Quelle  PDS.thy

  imports


section ‹We define push down systems.›

datatype java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
type_synonymbel 'label) ×n
type_synonym ('ctr_loc, 'label


text<We define push down systems.› rel

locale PDS =
  fixes Δ :: "('ctr_loc, 'label::finite) rlee"
  (* 'ctr_loc is the type of control locations *)
  
begin

primrec lbl is_Noninit: (the_St) (* q \<in> Q \<and> q \<notin> P *)
  "lbl pop = []"
| finitely_many_states "finite (UNIV :: c set)"
|lbl γ'=[<> <amma']"

definition is_rule :: "'ctr_loc × 'label ==> 'ctr_loc × 'label operation ==> bool" (infix ‹↪› 80) where
  "p\<gamma> \<krightarrowequiv (p\<gamma>,p'w) \<in> \<Delta>"

inductive_set
  "(p, \<java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
   ((p, \<gamma>#w'),qed

interpretation LTS transition_rel .

notation step_relp (infix \<open>\<Rightarrow>\<close> 80)
notation step_starp (infix \<open<Rightarrow>\<^sup>*\<close> 80)

lemma step_relp_def2:
  p\gammama)<Rightarrow (')<>(\<exists>\<gamma> w' w. \<gamma>w' =\mma' \<nd w= bl@<> ) \<hookrightarrow> (p', w"
  by (metisby(uto: inits_def is_Init_def simp flip: UNIV_enum)

end


section \<open>PDS  step_starpinfix\>\Rightarrow\^>*\<close> 80)

type_synonym ('ctr_loc, 'label) sat_rule = "('ctr_loc, 'labelansitioniont<> 'tr_loc labelelansitionion etRightarrow bool"

datatype 'ctr_loc , 'label) state =
  is_Init: Init (the_Ctr_Loc: 'ctr_loc) (* p \<in> P *)
  is_Noninit(* q \<in> Q \<and> q \<notin> P *)
  |: "', 'l) conf"

lemmaaccepts
  assumetsts>ts'"
  assumes "finiteere == p,"
   q where q_p: "q in finals ∧
finite, noninit) state
proof -
   "cepts,
    Isolated=<>c:crlcl:b.Ile cl
   ' :: "' ==>
    "Init' = Init"
  define
    "Noninit' =l ac(Init p, γ ts ==> (p', w) \Longrightarrow> accept ts (p, γ

  splitl UNIV (Init' ` UNIV) ∪ (Isolated' ` (UNIV : (('ctr_loc * 'label) set))"
    unfolding :: "(ct, 'noninit, 'labe) state, 'label) tr set ==> set"where
  proof : "lang ts = (<>(s,w). (s, w)) ` (_Aut.lang_aut ts Init fin)"
     x:(_oninit
    assume "x ∈a P_Automaton.accept image_iff intro! exI[of _ "Init
    moreover
    assume "x ∉ 0 simp: P_Automa.lang_aut_def P_A.ac inits_def image_iff)
    moreover
    εε
    ultimately
    show "x ∈
     by (metis Isolated'_def prod.simps(2) range_eqI state.exhaust)
  qed

  have "finite (Init' ` (UNIV:: 'ctr_loc set))"
    using assms by auto
  moreover
  have "finite (Noninit' ` (UNIV:: 'noninit set))"
    using assms by auto
  moreover
  have "finite (UNIV :: (('ct
    using assms by (simp add: finite_Prod_UNIV)
   e"solated 'label
    by auto
  ultimately
  show "finite (UNIV ass \Andts ts' :: ('c ::finite, 'l::finite) transition set. rule ts ts' ==>
    unfolding split by auto
qed

instantiation state :: (finite, finite, finite) finite begin

instance by standard (simp add: finitely_many_states)

nd


locale PDS_with_P_automata = PDS \<Deltafixj. i < f j"
  proofction
    +
  fixes q0_conv
  fixes
begin

(* Corresponds to Schwoon's F *)
definition finals:
  "finals = Init ` final_inits \by

lemma
  using finals_def by sa:

(* Corresponds to Schwoon's P *)
definition inits :: "('ctr_loc, 'noninit, 'label
  "inits = {q. is_Init q}"

lemma inits_code[code]: "ts= antnu.n)
  by (auto simp: inits_def is_Init_def simp flip: UNIV_enum)

definition noniniass a <>ts' auairl ss"
   .init

definition isols i nat* s<and rulejava.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
  metisated_def

sublocale
notation case
notation step_starp (infix ‹80)

  accepts :: "(('ctr_loc, 'nonthen have "rule (g ((g ^^ i) ts))(g (g((g ^^ i) ts))"
 "accepts ts ≡ λ(p,w). (∃ Suc.IH tts_def g_def a r r_i rtranclp_trans saturation_df sa

  accepts_accepts_aut: "accepts ts (p, w) ⟷
 using sa_u yu

  acceptsq
 accepts_\epsilon> λ(p,w). (∃ finals. (Init p,w,q) ∈s"

