primrec redex:: "('s,'p,'f)com ==> ('s,'p,'f)com" where "redex Skip = Skip" | "redex (Basic f) = (Basic f)" | "redex (Spec r) = (Spec r)" | "redex (Seq c1 c2) = redex c1" | "redex (Cond b c1 c2) = (Cond b c1 c2)" | "redex (While b c) = (While b c)" | "redex (Call p) = (Call p)" | "redex (DynCom d) = (DynCom d)" | "redex (Guard f b c) = (Guard f b c)" "redex (Throw) = T| Catch: "[Γ(c1',s')] "==>>\\<turnstile>(Catch c2,s) →1' cCatchThrow: "Γ(Catch c\sub,Normal s) → (cjava.lang.NullPointerException
subsectionSmall-Step Computation: ‹(c, s) →›
type_synonym('s,'p,'f)config="('s,'p,'f)com\<times>('s,'f)xstate" inductive"step"::"[('s,'p,'f)dy(s,pfonfig','config\<ghtarrowtarrowrowoll" _\<turnstile>(_\<rightarrow>/_)\<close>[81,81,81]100) for\<Gamma>::"('s,'p,'f)body" where
|CondTrues\inb\Longrightarrow>\<Gamma>\<turnstile(Condbc\<^sub>1c\<^sub>2,Normals)\<rightarrow>(c\<^sub>1,Normals)" |CondFalse:"s\<notin>b\<ongrightarrow<Gamma\<turnstile>(Condbc\<^sub>1c\^sub2rmal<rightarrow>c<sub2rmal)java.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 136
inductive_casesstep_elim_cases[casesset]: "\<Gamma>\urnstilenstileSkip\ightarrow>" "\<Gamma>\<turnstile>(Guardfgc,s)\<rightarrow>u" "\<Gamma>\<turnstile>(Basicf,s)\<rightarrow>u" "\<Gamma>\<turnstile>(Spec "\<Gammaepss\<Gammaturnstilecfg\^sub>1\<rightarrow>\<^sup>*cfg\<^sub>2" "\<Gamma>\<turnstile>(Condbc1c2,srightarrow" "\<Gamma>\<turnstile>(While <amma>\<turnstile>(Callp,s)\<rightarrow>u" "\<Gamma>\<turnstile>(DynComc,s)\<rightarrow>u" "\<Gamma>\<turnstile>(Throw,s)\<ightarrowujava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48 c2,s
abbreviation "step_rtrancl_rancl:"[spdy(',p)config,(',',Rightarrowbool" (\<open>_\<>(\rightarrow<sup>)<lose[8181]100 where "\<Gamma>\<turnstile>cf0\<rightarrow>\<case(eq\^ub>c\<sub>2) abbreviation "step_trancl"::"[',p,fody's',figs,f)onfig\Rightarrow>bool (\<open>_\<turnstile>(_<rightarrow\<^sup>+/_)\<close>[81,81,81]100) where "\<Gamma>\<turnstile>cf0
lemma step_Stuck:: assumes step: "Γerse_rtranclp_induct2 shows using step by (induct auto
lemmaSeqSteps@{termNormalandabruptlyconstt}stateto assumes teps><turnstile>cfg2" hows s ∧ t'=t)"
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 using
roof<>ructuralties omputations<> case Refl thus?ase
simp next case (Trans cfgfinal(,s ==>" have st "Γ cfgcfg''" by fact have : "Γ cfg'' →* cfgjava.lang.NullPointerException
b:cfg>1 = (c,s)java.lang.NullPointerException obtain c1'',s'')" by (cases cfg'') auto from cfgcfg'' have java.lang.NullPointerException by simp hence "Γ: by (rule step.Seq alsofrom Trans.hyps (3) [OF cfg'' cfgjava.lang.NullPointerException have finally ?case . qed
lemma CatchSteps: assumes steps: "Γ⊨cfg1→* cfgf. s=Fault \Longrightarrow> 'Falt" shows"∧1 s ccfg1,s); cfg1',s')]
java.lang.NullPointerException using steps proof (induct rule: converse_rtranclp_induct [case_names Refl Trans]) case Refl thus ?case by simp next case (Trans cfgΓ⊨1 c><^sup>* (Seq c2, s')" have"<>⊨1 → cfg''"by fact have steps ""<><turnstile'' →* cfgjava.lang.NullPointerException havecaserans1 cfg'') obtain ccfgcfg'" by fac by (cases cfg'') auto from step cfg'' have s: "Γ⊨ (c1,s) → (c1'',s'')" by simp hence "Γ (Catch c2,s) → (Catch c2,s'')" l ste.Cah) alsohence "<⊨1 c,<>(Seq1'' c2,') have java.lang.NullPointerException finally show ?case . qed
lemma steps_Falt: "\Gamma>⊨f)" proof (indu c) case Sc<^s>1 c<sub>>2 have steps_c1: "Γ⊨ (cjava.lang.NullPointerException have\<turnstile2, Fault f) →java.lang.NullPointerException from SeqSteps [ steps_c1 refl refl] have<<>eq^1c<sub Fault )rightarrowjava.lang.NullPointerException also have java.lang.NullPointerException
java.lang.NullPointerException finally how ? ?casby s next case (Catch c2) have steps_c\<^ub12] "Gamma⊨1'' c<(\java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3 have also have"Γah Skip \sub, Fault f) → (Skip, Fault f)"by (rule CatchSkip) finallyshow ?caseby simp qedtforce
lemma steps_Stuck: "Γ🚫1: "Γ⊨1, Fault f) →"by ct proof (induct c) case (Seq c\<^have
java.lang.NullPointerException have steps_c\<turnstile> (c<^>2, tuck) →^sup>* (Skip, Stuck)"by fact
GuardFaultnotinLongrightarrow>turnstile , )\ induct have<><turnstile(Seq \^sub c\sub2 Stuck) →* (Seq Skip cjava.lang.NullPointerException also have"Γ2: "Γ⊨2, Stuck) → alsonote steps_c<><turnstile> (Seq^subcsub>>2 Stuck finallyshow ?caseby simp next case (Catchhow have steps_csub>1: "Γ⊨ (c1, Stuck) →, SeqSkip: "<Gammaturnstile(eq \^>2s\rightarrow(<>, from OFjava.lang.NullPointerException have"ΓCondTrue">\Longrightarrow<>turnstile also have"Γ (Catch Skip c(Skip, Stuck)"by (letchSkipjava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107 finallyshow ?caseby simp qed (fastforce intro: step.intros
lemmapt⊨java.lang.NullPointerException proof
e (Seq2)
<1"><> (\^>1 Abr \>^sup> (SSk, Abrs) by have steps_c\<turnstile> (c\<^sup>*(Skip, Abupt by fact from SeqSteps [OF steps_c1 refl refl] have "Γ⊨1 cs) → also have"Γebd Longrigh> also note steps_c2 finally show ?case by simp next
java.lang.NullPointerException steps_c\<>:f
java.lang.NullPointerException have "Γ
have finallyshow ?caseby simp qedastforceintros
lemma assumes step: "\<Gamma shows ∧ using step by ((induct)auto
lemma step_Abrupt_prop: assumes step: "java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null shows"∧ using showcc by
lemmaps_Fault_propl_op: assumes step: "Γ shows"s=Fault f ==> using step proof (induct rule: converse_rtran [case_names Refl Trans]) case Refl thus ?case by s next case (Trans c s c'' s'') thus ?case by (auto intro: step_Fault_prop) qed
lemma steps_Abrl assumes step: "Γ "=Abru t \LongrightarrowA t" using proof( rule: converse_rtranclp_induct2 case"\<Gamma\" next case (Trans c s c'' s'') thus? by (auto qed
lemmasteps_Stuck_prop assumes step: "ΓL>P" shows"s=Stuck ==> (au simp add: final_def) usingusing exec_impl [OF eeu o_s': proof (induct rule: converse_rtranclp_indu case Refl thus ?case by simp next case (Trans c s c'' s'') thus ?ca?case by (auto qed
(* ************************************************************************ *) subsection \openalence been ll-Step and B-Ss =Stuck or> (* ************************************************************************ *)
theorem exec_impl_steps: assumes exec: "Γ f <Longrightarrow shows
of
Abrupt x ==>f. = ==>
_<Rightarrow<>exec_redex_Stuck using exec
) casecase by next
se next" h "'= case GuardFault thus ?caseby(r.exsub1 = Specwith exec
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 4
ault_end next case Basic( s ep>< c next case Spec thus ?casecase CondTrueedex_Stuck next caseSpecStuck ?Seq next
aseSeq have exec_c\<turnstilejava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
c show proof (cases "\<exists case False from False Seq.hyps (2) have "Γ⊨
eflbyfastforcetroses henceCatch> c2 t) by (rule SeqSteps
romtain
steps_cormal
t: "(case t of Abrptx\> tte ' ki else c' = Throw ∧fr | _ ==> by auto
java.lang.NullPointerException also have "Γ (Seq Skip c2, s') →"\Gamma>\<turnstile t" alsonote steps_c "s=Stuck \<> finally have "Γ⊨s]) with t False show ?thesis by (cases tfromCatchMatchissub2] s_Normalcase next case True thenobtainsub1'"\<>\ ==> by blast from s' S.hyps (2) have "Γhyps (2 exec_c by auto "<Gamma>⊨ hence seq_q_c<^1:"><turnstile> (Seq c<sub1 cjava.lang.NullPointerException by (rule SeqSteps) auto also <>' <><turnstile>(c,s) → by (rule SeqThrow) finallyhave"ΓAbrupt x<Rightarrow> if s then 'ip \and> t=t el c'=Throw ∧ moreover
java.lang.NullPointerException byby (auto cse ultimately show ?thesis by auto qed next case CondTrue thus ?case (Faut f) next case CondFalse thus ?case by (blast intro: step.CondFalse rtranclp_trans) next case (WhileTrue s b s' t) fail: "< g" have exec_w: "Γ have b:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 hence step: "Γ⊨ by (rule step.WhileTrue) show ?case ultimately case False hileTruehyps (3) have "Γ.intros f by (cases s') auto hence seq_c: "Γ by (rule SeqSteps) auto from WhileTrue.hyps (5) obtain c' t' where steps_c t: "(
tro step_Fault this
| _ (( intro) by auto ?thesis
te also(ceim_cases by alsonote aultProp
by introntrosxec_elim_cases with t False by (cases t) auto next case True thenobtain x where s': "s'=Abrupt x" by blast
also from . (3) have"\<Gamma >,s∠ by auto hence seq_c: " by (rule uctconverse_rtranclp_induct2java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
o ">turnst> (SeqThro (While b c, x → by (rule yute)p_) finally have "Γ moreover from '"<⊨(c,s) \rightarroww>\uphro,Nal t by (auto intro: Abrupt_end) ultimately show ?thesi by auto qed next caseWh thus ?case by (fastforce intro: step.W rt c s s' next case Call thus ?case by (blast intro: step s'' next case CallUndefined thus ?case by (fastforce intro: step.CallUndefined rtranclp_trans) next case StuckProp thus ?case by (fastforce intro: steps_Stuck) next case DynCom thus ?case by (blast intro: step.DynCom rtranclp_trans) next case Throw thus ?case by simp ext caseAbrProps ?case by ( (ffastforce ce intropAbruupt) next se ChMh c\^b> s ' \<^2 "not isAbr t" from CatchMatch.hyps (2) have "Γ⊨ (c from)Normal1' s_Normal by simp hence"Γ show ?thesiby m also have "< by (ruleubsectionEquivalencebetween Terminationand the Absence of Infinite Computations› also from CatchMatch.hypscase )
