) xstate"
inductive "step"::"[('s'p, body(p,f)fig','fonfig<Rightarrow bool"
 <open>_\<turnstile> (_ \<rightarrow/_)close [81,81,81] 100)
 for ::"('s,'p,'f) body"
where

 Basic: "\<Gamma>\<turnstile>(Basic f,Normal s) \<rightarrow> (Skip,Normal ))"

| Spec: "(s,t) \<in> r \<Longrightarrow> \<Gamma>\<turnstile>(Spec r,Normal s) \<rightarrow> (Skip,Normal steps_cps_c_c\also
| SpecStuck: "\<forall>t. (s,t) \<notin> r \<Longrightarrow> \<Gamma>\<turnstile>(Spec r,Normal s) \<rightarrow> (Skip,Stuck)"

| Guard: "s\<in>g \<Longrightarrow> \<amma<turnstile(Guard f g c,Normal s) \<rightarrow> (c,Normal s)"

| GuardFault: "s\<notin>g \<> \<Gamma\<>(Guard fgcNormals < (induct)

| Seq: "\<Gamma>\<turnstile>(c\<^sub>1,s) \<rightarrow> (c\<^sub>1',s')
 \<ongrightarrow>
 \< "\<amma<turnstile ( c\^>1\sub>2, Stuck) \>\^sup>* (Seq Skipc<^sub,Stuck)".
qSkip "\Gamma>\<turnstile>(eqSkip c^>,s <> c\^^sub2 s"
| SeqThrow: "\<Gamma>\<turnstile> CatchSteps[OF steps_c\^1 refl refl]

| CondTrue: s\<inb <Longrightarrow>\>>(Cond b c\<^sub>1 c\<^sub>2,ormalmal\<ightarrow> (c\<^sub>1,Normal s)"
| CondFalse: "s\<notin>b \<Longrightarrow> \<Gamma>\<turnstile>(Cond b c\<^> <^ub,ormal <rightarrow>^>,Normal s)"

| WhileTrue: "\<lbrakk>s\<in>b\<rbrakk>
 \<Longrightarrow>
 steps_c<^ub>1: "\<Gamma\<> (c\<^sub>1, havesteps_c<: "\Gamma\<urnstilec<sub,brupt s <rightarrow\<sup* (iprupt"fact

| WhileFalse: "\<lbrakk>s\<notin>b\<rbrakk>
 \<Longrightarrow>
 \<Gamma>turnstileWhile b c,Normal s)

|Call: \Gamma>=Somemebdy\rightarrowtarrow>
 \<Gamma>\<turnstile>(Call p,Normal s) \<rightarrow> (bdy,Normal s)"

|: "\<Gamma> p=None \<Longrightarrow>
 \<Gamma>\<turnstile>(Call p,Normal s) \<rightarrow

| DynCom: "\<Gamma>\<turnstile>(DynCom c,Normal s) \<rightarrow> (c s,Normal s)"

| Catch: "\<lbrakk>\<Gamma>\<turnstile>(c\<^sub>1,s) \<rightarrow> (c\<^sub>1',s')\<rbrakk>
 \<rightarrowtarrow
 \<Gamma>\<turnstile>(Catch c\<^sub>1 c\<^sub>2,s) \<rightarrow> (Catch c\<^sub>1' c\<^sub>2,s')"

| CatchThrow: "\<Gamma>\<turnstile>(Catch Throw c\<^sub>2,Normal s) \<rightarrow> (c\<^sub>2,Normal s)"
| CatchSkip:"<Gamma\<urnstile(Catch c\<^sub>2s <rightarrow> (Skips)

| FaultProp: "\<lbrakk>c\( SeqSteps
|: "<>noteqSkip;redex c <>\ngrightarroww Gamma\<turnstile>(c,Stuck) \<rightarrow(kip)
| AbruptProp: "\<>c\noteqp wss=Stuck\Longrightarrow '=Stuck"

lemmas step_induct otes_c\c>2
Basic Spec SpecStuck Guard Seq SeqSkip SeqThrow ave \amma<(SeqSeq c\<^sub>1 c\<^sub>2, Normal s) \<rightarrow>\<^sup>* (c', t')".
WhileTrue Call CallUndefined DynCom Catch CatchThrow CatchSkip
FaultProp StuckProp AbruptProp, induct set]

inductive_cases step_elim_cases [cases set]:
"\<Gamma>\<turnstile>(Skip,s) \<rightarrow> u"
"\<Gamma>\<turnstile>(Guard f g c,s) \<rightarrow> u"
"\ "s=Stuck java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
"Gamma>\<turnstile>(Spec r,s) \<rightarrow> u"
\><turnstile>(Seq c1 c2,s) \<rightarrow> u"
"\<Gamma>\<turnstile>(Cond b c1 c2,s) \<rightarrow> u"
"\<Gamma>\<turnstile>(While b c,s) \<rightarrow> u"
"Gamma(Calls<rightarrowjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
\<>\<turnstile>(DynCom c,s) \rightarrow u
"\<Gamma>\<turnstile>(caset of
"<ammaturnstile>( c1 c2s <>u"

inductive_cases step_Normal_elim_cases [cases set]:
"\<Gamma>\(eeps)ousing exec
\Gamma<turnstile(uard d ,Normal) <ghtarrow
"Gamma\>(Basicf, s) \rightarrow>u
"\<Gamma>\<turnstile>java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
\<\<turnstile>(Seq c1 c2,Normal s) \<rightarrow> u"
"\<Gamma>\<turnstile>(Condb c2,Normal )\rightarrow "
"\<Gamma>\urnstile>(While b cmal\rightarrowujava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
"\<amma>( p,Normal s) \<rightarrow> u"
Gamma>\rnstilenCom,al<> u"
"\<Gamma>\<turnstile> (c\<^>1, Normal s ycasesauto
"\<Gamma>\<turnstile>(Catch'True)

text \<open>The final configuration is either of by auto
termination, or @{term(ThrowNormal)ncaseprogram tartedted in
a @{term "Normal"} state and terminated abruptly. The @{const "Abrupt >\turnstile>(Seq (While b c x)\rightarrowThrow, Normalx"
modeltermination in contrast to thebig-step. Only \<^ub1"\<><turnstile (\^sub> s) \rightarrow>\<^up*Skip Abrupt s)byjava.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 115
program startstforceAbrupt
ate<close>

definition: "('s,pf) \Rightarrow wherejava.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
"final cfg = (fst cfg=Skip \<or>( henceby(ductto

abbreviation
"step_rtrancl: [s)y(',f (s,,) nfig <Rightarrow boolljava.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
 (\open<turnstile (_<rightarrow>^>/ _)\<close> [81,81,81] 100)
java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 6
usingw java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
abbreviationbreviation
"step_trancl": caseatchMisshMiss cjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0
 ( ep:\<Gamma\turnstile>withCatchMisschMissypsps)
where
 "<Gamma\<turnstile>cf0 \<rightarrow>\<^sup>f1 "\<Gamma><turnstile Catchc<sub1c\^>2 s\<ightarrow\^sup>*atch\^sub>2, "

(* ************************************************************************ *)
subsectionjava.lang.NullPointerException
(* ************************************************************************ *)

 "rupt<
 apply (induct c)
 apply aass exec: "\Gamma< induct
 done case

lemma no_step_final
 assumes step: "Γ
 shows "final<
using step
by inductto

lemmastep_final
 assumes step: "Γ
 shows "final cfgstep<Gammaturnstile> (cjava.lang.NullPointerException
using step
 by (cases cfg, cases cfg') (auto induct

lemma step_Abrupt:
 assumes
 shows "∧⊨
usingsubsection <>Equivlence betetw Small-Step anBigk \ >
by (induct) auto

lemma step_Fault:
 assumes step: "Γ
 showsf. s=Fault==>
using step
by (induct) auto

lemma step_Stuck:
 assumes step: "Γ (c, s) →
 s🪙| _ \Rightarrow> c'=Skip \and e:
using step
by (induct) auto

lemma SeqSteps:
 assumes steps: " cases : exec elim)
 shows
 Longrightarrow Γ1 c_^sub cjava.lang.NullPointerException
sing
proof seq_c"<a Seq
 case Refl
 thus ?case
 by simp
next
 case (Trans cfg_:elcas
 have step: " WhileTruec'here
 havesteps_cjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 havecfg<1, s)" and cfg₁', s')" by fact
 obtain'where'c^ub'"
 by (ases cfg'') auto
java.lang.NullPointerException
 have "Γ
 by
 hence java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 by (rule step
 also from Trans.hyps (3) [OF cfg'' cfg\^finally<Gamma<>* (c' t).
 have "Γ⊨ (Seq c₁' c₂, s'') →s
 finallynlly shocas.
qed

lemma CatchSteps:
 assumes steps: "Gamma>cfg₂"
 shows "∧
 ==>
using steps
proof (induct')
 caseRefl
 thus ?case
 by simp
next
 caserans g'
 have step
 have)maljava.lang.NullPointerException
 have cfg⊨ ==>
 obtain cjava.lang.NullPointerException
 by
 moreover
(<,s)<> auto:Abrupt_end
 by simp
 hence "Γ show ?thesis
 by (rule step.Catch)
 also by (autoro: st_Abrupt_end)java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
 have next
 finally show ?case .
qed

lemma steps_Fault: "Γ\redex_csub>
proof (induct
 case (Seq c\ :java.lang.NullPointerException
 have steps_c_\<Gammaturnstile thuscase ( intro.DynCom rtranclp_trans)
 have steps_csub2: java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 from SeqStep [O s\^>1 ref re]
 have "Γcase CatchMatch:Fault_endjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 also
 have "Γ⊨ intr: execint b mp
java.lang.NullPointerException
 finally show ?case by simp
next
 case (Catch c₁ = Spe \<(\
java.lang.NullPointerException
 from CatchSteps [OF st auto
java.lang.NullPointerException
 also
 have "Γ
 finallyshow simp
qed (fastforce

lemma steps_Stuck: "Γ (c, Stuck) →* (Skip, Stuck)"
proof (induct c)
 case (SeqGamma⊨1, Normal s) →<sup, t)
 have steps_c"🚫
java.lang.NullPointerException
java.lang.NullPointerException
 have{
 also
 have "Γjava.lang.NullPointerException
 also notesub
 finallyintro
next
 case (uto
 have steps_c exec_impl_steps_Normal_Abrupt
 from CatchStepssteps_c^sub1refl]
 have "Γ⊨ (au shows "Gamma,al\^* (Throw, Normal t)"
also
 have "Γ⊨ (Catch Skip cjava.lang.NullPointerException
 finally show ?case by simp
qed (fastforce intro: step.intros)+

lemma steps_Abrupt: "Γ
proof (induc
java.lang.NullPointerException
java.lang.NullPointerException
 have steps_cintro:A)
 from SeqSteps [OF s
 have "Γ
 also
 have\Gammaturnstile (Seq Skip cjava.lang.NullPointerException
 also note steps_c<have
 finally howy mp
next
 case (Catch c_{upt_end}:proofinduct
 havesteps_cby Skip ?case
 fromjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 have<><> ch1 c_\<^sup casesek
 also
 have"s'=Stuck
 owcsby
java.lang.NullPointerException

lemma
 assumes step: "Γ⊨
 shows "\<And
using step
by (induct) auto

lemma step_Abrupt_prop:
 assumes st"amma🚫
 shows "∧r ==>
using step
by (i)ase (Se

lemma step_Stuck_prop:
 assumes step: "Γ>qed

using step WhileFalse thuscase \urnstilestileredex c,s⟩ intro.intros:exec_Normal_elim_cases
by (induct) auto

lemma auto elim
 assumesjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 showslt> s'=Fault f"
using step
proof (induct rule: converse_rtranclp_induct2 [case_names Refl Trans])
 case Refl thu thus ?case by simp
next
 case (Tra s s'')
java.lang.NullPointerException
 by (auto intro: step_Fault_prop)
qed

lemma sfrom Seq.hyps (4) obtain c' t' where
 assumes step: "Γ⊨2
 shows "s=Abrupt t ==> s'=Abrupt t nots_Normal this
using step
proof (induct rule: converse_rtranclp_induct2 [case_names Ref Abpt \Rightarrow if s'= he '=Sk
 case Refl thus ?case by simp
nextrom '[siplifie Norm]
 case (Trans c s c'' s'')
 thus ?case
 by (auto intro: step_Abrupt_prop)
qed

lemma steps_Stuck_prop:
 assumes step: "Γ⊨ (c, s) →⊨assume<2:<><> ==>
 showsLongrightarrow'kjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
usingstep
prooftenverse_rtranclp_induct2rans)
 case Refl thus ?case by simp
nextmp
 case (Trans c s c'' s'')
 thus ?case
 by (auto intro: step_Stuck_prop)
qed