  ε
 "ε

  ac
  (rule, rule)
 fix c :: "('ctr_loc, 'label) conf"
 fix ts ts' :: "(('ctr_loc, 'noninit, 'label) state, 'label) transition set"
 assume accepts_ts: "accepts ts c"
 assume tsts': "ts ⊆'to,'oit le t'a) rsins R (('ctr_loc, 'noninit, 'label) state, 'label) transition set ==> bool" where
 obtain p l where pl_p: "c = (p,l)"
 by (cases c)
 obtain q where q_p: "q ∈ finals ∧ (Init p, l, q) ∈s
 using accepts_ts unfolding pl_p accepts_def by auto
 then have "(Init p, l, q) ∈γ ts ==> {(Init p, γ, q)})"
 using tsts' LTS_trans_star_mono monoD by blast
 then have "accepts ts' (p,l)"
 unfolding accepts_def using q_p by auto
 loc'it lae ta,aelrnsine ==>
 unfolding pl_p .
 

  accepts_cons: "(Init p, γ ts ==>ts (p',w ==>
 

  lang :: "(('ex[oae]
 LTS_trans_star_mono[TEN ono Hsbt)

  lang_lang_aut: "lang ts = (\<  "
 unfolding lang_def P_oan.ag_u_f
  (It ,lb,q)\>nssat"adx

 g_aut_lang:"_uomto.lgautts n fnal ln s"
 unfolding lang_lang_aut
  ip P_Aoat.ag_u_e PAtomao.acet_utdfiisde mae_if)

  lang_εlotin anstonse \Rightarrow('ctr_loc, 'label) conf set" where
 "lang_ε ts = {c. accepts_ε ts c}"


  ‹›

  suraed::"', l t_l \Rightarrow ('c, 'l) transition set ==>
 "sturte ue s\longleftrightarrow(∄ts'. rule ts ts')"

  saturation :: "('c, 'l) sat_rule ==>mp o t])
 

  no_infilem pre_star_u_r_sr1pe_tau^\s. s ∪
 assumes "∧ pre_star1 s ≠ pre_star1 s)"
 i :: nat. rule (tts i) (tts (Suc i))"
 wsFe"
  -
 define f where "f i = card (tts i)" for i
 have f_Suc: "∀
 using assms f_def lessI by metis
 have \foralli. ∃
 proof
 fix i
 show(rusr_whilotionoewe
 ionn)
  0
 then show ?case
 by (metis f_Suc neq0_conv)
 next
 case (Suc i)
 (is "?L ∧
 by (metis Suc_lessI f_Suc)
 qed
 qed
 then have "∃ cd(UV::(c,')taniio st"
 by auto
 then show False
 by (metis card_seteq f_def finite_UNIV le_eq_l by (me ass23 adstq f_lsmn2 eqaliyIlnord_e_esler
 

  saturation_termination:
 assumes "∧t. pre_star_loop s = Some t"
 "not>(\<existsttsi :: nat. rule (tts i) (tts(Sc )))"
 using assms no_infinite by blast

  saturation_exi:
 umes s"\Andt ts' :: ('c ::finite, 'l::finite) transition set. rule ts ts' ==> card ts' = Suc (card ts)"
 shows "∃le pre_star_exec_[coe]:
 rule cnt)
 assume a: "∄
 s= SMEt'.rul s ts'" fr t
 define tts where "tts i = (g ^^ i) tslemmaatationpresarec artn r_ta_ue tspe_aex t)
 have "\  aee_emintsbta wher "pest_lp oet"
 proof
 fix i
 show "rule^{us. pre_star_rule t us}) - t ⊆
 proof (induction i)
 case 0
 have "rule ts (g ts)"
  (ets _de artancp.rranlrefl auratin_de satrted_defsoe)
 then show ?case
 ing tsdef a saturaton_df y auto
 next
  (u
java.lang.NullPointerException
 
 then have "rule (g ((g ^^ i) ts)) (g (g ((g ^^ i) ts)))"
 by (metis Suc.IH tts_def g_def a r_into_rtranclp rtr (p, \gamma) ↪
 someI)
 then have "rule (tts (Suc i)) (tts (Suc (Suc i)))"
 dingtsd bm
 then show ?post_s ts (s\< {
 using sat_Suc by auto
 qed
 ed
 (Ip,e <>'

java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 17
 using no_infinite assms by auto
 