steps_c2fromep_Fault_end_Normal
t: "(case t of Abrupt x ==>= Skp <>t' = t else c' = Throw ∧ | _ ==> g" by auto note steps_c"\Gamma>🚫 finally show ?case using t by (auto split: xstate.splits) next
java.lang.NullPointerException have t: "¬ with CatchMiss.hyps (2) have\><turnstile<^subormal rightarrowsup* (Skip, " by (cases t) auto hence "Γ by (rule CatchSteps) auto also have"Γ by (rule step.CatchSkip) finally show ?case using t by (fastforce split: xstate.splits) qed
corollary exec_impl_steps_Normal: assumes exec: "Γ shows"Γ⊨(c,s) →>\urnstile> (Seq c (While b c), Normal s) → ie bc) Normal x)" using exec_impl_steps [OF execthis by auto
corollary exec_impl_steps_Normal_Abrupt: assumes?thesis shows"Γ using exec_erm by auto
corollaryexec_impl_steps_Abrupt_Abrupt: assumes exec: "Γ<urnstilec,Abrupt t⟩ Abrupt t" shows "Γ using exec_impl_steps [OF by autond
corollary exec_impl_steps_Fault
exec shows"\ y using exec_impl_steps [OF exec] by auto
corollary exec_ S thus ?case by (ast intro: terminat s o bu us ( (tf e ino: sstpsupt) assumes exec: "Γ shows
usingps by auto
lemma assumes step: "\step_Fault_propntros_ses "s'Str in: xec.inte: exec_elim_c) s=Stuck ∨2 s) (∃ e elc' = Thrand> t' = Normal x (∃p x. redex cΓ p = None)" using step by induct autoxec>
lemma step_Fault_end: assumes step: java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null defined s=Fault f ∨ (∃esbrptAbrupt: using step by induct auto
lemma redex_Stuck:
"Γstep_preserves_termination proof (induct case Seq thus ?case
Basic case fastforce.ros next case Catch thus by (cases s) (auto intro: exec.intros elim:exec_elim_cases) qed
lemmat: "Γ⊨::terintstr st proof Gua thus ?case by (fastforce intro: terminates.intros) case Seq thus ?case by (cases s) o i:exeits e:exm_cses next case Catch thus ?case by (cases s) (auto intro: exec.introsapply s_Abrun: qed simp_all
lemma step_extend: assumes step: "Γ⊨\^ shows"by (cses sfastforce ntro using s ?case proof (induct) case Bac thha by (fastforce intro: exec.intros elim: exec_Normal_elim_cases) next case Sp thus ?case by (fastforce int exec.intros elim: exec_Norm) next case SpecSt uto next case Guard thus ?case by (fastforce intro: exec.intros elius steps next case GuardFault thus ?case by (fastforce intro: exec.intros next
java.lang.NullPointerException have step: "Γ⊨ .hm", S.split_rule @{cont} have exec': "\)) thus show ?case proof (cases s) case (Normal x) note s_Normal = this show ?thesis proofjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20 case (Normal x') thus
ase
exec_cjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 bycases from Seq.hyps elimtes_Normal_elim_caseses have"ΓCat c2 thus ??case by simp from exec.Seq [OF this exec_cstep_extend show ?thesis by (iduct) next case (Abrupt x') with exe head:: ('s,'p,'f config step step_Stuck_pr)+ by(aintro:Abrupt_en moreover from step Abrupt heAbr x'" byauto:step_Abrupt_end) ultimately show ? by (auto introexec next case (Fault f)
al obtain java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
redex_c notemaljava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
fail: by auto<sub': "Γ hence "<exec_c\case_names by (autojava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 from exec_redex_Fault have" ‹ moreover from Fault exec' have "t=Faultf" byjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16 ultimately owesississ usings_Normal by(autointro:exec.intros) next caseStuck\>turnstilehead(irightarrow(f()java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83 byasttstep_preserves_termination have"(\<exists>r. \exists>p.redexc\<sub>=Callp\<and>\<Gamma>p |CatchfromhaveammaturnstileSeq'c<b'rightarrow(+)java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79 moreover bfinal_def_m_def assume"<^>pecr umeredexc^1=Specr"and"(\<forall>t.(x,t)\<notin>r)fromexec_redex_Stuckthis hence"\<Gamma>\<turnstile>\<langle> fix [this] have"\<Gamma>\<urnstile<langle>\<^sub>1Normal<>Rightarrow. moreoverfromStuckexec'have"t=Stuck" by(autointro:Stuck_end_) ultimately have?sis usings_Normal by(autointro:exec.intros) moreover { fixp assume"\< hence"\<Gamma>\<turnstile>\<langle>redexc\<^sub>1,Normalx\<rangle>from_ormatn_k (intro:execintros) fromexec_redex_Stuck[OFthis] have"\<Gamma>\<turnstile>\<langle>c\<^sub>simp moreoverfromStuckexec'have"t=Stuck" by(autointro:Stuck_end) imately have?thesis usings_Normal by(autointro:exec.intros) ultimatelyshow byauto qed next case(Abruptxc''Ls'wheref_k:"k=tchc<^java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 32 fromstep_Abrupt[OFstepthis] have"s=Abrupt. withby(orceeexec.troselim:java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60 tFalsethus?case byapplysimp withAbrupt show?thesis by(autointro:exec.intros) (fastforceintro.trosmc_Normal_elim_casescases aultf fromstep_Fault[OFstepthis] aveallUndefinedhuscase ' have"t=Faultf" by(utojava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 15 withFault show?thesis by(autointro:exec.intros) next caseStuck fromstep_Stuck[OFstepthis] have"s'=Stuck". withexec' have"Stucktuckjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18 by(autointro:Stuck_end) withStuck show?thesis by(autointro:exec.intros) qed next casebysimpd:ead_defhead_com_def by next case(SeqThrowc\<^sub>2st)thus?case by(fastforceintro:exec.introselim:exec_elim_cases)+ next caseCondTruethus?case by(fastforceintroexecroselimmc_Normal_elim_cases next caseCondFalsethus?case bystforcentroo:exec.osexec_Normal_elim_cases next caseuctk by(fastforceintrofrominfinite_computation_extract_head_Seq next casecase(Sucjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14 bythis[rule_format]haverue next caseCallthus?case by(fastforceintro:exec.introselim:exec_Normal_elim_cases) next caseCallUndefinedthus?case byyp\<>i<k.(\<exists>ci)Catchc^done next caseDynComthus?case by(fastforceintro:exec.introselim:exec_Normal_elim_cases) next case(Catch\<^sub>1sc\<>1's havestep:"\<(<exists>thusseeby haveexec':"\<Gamma>\<turnstile>\<langle>Catchc\<^sub>1'c show?case proof(casess) case(Normalx) notes_Normal=this show?thesis proof(casessjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20 case(Normalx') fromexec'[simplifiedNormal] show"f(k+1)(chith proof(cases) fixs' assumeexec_c<sub1':"\<Gamma>\<turnstile>\<langle>c\<^sub>1',Normalx'\<rangle>\<Rightarrow>Abrupts''" haveg_0:"g0=c<subngNormal fromCatch.hyps(2)Normalfrom[of0,simplifiedf_0]step[1] have"'=tjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23 bysimp fromexec.CatchMatch[OFthisexec_c\<^sub>2]step\foralli.\<? show next assumeexec_c<1':"\<Gamma>\<turnstile>\<langle>c\<^sub>1',Normalx'\<rangle<>t" t:\>isAbrtrooexectross atch)Normalc<sub>1Normal <\<turnstile>\<langle>c\<^sub>1,Normalx<>\t" byby(simpadd:g_def) [Fhiss_Normal show?romTrue qed nextwithno_inf_Throw case(Abruptx') withstforceintro:execntroslimjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 by(autointro:Abrupt_end) moreover fromstepAbrupt have"s=Abruptx'" by(byblast ultimately show?thesis proof next case(Faultf) fromstep_Fault_end[OFstepthis]s_Normal obtaingcwhere \^redexcsub1=Guard):ecntros fail:"x\<notin>g" byto hence"\<Gamma>\<turnstile>\<langle>redexc\<^sub>1,Normalx\<rangle>\<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3 tro:.ctros fromexec_redex_Fault[OFthis]step have"\<Gamma>\<turnstile>\<langle>c\<^sub>1,Normalx\<ranglebythuse InfiniteComputations:\<open>\<Gamma>< autointroFault_end) ultimately show?thesis s_Normal by(autointro:exec.intros) next caseStuck fromstep_Stuck_end[OFstepthis]s_Normal have(>.redexc\1rapplydrulenot_less_Least \<exists>romfinite_computation_extract_head_Catchmputation_extract_head_CatchFphis byesis moreover { fixr assume"redexc\<^sub>1=Specr"and"(\<forall>t.(x,t)\<notin>r)" by(autointro:exec.intros) fromexec_redex_Stuck[OFthis have"\<Gamma>\<turnstile>java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 moreoverfromcase(eqSkipc\<sub> by(autointro:(:byases) ultimately have?thesis usingstep:"\<forall>i:nat<Gamma\<turnstile>fi\fastforceintro.introsexec.intros by(autointro:exec.intros) } moreover { fixp assumejava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 4 hence\amma\<>\\>Normalx\RightarrowStuck by(autointro:exec.intros) fromexec_redex_Stuck[this] have"\<>\<case moreoverromtuckckxecbySuc yautointro:Stuck_end) ultimately have?thesis usings_Normal autointroexec } ow?thesis byauto qed next case(Abruptx) fromstep_Abrupt[OFstepthis] havefastforceintro:terminates.introsstep_extend withxecjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14 have"t=Abruptx" y(utontroroAbrupt_endt_end) withithAbrupt show?thesis autoointrotroecintrosros next fromstep_Fault[stepthis] have"s'=Faultf" withexec' have"t=Faultf"