(* ************************************************************************ *): Gamma<turnstile> ⟨1',Normal x'⟩ t"
subsection ‹
(* ************************************************************************ *)

theorem exec_impl_steps:
 assumes exec: "java.lang.NullPointerException
java.lang.NullPointerException
(case t of x')
Abruptr \ghtarrow=tthen c'=Sk<and'
| _ ==> c'=Skip ∧
exec
(induct)
?ase
by simp

case Guard thus ?case by (blast intro: step.Guard rtranclp_trans)

case GuardFault thus ?case by (fastforce intro: step.GuardFault rtranclp_trans)

case from Fault exec' have "t=Fault f"

case tus ? by fastfino: stepstep.BasiBsic rtranclp_tras)

case Spec thus ?case by (fastforce intro: step.Spec rtran pecSckhuss ?caase

case SpecStuck thus ?case by (fastforce intro: step.SpecStuck r cas Guuarus ?case

case (Seq cintro: e y (frcentro: exec.ioelim: exec_Nor_lim_ca)
java.lang.NullPointerException
java.lang.NullPointerException
show?case
(case "∃
casese
from False Seq.hyps (2)
have "Γ
by (cases s') auto
hence seq_c_Gamma⊨c_<> s an
by (rule SeqSteps) auto
from Seq.hyps (4) obtain c' t' where
steps_c_moreover
t: "(case t of
Abrupt x ==>
else '=assume ""redex c\<>1
| _ ==>
by auto
java.lang.NullPointerException
also have " "Γ (Seq Skip c_:Abrupt_end)
also note steps_c_moreover
java.lang.NullPointerException
with t False show ?thesis
by (case t)auto
next
case True
then obtain x where s': "s'=Abrupt x"
by blast
from s' Seq.hyps (2)
have "\<<Gamma
g c where
java.lang.NullPointerException
by (rule SeqSteps) auto
also have "Γ have ?thesis
by (rule SeqThrow)
java.lang.NullPointerException
moreover
java.lang.NullPointerException
by (auto intro: Abrupt_end)
ultimately show ?thesis
by auto
qed

case CondTrue thus ?case from Fault exec' have"t=F f"

case CondFalse thus ?case by (blast intro: step.CondFalse rtranclp_trans)

case (WhileTrue s b c s'
have exec_c: "Γ ⟨c,Normal s⟩
have exec_w: "Γ intro: fr step_Stuck_end [OF stepstep this] s_Ns_Normal
have b: "s ∈
e sttep: "Γ (Whileb ,Norma s) →
byby (rule step.WhileTrue)
show ?case
proof (cases "∃. (x, ) \notin r"
case False
from False uto int exec.intos)
have "Γautointro: execintros)
by (cases s') auto
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
by (rule SeqSteps) auto
from rom exec_dex_t [OF this]
steps_c(While b c, s') →* (c', t')" and
t: "(case t of
Abrupt ==>
else c' = Throw ∧
| _ ==>
by auto
note step also note seq_c
also have "Γby (auto intro:exectro
by (rule step.SeqSkip)
java.lang.NullPointerException
finally have "Γ⊨ \<Gamma\1 ormal s→t)
with t False show ?thesis
by (cases t) auto
next
case True
thenobtain wheere : "s'=Abrx"
t
note step
also
from s' WhileTrue.hyps (3)
ve<>\
by auto
hence
"\<Gamma<* (Seq Throw While , Normal x)"
by (rule Seboino: int)
also have "Γ⊨
by (rule SeqThrow)
finally have "Γtesis
moreover
frome_i
by (auto intro: Abrupt_end)

assume "redex c\<corollary
\<Gamma\

case WhileFalse thus ?case by (fastforce intro: step.WhileFalse rtrancl_trans)

case Call thus ?case by assume : "Γ

case CallUndefined thus ?case by (fastforce intro: step.CallUndefi step

case StuckPrnext

case DynCom thus ?case by (blast intro: step.DynCom rtranclp_trans)

case Throw thus ?case by simp
with Abrupt
case Aruphus se(y (stforc iro: stes_Abbrupt)

(CachMatch cFault f".
from CatchMatch.hyps (2)
⊨
by simp

by rule Ca case (Seq c\^>1 s c🚫

by (rule step.CatchThrow)

step_Fault_prop step_Sro:.intros: exec_im_cae)
steps_c_tros: exec_elim_acaaes)
t: "(case t of
Abrupt x \<>case
e ese c' =Trow \< \ p \and s=Normal x ∧
| | _ \<Rightarrow t' = t)"
by auto
steps_c\^2 mal_elim_casesi_cs+
finally show ?case
using t t
aut pli: xtae.plit)

case (CatchMiss c₂)
have t: "¬

have "Γ s
by (cases t) auby (atfc ir:erminatsitos a(c' s) n "<Gamma\c',s'⟩ t" by fact+
hence "Γ
by (rule CatchSteps) auto
also
have "\<Gammahrow_impl_exec
by (rule step.CatchSkip)
finally show ?case
using t
fastforce ixsaeplts

exec_impl_steps_Normal:
assumes
shows "Γ(c,s) →
exec_impl_steps [OF exec]

exec_impl_steps_Normal_Abrupt:
assumes e "\<\<r> ==>
not_"\lbrakk🪙f 0 =
exec_impl_steps [OF exec]
auto

exec_impl_stps_upt_brupt:
assumes exec: "Γ⊨⟨
shows "Γ exec_redex_Stuck:
step_preserves_termination:
ato

exec_impl_steps_Fault:
assumes exec: "Γ⊨
\Gamma>⊨ ?case
[OF exec]
auto

execex_Fault:
(f intr: trmnate.introst
shows"🚫) auto i exec.innros elixxec_elim_cass)

auto

stepArupt_en
assumes step:: "\Gamma⊨1, s) →1', s')"
shows s'=Abrupt x ==>
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10

step_Stuck_end:
assumes step: "ΓBasic us ?case
case SStuckPopths?aey fsfreitro erinate..ntos)
s=Stuck \<ornext
java.lang.NullPointerException
(∃
step
inductauto

step_Fault_end:
java.lang.NullPointerException
shows "s'=Fault f ==>
s=Fault f \<orbyterminates.intros exec.intros
java.lang.NullPointerException
step
induct auto

>
<>\
induct c)
case Seq
thus ?cathus cas
by (cases s) (auto intro: exec.intros elim:exec_elim_cases)

case Catch
thus ?case
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
simp_ how"Γ s')

exec_redex_Fault:
Γ⟨red c,s⟩ Fault f ==>⊨
(induct c)
case Seq
thus ?case
by (cass s) (autointro: : exec.intros elim:exec_elim_cases)

exec_c CallUndef
thus ?case
by (cases s) (auto intro: exec.intros elim:exec_elim_cases)
simp_all

step_extend:
assumes step: "Γ⊨
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
step
(induct)
case Basic thus ?case
by (fastforce intro: exec.intros elim: exec_Normal

case Spec thus ?case
by (fastforce intro: exec.intros elim: exec_Normal_elim_cases)
java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 4
case apply cla
by (fas hav "s=brupt x'"

case Guard thus ?case
by (fastforce intro: exec.intros elim: exec_Normal_elim_cases)

case GuardFault thus ?case
by (fastforce intro: exec.intros elim: exec_Normal_elim_cases)

case (Seq c₁ s c_: exec.intros)
java.lang.NullPointerException
have exec': "Γ⊨
show ?ca
proof cses s)
case (Normal x)
note s_Norma = this
show ?thesis
proof (cases s')
cse Nrmalx'
from exec' [simplified Normal] obtain s'' where
exec_c\<^>1
exec_c\^sub>2: "Γ ⟨: "\Gamma>⊨* (c',s)"
by cases
java.lang.NullPointerException
have "Γ "Γc↓ Γ⊨
by simp
from exec.Seq [OF this exec_c₎
case Re Rl thus?cse . (is "∀
next
case (Abrupt x')
with e have "t=Abrupt x x'"
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
moreover
from step Abrupt
have "s=Abrupt x'"
by (auby (auto intr "\><k.
ultimately
show ?thesis
by (auto intro: exec.intros)
next
case (Fault f)
from step_Fault_end [OF step thistes_ho?tess
obtain g c where
redex_c_\⊨)
fail: "x ∉ik. \\eists>c' s'.'. f (i + 1) = (Seq c' c🚫
by a show ?case
hence "Γ
by (auto intro: exec.intros)
from exec_redex_Fault [OF this]
have "Γ⊨ ⟨(asdest s)
moreover from Fault exec' have "t=Fault f"
on head_com:: "('s,'p,'f) from hyp [ul_ora,of "k -1"] f0
java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 16
show ?thesis
using s_Normal
by (auto intro: exeintros)
next
case Stuck
from have "Γ s') \rightarrow
"(🚫
java.lang.NullPointerException
by auto
moreover
{
fix r
assume " "redex c\sub= Spec " a a
"Γ clarify
o intro: exec.intros)
from exec_redex_[OF t
have "Γ (auto simp add: redex_Seq_False final_def)
moreover from Stuck exec' have "t=Stuck"
?
ultimately
have ?thesis
using s_Normal
by (auto intro: exec.intros)
}
moreover
{
fix p
assume "redex c🚫>c\\1,Nox\rangle<>Stuck
e\shows"∀C c' c🚫head (f i) →⊨ ) \<rightarrow Stuckend)
by (auto intro: exec.intros)
from exec_redex_Stuck [OF this]
have "Γ⊨∀ thesis
moreover from Stuck exec' have "t=Stuck"
by (auto intro: Stuck_end)
ultimately
have ?thesis
using s_Normal
by (auto i }
}case (Suc k)
ultimately show ?thesis
by auto
qed
next
case (Abrupt x)
from step_Abrupt [OF step this]
have "s'=Abrupt x".
with exec'
have "t=Abrupt x"
by (auto intro: Abrupt_end)
with Abrupt
show ?thesis
by (auto intro: exec.intros)
next
case (Fultimately
from step_Fault [OF step this]
have "s'=Fault f".
with exec'
have(rule le_Sucases)
by (auto intro: Fault_end)
with Fault
show ?thesis
by (auto intro: exec.in by }
next
case Stuck
from step_Stuck [OF step this]
have "s'=Stuck".
with exec'
have "t=Stuck"

with Stuck
show ?thesis
by (auto intro: exec.intros)
qed

java.lang.NullPointerException
by (cases s) (fastforce intro: exec.intros elim: exec)+

case (SeqThrow c_' x".
fastforce intro: exec.intr exec_elim_cases)+

case CondTrue thus ?case
by (fastforce intro: exec.inro eli:xecormlelicass

case CondFalse thus ?case
by (fastforce intro: apply blast

case WhileTrue thus ?case
by (fastforce intro: exec.intros elim: exec_Normal_elim_cases)

case WhileFalse thus ?case
y (fastore intr: xeitros limelim: exc_Normal_lim_c

case Call thus ?case
by (fastforce intro: exec.intros elim: exec_Normashow show cas

thusca
by (fastf

case DynCom thus ?case
by (fastforce intro: exec.intros elim: exec_Normal_elim_cases)

java.lang.NullPointerException
java.lang.NullPointerException
have ': "Γ (cases k) auto
show ?case
proof (cases s)
case (Normal x)
note s_Normal = this
show ?thesis
proof (cases s')
case (Normal x')
from exec' [simplified Normal]
show ?thesis