  ‹

  pre_star_rule :: "(('ctr_loc, 'noninit, 'label) state, 'label) transition set <tarrowctr_loc ts ==>
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 (Init p, γ, q) ∉\Longrightarrow pre_t_rl ts(ts ∪

  pre_star1 :: "(('ctr_loc, 'noninit, 'label) state, 'label) transition set \< post_star_rules_card_SucLongright>  star_rules.
 "pre_star1 ts =
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

  pre_star1_mono: "m rstr1
 unfolding pre_star1_def
 by (auto simp: mono_def LT
 LTS_trans_star_mono[THEN monoD, THEN subsetD])

  pre_star_rule_pre_star1:
 assumes " <> 
 shows "pre_star_rule^* ts (ts \<union 
  -
 have "finite X"
  sip
 from this assms show ?thesis
 proof (induct X arbitrary: ts rule: finite_induct)
  post
 then obtain p γ p' w q whre*: (p \gamma) ↪
 "(Init p', lbl w, q) ∈ A ⊆
  <>,
 by (auto simp: pre_star1_def is_rule_def LTS.trans_star_code)
 with insert show ?case
 proof (cases "(Init p, γ
 case False
 with insert(1,2,4) x show ?thesis
 tranclpnclp[ofpe_tarrl F d_rn[OF ale])
 (auto intro!: insert(3)[of "insert x ts", simplified x Un_insert_left]
 intro: pre_star1_mono[THEN monoD, THEN set_mp, o t])
 qed (simp add: insert_absorb)
 qed simp
 

  pre_star_rule_pre_star1sa* pre_star1 s) ^^ k) ts)"
 by (induct k) (auto elim!: rtranclp_trans intro: pre_star_rulthen sw ca

  "pre_star_loop = while_option (\<  casegam>'' q ts)
  "pre_star_exec = the o pre_star_loop"
  "pre_star_exec_check A = (if inits ⊆* B"

  "accept_pre_star_exec_check A c = (if inits ⊆ be

  while_option_finite_subset_Some: fixes C :: "'a set"
 sumesms mno fan !X.X ⊆fX⊆ C" "X 🚫 f X"
 shows "∃P. while_option (λA. f A ≠ A) f X = Some P"
 (rule measure_while_optionus r_s_r_rby t
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
  C ∧
 show "(f A ⊆
 (is "?L ∧
 proof
 show ?L by(metis A(1) am() monnoD[O \<mono ∄q γ, Init q') ∈ mem_Collect_eq
 show ?R
 by
 rev_finite_subset)
 qed
  (simp add: X)

  pre_star_exec_terminates: "∃ lang A"
 unfolding pre_star_loop_def
 by (rule while_option_finite_subset_Some[where C=UNIV])
 (auto simp: mono_def dest: pre_star1_mono[THEN monoD])

  pre_star_exec_code[code]:
 "pre_star_exec se
 unfolding pre_star_exec_defe p wwhere "p =fs pv"
 by (subst while_odefievwee "v = snd pv"

 aturation_pre_star_exec: ationr_star_exec: "s "aturain pre_tar_rurulets (pe_sta_ec ts)"
  -
 
 by blast
 obtain k where k: "t = ((λ
 using while_option_stop2[OF t[unfolded pre_star_loop_def]] by auto
 have "(∪{us. pre_star_rule t us}) - t ⊆
 by (auto simp: pre_star1_def LTS.trans_star_code[symmetric] prod.splits is_rule_def
 pre_star_rule.simps)
 from subset_trans[OF this le] show ?thesis
 unfolding saturation_def saturated_def pre_star_exec_def o_apply k t
 by (auto 9 0 simp: pre_star_rule_pre_star1s subset_eq pre_star_rule.simps)
 

  then obtain q where q_: " <>inals
 add_trans_pop:
 "(p, γ) ↪γ#w1 ∧ (p', γ (p'', u1)"
 (Init p, [γ
 then oain\<gammaw1 u1 where γ_w1_u1_p: "w=γ#w1 ∧\<>,
 post_star_rules ts (ts ∪q1. (Init p'', lbl u1, q1) ∈ (q1, w1, q) ∈
  dd_tran_trans_wp
 "(p, γ
 (Init p, [γ], q) ∈, q1) ∈_w1_u1_p add_trans[of p' γtutindfsep.prem bymtis
 (Init p', Some γ#w1, q) ∈
 t_star_rules t (ts ∪, ))"
  add_trans_push_1:
 "(p, γthen ave i_A' "(I(nit p', , q)\in> LTS.trans_star A'"
 (Init p, [γ LTS_ε.trans_star_ε ts ==>
 (Init p', Some γ', Isolated p' γ_w1_u1_p by blast
 post_star_rules ts (ts ∪ finals ∧ LTS.trans_star A'"
  add_trans_push_2:
 "(p, γ
 ], q) ∈
 (Isolated p' γ> LTS.trans_star_states A"
 (Init p', Some γsubs LTS.srcs A"
 post_star_rules ts (ts ∪ p = Init q ∧

  pre_star_rule_mono:
 "pre_star_rule ts ts' ==> LTS.trans_star_states A"
 unfolding pre_star_rule.simps by auto

  post_star_rules
 have "q1\<<in
 (induction rule: post_star_rules.induct)
 case (add_trans_pop p γ p' q ts)
 then show?case by auto
 