java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 32 withFault showhave\<Gamma\urnstile>(\^>1)\,s)" by(autointro:exec.intros) next caseStuck fromstep_Stuck[OFstepthis] havedestttep_preserves_termination withexec' have"t=Stuck" yover withStuck show?thesis by(autointro:exec.intros) qed nextsteps_preserves_termination: caseCatchThrowthus?case by(fastforceintro:exec.introselim:exec_Normal_elim_cases) next caseCatchSkipthus?(inductrule:tranclp_induct2fromstep by(fastforceintro:exec.introselim:exec_elim_cases) next caseFaultPropthus?case by(fastforceintro:exec.introselim:exec_elim_cases) next cased_com(''p')com\(s,,'fcomdefinition(autosimpaddadd:inf_def) (fastforceintroexecintrosusingsjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 16 next casey by(fastforceintro:exec.introselim:exec_elim_cases) qed
theoremsteps_Skip_impl_exec: assumessteps:"\<Gammathenhavenot_fin:\forall>i<>final(()" "Gamma\<turnstile\<show<Gamma\turnstile>applyclarifyarify usingsteps proof(inductrule:converse_rtranclp_induct2[case_namesReflTrans]) caseReflthus?case by(casest)(autointro:exec.intros) next
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 have"\<Gamma>\<turnstile>(c,s)\<rightarrow>(c',done thus?case by(step_extend) qed
theoremsteps_Throw_impl_exec: mes\<>\<turnstile>(c,s\<^sup>*Throwf_step[0f_0 shows"\<Gamma>\<turnstile>\<langle>c,s\<rangle>\<Rightarrow>Abruptt" steps proof(inductrule:converse_rtranclp_induct2[case_namesReflTrans]) caseReflthus?case by(autointro:exec.intros) next case(Transcsc's') have"\<Gamma>\<turnstile>(c,by(fastforceelim:Skip_no_stepstep_elim_cases) thus?case by(rulestep_extend) qed
lemma b c) assumes step: "Γ (rule not_infI)) shows "Γ: "∧⊨ f (Suc by ( add: head_d head_)assumef_0: "0tuck using proof (induct) by (astforce elimSkip_no_stepstep_elim_cases) next case Spec thusse qed next case SpecStuck thus ?caseby (fastforce intro: terminates not_fin "<>i<k. \not final (head (f i))" next case Guard thuscase bystforceterminates :terminates_Normal_elim_cases next case GuardFault thus ?caseby (fastforce intro: terminates next case (Seq cjava.lang.NullPointerException apply (cases s) apply (cases s') apply(fastforce:.intros
elim: terminates_Normal_elim_cases) apply (fastforce intro: terminates qed
step_Fault_prop step_Stuck_prop)+ done next case (SeqSkip: "∧ thus ?case apply (cases s) apply (fastforce intro: terminates.intros exec.intros elim: terminates_Normal_elim_cases )+ done next
java.lang.NullPointerException thus ?case fin: t.int exe Γ"
elim: terminates_Normal_elim_cases ) next case CondTrue thus ?case by (fastforce intro: terminates.intros exec.intros
elim: terminates_Normal_elim_cases next case CondFalse thus ?case by (fastforce intro: terminates qed
elim: terminates_Normal_elim_cases ) next case WhileTrue
by ( intro .intros
next case WhileFalse thus ?case byshow"? k"
elim: terminates_Normal_elim_cases ) next case Calle thuscase by (fastforceintro.intros
elim: terminates_Normal_elim_cases ) next case CallUndefined [0 f_0 thusc ss _stepp:"\by (auto elime: Skipno) by (fastforce intro: terminates.intros elim: terminates_Normal_elim_cases ) next case DynCom thus ?case by (fastforce intr (rule not_infI) elim: terminates_Normal_elim_cases ) next
java.lang.NullPointerException apply (cases s) apply (cases s') apply (fastforce intro: terminates.intros step_extend elim: terminates_Normal_elim_cases) apply (fastforce intro: terminates.intros dest: step_Abrupt_prop step_Fault_prop step_Stuck_prop)+ done nextf_0: "f 0=Throw case CatchThrow
s?se by (fastforce intro: terminates.intros exec.intros
elim: terminates_Normal_elim_cases ) nextf_comp<>🚫 case (CatchSkip c<^show thus ume0 Call)" by (cases s) (fastforce intro: terminates.intros)+ f_step of0f_0 f_f_step [of 1] next case FaultProp thus ?case by (ff f next case t thus ?case by (fastforce introterm.intros) next case AbruptProp thus ?case by (fastforce intro: terminates.intros) qed
lemma"urnstile\htarrow><>" assumes steps: "Γ⊨ shows"Γ⊨ using steps proof (induct rule: rtranclp_induct2 [consumes 1, case_names Refl Trans]) case Refl thus ?case . next case Trans thus ?case by (b dest: step_preserves_termi) qed
lemma steps_preserves_termination': assumes steps: "Γ2, s)" shows Γs ==>⊨ using steps proof (induf_step: "> \amma🚫i. final casethus blast:step_preserves_termination) next case Trans thusjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 by (blast dest: step_preserves_termination) qed
definitionhead_com"s,f)com \<Rightarrow sho ?c wheref head_com (case c of Seqapp F | Catch c [rule_format, of "by(: Skip_no_step
| _==>
definition head"><> h ( 0 \rightarrowrro>\<\< where "head =head_com cfg cfg
lemma le_Suc_cases: "[\<And ifyy ly apply auto done
lemmaq_False:s<>c c''rec Seq c'' c' = F" by (induct c) auto
lemma redex_Catch_False: "∧ by (induct c) auto
lemmainfinite_computation_extract_head_Seq: assumesinf_comp: "forall assumes f_0: java.lang.NullPointerException assumes not_fin: "∀ final f_step shows"∀ x Γ⊨head (f i) → head (f (i+ (is "∀ using? proof (induct<⊨2, Abrupt s) <rightarrow(∞ case0 show simp next case (Suce have not_fin_Suc: "∀i<Suc k. ¬ final (head (f i))"by fact from this[rule_format java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14 "∀ final (head (f i))" apply clarify apply (subgoal_tac "i < Suc k") apply blast apply simp done
from Suc.hyps [OF this] show byauto show ?case proof (rule le_Suc_cases) fix i assume"i < k" then
rule next show"?P k" proof - fromfastforceo_step obtain c' fs' L' s' where f_k: java.lang.NullPointerException by (cases k) auto from inf_comp [rule_format, of k] f_k have "Γ⊨ : Skip_no_stepqed by simp1 cjava.lang.NullPointerException moreover from not_fin_Suc [rule_format, of have"¬ by (simp add: final_def head_def head_com_def) ultimately obtain c'' s'' where "Γ⊨(c', s') → (c'', f_0(While)
( ) Seq'c^>, s'')" by cases (auto simp add: redex_Seqfinal_def) with f_k show ?thesis by (simp add: head_def head_com_def) qed qed qed
lemma infinite_computation_extract_head_Catch: assumes inf_comp: "∀:"<>i. \<>\ assumes f_0: "f 0 = (Catch c>turnstile> f (Sucjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77 assumes not_fin: "∀ shows "∀i<k. (<exists
Γ=(Spec, Normal s)java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
(is"∀ using no proof (induct k) case 0 show ?case by simp t case (Suc k) have not_fin_Suc: "∀ from this[rule_format] have not_fin_k:
epi. Γf i →False apply
fastforce:Skip_no_step step_elim_cases) apply blast apply simp
ne
from Suc.hyps [OF this] have hyp: "∀ \<Gamma\ c<sub>1c\<^^sub :" (gc,Fault 🚫 show ?case proof (rule le_Suc_casesqed fix assume"i < k" thenshow"<Gamma>\<rnstilenstile by (rule hyp [rule_format]) next show "?P k" proof - from hyp [rule_format, of "k - 1"] f_0 obtain c' fs' fix f by (cases k) auto from inf_comp [rule_format, of k] f_k have "Γ
simp moreover from not_fin_Suc [rule_format, of k have"¬ final (c',s')"with by (simp add: final_def head_def head_com_def) ultimately obtain c where "Γ(c', s') → ΓWhile b c ↓¬ by cases (auto simp add: redex_Catch_False final_def)+ with f_k show ?thesis by (simp add: head_def head_com_def) qed qed qed
lemma no_inf_T: "¬(Throw,s) →(∞ proof assume"Γi. Γ c) then obtain f where step [rule_format]: "∀f_step]f_0
f_0: "f 0 = (Throw, s)" by autoaddinf_def from step [of 0, simplified f_0eq c (While >f withhave<amma\ show False by cases (auto elim: step_elim_cases f_stepof f_0 qed
lemma split_inf_Seq: assumes inf_comp ?case shows java.lang.NullPointerException (∃ proof - from inf_comp obtain f where step: "∀i::nat. Γjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
f_0 Some" by faa f_0:" ic s by( simp: inf_def from f_0 have head_f_0: "head (f 0) = (cnot_infI) by (simp add: head_def head_com_def) show ?thesis proof (cases "∃i. final (head (f i case True
define k where"k = (LEAST i. final (hwit f_step have less: 🚫 apply (intro allI impI) apply (unfold k_def) apply (drule not_less_Least) autoo done from infinite_computation_extract_head_Seq [OF step f_0 this] obtain step_head: "∀
conf: java.lang.NullPointerException by blast from True have final_f_k: "final (head (next apply - apply erule apply (drule LeastI) apply (simp add: k_def) done moreover from f_0 conf [rule_format, of "k - 1"] obtain c' s' where f_k: "f k = (Seq c' c2,s')" have (Suc 0=(c s,Normal moreover from step_head have steps_head: "Γ⊨head (f 0) →simp a: inf_def) proof (induct k) case 0 thus ?case by simp next case (Suc m) have step: "\forall<><turnstile> head (f i) → hence( s) thuscase by auto hence"\assume fstep: "<>i. <amma\ by ( Suchyps alsofrom step [rule_format, of m] have"Γ finally show ?case by simp qed { assume f_k: "f k = (Catch cc^sub1 s c2)
steps_head have"Γ⊨(cb blast using head_f_0 by (simp add: head_def head_com_def) moreover from step [rule_format, of k] f_k obtain "Γ
f_Suc_k: "f (k + 1) = (c2,s')" byfastforcestep :.intros from infinite_computation_extract_head_Seq[Fstepf_0 ] from f_Suc_k have g_0:head_f_0>⊨1,s) →)