fix s''
assume exec_c_"Γ', '')" a
assume exec_c₍∃
java.lang.NullPointerException
have "Γ "\forall::nat. Γf i →) and
by simp
from exec.CatchMatch [OF this exec_c_{e_computation_extract_head_Catch}
?thesis by simp
from f_0 have heahead_f_0: "head (f 0) = (c\<^>1
java.lang.NullPointerException
assume t: "¬
from Catch.hyps (2) Normal exec_c_intro:.introselim: exec_No
have "Γ⊨ no defin k wh "k = (LEASTi.ina(hea (f i))
by
from exec.CatchMiss [OF this t] s_Normal
show ?thesis by simp
(sint:eec.iroselm:exe_)
next
case (Abrupt x')
with exec' have "t=Abrupt x'"
by (auto intro:Abrupt_end)
moreover
 infinite_computation_extract_head_Seq[
have "s=Abrupt x'"
by (auto intro: step_Abrupt_end)
ultimately
show ?thesis
by (auto intro: exec.intros)
next
case (Fault f)
from step_Fault_end [OF step this] s_Normal
obtain g c whlarifyapply -
redex_c\<^>1
failapply (drule LeasI)
by auto
java.lang.NullPointerException
by (auto intro: exec.intros)
from exec_redex_Fault [OF this]
have "Γ⊨ ⟨
moreover from Fault exec' have "t=Fault f"
by auto intro: Fault_)
ultimately
show ?thesis
using s_Normal
(auo intro exec.intros)
next
case Stuck
from step_Stuck_end [OF step this] s_Normal
have "(∃ [rule_format])
(\exists n case 0th ?cae y sip
by auto
moreover

fix r
assume "redex c_i<m.
hence "Γuto
y (auto introo: ex hence"\\Gamm>\<turnstilernstile* head (f m)"
from exec_redex_Stuck [OF this]
java.lang.NullPointerException
moreover from Stuck exe proo cases s')
not_fin_Suc [rul_for of k]
ultimately
have ?thesis
using s_Normal
by (auto intro: exec.intros)
}
f_k: "f k = (Seq Skip c) (Ca cwit s
{
fix p
assume "redex c_: head_def hea) fix 'java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
hence "Γ⊨from step[rule, of ] f_k
by (auto intro: exec.intros)
from exec_redex_Stuck [OF this]
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
Stuck ex ve "t="
by (auto intro: Stuck_end)
ly
have ?thesis
using ing_N
(auto intr: exec.intros)
}
ultimately show ?thesis
by auto
qed
next
case (Abrupt x)
from step_Abrupt [OF step this]
ve "'=Abrupt x".
with exec'
have "t=Abrupt x"
by (auto intro: Abrupt_end)
with Abrupt
java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 16
by (auto intro: exec.intros)
next
case (Fault f)
from step_Fault [OF step this] moreover
have "s'=F{
exec'
have "t=Fault f"
by (auto intro: Faultend)
with Fault
show ?thesis
(autointo: x e.itos)
next
case Stuahyps (2 No exec_\^1' s_N
from step_Stuck [OF step this]
have "s'=Stu allI impI)
with exe have "Γ\rangle 🚫
have "t=Stuck"
by (auto intro: Stuck_end)
with Stuck
show ?thesis
by (auto intro: exec.intros)
qed

case CatchThrow t (simp add: k_de)
exec.i eli:

CatchSkipse
moreover

case FaultProp thus ?ca
by (fastforce intro: exec.intros elim: exec_elim_cases)

case StuckProp thus ?case
by (fastforce intro: exec.intros elim: exec_elim_cases)

case AbruptProp thus ?case
proof

(rul Suc.)
steps: "\<Gammaalso m
shows "Γ simp
steps
(induc rom infinite_computation_extract_head_Seq [OF step f_0 this ]
case Refl thus ?case
by (cases t) (auto intro: exec.intros

case (Trans c s c' s')
have "Γ
thus ?case
by (rule step_extend)qe

steps_Throw_impl_exec:
assumes steps: "Γ⊨
shows "Γ⟨ A t"
steps
duct ule conve_trncpinuct2cs_aes Refp fom _u
case Refl thus ?case
by (auto intro: exec.intros)

case (Trans c s c' s')
have "Γ⊨
e
by rule step_ex)

(* ************************************************************************ *)
subsection Computations\Gamma\}
(* ************************************************************************ *)

definition inf:: assumes: s'Normalf auto:Fault_end)
(‹
"\<Gamma

==>T
:inf_def)))

(* ************************************************************************ *)
subsection ‹
(* ************************************************************************ *)

lemmahave(<rredex1 =Specapply(drule)
 assumesby simp add)
 shows"(> fro infinte_commp [O tep f_0 ]
using step
proof (induct)
 case Basic thus ?case by (fastforce intro: terminates.intros)
nextobtain ste: "forall>turnstile i \rightarrowhead+ 1)" and
 case Spec thus ?case by (fastforce intro: terminates.intros)
next
 case SpecStuck thus ?case by (fastforce intro: terminates.intros)
next
 case Guard t onf: "<have
 by (fastforce intro: terminates.introsfix
next
 case GuardFault thus ?case by (fastforce introapply -

 case (Seq cjava.lang.NullPointerException
 apply (cases s)
 apply (cases)
 apply (fastforce intro: terminates "\Gamma⊨ head (f (k + 1))" by
 elim: terminates_Normal_elim_cases
 applyforceest
 step_Fault_prop)
 done
next
 case(eqSkip^sub
 thus ?case
 apply (cases s)
 apply intro ( kauto
 elim: terminates_Normal_elim_cases )+
 done
next
 case (SeqThrow cjava.lang.NullPointerException
 from inf_comp f where
 by( : terminatesexec
 elim )
next
 case CondTrue
 thus ?case
 by (fastforce intro: terminates exec.java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
 elim: terminates_Normal_elim_cases )
next
 case CondFalse
 thus ?case
 by (fastforce intro: terminates.intros
 :terminates_Normal_elim_cases
java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 4
 case WhileTrue
 thus ?case
 by (fastforce "\forall< Γhen "ammaturnstile<redex<ub \angle\Rightarrow>""
 elim: terminates_Normal_elim_cases
next [Fthis
 caseWhileFalse
 thus
 by (fastforce "<<turnstile> head (f 0) \rightarrow<su>* ead (f m)"
 elim: terminates_Normal_elim_cases )
next
 case Call
 thus ?case
 by (fastforce intro: terminates.intros
 elim: terminates_Normal_elim_cases head(m+1)byjava.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
next
 case CallUndefined
 thus obtainby(uto: {
 by fastforce:terminatesjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
 elim: terminates_Normal_elim_cases )
next
 caseDynCom
 thuscase thuscase simp
 by (fastforce intro: terminates.intros
 elim:by( : head_def
next
 caseCatch1sc\> 'c<subcase
 apply (cases s)
 apply (cases s')
 y (fastforce terminatesintros
 elim: terminates_Normal_elim_cases)
 apply ( intro : step_Abrupt_prop
 step_Fault_prop step_Stuck_prop)+
 done
next
 case CatchThrow
 thus ?case
 by (fastforcey(autontroroAbrupt_endt_end
 elim: terminates_Normal_elim_cases finallyw mp
nextwith
 case (CatchSkip {
 thus
 byautointrointros
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 4
 case FaultProp thus ?case by (fastforce
next
 case StuckProp thus ?case by (fastforce intro: terminates.introsby( no_step_finalsimp)
next
 case AbruptProp thus ?case by (fastforce intro: terminates.intros}
qed

lemma steps_preserves_termination:
 assumes: "Γ(c,s) \< assume,ss')"java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
 shows "\ (rule no_s') (auto simp add: final_def)
using steps
proof (induct rule: rtranclp_induct2 [consumes 1, case_names Refl Trans])
 case Refl thus ?case . fix x x
next
 case Trans
 thus ?case
blast des:ste)
qed

ML ‹>) byreove
 ML_Thms.bind_thm (" ", Split_Rule.split_rule @{context}
(Rule_Insts.read_instantiate @{context}
(("a", 0,Positi.none), "(aa,aab)"),, ((("b" 0), P Position.none),
@{thm tranclp_induct}));
› intro

lemma'
 assumes steps: "Γ⊨(c,s) →from f_Suc_k
 shows "<Gammac↓s \Longrightarrow\Gamma>⊨g (java.lang.NullPointerException
using steps(mp
proofrule step
 case Step thus ?case by blast: step_preserves_termination
next
 case Trans g_0 \amma>urnstile2,s') → …
 y(uto
 byultimately
qed

definitionhead_coms,p <Rightarrow>(',p'fcom"java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
where
"head_com c fastforce:.intros '
 (case c of
 Seq c_have: "∀ final (head (f i))"
 | Catch byblast
 | _ ==>

definition head:: "('s,'p,'f) con from notot_f
 where "head cfg = (head_com (fst cfg), sndFalse

lemma le_Suc_cases: "[∧i. [i < k] ==> P i; P k] ==> ∀
 applya clar
 apply (case_tac "i=k")
 apply auto
 done

lemmaredex_Seq_False: "∧ c''c') = "
 by (induct c) auto

lemma redex_Catch_False: "∧


lemma infinite_computation_extract_head_Seq:
 assumes inf_comp: "∀
 assumes f_0: "f 0 = (Seq₂,)"
 assumes not_fin: "∀i<k. ¬ final (head (f i))"
 shows "∀rule
 Γcase
 (is "∀
using not_fin
proof (esteps: "Gamma(cs) →*(Throwfrom [[of] f_0
 case 0
 show ?case by simp
next
 case (Suc k)
 have not_fin_Suc:
 "forall><Suc . <no fial (head ( ste
 from this[rule_ormat]havee n_fin_kk:
 "∀ final (head (f i))"
 apply clarify
 apply (subgoal_tac "i < Sucby simp
 apply blast
 apply simp
 done

 from Suc.hyps [OFbyrule)
 have hyp: "∀i<k. (∃assume _0: " = (Spec)"
 Γ False
 show ?case
 proof (rule le_Suc_c qed
 fix i
 assume "i < k"
 then show "?Pi"
 by (rule hyp [rule_format])

 show "?P k"
 roof
 from hyp [rule_format, of " 1]
 obtain c False
 by (cases k) auto
 from inf_comp [rule_format, of k] f_k
 have "Γ1 c_>,St) →
 by simp
 moreover
 from not_fin_Suc [rule_format, of k] f_k
 have "¬i. Γ ( )
 by (simp f_0
 ultimately
 obtain' where
 "Γ
 "
 by cases (auto simp add:show ?case
 with f_k
 show ?thesis
 simphead_com_defassumef_0"f (S,S)"
 qed
 qed step
qed

lemma infinite_computation_extract_head_Catch elim)
 assumes inf_comp: "∀ thus ?case by (fastforce intro: ter qedqed
 assumes f_0: "f 0 = (Catch cjava.lang.NullPointerException
 assumes: \forall<> al
owsi<k. (∃ ?
 Γelim
 (is "∀
using not_fin
proof (induct k)
 case
 show ?case by simp
next
 case (Suc k)
 have not_fin_Suc:
 "∀ intro:terminatesintros
 show False
 "∀i<k. ¬ final (head (f i))"
 apply clarify
 apply (subgoal_tacdone
 apply blast
 apply simp
 done

 from False
 have:🚫
 turnstilefi+java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
 show ?
 proof (rule le_Suc_cases)
 fix i
 assume "i < k ix f
 show "P i
 by (rule hyp [rule_format])
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 6
showP
 proof -
 fromhyprmat 1]_
 obtainCall
 by (cases k) auto
 from inf_comp ?
 have "Γ i terminates.in
 by simp
 moreover
 from not_fin_Suc [rule_format, of k] f_k
 have "¬ final (c',s')"
 by (simp add: final_def head_def head_com_def)
 ultimately
 obtain c'' s f_sstep:" autokip_no_step
 "Γ⊨
 asassumef_: "0 Basic
 by [0] ]
 DynCom
 show ?thesis
 by simphead_com_def
 qed
 qed
qed

lemma (chsub1 s cjava.lang.NullPointerException
proof
 assume "\< apply
 then obtain f where
 step [rule_format]: "∀i::nat. \
 f_00 =(, s)"
 by (auto simp add: inf_def)
 from step [of 0, simplified f_0] step [of 1]
 show hus ?case
 by cases (auto elim: step_elim_cases)
qed