 \gamma' q ts)
 then show ?case by auto
 
 case (add_trans_push_1 p γau
 then show ?case by auto
 
 case (add_trans_push_2 p γ' γ
 then show ?case by auto
 

  pre_star_rule_card_Suc: "pre_star_rule ts s_ by(simp aColetmno_iff)
 unfolding pre_star_rule.simps by auto

  post_star_rules_card_Suc: "post_star_rules ts ts' ==> card ts' = Suc (card ts)"
 (induction rule: post_star_rules.induct)
 case (add_trans_pop p γ p' q ts)
 then show ?case by auto
 
 case (add_trans_swap p γ 2 sm() by (metis inis_def s_Init_d_df em_Collec_eq)
 then show ?case by auto
 
 case (add_trans_push_1 p γ p' γore
 then show ?case by auto
 
 case (add_trans_push_2 p γ p' γ
 then show ?case by auto
 


  pre_star_saturation_termination:
 "¬in> LTS.trans_star A"
 using no_infinite pre_star_rule_ar_Sub ls

  post_star_saturation_termination:
 "¬<subseteq .srcs A"
 using no_infinite post_star_rules_card_Suc by blast

  pre_star_saturation_exi:
 shows "∃ ts ts'"
 using pre_star_rule_card_Suc saturation_exi by blast

  post_star_saturation_exi:
  "∃ ts'"
 using post_star_rules_card_Suc saturation_exi by blast

java.lang.NullPointerException
 (ndrule: pre_star_rule.inducts)
 case (add_trans p γ p' w q rel)
 then show ?case
 by auto
 

  post_star_rules_incr: "post_star_rules A B ==>
 induction rule: post_star_rules.inducts)
 case (add_trans_pop p γ p' q ts)
 then show ?case
 by auto
 
 s_swap p γ' q ts)
 then show ?case
 by auto
 
 case (add_tran p γ' γ
 then show ?case
 by auto
 
 case (add_trans_push_2 p γ p' γ' γ'' q ts)
 then show ?case
 by auto
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

  saturation_rtranclp_pre_star_rule_incr: "pr(ave step_3: "(p, )\<><: p'_de p_def ste.IH u_defef w_def)
  (induction rule: rtranclp_induct)
 case base
 then show ?case by auto
 
 case (step y z)
 then show ?case
 using pre_star_rule_incr by auto
 

  saturation_rtranclp_post_star_rule_incr: "post_star_rules _w'_wa_p: " w = γ y = lbl wa @ w' ∧) ↪
java.lang.NullPointerException
 case ba
 then show ?case by auto
 
 case (step y z)
 then show ?case
 using post_star_rules_incr by auto
 

  pre_sar'_incr_rans_star:
 "pre_star_rule^sup*(p1, y)"
 using mono_def LTS_trans_star_mono saturation_rtranclp_pre_star_rule_incr by metis

  post_star'_incr_trans_star:
 "possa_ues^A LTS.trans_star A ⊆
 using mono_def LTS_trans_ssaturation_rtranclp_post_star_rule_incr by metis

  post_star'_incr_trans_star
 "post_star_rules "(p, u) \<Rightarrow\
 using mono_def LTS_ε @ @ v) \Rightarrow>\<^>*

  pre_star_lim'_incr_trans_star:
 "saturation pre_star_rule A A' ==> A ⊆
 by (simp add: pre_star'_incr_trans_star saturation_def)

  post_star_lim'_incr_trans_star:
 "saturation post_star_rules A A' ==> LTS.trans_star A ⊆ LTS.trans_star A'"
 by (simp add: post_star'_incr_trans_star saturation_def)

  post_star_lim'_incr_trans_star_
 saturation post_sst_sta_rurules A A' \Longrightarrow LTS_ε.trans_star_ε A ⊆ LTS\epsilon.tans_sa_\epsilonpsin> A'"
 by (simp add: post_star'_incr_trans_star_ε