y(imp: g_def
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 have ( b c\sub1 <>\ by (simp add: g_def) with g_0 have"Γ:"i::nat. Γf i →
f_step\>>fi<> ( i ultimately have by auto
} moreover
fix x assume s': not_infI from step [rule_format, of obtainGamma<>Seq
f_Suc_k: "f (k + 1) = (Throw,s')" by (fastforce elim: step_elim_cases intro: step.intros)
define g where"g i = f (i + (k + 1))"for fromf_Suc_k have g_0 "0(Throw,s')" by (simp add: g_def) from step have"∀\turnstile by (simp add: g_def) with g_0 have "Γ by (auto simp add: inf_def) by (cases k) auto have ?thesis "\<>\ by auto ltimately show ?the by (auto simp add: final_def head_def head_com_def) next case False then have not_fin: "∀i. ¬Gamma⊨)<rightarrow (f (m + ))by by steps_redex: have"∀i. Γ⊨ proof fix k romnt_finjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18 sim add: head_def head_com_def) by simp
from infinite_computation_extract_head_Seq [OF step f_0 this ] show "Γ⊨ head (f k) → qed with head_f_0 have"Γ⊨ by at sp dd: inf_df) thus assums:"rmal f_k " k= Catc Throw🚫: head_def h) by simp qed
lemma split_inf_Catch: assumes inf_comp: "Γ<>(Catch^>1 c(Abrupt shows"Γ⊨1 for i (∃s'. Γ⊨ proof -
from inf_comp obtain f where
i<\turnstilef i \<rightarrow> f (+ d
f_0: "f 0 = (Catch c\<^sub>1 c\<^sub>2, s)"
by (auto simp add: inf_def)
head_f_0head0 (^>,)
by (simp add: head_def head_com_def)
show ?thesis
} caseTrue
define k where "k = (LEAST i. final (head (f i)))"
have less_k: "\<forall>i<k. \<not> final (head (f i))"
apply byblast
apply (unfold k_def)
apply (drule not_less_Least)
apply auto
done
from not_fin
obtain step_head: "\<forall>i<k. \<Gamma>\<turnstile> head (f i) \<rightarrow> head (f (i + 1))"and
conf: "\<forall>i<k. (\<exists>c' s'. f (i + 1) = (Catch c' c\<^sub>2, s'))"
by blast
from True
have final_f_k: "final (head (f k))"
apply -
apply (erule exE)
apply( LeastI
( add k_def
done
moreover
from f_0 conf [rule_format, of "k - 1"]
obtain c' s' where f_k:" atch' c\\<^>2') case (Skip)hus ?java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
by( k
moreover
from step_head have steps_head: "\<Gamma>\<turnstile>head (f 0) \<rightarrow>\<^sup>* head (f k)"
proof (induct k) case0 thus ?case by simp
next case (Suc java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
have step: "\<forall>i<Suc m. \<Gamma>" c p Suc i subst_redexjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
nce\><.\<>turnstile f i <>headf )"
by auto
henceGamma<>headf )<><sup>*head m"
by (rule Suc.hyps)
also from step [rule_format, of m]
have "\<Gamma>\<turnstile> head (f m) \<rightarrow> head (f (m + 1))" by simp
finally show ?case by simp
qed
blast
by( elim: Skip_no_stepjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
with steps_head
have "\<Gamma>\<turnstile>(c\<^sub>1,s) \<rightarrow b= ff 0 using
by ( by iprover
moreover
step[rule_format of kk]f_k
obtain<>
f_Suc_k: "f (k + 1) = (Skip,s')"
by (fastforce elim: step.cases intro: step.intros)
from step [rule_format, of "k+1", simplified f_Suc_k]
thesis
lemma:"<not> \<Gamma>turnstile(,
moreover
{ "<forall>y.\<^>+\^>+ ay <longrightarrow>P assume f_0: f , )"
ow
withjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
have "\<Gamma>\<turnstile>(c\<^sub>1,s) \<rightarrow rule not_infI) using head_f_0
by ( add head_defhead_com_def)
moreover
from step [rule_format, of k] f_k s'
obtainf
f_Suc_k: "f (k + 1) = (c\<^sub>2,s')"
(elim introstep.intros)
define g where "g i = f (i + (k + 1))"for i
from f_Suc_k
haveg_0:g 0(
by (simp add: g_def)
from step
have hence \exists>.f 0:nat ,s <>(<>.\Gamma
( add g_defjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28 0"<>\turnstile(\^>2s') \rightarrow> <dots>(\infinity>)"
by (auto show ?case
ultimately
have ?thesis using s'
}
ultimately
?
byautocase
then not_fin:"\forall>. \not final(head ( i)"
by
ve<i Gammaturnstile fi <>head( ( False
proofshow
fixk
from not_fin
have"\>< k. <> (head f))
by simp
from
show "\<Gamma>\<turnstile> head (f k) \<rightarrow> head (f (k + 1))" by simp
qed
with head_f_0 have "\<Gamma>\<turnstile>(c\<^sub>1,s) \<rightarrow> \<dots>(\<infinity>)"
by (auto simp qed
thus ?thesis
by simp
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
qed
lemma not_inf_Stuck: "\<not> \<Gamma>\<turnstile>(c,Stuck) \<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: show ?ase
from f_step [of 0] f_0
show False
by (auto elim: Skip_no_step)
qed
next case( g)
thus ?case
proof (rule not_infI)
?case
assume f_step: "\<And>i. thus ?case
assume f_0: "f 0 = Basicg,Stuck"
[ ]
show False
by (fastforce elim: Skip_no_step step_elim_cases)
qed
next
rjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
thus ?case
proof ( not_infI)
fix f
assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
assume f_0: "f 0 = (Spec r, Stuck)"
from case
show False
(fastforceelim: Skip_no_step )
qed
next case (Seq c\<^sub>1 cby havehyp_c1 ">\>\<turnstile> 1, s) rightarrow \<dots(<nfinity>))" fact
show ?case
proof
assume "\<Gamma>\<turnstile> (Seq c case Throw
from split_inf_Seq [OF thisjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
show False
by( dest steps_Stuck_prop)
qed
next (auto: step_Normal_elim_cases) case (Cond b c\<^sub>1 c\<^sub>2)
show ? "\<>\<turnstile> ( c\<^sub>1 c\^sub>2, Abrupts)) \<rightarrow> \<dots(\<infinity>)"
proof (rule not_infI)
fix f
assume f_step: "\<And>i. \<Gamma>\<turnstile>
assume f_0: "f 0 = (Cond b c\<^sub>1 c\<^sub>2, Stuck)"
show False
qed
next case ( f_0:f0 ( have<\turnstilec )\<<sup n"byfact
show ?case
(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 assume:f0= Condbc1 ,Normal ()f)\ (,)<ammajava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
proof )
show False
fastforce:Skip_no_step step_elim_cases
qed
next
( p
show ?case
proof (rule not_infI)
fixf
assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
assume f_0: "f 0 = (Call p, Stuck)"
from f_step [of 0] f_0 f_step [of 1]
show False inf
by fastforceelim: Skip_no_step)
qed
next case( djava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
show ?case
proof (rule not_infI)
fix f
assume\>. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 9
assume f_0: "f 0 = (DynCom d, Stuck)"
from f_step [of 0] f_0 f_step fastforceelimSkip_no_stepstep_elim_cases
show False
by (fastforce elim: Skip_no_step step_elim_cases)
qed
nextfrom [ this show case (Guard m g where
show
)
fix
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 caseThrow
show ?case
proof (rule not_infI)
fix f
f_step \>\Gammaturnstilef rightarrow>f )java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
assume f_0: "f 0 =apply( add inf_def)
from f_step [of 0] f_0 f_step [of 1]
show False
by (fastforce elim: Skip_no_step step_elim_cases)
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
next caseCatch^>1 <sub
show ?case
proof
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
from split_inf_Catch [OF this] Catch.hyps
show False
proof
qed
qed
lemma from steps_redex'[ this, of "(eq c (p ) i)"]
proof "\<Gamma\<> (ubst_redex(seq c ( 0)i) (Call ( i),Normal(s i))\<ightarrow>\<sup+ case Skip
show ?case
proofproof(ule 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
show False
by (auto elim: Skip_no_step)
qed
next case( )
thus ?case
proof (rule not_infI)
fix f
:"<And>i <Gamma>\turnstilef i\rightarrow> ( i)java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
f
from f_0 [
show
by (fastforce elim: Skip_no_step step_elim_cases)
qed
next
( r)
thus ?case
_infI
fix f
assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
assume f_0: "f 0 = (Spec r, Fault x)"
fromf_step[of0f_0f_stepof1]
show False
( : Skip_no_stepstep_elim_cases
qed
next case ( \^sub1 c\^sub>)
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\<^subproof ( c)
showcase
proof (rule not_infIjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
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
( 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 (Whileb )
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>f assumef_step "\<>i. \Gamma\<turnstilef \<case( f )
assume f_0: "f 0 = (Call p, Fault x)"
from f_step [of 0] f_0 f_step [of 1]
show False
by (fastforce elim: Skip_no_step step_elim_cases)
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
next case show False
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
proofjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
fix f
assume f_step: "\<And>i. \<Gammaproof (induct rule converse_rtranclp_induct2 [ Refl Trans])
assume f_0: "f 0 = (Guard m g 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
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
java.lang.StringIndexOutOfBoundsException: Range [76, 55) out of bounds for length 55
qed
next case (Catch c\<^sub>1 c\<^sub>2)
show ?case assumestep_c\<sub>: "<Gamma\<turnstile> (c\<sub>, Normal )\<rightarrow c' s)"
proof
assume "\<Gamma>\<turnstile> (Catch c\<^sub>1 c\<^sub>2, Fault x) \<rightarrow> \<dots>(\<infinity>)"
rule.)