lemma split_inf_Seq:
 assumes in: "\Gamma<urnstile(
 showsturnstile<>,s <ightarrow>(<nfinity<or
 (∃s'. Γ\fromf_step
proof -
 from inf_comp obtain f where
 show
 f_0sub c_"⊨ Γc'<>s"
 by (auto simp add: inf_def)
 from have head_f_0: "headf_step.<🚫
 ( : step_elim_cases
 show ?thesis
 proof (cases "∃ (head (f i))")
 casejava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 define k where "k = (LEAST i. final (head (f i)))"
have case
 apply (intro allI impI)
 applyunfold
 apply (drule not_less_Least)
 apply auto
 done
 from infinite_computation_extract_head_Seq [OF step f_0 thisf_step[ofstep
 obtain step_head: "
 conf: "∀
 by blast
 from True
definition head_com:: "('s,p,'f) com ==>('s,'p,'f) com"
 apply f_stepAndi. Γ> i → f (Suc i)"" c =
 apply (erule exE f_0: "f 0 =(Guard m g c, Fault x)"
 apply (drule )
 applyshowFalse
 done
 moreover
 fromf_0 confule_formatk-1 fastforceSkip_no_step)
 obtain c' s' where f_k: "f k = (Seq c' c_| <Ri> c)"
 by (cases k) auto ?case
 moreover
 from step_head have steps_head: "Γ⊨head (f 0) →^qed
 proof (induct k)
 case 00 thus ?case b by simp
 next
 ase(um)
 have step: "∀>\>
 hence
 by auto
 hence "\Gamma\turnstile> head (f 0)<righta>w^sup>* head (f m)"
 by (rule Suc.hyps)
 also from step [rule_format, ofwhere cfg =( (fst), snd)"
 have \\<turnstile> head (f m) →
 y ?case by simp

 by (autdest: stault_prop)
 ssume _kfk = Seq Skip c\^>2 aclarif
 with steps_head
 have pply(c
 using head_
nduct
 moreover
 fromsho?case[o
 obtain "Γ⊨(Seq Skip cassume"\And>ilem redex_Seq_Fal"' c' dexSeq' ) False
 f_Suc_k
 by (fastforce elim: step.cases intro: step.intros)
 define g where "g i = f (i + (k + 1q
 from f_Suc_k
 have g_0: "g
 by (simp add: g_def)
 thus
 have "∀
 by roof (rue not_i)
 with g_0 have "Γ⊨(cjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
 by imp
 ultimately

 by auto
 }
 moreover
 {
 fix
 assume s': "s'=Normal x" and f_k: "f k = (Seq Throw c\ f [of 0] f_0f_step [of ]
 from step [rule_format, of k] f_k s'
 obtain "Γ⊨
 f_Suc_k: "f (k + 1) = (Throw,s')"
 by (fastforce elim: step_elim_cases intro: step.intros)

assume<\turnstile (Seq1 c₎"
 have g_0: "g 0 = (Throw,s')"
 by (simp add: g_def)

 have "∀ : inf_def
 by (simp add: g_def)
 withg_0 🚫1 ( not_infI
 by (auto simp add: inf_def)
 with no_inf_Throw
 have ?thesis
 by auto
 }
 ultimately
 show ?thesis
 by (autofixf
 next
 case False
 then have not_fin
 by
 have "∀
 proof
 fix k
 from not_fin
 have "proofrulejava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
 by simpassume f_0 0 = b by(fastforce: step step)

from infinite_computation_extract_head_Seq [F step f_0 this
 show "Γ⊨
 qed
 with he have "Γ(cstep_elim_cases
 by
 thus ?thesis
 by simp
 qed
qed

lemma split_inf_Catch:
 assumes inf_comp: "Γ⊨(Catch c₁ c_show ?casease
 "Gamma>(c_dots>
 (∃s'. Γ⊨(c_fixf
proof -
 from inf_comp obtain f where
 step:∀⊨Guard
 f_0subst_redex
 by (auto simpby f_stepi.<<> i <ightarrowSuci)"
 from f_0 have head_f_0: "head (f 0) = (cjava.lang.NullPointerException
 by (simp add ?thesis
 show
 proofcases>. final (head (f i))")
 case True
 define k where "k =
 havelemmatep_redex
 apply (intro allI)
show
 apply (drule not_less_Least)
 apply auto
 done
 from infinite_computation_extract_head_Catch [OF step f_0 this]
 tain> <amma\ i
 : java.lang.NullPointerException
 by blast "∧
 from True
 have_ final (f k))"
 apply -
 apply (erule exE)

 apply (simp add: k_def)
 done
 moreover
 from f_0 conf [rule_format, of "k - 1"]
 obtain c' s' where f_k: "f k = (Catch c' cjava.lang.NullPointerException

 moreover
 from havesteps_head"<><turnstile>head (f 0) →* head (f k)"
 proof (induct k)
 case
 next
 case (Suc m)
 have step: "\<forallMLelim:ultim
 hence "∀i<m. hesis
 by auto
 hence "Γ⊨ head (f
 by (rule Suc.hyps)
 also from step [rule_format, of m]
 have "Γ head (f m) \rightarrow> head + 1" by simp
 finally show ?case by simp
 proof (r not_inI)
 {
 assumefi f
 with steps_head
 have "<><(c\^subsupjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 using head_f_0
 by(impfjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
 moreover
 from step [rule_format, of k] f_k
 obtain java.lang.NullPointerException
 f_Suc_k: "f (k + 1) = (Skip,s')"
 by (fastforce elim: step.cases intro: step.intros)
 from st [rule_fomat, o " +",simplif f_Suc_k
 have ?thesis
 by (rule no_step_final') (auto simp add: final_def)
 }
 moreover
 {
 fix x
 ume 's'=Norma x" and: atch<ub2, s')"
 with steps_head
 have "Γ⊨(cjava.lang.NullPointerException
 using head_f_0
 by (simp adddef head_com_def
 moreover
 from step [rule_format, of k] f_k s'
 obtain "Γ⊨(Catch Throw c_qed
 f_Suc_k: "f (k + 1) = (c_\turnstile c₂,s) → )
 by (fastforce elim: step_elim_cases intro: step.intros)
 define g where "g i = f (i + (k + 1)"i
 from f_Suc_k
 have g_0: "g 0 = (c₂,s')"
 by (simp add: g_def)
 from step
 have "∀i. Γ⊨ step: "∀::nat. \Gamma>⊨i+1" annd
 by (simp add: g_def)
 with g_0 have "\<Gamma>\<turnstile>(c\<^sub>2,s') \<rightarrow> \<dots>(\<infinity>)"
 by (auto: "head (f 0) = (\<sub1,s"
 ultimately
 have ?thesis
 using s'
 by auto

 ultimately
 show ?thesis
 by (auto simp add: final_def head_def head_com_def)
 next
 case False
 then have not_fin: "\<forall>i. \<not> final (head (f i))"
 blast
 have "\<forall>i. \<Gamma>\<turnstile>head (f i) \<rightarrow> head (f (i + 1))"
 proof
 fix k
fromjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 have "\<forall>i<(Suc k). \<not> final (head (f i))"
 by simp

 from infinite_computation_extract_head_Catch [OF step f_0 this drule)
 show "\<Gamma>\<turnstile apply (simpadd: k_def)
 qed
 with head_f_0 have "\<Gamma>\<turnstile>(c\<^sub>1,s) \<rightarrow> \<dots>(\<infinity>)"
 by (auto simp add: inf_def)
 thus ?thesis
 by simp
 qedjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
qed

lemma Skip_no_step: "Gamma\turnstile(Catch Skip c<sub2s)\rightarrow Skips'"java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
 apply'java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 apply (simp add: final_def)
 done

:\not\,)\rightarrowdots(<>"
proof (induct c)
 }
 ?
 proof
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assumef_0: "f 0=( Stuck"
 from f_step [of 0] f_0
ow False
 by (auto elim: Skip_no_step)
 qed
next
 case (Basic g)
 thus ?case
(rulenot_infI
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (Basic g, Stuck)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (Spec r)
 thuscase
 proof (rule not_infI)
x java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
assumefastforceelim:step_elim_cases:step.intros)
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by fastforceelim: step_elim_cases
 qed
next
 case (Seq c\<^sub>1 c\<^sub>2)
 show ?case
java.lang.StringIndexOutOfBoundsException: Range [25, 7) out of bounds for length 7
 assume "\<Gamma>\<turnstile> (Seq c\<^sub>1 c\<^sub>2, Stuck) \<rightarrow> \<dots>(\<infinity>)"
 from OF].hyps
 show False
 by (by(imp: g_def)
 qed
ext
 case (Cond 0have\Gamma\<>c<sub2,'\rightarrow \\infinity (imp : inf_def)
showcase
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume hence \not\exists. \<>i f( i ijava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed ultimately
next
 case (While b c)

 proof (rule not_infI)
 fix next
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc f0: " = (java.lang.StringIndexOutOfBoundsException: Range [19, 14) out of bounds for length 14
 assume f_0: "f 0 = by blast
 from f_step [of 0] f_0 f_step [of 1]
 False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (Call p)
 show ?case
 proof (rule not_infIhave "\foralli<Suck). \not final(( i"
 fix
 assume f_step: "\<And>i. by simpsimp
 assume f_0: "f 0 = (Call p, Stuck)"
 from f_step [of0] f_0 f_step [ 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (DynCom d)

 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \ next
 assume f_0: "f 0 = (DynCom d, Stuck)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (Guard m g c)
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (Guard m g c, Stuck)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case Throw
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (Throw, Stuck)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (Catch c\<^sub>1 c\<^sub>2)
 show ?case
 proof
 assume "\<Gamma>\<turnstile> (Catch c\<^sub>1 c\<^sub>2, Stuck) \<rightarrow> \<dots>(\<infinity>)"
 from split_inf_Catch [OF this] Catch.hyps
 show False
 by (auto dest: steps_Stuck_prop)
 qed
qed

lemma not_inf_Fault: "\<not> \<Gamma>\<turnstile>(c,Fault x) \<rightarrow> \<dots>(\<infinity>)"
proof (induct c)
 case Skip
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (Skip, Fault x)"
 from f_step [of 0] f_0
 False
 by (auto elim: Skip_no_step)
 qed
next
 case (Basic g)
 thus ?case
 proof (rule not_infI)
 fix f
 f_step \<> <Gamma<> <>f ( i"
 assume f_0: "f 0 = (Basic g, Fault x)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (Spec r)
 thus ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i show java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
 assume f_0: "f 0 = (Spec r, Fault x)"
 from f_step [of 0] f_0 f_step [of 1]
 showFalse
 by (fastforce elim: Skip_no_step step_elim_cases)

next
 case (Seq c\<^sub>1 c\<^sub>2)
 show ?case
 proof
 assume "\<Gamma>\<turnstile> (Seq c\<^sub>1 c\<^sub>2, Fault x) \<rightarrow> \<dots>(\<infinity>)"
 from split_inf_Seq [OF this] Seq.hyps
 show False
 by (auto dest: steps_Fault_prop)
 qed
next
 case (Cond b c\<^sub>1 c\<^sub>2)
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (Cond b c\<^sub>1 c\<^sub>2, Fault x)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (While b c)
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (While b c, Fault x)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (Call p)
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile proof (rule not_infI)
 assume f_0: "f 0 = (Call p, Fault x)"
 from
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
qed
next
 case (DynCom d)
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (DynCom d, Fault x)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (Guard m g c)
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
assumef_0"f 0 =(Guardm gc,Fault x)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case Throw
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (Throw, Fault x)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (Catch c\<^sub>1 c\<^sub>2)
 case
 proof
assumeGamma<>(Catch\^> <>,Faultx)<ightarrow> <ots>(infinity"
 from split_inf_Catch [OF this] Catch.hyps
 show False
 by (auto dest: steps_Fault_prop)
 qed
qed

lemma not_inf_Abrupt: "\<not> \<Gamma>\<turnstile>(c,Abrupt s) \<rightarrow> \<dots>(\<infinity>)"
proof (induct c)
 case Skip
 show ?case
 proof (rule not_infI)
 fix f
 assume:"\Andi.\Gamma\turnstilefi \> Suci)
 assume f_0: "f 0 = (Skip, Abrupt s)"
 from f_step [of 0] f_0
 show False
 by (auto elim: Skip_no_step)
 qed
next
 case (Basic g)
 thus ? assume : \<>. <><turnstilef i \rightarrowf (Suci)
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (Basic g, Abrupt s)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (Spec r)
 thus ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assumef_0:f0 (pecr,brupt s)
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (Seq c\<^sub>1 c\<^sub>2)
 show ?case
 proof
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 from split_inf_Seq [OF this] Seq.hyps
 show False
 by (auto dest: steps_Abrupt_prop)
 qed
next
 case (Cond b c\<^sub>1 c\<^sub>applyerule'java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 show ?case
 proof (rule not_infI)
 f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (Cond b c\<^sub>1 c\<^sub>2, Abrupt s)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (While b c)
 show?ase
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (While b c, Abrupt s)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (case Basic
 qed
next
 case (Call p)
 showcase
 proof (rule not_infI)
 fix f
 assume f_step:assumef_0 "f =( g )"
 assume f_0: "f 0 = (Call p,ep[f1
 from f_stepshow
 show
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (DynCom d)
 show case(Spec)
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (DynCom d, Abrupt s)"
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (Guard m g c)
 show ?case
 proof (rule not_infI)
 fix f
 by(fastforce elim Skip_no_stepstep_elim_cases
 assume f_0: "f 0 = (Guard m g c, Abrupt s)"
 fromf_step ]f_0 [of
 show False
 ( hyp_c1 <><Gamma<turnstile(Normal)rightarrow\<><nfinity>"byjava.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
 qed
next
 caseThrow
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (Throw, Abrupt s)"
 from f_step [of 0] f_0 f_step [of 1]
 show frombf_step]f_0
 by auto:steps_Stuck_prop)
 qed
next
 case autoelim)
 show ?case
 proof
assumeGamma< Catch<subinfinity>java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
 from split_inf_Catch [OF this] Catch.hyps