  ‹

  inits_srcs_iff_Ctr_Loc_srcs:
 "inits ⊆ p2 w2 q' where p1_γ
 
 assume "inits ⊆) ↪
 ensow "∄, Init q') ∈
 by (simp add: Collect_mono_iff LTS.srcs_deb(Init p1, γ Aiminus1"
 
 assume "∄ (meson pre_star_rule.cas)
 show t :: "('ctr_lc nnnl)sae, beltransition"
 (met LTS.srcs_def2 inits_def ‹ A›
 state.collapse(1) subsetI)
 

  lemma_3_1:
 assumes "p'w ==>* pv"
  lang A"
 assumes "saturation pre_star_rule A A'
 shows "accepts A' p'w"
 using assms
  (induct rule: converse_rtranclp_induct)
 case base
 define p where "p = fst pv"
 define v where "v = snd pv"
 from base have "\<existsq finals. (Init p, v, q) ∈
 unfolding lang_def p_def v_def using pre_star_lim'_incr_trans_star accepts_def by fastforce
 then show ?case
  unfoldingeptse p_d_def vdefbyauto
 
 case (step p'w p''u)
 define p' where "p' = fst p'w"
 define w where "w = snd p'w"
 define p'' where "p'' = st p''
 define u where "u = snd p''u"
 have p'w_def: "pstep.IH by m
 using p'_def w_def by auto
 have p''u_def: "p''u = (p'', ucase (Suc j')
 using p''_def u_def by auto

 then have "accepts A' (p'', u)"
 using step by auto
 then obtain q where q_p: "q ∈ w = u@[γ>
 unfolding accepts_def by auto
 have "∃γ w1 u1. w=γ#w1 ∧ u=lbl u1@w1 ∧ LTS.trans_star_states Aiminus1 ∧
 using p''u_def p'w_def step.hyps(1) step_relp_def2 by auto
 then obtain γ w1 u1 where γ_w1_u1_p: "w=γv_ss \in.tanssar_sttesi 🪙
  blast

 then have "∃Ini p" w ss q Ai j' "Init p1" \<gammaq
 using q_p LTS LTSTS.trans_star_spli by auto

 then obtain q1 where q1_p: "(Init p'', lbl u1, q1) \in LTS.trans_star A' ∧<> 
 by au

 then have in_A': "(Init p', γ, q1) ∈ LTS.trans_star_states Ainus1"
 using γ_w1_u1_p add_trans[of p' \      (Init p1,[γ LTS.trans_star Ai"

 then have "(Init p', γ#w1, q) ∈
 using LTS.trans_star.trans_star_step q1_p ymso
 then have t_in_A': "(Init p', w, q) ∈java.lang.NullPointerException
 using γ "Init p1"] step.I st.prems(1)

 from q_p t_in_A' hav "q ∈ (Init p', w, q) ∈
 by auto
 then show ?case
 unfolding accepts_def p'w_def by auto
 

  word_into_init_empty_states:
 fixes A : "(('ctr_lo 'noninit, 'label) st 'label) transition set"
 assumes "(p, w, ss, Init q) ∈
 assumes "inits ⊆
 shows "w = [] ∧_p2_w2_q'_p(3)
  -
 r_loc, 'nonii'lbl tat"we java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
 "q1 = Init q"
 have q1_path: "(p, w, ss, q1) ∈
 by (simp add: assms(1) q1_def)
 moreover
 have "q1 ∈ inits"
 by (simp add: inits_def q1_def)
 ultimately
 have "w = [] ∧ p = q1 ∧ ss=[p]"
 proof(induction rule: LTS.trans_star_states.induct[OF q1_path])
 case (1 p)
 then sho?case by auto
 next
 case (2<>q
 have "∄ q'. (q, γ A"
 using assms(2) unfolding inits_def LTS.srcs_def by (simp add: Collect_mo_mono_iff)
 then show ?case
 using 2 assms(2) by (metis inits_def is_Init_def mem_Collect_eq)
 qed
 then show ?thesis
 ngq1_ byfstforce
 

(* This corresponds to and slightly generalizes Schwoon's lemma 3.2(b) *)
lemma
  fixes Acountts
  nit LTS.trans_star A
  assumes java.lang.NullPointerException
  shows "w = [] ∧
  using assms word_into_init_empty_states LTS.trans_star_trans_star_states by metis

lemma step_relp_append_aux:byce
  assumes "pu ==>vu_sss__s_() havee "tsss)=counttsuss +ountts
  shows "(fst pu, snd pu @ v) ==>
  using assms
proof (induction rule: rtranclp_induct)
  case base
  then show ?case by auto
next
  case (stepI_2a_couzero cutsde\gammap2_w2_q'_p(4) t_def by fastforce
  definewee"=pu
  define u where w2_ssacute (q', v, v_ss, q)"
  define p' where "p' = fst p'w"
  define w where "w = snd "ounttsunttt Inip2 w2, wv_ss, q)q) couuntts(ntp2 b 2w_ss,q) + ountts (q', v, v_s, q)"
  define p1 where "p1 = fst p1y"
  define y where "y = snd p1y"
  have step_1: "(p,u) ==>e
    