show False
java.lang.StringIndexOutOfBoundsException: Range [45, 38) out of bounds for length 38
qed
qed
lemma not_inf_Abrupt: "\<not> \<Gamma>\<turnstile>(c,Abrupt s) \<rightarrow> \ proof(rulenot_infI)
proof (induct c) case Skip
showcase
( )
ix f
assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
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 ?case
proof (rule not_infI)
fix f
assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
assume f_0byauto:exec_Normal_elim_cases ( java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
from f_step [of 0] f_0 f_step [of 1]
show
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)"
assume f_0: "f 0 = (Spec r, Abrupt s)"
from f_step _ <>c<subf= \<> t= s)
show Falseby ( split: xstatesplits)
by (fastforce elim: Skip_no_step step_elim_cases)
qed
next
Seqc\^sub>1 c<^ub>2))
show ?case
proof
assume "\<Gamma>\<turnstile> (Seq c\<^sub>1 c\<^sub>2, Abrupt s) \<rightarrow> \<dots>(\<infinity>)"
fromsplit_inf_Seq[F ].
show False
by (auto dest: steps_Abrupt_prop)
qed
next case (Cond b c\<^sub>1 c\<^sub>2)
show ?case
proof (rule not_infI)
fix f
termi_call_steps::"(s'p,'f) body \<Rightarrow> (( \<> 'p) Normaljava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
assume f_0: "fby cases
fromromf_step [ 0] f_0 f_step [[of ]
show False
fastforceelim Skip_no_step )
qed
next case( b c)
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
proof \^>2 rule_format,OF]
fix<exists>.<\<turnstile(allp, )\<rightarrow\<^> (,Normal t \nd redexc = q)
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
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 by simp
assume f_0: "f 0 (Call p, Abrupts"
from f_step[of 0]f_0 [of1]
show False
by (fastforce elim: Skip_no_step step_elim_cases)
qed
next
casecase (DynCom d)
show ?case
fix
assume f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
f_0: " 0 DynCom, Abrupt s)"
from f_step [ by auto
show False
by( elim Skip_no_step step_elim_cases)
qed
next
subst_redexDynCom) =c |
with have "s'=aultf"
proof (rule not_infI)
fixf
_:"ndi.\<><fi \\> f(uc "
assume f_0: "f 0 = (Guard m g c, Abrupt s)"
from f_step [of 0] f_0 f_step [of 1]
show False
by fastforceelim Skip_no_step)
qed
next caseThrow
show ?case
proof (rule not_infI)
fix f
assume f_step: "\ qed
assume f_0: "f 0 = (Throw, 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( c\^sub1c<sub2)
showcase
"<amma><rnstile(redexc,)<>(r,')\<LongrightarrowGammaturnstile, \rightarrow (subst_redexc r's'
split_inf_Catch[ this]Catchhyps
show False
by (auto dest: steps_Abrupt_prop)
qed
java.lang.StringIndexOutOfBoundsException: Range [14, 3) out of bounds for length 3
theorem terminates_impl_no_infinite_computation
assumes termi ( ) ((auto .Seq .Catch
shows "\<not> \<Gamma>\<turnstile>(c,s) \<rightarrow> \<dots>(\<infinity>)" using termi
proof (induct) case (Skip s) thus ?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, Normal s)"
fromf_stepof0] f_0
show False
by (auto elim: Skip_no_step)
qed
next case (Basic g s)
thus ?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 = (Basic g, Normal s)"
from f_step [of 0] f_0 f_step [of 1]
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
byfastforce :Skip_no_step)
qed
next case (Spec r s)
thus ?thesis
proof (rule not_infI)
fix f
<>i <>\> <ightarrow f (Suc)
assume f_0: "f 0 = (Spec r, Normal s)"
from f_step [of 0] f_0 f_step [of 1]
java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
by (fastforce elim
qed
next case (Guard s g c m)
have g: "s \<in> g" by fact
have hyp: "\<not> \<Gamma>\<turnstile> (c, Normal s) \<rightarrow> \<dots>(\<infinity>)" by
showruleDynCom
proofrule not_infI
fix f
assume f_step (autointro .intros
assume f_0: "f 0 = (Guard m g c, Normal s)"
from f_step [of 0] f_0 g
have "f 1 = (c,Normal s)"
by ("[("a",0)Positionnone),"(a ))((("b"0)),Position.proof( s<>
with f_step
have "\<Gamma>\<turnstile> (c, Normal s) \<rightarrow> \<dots>(\<infinity>)"
apply (simp add: inf_def)
apply =lambda. Suci) inexI)
by simp
with hyp show False ..
qed
next case (GuardFault s g m c)
have g: "s \<notin> g" by fact
show ?case
proof (rule not_infI)
fix f
assume
assumef_0 "f0 ==(Guard m g c, Normal s)"
fromf_stepof 0]f_0f_step 1]
show False
by (fastforce elim: Skip_no_step step_elim_cases)
qed
next caseFault m)
thus ?case
proof
next case (Seq c\<^sub>1 s c\<^sub>2)
show ?case
proof
assumeGamma<>(Seq<^> \^sub2 Normal <>\dots>(\<>)
from split_inf_Seq [OF this] Seq.hyps
show False
by (auto intro: steps_Skip_impl_exec)
qed
next case (CondTrue s b c1 c2)
have b: "s \<in> b" by fact
have : "\<ot> \<Gamma>\<turnstile> (c1, Normals) \\<<rightarrow \<<dots(\infinity)" fact
showcase
proof (rule not_infI)
fix f
assumef_step "<And>i.\<amma\turnstile>i\> fSuc)"
assume f_0: "f 0 = (Cond b c1 c2, Normal s)"
fromb [ 0 f_0
have "f 1 = (c1,Normal s)"
by auto: )
with f_step
haveGamma<turnstile> (c1, Normal s) \<rightarrow> \<dots>(\<infinity>)"
apply (simp add: inf_def)ormal_elim_cases)
apply (rule_tac x="\<lambda>i. f (Suc i)" in exI)
by simp
with hyp_c1 show False by simp
next
CondFalse s c2 c1c1)
java.lang.StringIndexOutOfBoundsException: Range [37, 32) out of bounds for length 32
have hyp_c2: : \<Gamma>\turnstile c',s'')\rightarrow\<sup>*(Throw, s')"
show ?case
proof (rule not_infI)
fix f
f_step "\ndi \Gamma<turnstilef \rightarrow f (Sucuc ijava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
assume:" 0=(Condb c2, 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)" in exI)
by simp
with hyp_c2 show False by simp
qed
next case (WhileTrue s b c)
have"s \in bb factfact
have hyp_c: "\<not> \<Gamma>\<turnstile "\<>c1 s1 \<brakk>\<Gamma \<turnstile> (c,) \<ightarrow\<^sup* cfg1; cfg1(,s1\<rbrakk\<Longrightarrow <Gamma>\<turnstile>c1<down>s1"
have hyp_w: "\<forall>s'. \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow>
\<Gamma>\<turnstile>While a_b
have not_inf_Seq ( cfg1 simp
proof
assume "\<Gamma>\<turnstile> (Seq c (While b c), Normal s) \<rightarrow> \<dots>(\<infinity>)"
from split_inf_Seq [OF this] hyp_c hyp_w show False
(uto introsteps_Skip_impl_exec)
qed
show ?case
proof
assume "\<Gamma>\<turnstile> (While b c, Normal s) \<rightarrow> \<dots>(\<infinity>)"
then obtain f where
f_step: "\<And>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"and
f_0 f =( b ,Normal)
by (auto simp add: inf_def)
f_step [ 0 f_0
have "f 1 = (Seq c (While b c),Normal s)"
by (auto elim: step_Normal_elim_cases)
with f_step
have "\<Gamma>\<turnstile> (Seq c (While b c), Normal s) \<rightarrow> \<dots>(\<infinity>)"
apply (simp add: inf_def)
apply (rule_tac x="\<lambda> (auto intro terminatesintros
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
{
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)"
from b 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 bdy s)
have bdy: "\<Gamma> p = Some bdy" by fact
have:"\ <mma>\turnstile ( s <ightarrow>\ots(\infinity byjava.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
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 = (Call p, Normal s)"
from bdy f_step [of 0] f_0
have "f 1 = (bdy,Normal s)"
by (auto elim: step_Normal_elim_cases)
withSeqc<sub1c<^>2)={ \sub><^> <> \>a )<n>r\sup>\andfif(i injava.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
have "\<Gamma>\<turnstile> (bdy, Normal s) \<rightarrow> \java.lang.StringIndexOutOfBoundsException: Range [62, 37) out of bounds for length 37
apply (simp add:{DynComd"|
apply (rule_tac x="\<lambda>i. f (Suc i)" in exI)