 by (auto dest: steps_Abrupt_prop)
 qed
qed

theorem terminates_impl_no_infinite_computation:
 assumes termi: "\<Gamma>\<turnstile>c \<down> s"
 shows "\<not> \<Gamma>\<turnstile>(c,s) \<rightarrow> \<dots>(\<infinity>)"
using termi
proof (induct)
 Skip thus
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assumef_0:" 0 = Skip, "\Gamma>(,s\>^>*f"fact
 fromwithseq show"<Gamma>\<turnstile>(, s \rightarrow\^> f Suc n)
 show False
 by (autoproofrulenot_infI)
qed
next
 case (Basic g s)
 thus ?case
 proof not_infI
 f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 fi fi iny \><() <>^>x < Gamma<> \rightarrow<up}
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforceelim: Skip_no_stepstep_elim_cases)
 qed
next
 case (Spec r s)
 thus ?case
 proof (rulenot_infI
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0:by( elim Skip_no_step step_elim_cases)
 from f_step [of 0] f_0 f_step [of 1]
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed
next
 case (case Call)
 show?case
 have hyp: "\<not> \<Gamma>\<turnstile> (c, Normal s) \<rightarrow> \<dots>(\<infinity>)" apply (impaddinf_def
 show ?case
 proof (rule not_infI)
 fixjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (Guard m g c, Normal s)"
 by ( elim: Skip_no_step step_elim_cases
have1 =,java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 by (fastforce elim: step_elim_cases)
 with f_step
 have "\<Gamma>\<turnstile> (c, Normal s) \<rightarrow> \ case DynCom )
 apply (simp add: inf_def)
apply=\>.f(uci"in )
 by simp
 with hyp show False ..
 qed
next
 case (GuardFault s g m c)
 haveg"s\<>g"byfact
 show ?case
 proof (rule not_infI)
 fix f
 assume
 assume f_0: "f 0 = (Guard m g c, Normal s)"
 from g f_step [of 0] f_0 f_step [of 1]
 showFalse
 by( elim: Skip_no_step step_elim_cases)
 qed
next
caseFault c)
 thus ?case
by( )
next
 case (Seq c\<^sub>1 s c\<^sub>2)
 show ?case
 proof
 assume "\<Gamma>\<turnstile> (Seq c\<^sub>1 c\<^sub>2, Normal s
 from split_inf_Seq [OF this] Seq ?case
 show False
 byauto proof not_infI
 qed
next
 proof
 have b: "s \<in> b" by fact
 have hyp_c1: "\<not> \<Gamma>\ g where" assume"<Gamma\<turnstile ( b,ormal \<ghtarrow> \dots(<infinity>"
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (Cond b c1 c2, Normal s)"
 by(simp add: )
 have "f 1 = (c1,Normal s)"
 by (auto elim: step_Normal_elim_cases)
 with f_step
 have "\< f_0 " seq ) Call )
 apply (simp add: inf_def)
 apply (rule_tac x="\<lambda>i. f (Suc i)" in exI)
 by simp
 with hyp_c1 show False by simp
 qed
next
 case (CondFalse s b c2 c1)
 have b: "s \<notin> b" by fact
 have hyp_c2: "\<not> \<Gamma>\<turnstile> (c2, Normal s) \<rightarrow> \<dots>(\<infinity>)" by fact
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \ by(induct i)(auto simp simp add: redex_subst_redexred_c)
 assume f_0: "f 0 = (Cond b c1 c2, Normal s)"
 from b f_step [of 0] f_0
 have "f 1 = (c2,Normal s)"
 by (auto elim: step_Normal_elim_cases)
 with f_step
 have "\<Gamma>\<turnstile> (c2, Normal s) \<rightarrow> \<dots>(\<infinity>)"
 apply (simp add: inf_def)
 apply (rule_tac x="\<lambda>i. f (Suc i)" inassume:"\<And>. \<Gamma>\<turnstile>f i \\<rightarrow> f (Suc i)
 by simp
 with hyp_c2 show False by simp
 qed
next
 case (WhileTrue s b c)
 have b: "s \<in> b" by fact
 have hyp_c: "\<not> \<Gamma>\<turnstile> (c, Normal s) \<rightarrow> \<dots>(\<infinity>)" by fact
 have hyp_w: "\<forall>s'. \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow>
\<Gamma\turnstileWhile bc \downs'\and <not \Gamma\<> Whileb c, s' \rightarrow\<dots(<infinity"byfact
 have not_inf_Seq: "\<not> \<Gamma>\<turnstile> (Seq c (While b c), Normal s) \<rightarrow> \<dots>(\<infinity> qed
 proof
 assume "\<Gamma>\<turnstile> (Seq c (While b c), Normal s) \<rightarrow> \<dots>(\<infinity>)" by
 from split_inf_Seq [OF this] hyp_c hyp_w show
 by (auto intro: steps_Skip_impl_exec)
 qed
 show ?case
 proof
 assume "\<Gamma>\<turnstile> (While b c, Normal s) \<rightarrow> \<dots>(\<infinity>)"
 with f_step "\Gamma>\<turnstile> (c s Normal s)\<ightarrow> \<dots(<>)"
 f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)" and
 f_0: "f 0 = (While b c, Normal s)"
 ( simpadd)
 from f_step [of 0 apply ( x="<>i f(Suc i)" inexI)
 have "f 1 = (Seq c (While b c),Normal s)"
 by (auto elim: step_Normal_elim_cases)
 with f_stepr' "' in>redexesc'
 have "\<Gamma>\<turnstile> (Seq c (While b c), Normal s) \<rightarrow> \<dots>(\<infinity>)"
 apply (simp add: inf_def)
 apply (rule_tac x="\<lambda>i. f (Suc i)" in exI)
 by simp
 with not_inf_Seq show False by simp
 qed
next
 case (WhileFalse s b c)
 have b: "s \<notin> b" by fact
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (While b c, Normal s)"
 fromb f_step [of 0] f_0 f_step[f1
 show False
 by (fastforce elim: Skip_no_step step_elim_cases)
 qed Call ?case
next
 case (Call p bdy s)
 have bdy: "\<Gamma> p = Some bdy" by fact
 hyp \not \<><urnstilebdyNormal)<rightarrow\dots(\infinity fact
 show ?case
 proof (rule not_infI)
 fix f
 assumef_step:"\<And>i.\Gamma\turnstile>f i\<<> fSuc i"
 assume f_0: "f 0 = (Call p, Normal s)"
 from bdy ?
 havef 1=(,byfastforceintro .intros :step_elim_cases root_in_redexes)
 by (auto elim: step_Normal_elim_cases
 with f_step
 have "show False
 apply (simp add: inf_def)
 apply r:"r = Catch c\1 c<^>2 \\<r r \<n> c\<^subjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 by simp
 with hyp show False by simp
 qed
next
 case (CallUndefined p s)
 have no_bdy: "\<Gamma> p = None" by fact
 show ( c\sub1sc\^sub2))
 proof (rule not_infI)
 fix f
 assume show?ase
 assume f_0: "f 0 = (Call p, Normal s)"
 from no_bdy f_step [of 0] f_0 f_step [of 1] next
 show False
 by (fastforce elim: Skip_no_step]
 qed
next
 case (Stuck c)
 show ?case
 \^>:"<>\turnstile (uto intro: steps_Throw_impl_execjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
next
 case (DynCom c s)
 have"Gamma<turnstile (c\sub1 c\<sub2,s java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 show ?case
 proof (rule not_infI)
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
assumef_0: 0 = (DynComc Normal)
 from f_step [of 0] f_0
 have "by (inductc) auto intro: stepSeq step)
 by (auto elim: step_elim_cases)
 with f_step have "\<Gamma>\<turnstile> (c s, Normal s) \<rightarrow> \<dots>(\<infinity>)"
 apply (simp add: inf_def)
 apply "<Gamma\turnstile r ss)<rightarrow\^>* ('s'java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
 by simp
 with hyp
 show False by simp
 qed
next
 case (Throw s) thus ?case
 "<>\turnstile> subst_redex cr',s'' \ightarrow\<^> ((subst_redexsubst_redexcr' ')
 fix f
 assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
 assume f_0: "f 0 = (Throw, Normal s)"
 from f_stepf_step[ 0]
 show False
 by( step_elim_cases)
 qed
next
 case (Abrupt c)
 show ?case
 by (rule not_inf_Abrupt)
next
 case (Catch c\<^sub have "\<>\<turnstile> (subst_redex c r s) \<rightarrow> (subst_redex c r'' s')"java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90
 show ?case
proof
 assume "\<Gamma>\<turnstile> (Catch c\<^sub>1 c\<^sub>2, Normal s) \<rightarrow> \<dots>(\<infinity>)"
 from[ ]Catch
 show False
 by (auto
 qed
qed

definition
termi_call_steps :: "('s,'p,'f) body \<Rightarrow> (('s \<times> 'p) \<times> ('s \<times> 'p))set"
where
"termi_call_steps \<Gamma> =
{((t,q),(s,p)). \<Gamma>\<turnstile>Call p\<down>Normal[(("a"0,Position),(aa,ab") (("b, ),Position) "babb) []
\<existsc.\<><turnstile>Call, s <rightarrow<sup+ c,ormal)\ thmtrancl_induct)java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

primrec subst_redex:: "('s,'p,'f)com \<Rightarrow> (lemma steps_redex'
where
" c= "java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
"subst_redex (Basic f) c = c" |
"subst_redex (Spec r) c = c" |
"subst_redex (Seq c\<^sub>1 c\<^sub>2) c = Seq (subst_redex c\<^sub>1 c) c\<^sub>2" |
"subst_redex (Cond b c\<^sub>1 c\<^sub>2) c = c" |
"subst_redex (While b c') c = c" |
"subst_redex (Call p) c = c" |
"subst_redex (DynCom d) c = c" |
"subst_redex (Guard f b c') c = c" \subst_redex)>subst_redex,s)java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
"subst_redex (Throw) c = c" |
subst_redexCatch>\^>) c\^>)c<>2java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87

caseTrans r'')
 "subst_redex c (redex c) = c"
 by (induct c) auto

lemma redex_subst_redex: "redex (subst_redex c r) = redex r"
 by (induct c) auto

lemma step_redex':
 shows "\<Gamma>\<turnstile>(redex c,s) \<rightarrow> (r',s') \<Longrightarrow> \<Gamma>\<turnstile>(c,s) \<rightarrow> (subst_redex c r',s')"
by (induct c) (auto intro: step.Seq step.Catch)

lemma where
 shows "\<Gamma>\<turnstile>(r,s) \<rightarrow> (r',s') \<Longrightarrow> \<Gamma>\<turnstile>(subst_redex c r,s) \<rightarrow> (subst_redex c r',s')"
by (induct c) (auto intro: step.Seq step.Catch)

steps_redex
 assumes steps: "\<Gamma>\<turnstile> (r, s) \<rightarrow>\<^sup>* (r', s')"
 shows "\<And>c. \<Gamma>\<turnstile>(subst_redex c r byiprover
using steps
proof (induct rule: converse_rtranclp_induct2 [case_names Refl show
 case Refl
showGamma\turnstile(subst_redex ' s' <rightarrow<sup* subst_redexcr' s)java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
 by simp
next
 case (Trans (uleexI =]rule)
using java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 step_redex [OFthis
 have "\<Gamma>\<turnstile> (subst_redex c r, s) \<rightarrow> (subst_redex c r'', s'')".
 also
 have "\<Gamma>\<turnstile> (subst_redex c r'', s'') \<rightarrow>\<^sup>* (subst_redex c r', s')" by fact
 finally show ?case .
qed