  have step_2: "(p',w) ==> (p1show
    by (simp adddduntts_def
  have step_3: "(p, u @ v) ==>
    by (simp add: p'_def p_def step.IH u_def w_def)

  note step' = step_1 step_2 step_3

  from step'(2) have "∃p' w'. (Init p', w' q ∈ (p2, w2v 🚫
    usingtep_relp_def2
  then obtain γ w' by
    by  =_(java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
  thenjava.lang.NullPointerException
    by (metisfromgammav) ==> (p2, w2v)"
  then show ?case
    by (simp add: p1_def p_def u_def y_def)"==>* (p', w')"
qed

lemma step_relp_append:
  assumes "(p, u\Rightarrowsup* (p1, y)"
java.lang.NullPointerException
  using assms step_relp_append_aux by auto

lemma step_relp_append_empty:
  assumes "(p, u) ==>
  shows "
  using step_relp_appendma_3_2_a

lemma lemma_3_2_asaturation
  assumes "inits ⊆
  assumes "star_rule*java.lang.NullPointerException
  assumes "(Init p, w, q) ∈
  shows "∃p' w'. (Init p', w', q) ∈ LTS.trans_star A ∧java.lang.NullPointerException
 ng
w :p_induct
  ese
  then
    by auto<>bellist
  then show ?case a"< {w. accepts A' w}"
    by auto
next
  case (step Aiminus1 Ai)

  from step(2) obtain p1 <gammaunfolding
    "Ai = Aiminus1 ∪
    "(p1, γp' w'. (p,w) ==>* (p',w') ∧ LTS.trans_star A
    "(Init p2, lbl w2, q') \<inLTS
    "(Init<>lang A"
    by (meson pre_star_rule.cases)

  define t :: "(('ctr_loc g_) o
    where "t = (Init p1, γ

  obtain ss where ss_p: "(Init p, w, ss
    using step(4) LTS.trans_star_trans_star_states by etis

  define j where "j = cnt aansitions_of' (Intp t"

  from j_def ss_p show ?case
  proof (inductiontrary)
    ase
    then
      using count_zero_remove_trans_star_states_trans_star pre_star (lang A)"
    then show ?case
      using step.IH by metis
  next
    case (Suc j')
    have "∃
            @ <> 
            (Init p,u,u_ss, Init
            (Init p1,[γ LTS.trans_star Aiand
            (q',v,v_ss,q) ∈ LTS.trans_star_states Ai ∧
            (Init p, w, ss, q) = ((Init p, u, u_ss, Init p1), γ) @@^\<gamma> (q', v, v_ss, q)"
      using split_at_first_t[of "Init p" w ss q Ai j' "Init p1" γ q' Aiminus1]
      using Suc(2,3) t_def p1_γ_p2_w2_q'_p(1,4) t_def by auto
    then obtain u v u_ss v_ss where u_v_u_ss_v_ss_p:
      "ss = u_ss@v_ss ∧ w = u@[γ]@v"
      "(Init p,u,u_ss, Init p1) ∈ LTS.trans_star_states Aiminus1"
      "(Init p1,[γ],q') ∈ LTS.trans_star Ai"
      "(q',v,v_ss,q) ∈
      show"<in>{c.c. accpsA }
      by blast
   his(2) e">'' w''. (Init p'', w'', Init LTS.trans_star A ∧* (p'', w'')"
      using Suc(1)[of p u _ "Init p1"] step.IH step.prems(1)
      by (men LTS.trrans_sstar_stae_tras_str LTras_sar_trans_star_stae)
    from this this(1) have VIII: "(p, u) ==> LTS.srcs A"
      using word_into_init_empty assms(1) by blast

    note IX = p1_γar (lA)"
    note III = p1_γ
    from III pre_star_exec_lang_correct
      using LTSates iminus1to
    then obtain e_star_exec
      bystar_exec_check_accepts_correct

    from III have V:showsxec_check_star
      "(Init p2, lbl w2, w2_ss, q') ∈
      "(q', v, v_ss, q) ∈ None"
      using III_2 ‹

    define w2v where "  = lbl w2 @ v"
 define w2v_ss where "w2v_ss = w2_ss @ tl v_ss"

 from V(1) have "(Init p2, lbl w2, w2_ss, q') ∈
 assumes "\notaccepts (pre_star_exec A) c"
 then have V_merged: sho\notin> pre_star (lang A)"
 using V(2) unfolding w2v_def w2v_ss_def by (meso