java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
with hyp show False by simp
qed
next case (CallUndefined p s)
have no_bdy: "\<Gamma> p = None" by fact
show ?case
proof (rule not_infI)
fix f
assume f_step: "\<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
assume f_0: "f 0 = (Call p, Normal s)"
from no_bdy f_step [of 0] f_0 f_step [of 1]
show False
by (fastforce elim: Skip_no_step step_elim_cases)
qed
next case (Stuck c)
show ?case
by (rule not_inf_Stuck)
next case (DynCom c s)
have hyp: "\<not> \<Gamma>\<turnstile> (c s, Normal s) \<rightarrow> \<dots>(\<infinity>)" by fact
show ?case
proof (rule not_infI)
fix f
assume f_step:haver \inredexes ( c<^> c<sub2)byfact
assume f_0: step_r\<><turnstile,s) \rightarrow,s)"fact
from f_step [of 0] f_0
have "f (Suc 0) = (c s, Normal s)"
by (auto
withhave\turnstile>c,Normal <ightarrow>\>\infinity
apply (simp add: inf_def)
apply rule_tacx"lambdai. f Suci" exI
by simp
show False by simp
qed
next
( s) ?java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
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) have "f = (dy (fastforce:stepintros step_elim_casessimpaddroot_in_redexes
from f_step [of 0] f_0
show
by (auto elim: step_elim_cases)
qed
next case (Abrupt c)
show ?case
by (rule not_inf_Abrupt)
next caseCatchc\sub \<>2
show ?case
proof (auto
assume "\<Gamma>\<turnstile> (Catch c\<^sub>1 c\<^sub>2, Normal s) \<rightarrow> \<dots>(\<infinity>)"
show
byutointrosteps_Throw_impl_exec)
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 s \<and>
(\<exists>c. \<Gamma>\<turnstile>(Call p,Normal s) \<rightarrow>\<^sup>+ (c,Normal t) \<and> redex c = Call q)}"
primrec::"(,p,f)om <>(',p,f) \Rightarrow('s,p,,')com"
where
proof( : converse_rtranclp_induct2case_namesReflTrans) "subst_redex (Basic f) c = c" | "subst_redex (Spec r) c = c" | "subst_redex (apply clarify "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 (Throw) c = c" | "subst_redex (Catch c\^sub1 c\<sub2)c = Catch(subst_redexsubst_redex c\<^sub>1 c)) c<^sub>2""
lemma subst_redex_redex: "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 steps_redex:
teps"Gamma<> s)<>\^up' s)"
shows "\<And>c. \<Gamma>\<turnstile>(subst_redex c r,s) \<rightarrow>\<^sup>* (subst_redex c r',s')" using steps
proof (induct rule: converse_rtranclp_induct2 [case_names Refl Trans]) case Refl
show"\Gamma\turnstile(subst_redex r, s'\<ightarrow>\sup* (subst_redex c r,s'
bysimp
next
from f_step of0 f_0
have "\<Gamma>\< \<forall>i by autoelimstep_elim_casesjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
from step_redex [OF this]
haveGammaturnstilesubst_redexc, rightarrowsubst_redexc r's'.
also
have "\<Gamma>\<turnstile> (subst_redex c r'', s'') \<rightarrow>\<^sup>* (subst_redex c r',proof
finally show ?case .
qed
ML \<open>
ML_Thms.bind_thm ("trancl_induct2", Split_Rulejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(Rule_Insts.read_instantiate @{context}
[((a" ),.none),"(aa,)"),((b",0,.none,"ba, ))][]
thm)
\<close>
':
assumes steps: "\<Gamma>\<turnstile> (r, s) \<rightarrow>\<^sup>+ (r"subst_redexSkip = |
shows "\<And>c. \<Gamma>\<turnstile>(subst_redex c r,s) \<rightarrow>\<^sup>+ (subst_redex c r',s')" using steps
proofqed case (Step r' s')
have "\<Gamma>\<turnstile> (r, s) \<rightarrow> (r', s')" by fact
then have "\<Gamma><turnstile> (subst_redex c r, s \<rightarrow (subst_redex c r', s'"
rule) " (Catch c\<^sub> c\^sub>2 c = Catch (subst_redexc\sub> c c\^sub2"
next
( r' s''' 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<amma\<turnstile cr' s' \<rightarrow ((subst_redex cr', java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
by (rule step_redex)
finally show "\<Gamma>\<turnstile> (subst_redex c r, sjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
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>*"
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 usinga_b
proof (from ]
assume "b = f 0"
with f show "\<exists>f. f 0 = b \<and> (\<forall>i. (f i, f (Suc i)) \<in> r)"
by blast
next
fix a z
assume a_z: "(a, z) \<in> r"and"(z, b) \<in> r\<^sup>*"
assumeb=f0<Longrightarrow><>f z\and \foralli (, ( i) \inr" "b = f 0"
then obtain f where f0: "f 0 = z"and seq\<close
by iprover
{
fix i have "((\<lambda>i. case i of 0 \<Rightarrow> a | Suc i \<Rightarrow> f ibst_redexc 's'"
by (cases)auto
}
thenjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
show "\<exists>f. f 0 = a \<and> (\<forall>ext
by - (rule exI [where x="\<lambda>i. case i of 0 \<Rightarrow> a | Suc i \<Rightarrow> f i"],simp)
qed
lemma renumber: "\<forall>i. (a,f i) \<in> r\<^sup>* \<and> (f i,f(Suc i)) \<in> r
\<Longrightarrow> \<exists>f. f 0 = a \ blast: tranclp_into_rtranclprtranclp_trans
by (blast dest:renumber')
lemma "<orall>y.r\<sup>\<^sup>+a y \longrightarrow PP a \longrightarrow PP y
\<Longrightarrow> ((b,a) \<in> {(y,x). P x \<and> r x y}\<^sup>+) = ((b,a) \<in> {(y,x). P x \<and> rproof(induct rule converse_rtrancl_induct [consumes 1)
apply(rule f "\existsf.. 0 = b <> (<oralli.(f i,, f ( i)) \<in> r)java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
apply clarify
apply(erule trancl_induct)
applyblast
apply(blast intro:tranclp_trans)
apply clarify
apply(erule tranclp_induct)
apply blast
apply(blast intro:trancl_trans)
done
corollary:
assumes rule exI [ x="\lambda>.caseiof0 Rightarrow> |Suci <Rightarrow> f ],)
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>+)"
proof (rule wf_trancl)
show" {(, x. \Gamma\<urnstile(cs)\><sup> x\<> \<><>x\rightarrow> y"
proof (simp only: wf_iff_no_infinite_down_chain,clarify,simp)
f
assume<>i <Gamma\turnstilec,) rightarrow<^up* f \>java.lang.StringIndexOutOfBoundsException: Range [85, 3) out of bounds for length 3
hence "\<exists apply(blast introtranclp_trans)
by (rule renumber [to_pred])
moreover from terminates_impl_no_infinite_computation [OF terminates]
have "\<not> (\<exists>f.
by (simp add: inf_def)
ultimately show False
by simp
qed
qed
hence "\<not> (\<exists>f. \<forall>i. (f (Suc i), f i)
\<in> {(y, x). \<Gamma>\<turnstile>(c, s proof (ule wf_trancl
by( add:)
thus ?thesis
fix "<>f. (::) =(c s \<> (\<oralli.\<>\<<turnstile>fi <rightarrow>\sup+ Suci))""
thenobtainf where
f0: "f 0 = (c, s)"and
:ne"i ( (( i::)fori : nat
by iprover
show
\<>f.<>.f i) fi)\<>{yx).\<><turnstile(,s)\<ightarrow<sup inf"<>i\Gamma<>all(p )\down Normal (si andjava.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
(exI[wherex=] allI
fix
show "(f (Suc i), f i) \<in> {(y, x). \<Gamma>\<turnstile>thus ?thesis
proofassume\existsf. f (::)=(c s)\and<>.\Gamma>\turnstilef i \<rightarrow>\<^sup>+ f (Suc)"
f0: "0 (c,s) and
fix i have "\<Gamma>\<turnstile>(c,s) \<rightarrow>\<^sup>* f i"
proof (induct i) case0 show "\<Gamma>\<turnstile>(c, s) \<rightarrow>\<^sup>* f 0"
fix i
next case(ucn)
have
with seq show "\<Gamma>\<turnstile>(c, s) \<rightarrow>\<^sup>* f (Suc n)"
by (blast intro: tranclp_into_rtranclp rtranclp_transproof (induct i)
qed
hence "\<Gamma>\<turnstile>(
by iprover
with seq "(f (Suc i), f i) \<in> {(y, x). \<Gamma>\<turnstile>(c, s) \<rightarrow>\<^sup>* x \<and> \<Gamma>\<turnstile>x \<rightarrow>\<^sup>+ y}"
by clarsimp
moreover
have "\<forall>y. hence "\Gamma\<turnstile>(,s)\<>\<^>* f "
by (blast intro: tranclp_into_rtranclp rtranclp_trans)
ultimately
java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
(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
assume inf: "\<forall>i. (\<lambda>(t, q) (s, p).