ML \<open>
 ML_Thms.bind_thm ("trancl_induct2", Split_Rule.split_rule @{context}
 (Rule_Insts.read_instantiate @{context}
 [((("a", "b 0 \<Longrightarrow> \existsf.f 0= <and> (\forall>i.( i f Suc) \in> )java.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
 @{thm trancl_induct}));
<

lemma steps_redex':
 assumes steps: "\<Gamma>\<turnstile> (r, s) \<rightarrow>\<^sup>+ (r', s')"
cr's')"
using steps
proof (induct rule: tranclp_induct2
 case (Step r by (cases i)auto
 have "\<Gamma>\<turnstile> (r, s) \<rightarrow> (r', s')" by fact
 thencase showthen
 by (rule step_redex)
 then show "\<Gamma>\<turnstile> (subst_redex c r, s) \<rightarrow>\<^sup>+ (subst_redex c r', s')"..
ext
 case (Trans r' s' r'' s'')
 have "\<Gamma>\<turnstile> (subst_redex c r, s) \<rightarrow>\<^sup>+ (subst_redex c r', s')" by fact
 also
 have "\<Gamma>\<turnstile> (r', s') \<rightarrow> (r'', s'')" by fact
 hence "\<Gamma>\<turnstile> (subst_redex c r', s') \<rightarrow> (subst_redex c r'', s'')"
 by (rule
 finally show "\<Gamma>\<turnstile> (subst_redex c r, s) \<rightarrow>\<^sup>+ (subst_redex c r'', s'')" .
qed

primrec seq:: "(nat \<Rightarrow> ('s,'p,'f)com) \<Rightarrow> 'p \<Rightarrow> nat \<Rightarrow> ('s,'p,'f)com"
where
"seq c p 0 = Call p" |
"seq c p (Suc i) = subst_redex (seq c p i) (c i)"

lemma renumber':
 assumes f: "\<forall>i. (a,f i) \<in> r\<^sup>* \<and> (f i,f(Suc i)) \<in> r"
 assumes a_b: "(a,b) \<in> r\<^sup>*"
 shows " \<orall \+ay<> Pa<> Py
using a_b
induct:converse_rtrancl_inductconsumes ]
 assume "b = f 0"
with show\>f f0 b \and\<> fi, f Suc \in> r"
 by blast
next
 fix a z
 assume a_z: "(a, z) and> <Gamma\turnstilex <rightarrow\<sup+ }java.lang.StringIndexOutOfBoundsException: Index 148 out of bounds for length 148
 assume "b = f 0 \<Longrightarrow> \<exists>f. f 0 = z \< apply blast
 "b = f 0"
 then obtain f where f0: "f 0 = z" and seq: "\<forall>i. (f i, f (Suc i)) \<in> r"
 by iprover
 {
 fix i have "((\<lambda>i. case i of 0 \<Rightarrow> a | Suc i \<Rightarrow> f i) i, f i) \<in> r"
 using seq a_z f0
 by (cases i) auto
 }
 then
 show " terminates_impl_no_infinite_trans_computation
by-(ruleexI wherex"\<lambdai.case 0 \<> a \Rightarrow>f i"simpjava.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102

lemma renumber:
"\<forall>i. (a,f i) \<in> r\<^sup>* \<and> (f i,f(Suc i)) \<in> r
\<Longrightarrow> \<exists>f. f 0 = a \<and> (\<forall>i. (f i, f(Suc i)) \<in> r)"
 by (blast dest:renumber')

lemma wf{(y x)<><>c,)\rightarrow\^> and\<amma\turnstile <rightarrowy}
 "\<forall>y. r\<^sup>+\<^sup>+ a y \<longrightarrow> P a \<longrightarrow> P y
 \<Longrightarrow> ((b,a) \<in> {(y,x). P x \<and> r x y}\<^sup>+) = ((b,a) \<in> {(y,x). P x \<and> r\<^sup>+\<^sup>+ x y} fixf
 assume "\<forall>i. \Gamma>\<turnstile>(cs \<>\<^>* fi\and
apply clarify
apply(erule trancl_inductjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 apply blast
apply( :tranclp_trans
apply clarify
apply(erule tranclp_induct)
apply blast
apply(blast intro:trancl_trans)
one

corollary terminates_impl_no_infinite_trans_computation:
assumes terminates: \<><urnstilec\\<>"
shows "\<not>(\<exists>f. f 0 = (c,s) \<and> (\<forall>i. \<Gamma>\<turnstile>f i \<rightarrow>\<^sup>+ f(Suc i)))"
proof -
 have "wf({(y,x). \<Gamma>\<turnstile>(c,s) \<rightarrow>\<^sup>* x \<and> \<Gamma>\<turnstile>x \<rightarrow> y}\<^sup>+)"
 ( )
 show " (\<>c. <Gamma>\turnstile ( p Normal (simpaddwf_iff_no_infinite_down_chain
 java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
 f
 assume "\<forall>i. \<Gamma>\<turnstile>(c,s) \<rightarrow>\<^sup>* f i \<and> \<Gamma>\<turnstile>f i \ assume"\exists.f(:at (,s)\and<>. Gamma\turnstile> \rightarrow\^> f( ))
 hence "\<exists>f. f (0::nat) = (c,s) \<and> fwhere
 by (rule renumber [to_pred])
 moreover from terminates_impl_no_infinite_computation [OF terminates]
 have "\<not> (\<exists>f. f (0::nat) = seq: pwherepi =(snd (f):b)"for
 by (simp add: inf_def)

 by simp"existsf.\foralli ( (Suci, i \in {, x) \Gamma\<turnstile>(, s \<ightarrow>\^> haveinf:"i \>turnstile (p \>Normal(s <and
 qed
 qed
"\not> \exists>f\<forall>i ( proof( exI[where=]allI)
 \<in> {(y, x). \<Gamma>\<turnstile>(c, s) i
 by (simp add: wf_iff_no_infinite_down_chain)
thusthesis
 proof (rule contrapos_nn)
 "<>f 0:at (,s <> (\foralli <Gamma>turnstile> ii i)"
 then obtain f where
 f0 " =c,s)and
 seq: "\<forall>i. \<Gamma>\<turnstile>f i \<rightarrow>\<^sup>+ f (Suc i)"
 by iprover
 show
 "\<exists>f. \<forall>i. (f (Suc i), f i) \<in> {(y, x). \<Gamma>\<turnstile>(c, s) \<rightarrow>\<^sup>* x \<and> \<Gamma>\<turnstile>x \<rightarrow> y}\<^sup>+"
 proof (rule exI [where x=f],rule allI)
 fixjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 show "(f case (uc njava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24

 {
 fix i have "\<Gamma>\<turnstile>(c,s) \<rightarrow>\<^sup>* f i"
 proofinduct )
 case 0 show "\<Gamma>\<turnstile>(c, s) \<rightarrow>\<^supjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 by (simp add: f0)
next
 case (Suc n)
 have "\<Gamma>\<turnstile>(c, s) \<rightarrow>\<^sup>* f n" by fact
 with seq show "\<Gamma>\<turnstile>(c, s) \<rightarrow>\<^sup>* f (Suc n)"
 by (blast intro: tranclp_into_rtranclp rtranclp_trans)
 qed
 }
hence<>\<turnstilecs)\rightarrowsup i
byjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 with seq have
 "(f (Suc i), f i) \<in> {(y, x). \<Gamma>\<turnstile>(c, s) \<rightarrow>\<^sup show ?thesis
 by clarsimp
 moreover
 \forally.\Gamma\turnstile>fi\rightarrow\^+y<><><>c,s < by(subst
 by (blast intro: tranclp_into_rtranclp rtranclp_trans)
 ultimately
 qed
 by (subst lem )
 qed
 qed
 qed
qed

theorem wf_termi_call_steps: "wf (termi_call_steps \<Gamma>)"
proof (simp only: termi_call_steps_def wf_iff_no_infinite_down_chain,
 clarify,simp)
 fix f
 by blast
 \<Gamma>\<turnstile>Call p \<down> Normal s \<and>
 (> Gamma>Callp, \rightarrow<sup+(c, Normal t)\<and c Callq)
( Suc))(f i"
 define s where "s i = fst (f i)" for i :: nat
 define p where "p i = (snd (f i)::'b)" for i :: nat
 from inf
 have inf': "\<forall>i. \<Gamma>\<turnstile>Call (p i) \<down> Normal (s i) \<and>
 (\<exists>c. \<Gamma>\<turnstile> (Call (p i), Normal (s i)) \<rightarrow>\<^sup>+ (c, Normal (s (i+1))) \<and>
 redex c = Call (p (i+1)))"
 apply -
 apply (rule allI)
 apply (erule_tac x=i in allE)
 apply (auto simp add: s_def p_def)
 done
 show False
 proof -
 from inf'
ve "\exists>c.forall>i\<Gamma><urnstileCall ( )<> Normal (s i)<andjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
 \<Gamma>\<turnstile> (Call (p i), Normal (s i)) \<rightarrow>\<^sup>+ (c i, Normal (s (i+1))) \<and>
 redex (c i) = Call (p (i+1))"
 apply -
 apply (rule choice)
 by blast
 then obtain c where
 termi_c: "\<forall>i. \<Gamma>\<turnstile>Call (p i) \<down> Normal (s i)" and
 steps_c: "\<forall>i. \<Gamma>\<turnstile> (Call (p i), Normal (s i)) \<rightarrow>\<^sup>+ (c i, Normal (s (i+1)))" and
 red_c: "\<forall>i. redex (c i) = Call (p (i+1))"
 by auto
 define g where "g i = (seq c (p 0) i,Normal (s i)::('a,'c) xstate)" for i
 from red_c [rule_format, of 0]
0 ( p0 Normal0)"
 by (simp add: g_def)
 moreover
 {
 fix i
 have "redex (seq c (p 0) i) = Call (p i)"
 ( i) autosimp add redex_subst_redex)
 from this [symmetric]
 have "subst_redex (seq c (p 0) i) (Call (p i)) = (seq c (p 0) i)"
 by (simpapply (ule choice
 } note subst_redex_seq = this
 have "\<forall>i. \<Gamma>\<turnstile> (g i) \<rightarrow>\<^sup>+ (g (i+1))"
 proof
 fixjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 from steps_c [rule_format, of i]
 have "\<Gamma>\<turnstile> (Call (p i), Normal (s i)) \<rightarrow>\<^sup>+ (c i, Normal (s (i + 1)))".
 from steps_redex'definegwhere "gi seq c Normal :(,c)xstate) i
 have "\ from [rule_format of0
 " = ( 0) s)"
 hence "\<Gamma>by(simpadd g_def)
 (seq c (p 0) (i+1), Normal
 by (simp add: subst_redex_seq)
 thus "<Gamma\turnstile g i) \>\^>+( (+1))
 by (simp addhaveredexseqcp ))= ( )
 qed
 moreover
 from terminates_impl_no_infinite_trans_computation [OF termi_c [rule_format, of 0]]
 i
 False
 by fromsteps_c [ of]
 qed
qed

lemma no_infinite_computation_implies_wf:
 assumes not_inf: "\<not> \<Gamma>\<turnstile> (c, s) \<rightarrow> \<dots>(\<infinity>)"
 shows "wf {(c2,c1). \<Gamma> \<turnstile> (c,s) \<rightarrow>\<^sup>* c1 \<and> \<Gamma> \<turnstile> c1 \<rightarrow> c2}"
proof (simp only: wf_iff_no_infinite_down_chain,clarify, simp)
 fix f
 \forall\Gamma\turnstile>c,s)\<><sup* i <><Gamma\turnstile i\<> Suc)java.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 130
 hence "\<exists>f. f 0 = (c, s) \<and> (\<forall>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i))"
 by( renumber[to_pred])
 moreover from not_inf
 have "\<not> (\<exists>f. f 0 = (c, s) \<and> (\<forall>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)))"
 simpaddinf_def)
 ultimately show False
 by simp
qed

lemma not_final_Stuck_step: "\<not> final (c,Stuck) \<Longrightarrow> \<exists>c' s'. \<Gamma>\<turnstile> (c, Stuck) \<rightarrow> (c',s')"
by (induct c) (fastforce intro: step.intros simp add: final_def)+

lemma not_final_Abrupt_step:
 "\<not> final (c,Abrupt s) \<Longrightarrow> \<exists>c' s'. \<Gamma>\<turnstile> (c, Abrupt s) \<rightarrow> (c',s')"
by (induct c) (fastforce intro: step.intros simp add: final_def)+