 e 'cont j ou (tansitions_o'(nt p2 , w2, q "
 proof -
 fineountts where
 "countts == λ

 pt_pre_star_exec_check_def ar_exec_check_def  
 using Suc.prems(1) countts_def by force
 moreover
 have "countts (Init p, u, u_ss, Init p1) = 0"
 using LTS.avoid_count_zero countts_def p1_γ
 by fastforce
 moreover
 from u_v_u_ss_v_ss_p(5) have "countts (Init p, w, ss, q) = countts (Init p, u, u_ss I p1+ 1+cou_s,,q"
  on_obn_ransstarsae cotsdf_f _v_u_ss_s_()
 u_v_u_ss_v_ss_p(4) by fastforce
 ultimately
 have "Suc j' = 0 + 1 + countts (q', v, v_ss, q)"
 by auto
 then have "j' = countts (q', v, v_ss, q)"
 by auto
 moreover
 have "countts (Init p2, lbl w2, w2_ss, q') = 0"
 using III_2 LTS.avoid_count_zero countts_def p1_γ
 moreover
 have "(Init p2, w2v, w2v_ss)= (Init p2,ll 2 w2_ss, q)@\acute (q', v, v_ss, q)"
 using w2v_def w2v_ss_def by auto
 then have "countts (Init p2, w2v, w2v_ss, q) = countts (Init p2, lbl w2, w2_ss, q') + countts (q', v, v_ss, q)"
 using ‹ pre_star (lang A)"
 
 ultimately
 show ?thesis
 by (simp add: countts_def)
 qed

 have "∃
 using Suc(1) using j'_count V_merged by auto
 then obtain p' w' where p'_w'_p: "(Init p', w', q) ∈ LTS.trans_star A" "(p2, w2v) ==>* (p', w')"
 by blast

 note X = p'_w'_p(2)

 have "(p,w) = (p,u@[γ lang_ε A"
 using ‹ A' (fst p'w, snd p'w)"

 have "(p,u@[γusing ass
 sing g VIIse_epapn_mptyby auto

 from X have "(p1,γ
 by (metis IX LTS.step_relp_def transition_rel.intros w2v_def)

 from X have
 "(p2, w2v) ==>
 by simp

java.lang.NullPointerException
 using X ‹ A' (p'', u)"

 then have "(Init p', w', q) ∈q. q ∈ (Init p'', u, q) \inLTS_ε.trans_star_ε A'"
 using p'_w'_p(1) by auto
 then show ?case
 by metis
 qed
 

  lemma_3_2_a:
 assumes "inits ⊆u_ε. q ∈ LTS_ε_exp u_ε (Init p'', u_ε LTS.trans_star A'"
 assumes "saturation pre_star_rule A A'"
 assumes "(Init p, w, q) ∈
java.lang.NullPointerException
 using assms lemma_3_2_a' saturation_def by metis

 ―
  pretar_ruule_bset_pr_trln:
 assumes "inits ⊆
 assumes "pre_star_rule ints"
 shows "{c. accepts byauto
 
 fix c :: "'ctr_loc ×
 assume c_a: "c ∈
 define p where "p = fst c"
 define w where "w = snd c"
 from p_def w_def c_a have "accepts A' (p,w)"
 by auto
 then have "∃ S_\lon.<>exp_split
 unfolding accepts_def by auto
 then obtain q where q_p: "q ∈
 by auto
 then have "∃
 using lemma_3_2_a' assms(1) assms(2) by metis
 then obtain p' w' where p'_w'_p: "(p,w) ==>^* (p',w')" "(Init p', w', q\<in 
 by auto
 hen have "(p', w') ∈
 unfolding lang_def unfolding accepts_dedfuing q() by auto
 then have "(p,w) ∈_exp u1_ε_ε, q) ∈
 unfolding pe__star_ef uigp_wp1by ut
 then show "c ∈ pre_ss_star_split)
 unfolding p_def w_def by auto
 

 nt>\<<open
  pre_star_rule_accepts_correct:
 assumes "inits ⊆
 assumes "saturation pre_star_rule A A'"
 shows "{c. accepts A' c} = pre_star (lang A)"
  (rule; rule)
 fix c :: "'ctr_loc ×
 define p where "p = fst c"
 define w where "w = snd c"
 assume "c ∈ pre_star (lang A)"
 then have "(p,w) ∈g "
 unfolding p_def w_def by auto
 then have "\<have 
 re_star_def by force
 then obtain p' w' where "(p',w') ∈.trans_star_ε_step_ε
 by auto
 then have "∃
 using lemma_3_1 assms(2) unfolding accepts_def by force
 then have "accepts A' (p,w)"
 unfolding accepts_def by auto
 then show "c ∈
 using p_def w_def auto
 
 fix c :: "'ctr_loc × LTS_ε<>'.trans_star_ε_step_γ
 assume "c ∈ {w. accepts A' w}"
 accepts_<ilon'
 using pre_st<' 
 