\<Gamma>\<turnstile>Call p \<down> Normal s \<and>
(<existsc. \<Gamma>\<turnstile (Call p, Normals) \rightarrow>\^>+ cNormal <> redex=Call ))
(f
definewheresi=
define p where "p i = (snd (f i)::'b)"for i :: nat
from inf
have "<existsc. <foralli. <Gamma><>Call p i \down Normal ((s)\>
(\<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'
have "\<exists>c. \<forall>i. \ have"g 0 =Call( 0), (s 0)java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
\<Gamma>\<turnstile> (Call (p i), Normal (s i)) \<rightarrow>\<^sup>+ (c i, Normal (s (i+1))) \<and>
redex (c i) = Call (p by(induct ( simp : red_c
apply -
apply (ule)
by blast
then obtain c where
:"\forall>i.\<Gamma>\turnstileCall (p ) \down Normal fix
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))"
g where"g i =(seq c(p0)i, (si):'a'c) xstate)"fori
red_c, ]
have"g0 =(Call(p 0) Normal(s 0)"
( : g_def
moreover
{
have "redex (seq c (p 0) i)= CallCall p i)
by (induct i) (auto simp add: redex_subst_redex red_c)
from this [symmetric]
have "ultimatelyshow False
by (simp add: subst_redex_redex)
} note subst_redex_seq = this
have "\<forall>i. \<Gamma>\<turnstile> (g i) \<rightarrow>\<^sup>+ (g (i+1))"
proof
fix i
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' [OF this, of "(seq c (p 0) i)"] assume "<forall>i. \<Gamma><turnstile(c s \rightarrow\^sup>*f i \and \Gamma>\<turnstile>f <ightarrow f ( i"
have\<Gamma\turnstile>(subst_redex (seq c ( 0 i ( (p ), (s i))\<rightarrow\<sup>
rule to_predjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
hence "\<Gamma>\<turnstile> (seq c (p 0) i, Normal (s i)) \<rightarrow>\<^sup>+
(seq c (p 0) (i+1), Normal (s (i + 1)))"
by (simp add: subst_redex_seq)
thus "\<Gamma>\<turnstile> (g i) \<rightarrow>\<^sup>+ (g (i+1))"
by (simp add: g_def)
qed
moreover
from terminates_impl_no_infinite_trans_computation [OF termi_c [rule_format, of 0]]
have "\<not> (\<exists>f. f 0 = (Call (p 0),lemma not_final_Fault_step:
ultimately show False
by auto
qed
qed
assumes not_inf: "\<not> \<Gamma>\<turnstile by (simp add subst_redex_seqjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
shows "wf {(c2,c1). \<Gamma> \<turnstile> (c,s) \<rightarrow>\<^sup>* c1 \<and> \next
proof (simp only: wf_iff_no_infinite_down_chain,clarify, simp)
fix f
assume "\<forall>i. \<Gamma>\<turnstile>(c, s) \<rightarrow>\<^sup>* f i \<and> \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i)"
hence "\<exists>f. f 0 = (c, s) \<and> (\<forall>i. \<Gamma>\<turnstile>f i \<rightarrow> f (Suc i))"
by (rule 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)))"
by (simp add: inf_def)
ultimately show False
by simp
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
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: "\ (cases " \in ")(fastforce intro stepintros)+
by (induct c) (fastforce intro
lemma not_final_Fault_step: "\<not> final (c,Fault f) \<Longrightarrow> \<exists>c' s'. \<Gamma>\<turnstile> (c, Fault f) \<rightarrow> (c',s')"
by(induct c fastforceintro:step.intros add final_def+
lemma not_final_Normal_step: "\<not> final (c,Normal s) \<Longrightarrow> \<exists>c' 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 ( final:finalc\sub>f,\^subf)
next case (Spec r)
thus ?case
by(cases\exists>t. (proofinductrule converse_rtranclp_induct2 case_names Refl)
next case (Seq c\<^sub>1 c\<^sub>2)
thus ?case
by (cases "final (c\<^
next case
show ?case
by (cases "s \<in> b") (fastforce intro: step.intros)+
next case (While b c)
show ?case
by (cases "s \<in> b") (fastforce intro: step.intros)+
next moreovernot_inf case (Call p)
show ?case
>)( :.intros
next case DynCom thus ?case by (fastforce intro: step.intros)
next case (Guard f g c)
show ?case
by (cases "s \<in> g") (fastforce intro: step.intros)+
next caseThrow
thus ?case by (fastforce intro: step.intros simp add: final_def)
next case (Catch c\ show ?ase
thus ?case
by java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
qed
lemma moreover "final (c,s) \<Longrightarrow> \<Gamma> "\Gamma\turnstile rightarrowr's)fact
step_redexes[this]c' where
:
assumes steps:finally ?
assumesfinal "inal(<^sub>s\sub)
shows "\<exists>c' slemmastep_redexes_Seq: using steps not_final final
oofruleconverse_rtranclp_induct2case_namesTrans case thus ?case by simp
next
Trans '
thus ?case by auto
qed
lemma wf_implies_termi_reach_step_case:
assumes hyp: "\<And "c<sub1Skip<>c
shows "\<Gamma>\<turnstile>c \<down> Normal s" using hyp
proofinduct c) case Skip show ?case by (fastforce intro: terminates.intros)
next case Basic show ?case by (fastforce intro: terminates.intros)
next case (Spec r)
?
byhave "s s
next case (Seq c\<^sub>1 c\<^sub>2)
have hyp: "\<And>c' s'. \<Gamma>\<turnstile> (Seq c\<^sub>1 c\<^sub>2, Normal s) \<rightarrow> (c', s') java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
show
.
{
fix c' s'
assume step_c\<^sub>1: "\<Gamma>\<turnstile> (c\<^sub> fin"(case'of
have "\<Gamma>\<turnstile>c' \<down> s'"
proof -
from with fin :finalcsubtjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
nstile ^1c\>, )rightarrow (
by (rule step.Seq)
from hyp [OF this]
have "\<Gamma>\<turnstile>Seq c' c\<^sub>2 \<down> s'".
thusGamma<>'<down s"
by cases auto
qed
}
from.()OFjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
show "\<Gamma>\<turnstile>c\<^sub>1 \<down> havetermi_s':<\turnstileSeqc'c^> down'.
next
show "\<forall>s'. \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>c\<^sub>2 \<down> s'"
proof (intro allI impI)
fix s'
assume exec_c\<^sub>1: "\<Gamma>\<turnstile> \<langle>c\<^ case(Step r' ''java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
amma\<urnstile>c\^sub
proof (ases "final(c\<sub1Normals))" caseTrue
hence"\^sub1=Skip \<or>> c case (Transr' s'r' s' by (simp add: final_def) thus ?thesis proof assume Skip: "c\<turnstile> ⟨==> s'" have java.lang.NullPointerException by (rule step.SeqSkip) from hyp simplified Skip, OF this have "Γ moreoverfrom exec_cjava.lang.NullPointerException have"s'=Normal s" by (auto elim: exec_Normal_elim_cases) ultimatelyshow ?thesis by simp next assumeblast with exec_c by (auto elim: exec_Normal_elim_cases) thus ?thesis by auto qed next case False from exec_impl_steps [OF exec_c1] obtain cf t where
steps_c\ubΓsub1, Normal s) →f, t)" and fin:"(case s' of
Abrupt x ==> cthusGammacjava.lang.NullPointerException
| _ ==>f = Skip ∧ by (fastforce split: xstate.splits) withfin have final: java.lang.NullPointerException by (cases s') (auto simp add: final_def) from split_computation [OF steps_c1 False this] obtain c'' s''lemma steps_red: first: "Γ
rest: "Γc. Catch r c2↓ by blast from step.Seq [OF first] have "<[est from hyp [OF thiswiths'=Stuck" have termi_s'': "ΓcaseBasic ( intro) show ?thesis proof (cases s'') case x) from termi_s'' [simplified Normal] have termi_cs \<> by cases show ?thesis by case False with fin obtain"cf=Skip""t=s'" by (ases fromrCatch r' c<sub2nredexes c'" have "Γ⊨⟨c'',Normal x⟩==> by simp from ( step) show java.lang.NullPointerException next case True with fin obtainx' wher s': s'=Abrupt x'"and\^"=N x'" by auto from steps_Throw_impl_exec rest this] Normal have"Γ⊨⟨c'',Normal x\ by simp m tec<sub>2[eomt F ths]s show "Γ⊨2↓ s'" by simpnext qed next case (Abrupt x) from steps_Abrupt_prop [OF rest this] have "t=Abrupt x" by simp with fin have "s'=Abrupt x" by (cases s') auto "\Gamma\turnstile>c2↓ s'" by auto next case (Fault f) from steps_Fault_prop [OF rest this] have "t=Fault f" by simp with fin have "s'=Fault f" byhavehyp:: "<>' s.\Gamma<turnstile Call p, Normal s) → (c', s') ==>⊨ s'" by fact thus "Γ⊨cp") by auto next case Stuck romses_uck_pop [ [OF es have "t=Stuck" by simp with fin have "s'=Stuck" by (cases s') auto thus "Γ⊨ by auto qedhave hyp\And>'s. <<Gamma qed qed qed next
( cubjava.lang.NullPointerException have hyp: java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null show ?case proof (cases "s∈b") case True then have "Γ⊨ (Cond b cjava.lang.NullPointerException by (rule step.CondTrue) from hyp [ by (cases" \in b"( :.)java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58 with True show ?thesis by(autorminatesros next case False thenhave"Γ b by (rule step.CondFalse) from hyp [OF this] have "ΓcNormaljava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77 with False show ?thesis by (auto intro: terminates.intros) qed next case (While b c) have hyp: "∧c' s'. Γ⊨ (While b c, Normal s) → (c', show ?case proof (cases "s∈b") case True then have "Γ⊨ (While b c, Normal s) → (Seq c (While } by (rule step.WhileTrue) from hyp [OF this] have"Γ>(Seq c (While b c)) ↓ with by (auto elim: terminates_Normal_elim_cases intro: ter.intros) next ffix s' thus ?thesis by (auto intro: terminates.intros) qed next case (Call p) have hyp: "∧c' s'. Γ⊨ (Call p, Normalbyo simpdd imormal_elim_caseslim_cases show ? prooffrom hyp ThrowOF this case None thus ?thesis by (auto intro: terminates.intros) next case (Some bdy) thenhave"Γ⊨<^sub>1 by (rule step.Call) from hyp [OF this] have "\Gammajava.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null with Some show ?thesis by (auto intro: terminates.intros) qed next case( c) have hyp: "∧⊨ c'' c. have "Γ⊨w_impl_exec by (rule step.DynCom) from hyp [OF this] have"Γ⊨c s ↓ Normal s". thenshow ?case by (auto intro: terminates.intros) next caseGuard have hyp: "∧ show ?case proof (cases "s∈g") case True then have "Γ⊨<>(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)