not_final_Fault_step
 "\<not> final (c,Fault f) \<Longrightarrow> \<exists>c' s'. \<Gamma>\<turnstile> (c, Fault f) \<rightarrow> (c',s')"
by (induct c) (fastforce intro: step.intros simp add: final_def)+

lemma not_final_Normal_step:
grightarrow\existsc's' \<Gamma>\<turnstile> (c, Normal s) \<rightarrow> (c',s')"
proof (induct c)
 case Skip thus ?case by (fastforce intro: step.intros simp add: final_def)
next
 case Basic thus ?case by (fastforce intro: step.intros)
next
 case (Spec r)
 thus ?case
 by (cases "\<exists>t. (s,t) \<in> r") (fastforce intro: step.intros)+
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 case (Seq c\<^sub>1 c\<^sub>2)
 thus ?case
by c\sub,Normal" (fastforce : .ntros add )
next
 case (Cond b c1 c2)
 java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
 by (cases "s \<in> b") (fastforce intro: step.intros)+
next
 case (While b c)
 show ?case
 by (cases "sqed
next
 case (Call p)
 show ?case
 by (cases "\<Gamma> p") (fastforce intro: step.intros)+
next
 case DynCom thus ?case by (fastforce intro: step.intros)
next
 case (Guard f g c)
 show ?case
 by("\<>g)(fastforce:.intros)
next
 case Throw
 thus ?case by (fastforce intro: step.intros simp add: final_def)
next
 case (Catch c\<^sub>1 c\<^sub>2)
 thus ?case
 by ( (induct) ( intro step. simpadd:)
qed

al_termi
"final (c,s) \<Longrightarrow> \<Gamma>\<turnstile>c\<down>s"
 by (cases s) (auto simp add: final_def terminates.intros)

lemma split_computation:
assumes steps: "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c\<^sub>f, s\<^sub>f)"
assumes not_final: "\<not> final (c,s)"
assumesfinal " (c\^subf,s)"
shows "\<exists>c' s'. \<Gamma>\<turnstile> (c, s) \<rightarrow> (c',s') \<and> \<Gamma>\<turnstile> (c', s') \<rightarrow>\<^supjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
using steps not_final final
( :converse_rtranclp_induct2case_names ]java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
 case Refl thus ?case by simp
next
 case (Trans c s c' s')
 thus ?case by auto
qed

lemma wf_implies_termi_reach_step_case:
assumes hyp: "\<And>c' s'. \<Gamma>\<turnstile> (c, Normal s) \<rightarrow> (c', s') \<Longrightarrow> \<Gamma>\<turnstile>c' \<down> s'"
shows "\<Gamma>\<turnstile>c \<down> Normal s"
using hyp
proof (induct c)
 case Skip show ?case by (fastforce intro: terminates.intros)
next
 case Basic show ?case by (fastforce intro: terminates.intros)
next
 case (Spec r)
 show ?case
 by (cases "\<exists>t. (s,t)\<in>r") (fastforce intro: terminates.intros)+
next
 case (Seq c\<
 have hyp: "\<And>c' s'. \<Gamma>\<turnstile> (Seq c\<^sub>1 c\<^sub>2, Normal s) \<rightarrow> (cjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 show?
 proof (rule terminates.Seq)
 {
 fix c' s'
assume\: "\Gamma\turnstile c\^>1 Normal )\<ightarrow c's)"
 have "\<Gamma>\<turnstile>c' \<down> s'"
 proof -
 moreover
have<>< '\>r',s')byfact
 by step_redexes r'obtainc'where
from OFthis
 have "\<Gamma>\<turnstile>Seq c' c\<^sub>2 \<down> s'".
 thus "\<Gamma>\<turnstile>c'\<down> s'"
 by cases auto
 qed
 }
 from Seq.hypsassumes final: "inal c\sub>f,<^>f)
 show "\<Gamma>\<turnstile>c\<^sub>1 \<down> Normal s".
 nextby (induct) fastforce intro stepintros simp :by :"<>\<>(r,,s)\<ightarrow(r',"
 "<>s'. <>\turnstile <langle>c\ ( : converse_rtranclp_induct2 [case_names Trans]java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
 proof (intro allI impI)
 fix s'
 assume exec_c\<^sub case (Trans c ssc's')
 show "\<Gamma>\<turnstile>c\<^sub>2 \<down> s'"
 proof
 case True
 hence"c\^sub>1=Skip \<or> java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 0
 by (simp add: final_def)
 thus ?thesis
 proof
 assume Skip: "c\<^ (induct c)
 have "\<Gamma>\<turnstile> \<existsc' <Gamma\<turnstile(,)\rightarrow><sup*(c's)\and r'c\^sub>2 <> redexes c"
 by (rule step.SeqSkipusing
 from proofinduct: converse_rtranclp_induct2 case_names Trans
have\><>\Normal.
 moreover from exec_c\<^sub>1showcase
 have "'Normal"
 by( elimexec_Normal_elim_cases
 ultimately
 next
 assume Throw: "c\<^sub>1=Throw"
withexec_c^>1s=Abrupts""
 by (auto elim: exec_Normal_elim_cases)
 thus ?thesis
 byauto
 qed
 next
 case False
 from exec_impl_steps [OF exec_c\<^sub>1]
 obtain c\<^>f t where
 steps_c\<^sub>1: "\<Gamma>\<turnstile> (c\<^sub>1, Normal s) \<rightarrow>\<^sup>* (c\<^sub>f, t)" and
 : s
 Abrupt x \<Rightarrow> c\<^sub>f = Throw \<and> t = Normal x
 | _ \<Rightarrow> c\<^sub>f = Skip \<and> t = s')"
 by (fastforce split: xstate.splits)
 with fin have final "final (c\<^>f,t)"
 by (cases s') (auto simp add: final_def)
 from split_computation [ r' "Seq 'c c\^sub2, srightarrow> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 obtain c'' s'' where
 first: "\<Gamma>\<turnstile> (c\<^sub>1, Normal s) \<rightarrow> (c'', s'')" and
 showcase
byjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 from step.Seq [OF first]
 have "\<Gamma>\<turnstile> (Seq c\<^sub>1 c\<^sub>2, Normal s) \<rightarrow from Seq.yps ( [OF this]
 from hyp [OF this]
 termi_s:\><> ' \\<down> s''"
 show ?thesis
 proof (cases s'')
 case (Normal shows "\And>. Seqrc<^ub>2 <> redexes c
 from termi_s\> exists<>turnstile, <><sup(''\and'c<sub2 <in redexesc"
 have ><urnstile\sub?
 ( " c<>, s"
 show True
 proof c<>1Skiporc 's r's')
 case False
 Trans c'
 by (cases s') auto
 from steps_Skip_impl_exec [OF rest [simplified this]] Normal
 have "Γc'',Normal x⟩
 by simp
 from term<> [rule_format, OF this]
 show "Γ⊨

 case True
 with fin obtain x' where s': "s'=Abrupt x'" and
 by auto
 from steps_Throw_impl_exec [OF rest [simplified this]] Normal
 have "\Gamma>\<turnstile c'',Normal x⟩
 by simp
 from termi_c₂ [rule_format, OF this] s'
 w \<>\
 qed
 next
 case (Abrupt x)
 from steps_Abrupt_pro[OF rest this]
 have "t= step<><le,
 assumes: "Catch r c\<^>2<^sub>1: "<><turnstile> (c_* (c_\<turnstilec, rightarrow (c',s')\and> Catch2 ∈ c'"
 by (cases s') auto
 "Gammac\sub2 ↓
 finfinalf,t)"
 next
 case (Fault f)
 from steps_Fault_prop [OF rest this]
 have "t=Fault f" by simp
 with fin have "s'=Fault f"
 by (cases s') auto
 thus "<⊨cs'"
 by auto
 next
 case Stuck
teps_Stuck_prop[F rest this]
 have "t=Stuck" by simp
 with fin have "sStuck
 by (cases s') auto
 thus⊨2 ↓s
 by auto
 qed
 qed
 qed
 qed
next
 case
 have hyp: "∧c' s'. Γ⊨o
 show ?case
 proof (cases "s∈b") by ba
 case True
 then have "Γ
 byrule.CondTrue
 from hyp [OF this] have java.lang.NullPointerException
 assassu steps: "Gamma⊨\^>(' s)java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
 by (auto [OF [simplified]] java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
 next
 case False
 then have java.lang.NullPointerException
 by (rule step.CondFalse)
 from hfrom termi_\^ [rl_fra, F this] '
 with False show ?thesis
 by (auto intro: terminates.intros)
 qed
next
 case (While b c)
 have hyp: "∧
 show ?case
 proof (cases "s∈b")
 case True
 then have "Γ
 by (rule step.WhileTrue)
 from hyp [OF this] have "Γ⊨(Seq c (While b c)) ↓ Normal s".thus\<><
 with True show ?thesis
 by (auto elim: terminates_Normal_elim_cases intro: terminates.intros)
 next
 case False
 thus ?thesis
 by (auto intro: terminates.intros)
 qed
next
 case (Call p)
 yp "\Andc s' <><>( Γc' ↓
 show ?case
 proof (cases "Γ
 case None
 thus ?thesis
 by (auto intro: terminates.intros)
 next
 case (Some bdy)
 then rom stp_tuck_prkpop[O r
 by (rule step.Call)
 from hyp [OF this] have "Γ⊨bdy ↓ Normal s".
 with Some show ?thesis
 by (auto intro: terminates.intros)
 qed
next
 case (DynCom c)
 : "<> ' \Gamma>⊨ (DynCom c, Normal s) → (c', s') ==>) java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 have "Γ
 by (rule step.DynCom)
 from hyp [OF this] have "Γ⊨ctarrow \^> sub2)
 then show ?case
 by (auto intro: terminates.intros)
next
 case (Guard f g c)
 have hyp: "∧c' s'. Γ⊨ (Guard f g c, Norma
 show ?case
 proof (cases "s∈g")
 case True
 then have "Γ⊨ (Guard f g c, Normal s) → (c, Normal s)"
 by (rule step.Guard)
 from hyp [OF this] have "Γ⊨c↓ Normal s".
 with True show ?thesis
 by (auto intro: terminates.intros)
 next
 case False
 thus ?thesis
 by (auto intro: terminates.intros)
 qed
next
 case Throw show ?case by (auto intro: terminates.intros)
next
 case (Catch c₁ (auintro: terminates.intros)
 have hyp: "∧
case
 proof (rule terminates.Catch have><turnstile> (Cond c₁ c₂, Normal s) → (c₂, Normal s)"
 {
 fix c' s'
 assume step_c₁: "Γ⊨2 ↓ s".
 have "Γ⊨c' ↓ s'"
 proof -
 from step_c₁
 have "Γ⊨
 by (rule step.Catch)
 from hyp [OF this]
java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 65
 thus "Γ
 by cases auto
 qed
 }
 from Catch.hyps (1) [OF this]
 show "Γ⊨c_\<turnstileb) Normal s".

 show "∀minates
 proofnext

 assume exec_cthus
 (nates
 proof (cases java.lang.NullPointerException
 case True
 with exec_c₁
 have Throw: "c₁=Throw"
 (auto simp add: final_defelim: exec_Normal_el)
 have "Γ⊨(Catch Throw c_case
 by (rule step.CatchThrow)
 from [simplified, OF]
 have "Γ⊨
 moreover from exec_c₁ Throw
 have "s'=s"
 by (auto elim: exec_Normal_elim_cases)
 ultimately show ?thesis by simp
 next
 case False
 from exec_impl_steps [OF exec_c\^]
 obtain c_f t where
 steps_c₁: "Γ⊨ (c_<>
 by (fastforce split: xstate.splits)
 from split_computation [OF steps_c₁ False]
 obtain c'' s'' where
 first: "Γ⊨sho?ca
 rest: "Γ⊨ (c'', s'') →)fastforce
 by (auto simp add: final_def)
 from step.Catch [OF first]
 have java.lang.NullPointerException
 from hyp [OF this]
 have "ΓCatchc^2 ↓ s''"
 moreover
 from steps_Throw_im [OF rest]
 have "Γ⊨DynCom
 moreover
 from rest obtain x where "s''=Normal x"
 by (cases s'')
 (auto dest: steps_Fault_prop steps_Abrupt_prop steps_Stuck_prop)
 ultimately show ?thesis
 by (fastforce elim: terminates_elim_cases)
 qed
 qed
 qed
qed

lemma:
assumes wf: "wf {(cfg2,cfg1). Γ \<turnstile (Guard f g c, Normal s) \<ightarrow
shows "∧c1 s1thus ( intro :final_def
using wf
proof (induct cfg1, simp
 fix c1 s1
 assume reach: "Γ1 c\sub)
 assume hyp_raw: "qed
 [Γ⊨
 ==> Γ⊨c2 ↓ s2"
 havehyp: "AndGamma>c1 →
 apply -
 apply (rule next
 apply assumption
 using
 apply simp
 apply (rule refl)
 done

 showc1 \>
 proof (cases s1)
 case (Normal s1')
 withp_casemal
 show ?thesis
 by auto
 qed (auto intro: terminates.intros)
qed

theorem no_infinite_computation_impl_terminatessplit_computation
 assumes not_inf: "¬
 "<amma⊨c↓s"
proof -
 from no_infinite_computation_implies_wf [OF not_inf]
 have wf: "wf {(c2, c1). Γ⊨(c, s) →^\>c1 c2}".
 show ?thesis
 by (rule wf_implies_termi_reach [OF wf]) auto
qed

corollary terminates_iff_no_infinite_computation:
 "Γ⊨c↓s = (¬ cases auto
 apply (rule)
 apply (erule terminates_impl_no_infinite_computation)
 apply (erule no_infinite_computation_impl_terminates)
 done