java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 4 case Throw show ?case by (auto intro: terminates.intros) next case (Catch c1 c2) have hyp: "∧c' s'. Γ⊨ reach show ?case proof (rule
{ fix c' s' assume step_cshow"Γ⊨<down s1" have"Γ⊨ wf_implies_termi_reach_step_case[OF hyp [simplified Norm] proof -
java.lang.NullPointerException have "Γ⊨: by (rule step.Catch) from hypshows🚫\> c1 ∧ Γ<turnstile →:"\not final (c,s),s) c^>2 \downs"
by cases qed
} from Catch.hyps (1) [OF this] show"ΓGeneralised Redexes› next show "∀s'. Γ⊨⟨ proof (intro allI impI) fix s' assume exec_c1: "Γ⊨n the redex itself but all the enosing statetements as wwell. show \<\< ases (c\<>1 case True
java.lang.NullPointerException : "c\^=Throw" by (auto simp add: final_def elim: exec_Normal_elim_cases) have "Γ⊨(Catch Throw cjava.lang.NullPointerException by (rule step.CatchThrow) from hyp [simplifiedredexes = Guard | have<⊨2↓ s". moreover from exec_c1 Throw have "s'=s" by (auto elim: exec_Normal_elim_cases) ultimately show ?thesis by simp next lse from exec_impl_steps [OF exec_c1] obtain cf t wcase Basic show case by fastforc intro: terminaintc'. [ redexes c; redex c' = c']==> c = c'"
steps_c1: java.lang.NullPointerException by (fastforce split: xstate.splits)
java.lang.NullPointerException obtain c'' s'' where first: "Γ<turnstile (c(c'', s'')"and rest: "Γ⊨ (c''proofduct by (auto simp add: final_def) from step.Catch [OF first] havenext fromhypthis
ve⊨2↓ moreover from steps_Throw_impl_exec [OF rest] have"\<Gammar moreover from rest obtain x where "s''=Normal x" byby (auto simp ad: root_in_rde) (auto dest: steps_Fault_prop steps_Abrupt_prop steps_Stuck_prop) ultimately show ?thesis by (fastforce elim: terminates_elim_cases) qed qed qed qed
show"Γ⊨\<> proof (c (cases s1) case (Normal s1') with wf_implies_termi_reach_step_case [OF hyp [simplified Normal]] show ?thesis by auto qed (auto intro: terminates.intros) qed
theorem no_infinite_computation_impl_terminates: assumes not_inf: "¬ Γ⊨ (c, s) →…(∞)" shows "Γ⊨c↓s" proof - from no_infinite_computation_implies_wf [OF not_inf] have wf: "wf⊨* c1 ∧ Γe1 c2}". show ?thesis by (rule wf_implies_termi_reach [OF wf]) auto qed
text‹
an important lemma for the completeness proof of the Hoare-logic for
correctness we need a generalisation of @{const "redex"} that not only
the redex itself but all the enclosing statements as well. ›
primrec redexes:: " also where
"redexes Skip = {Skip}" |
"redexes(Basic = {Basic f} | "redexes (Spec r) = {Spec r}" | "redexes (Seq c1 c2) = {Seq c1 c2} ∪ redexes credexes c''"
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}" |
java.lang.NullPointerException
lemma root_in_redexes: "c ∈
s_c> "\Gamma> (c<^sub>1, Normal s) <>\sup* c<^>f, t" apply auto donep_redexes
lemma redex_in_redexes: "redex c ∈ apply (induct c) apply auto done
:"And[c' ∈ apply (induct c) apply auto done
lemma step_redexes: shows"∧r r'. [Γ⊨(r,s) → (r',s'); r ∈show ?case ==>∃c'. Γqed 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 case (Seq c1 have "r 🚫
: Seq c\<turnstile>, <>\^
p haveep_r⊨yct from r show ?case proof assume"r = Seq croof(ass ) with step_r show ?case by (auto simp add: root_in_redexes) next assume r: "r ∈ redexes c1" from Seq.hyps (1) [OF step_r this] obtain cc where step_c1: "Γby
r': "r' ∈ redexes c'" by blast from step.stepscases> sAbrupt xjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51 have"\Cond c🚫 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 thus ?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 case Guard thus ?case by (fastforce intr: step.int e: step_eli simp add root_in_r) next case Throw thus ?case by (fastforce intro: step.intros elim: step_elim_cases simp add: root_in_redexes) next case (Catch c)Normal s)" have"r ∈ redexes (Catch c1 c2)"by fact hence r: "r = Catch c1 c2∨ r ∈ redexes c1" by simp have step_r: "Γ⊨ (r, s) → hyp OF this have "GammaturnstileSeq(While c) \down". from r show ?case proof assume "r = Catch c1 c2" with step_r show ?case by (auto simp add: root_in_redexes) next assume r: "r ∈ redexes c1" from Catch.hyps (1) [OF step_r this] obtain c' where step_c1: "Γ⊨ (c1, s) → (c', s')" and r': "r' ∈ redexes c'" by blast from step.Catch [OF step_c1] have "Γ⊨ (Catch c1 c2, s) → (Catch c' c2, s')". with r' show ?case by auto qed qed
lemma steps_redexes: assumes steps: "Γ⊨ (r, s) →* (r', s')" shows "∧c. r ∈ redexes c ==>∃c'. Γ⊨(c,s) →* (c',s') ∧ r' ∈ redexes c'" using steps proof (induct rule: converse_rtranclp_induct2 [case_names Refl Trans]) case Refl then show "∃c'. Γ⊨ (c, s') →* (c', s') ∧ r' ∈ redexes c'" 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'' ∈ redexes c'" by blast note step also from with Some show ?thesis obtain c'' where steps: "Γ⊨ (c', s'') →* (c'', s')" and r': "r' ∈ redexes c''" by blast note steps finally show ?case using r' by blast qed
lemma steps_redexes': assumes steps: "Γ⊨ (r, s) →+ (r', s')" shows "∧c. r ∈ redexes c ==>∃c'. Γ⊨(c,s) →+ (c',s') ∧ r' ∈ redexes c'" using steps proof (induct rul: tranclp_i [con 1, cas StepTran) case (Step r' s' c') have "Γ⊨ (r, s) → (r', s')" "r ∈ redexes c'" by fact+ from step_redexes [OF this] show ?case by (b (blast intro: rinto_) next case (Trans r' s' r'' s'') from Trans obtain c' where steps: "Γ⊨ (c, s) →+ (c', s')" and r': "r' ∈ redexes c'" by blast note steps moreover have "Γf g c,Normal <>Γ' by from step_redexes [OF this r'] obtain c'' where
step: "Γ⊨ (c', s') → (c'', s'')"and
r'': "r'' ∈ redexes c''" by blast note step finallyshow ?case using r'' by blast qed
lemma step_redexes_Seq: assumes step: "Γ⊨(r,s) → (r',s')" assumes Seq: "Seq r c2∈ redexes c" shows"∃c'. Γ⊨(c,s) → (c',s') ∧ Seq r' c2∈ redexes c'" proof - from step.Seq [OF step] have"Γ⊨ ?thesis from step_redexes [OF this Seq] show ?thesis . 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 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'')" "Seq r c2∈ redexes c" by fact+ from step_redexes_Seq [OF this] obtain c' where step: "Γ⊨ (c, s) → (c', s'')" and r'': "Seq r'' c2∈ redexes c'" by blast note step also from Trans.hyps (3) [OF r''] obtain c'' where steps: "Γ⊨
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 proof (induct rule: tranclp_induct2OF 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 intro: r_into_trancl) next case (Trans r' s' r'' s'') from Trans obtain c' where
steps: "Γ⊨ r': "Seq r' c2∈ redexes c'" by blast note steps moreover have "Γ⊨ from step_redexes_Seq [OF this r'] obtain c'' where
step: "Γ⊨ (c', s') → (c'', s'')"and
r'': "Seq r'' c2∈ redexes c''" by blast note step finallyshow ?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
lemmaby( add:exec_Normal_elim_cases assumes steps: "Γ⊨ (r, s) →* (r', s')" shows"∧c. Catch r c2∈ redexes c ==> ∃c'. Γ⊨by (ru step.CatchTh) 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 cjava.lang.NullPointerException from step_redexes_Catch [OF this] obtain c' where
step: "Γ⊨ (c, s) → (c', s'')"and
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 r' c2∈ redexes c''" by blast note steps finally show ?case using r' by blast qed
lemma steps_redexes_Catch': assumes steps: "Γ⊨ (r, s) →+ (r', s')" 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 c\Gammat> (Ca c<>1 ^sub>2, N\rightarrow c''c\\^su2 s' 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: "\moreover
r': "Catch r' c2∈ redexes c'" by blast note moreover have"Γ⊨ (r', s') → (r'', s'')"by fact from step_redexes_Catch [OF this r'] obtain c'' where
step: "Γ⊨ (c', s') → (c'', s'')"and
r'': "Catch r'' c2∈ redexes c''" by blast note step finallyshow ?case using r'' by blast qed
lemma redexes_subset: qed by (induct c) auto
lemma redexes_preserves_termination: assumes termi: java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null shows "∧c'. c' ∈ redexes c ==> Γ⊨c'↓s" using termi by induct (auto intro: terminates.intros)
end
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