(* ************************************************************************* *)
subsection ‹
(* ************************************************************************* *)

text ‹
an important lemma for the completeness proof of the Hoare-logic for
correctness we need a generalisation of @{const "redex"} that not only
tself blf butt all thhe enclosingstatements as well.
›

primrec redexes:: "('s,'p,'f)com ==> ('s,'p,'f) proof (cases "final<sub,Normal s)")
where
"redexes Skip = {Skip}" |
"redexes (Basic f) = {Basic f}" |
"redexes (Spec r) = {Spec r}" |
"redexes (Seq c₁ chaveThrowcjava.lang.NullPointerException
java.lang.NullPointerException
"redexes (While b c) = {While
"redexes (Call p) = {Call p}" |
"redexes (DynCom d) = {DynCom d}" |
"redexes (Guard f b c) ) ={Guard f b c}"
"redexes (Throw) = {Throw}" |
"redexes (Catch ch "ΓcNormal.

lemma root_in_redexes: "c ∈ redexes c"
 apply (induct c)
 apply auto
 done

lemma redex_in_redexes: "redex c ∈
 case Fa
 apply auto
 done

lemma redex_redexes: "∧c' ∈ redex
 apply (induct c)
 apply auto
 done

lemma step_redexes:
 shows " c
 ==> ∃c'. Γ⊨⊨1, Normal s) → and
proof (induct c)
 case Skip thus ?case by (fastforce intro: step.intros elim: step_elim_cases)
next
 case Basic thus ?case by (fastforce intro: step.intros elim: step_elim_cases)
next
 case Spec thus ?case by (fastforce intro: step.intros elim: step_elim_cases)
next
java.lang.NullPointerException
 have "r "ΓCatch c'' cs''".
 hence r: "r = Seq c₁ c₂ ∨
 by simp
 have step_r: "Γ⊨ (r, s) → (r', s')" by fact
 from r show ?case
 proof
 assume "r = Seq c₁ c₂"
 with step_r
 show ?case
 by (aauto simp add: root_in_rexes)
 next
 assume r: "r ∈ redexes c₁"
 from Seq.h (1 [OFstep_r this]
 obtain c' where
 step_c₁: "Γ⊨ (c₁, s) → (c', s')" and
 r': "r' ∈
 by blast
 from
 have "Γ
 with r'
 show ?case
 by auto
 qed
next
 case Cond
 thus ?case
 by (fastforce intro: step.intros elim: step_elim_cases simp add: root_in_redexes)
next
 case While
 thus ?case
 by (fastforce intro: step.intros elim: step_elim_cases simp add: root_in_redexes)
next
 case Call Call ththus ?case
 by (fastforce intro: step.intros elim: step_elim_cases simp add: root_in_redexes)
next
 case DynCom thus ?case
 by (fastforce intro: step.intros elim: step_elim_cases simp add: root_in_redexes)
next
 case Guard thus ?case
 by (fastforce intro: step.intros elim: step_elim_cases simp add: root_in_redexes)
next
 case Throw thus ?case
by (fatforce intro: ste.introselim: tep_elim simp add: root_in_redexes)
next
 case (Catch c₁ c₂)
 have "r ∈ redexes (Catch c_proofs1
 hence r: java.lang.NullPointerException
 by simp
 have step_r: "proof allI
 from r show ?case
 proof
 assume java.lang.NullPointerException
 with step_r
 show ?case
 by (auto simp add: root_in_redexes)
 next
 assume r: "r ∈show\^<s"
 from Catch.hyps (1) [OF step_r this]
 obtain c' where
 step_c₁: "Γ⊨ (c₁, s) →
 r': "r' ∈ redexes c'"
 by hencec<>=Skip="
 from step.Catch [OF step_c{(c2, c1). Γ(c, s) →^\<turnstil>c →
 have "Γ⊨
 with r'
 show
 by auto
 qed
qed

lemma steps_redexes:
 assumes steps: apply no_infinite_computation_impl_terminates
 shows "∧c. r ∈ redexes c ==> ∃
using steps
proof (induct rule: converse_rtranclp_induct2 [case_names Refl Trans])
 case Refl
 then
 show "∃c'. Γ⊨ (c, s') →java.lang.NullPointerException
 by auto
next
 case (Trans r s r'' s'')
 have "Γ⊨ (r, s) → (r'', s'')" "r ∈ redexes c" by fact+
 from step_redexes [OF this]
 obtain c' where
 step: "Γ⊨ (c, s) → (c', s'')" and
 r'': "r'' ∈
 by blast
 note step

 from Trans.hyps (3) [OF r'']
 obtain c'' where
 steps: "Γ⊨ (c', s'') → f) =Basicjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
 r': "r' ∈
 by blast
 note steps
 finally
 show ?case
 using r'
 by blast
qed

lemma steps_redexes':
 assumes steps: "Γ⊨ (r, s) →⁺ (r', s')"
 shows "∧ redexes c"
using steps
java.lang.NullPointerException
 case (Step r' s' c')
 have "Γ⊨ (r, s) → (r', s')" "r ∈ redexes c'" by fact+
 from ste [OF this]
 show ?case
 by (blast intro: r_into_trancl)
next
 case (Trans r' s' r'' s'')
 from Trans obtain c' where
 steps: "Γ⊨ (c, s) →java.lang.NullPointerException
 r': "r' ∈ redexes c'"
 by blast
 notesteps
 moreover
 have "Γ⊨ (r', s') → (r'', s'')" by fact
 from step_redexes [OF this r'] obtain c'' where
 step: "Γ⊨ (c', s') → (c'', s'')" and
 r'': "r'' ∈ redexes c''"
 by blast
 note step
 finally ow
 using r'' by blast
qed

lemma step_redexes_Seq:
 assumes step: "Γ⊨(r,s) → (r',s')"
 assumesSeq"Seqrc\^>2\in>redexes "
 shows "∃c'. Γ⊨(c,s) → (c',s') ∧ Seq r' c₂ ∈ redexes c'"
proof -
 from step.Seq [OF step]
 have "Γ⊨ (Seq r c₂, s) → (Seq r' c₂, s')".
 from step_redexes [OF this Seq]
 haver have<amma<
qed

lemma steps_redexes_Seq:
 assumes steps: "Γ (r s)\rightarrow* (r', s')"
 shows "∧c. Seq r c_bysim
 ∃c'. Γ step_r: "Γ (r, s) → (r', s')" by f fact
using steps
proof (induct rule: converse_rtranclp_induct2 [case_names Refl Trans])
 case Refl
 then show ?case
 byproof (casases s'')

next
 case (Trans r s r'' s'')
 have "Γ⊨ (r, s) → (r'', s'')" "Seq r cjava.lang.NullPointerException
 from step_redexes_Seq [OF this]
 obtain c' where
 step: "Γ<^sub>2:: \forall>t. t. \>turnsti> \langle>c'',Norm x\<>
 r'': "Seq r'' c₂ 'where
 by blast
 note step
 also
 from Trans.hyps (3) [OF r'']
 obtain c'' where
 stepsproof( "\existsx'. s'=Abrupt x'")
 r': "Seq r' c₂ ∈ redexes c''"
 by blast
 note steps
 finally
 show ?case
 using r'
 by blast
qed

lemma steps_redexes_Seq':
 assumes steps: "Γ⊨ (r, s) →⁺ (r', s')"
 shows "∧c. Seq r c₂ ∈ redexes c
 ==> ∃c'. Γ⊨(c,s) →⁺ (c',s') ∧ Seq r' c₂ ∈ redexes c'"
using steps
proofinduct consumescase_namesTrans
 case (Step r' s' c')
 have "Γ⊨ (r, s) → (r', s')" "Seq r c₂ ∈ redexes c'" by fact+
 from step_redexes_Seq [OF this]
 show ?case
 by (blast introshowGamma2 ↓ s'" .
next
 case (Trans r' s' r'' s'')
 from Trans obtain c' where
 steps: "Γ⊨ (c, s) →java.lang.NullPointerException
 r': "Seq r' c₂ ∈ redexes c'"
 by blast
 note steps
 moreover
 have "Γ⊨ (r', s') → (r'', s'')" by fact
 from step_redexes_Seq [OF this r'] obtain c'' where
 fromrest
 r'': "Seq r'' c₂ ∈ redexes c''"
 by blast
 note step
 finally show ?case
 using r'' by blast
qed

lemma step_redexes_Catch:
 assumes step: "Γ⊨(r,s) → (r',s')"
 assumes Catch: "Catch r c₂ ∈ redexes c"
 shows "∃c'. Γ⊨(c,s) → (c',s') ∧ Catch r' c₂ ∈ redexes c'"
proof -
 from step.Catch [OF step]
 have "Γ⊨ (Catch r c₂, s) → (Catch r' c₂, s')".
 from step_redexes [OF this Catch]
 show ?thesis .
qed

lemma steps_redexes_Catch:
 assumes steps: " by (cases s') auto
 shows "∧c. Catch r c₂ ∈ redexes c ==>
 ∃c'. Γ⊨(c,s) →^* (c',s') ∧ Catch r' c₂ ∈ redexes c'"
using steps
proof (induct rule: converse_rtranclp_induct2 [case_names Refl Trans])
 case Refl
 then show ?case
 by (auto)

next
 case (Trans r s r'' s'')
 have "Γ⊨ (r, s) → (r'', s'')" "Catch r c₂ ∈ redexes c" by fact+
 from step_redexes_Catch [OF this]
 obtain c' where
 step: "Γ⊨ (c, s) <rightarrowwith
 r'': "Catch r'' c₂ ∈ redexes c'"
 by blast
 note step
 also
 from Trans.hyps (3) [OF r'']
 obtain c'' where
 steps: "Γ⊨ (c', s'') →^* (c'', s')" and
 r': "Catch y auto
 by blast
 note steps
 finally
 show ?case
 using r'
 by blast
qed

lemma steps_redexes_Catch':
 assumes steps: "Γ⊨ (r, s) →java.lang.NullPointerException
 shows "∧c. Catch r c₂ ∈ redexes c
 ==> ∃c'. Γ⊨(c,s) →⁺ (c',s') ∧ Catch r' c₂ ∈ redexes c'"
using steps
proof (induct rule: tranclp_induct2 [consumes 1, case_names Step Trans])
 case (Step r' s' c')
 have "Γ⊨ (r, s) → (r', s')" "Catch r c₂ ∈ redexes c'" by fact+
 from step_redexes_Catch [OF this]
 show ?case
 by (blast intro: r_into_trancl)
next
 case (Trans r' s' r'' s'')
 from Trans obtain c' where
 steps: java.lang.NullPointerException
 r': "Catch r' c₂ ∈ redexes c'"
 by blast
 note steps
 moreover
 have "Γ⊨ (r', s') →1 c s'"byfact
 by blast
 note step
 finally show ?case
 using r'' by blast
qedshow case

lemma redexes_subset:"∧proof (cases "∈
 by (induct c) auto

lemma redexes_preserves_termination:
 assumes termi: "Γ⊨2 Normal s)s) → s)"
 shows "∧c'. c' ∈ redexes c ==> Γ⊨c'↓s"
using termi
by induct (auto intro: terminates.intros)

end

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