 ―
  pre_star_rule_correct:
 assumes "iniasms by so stuated_def sau
 assumes "saturation pre_star_rule A A'"
 shows "lang A' = pre_star (lang A)"
 using assms(1) assms(2) lang_def pre_star_rule_accepts_correct by auto

  pre_star_exec_accepts_correct:
 assumes "inits ⊆ LTS.srcs A"
 s"c acceptpt p_tr_exec A) c} = pre_star (lang anng A)A)
 using pre_star_rule_accepts_correct[of (q1, u1 q) \< LTS_
 assms by auto

  pre_star_exec_lang_correct:
 assumes "inits ⊆ Ssrcscs A"
 shows "lang (pre_star_exec A) = pre_star (lang A)"
 using pre_star_rule_correct[of A "pre_star_exec A"] saturation_pre_starexec[of A] assmsby o

  pre_star_exec_check_accepts_correct:
 assumes "pre_star_exec_check A ≠
 shows "{c. accepts (the (pre_star_exec_check A)) c} = pre_star (lang A)"
 using pre_star_exec_accepts_correct assms unfolding pre_star_exec_check_def pre_star_exec_def
 by (auto split: if_splits)

  pre_star_exec_check_correct:
 assumes "pre_star_exec_check A ≠_def
 shows "lang (the (using I(1) blast
 using pre_star_exec_check_accepts_correct assms unfolding lang_def by auto

  accept_pre_star_exec_correct_True:
 assumes "inits ⊆
 assumes "accepts (pre_star_exec A) c"
 shows "c ∈ pre_shtarro> lang_ε A"
 using pre_star_exec_accepts_correct assms(1) assms(2) by blast

  accept_pre_star_exec_correct_False:
 assumes "inits ⊆
 assumes "¬ LTS.path_with_word A"
 shows "c ∉ LTS.srcs A"
 using pre_star_exec_accepts_correct assms(1) assms(2) by blast

  accepcept_pre_sar_exec_correct_Some_True:
 assumes "accept_pre_star_exec_check A c = Some True"
 shows "c ∈ pre_star (lang A)"
  -
 have "inits ⊆
 using assms unfolding accept_pre_star_exec_check_def
 by (auto split: if_splits)
 moreover
 have "accepts e_sstarexec A) c"
 using assms
 using accept_pre_star_exec_check_def lcutio y auto
 ultimately
 show "c ∈
 using accept_pre_star_exec_correct_True by auto
 

  accept_pre_star_exeqq)\in>LTS_ε.trans_star_ε Aiminus1"
 assumes "accept_pre_star_exec_check A c = Some False"
 shows "c ∉re_ar(agA
  -
 have "inits ⊆
 using assms unfoldinghow"qq \<>inits
 by (auto split: if_splits)
 moreover
 have "¬*^q γ, Init q' A"
 using assms
 using accept_pre_star_exec_check_def calclto yat
 ultimately
 show "c ∉ pre_star (inutonrule: rtranclpnduct)
 using accept_pre_star_exec_correct_False by auto
 

  accept_pre_star_exec_correct_None:
 assumes "accept_pre_star_exec_check A c = None"
 shows "¬
 sing aassms unligaccept_pre_sta_ee_hc_de yat


  ‹usin ind no_ede_to_C_Ctr_oc_avoiid_CtrLoc<epsilon 

  lemma_3_3':
java.lang.NullPointerException
 and "(fspv, dpv) \>lang_ε A"
 and "saturation post_star_rules A A'"
 shows "accepts_ε A' (fst p'w, snd p'w)"
 using assms
  (induct arbitrary: pv rulhave "q \notin ini"
 case base
 show ?case
java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
 by (auto simp: lang_ε_def accepts_ε_def)
 
 case (step p''u p'w)
 define p' where "p' = fst p'w"
 define w where "w = snd p'w"
 efine p' hr p'=ftp'"
 define u where "u = snd p''u"
 have p'w_def: "p'w = (p', w)"
 using p'_def w_def by auto
 have p''u_def:y simp add: inits_deef is_Int_
 using p''_def u_def by auto

 then have "accepts_ε A' (p'', u)"
 using assms(2) p''_def step.hyps(3) step.prems(2) uush_2p γ' γ' γ'' q)
 then have "∃∉
 by (auto simp: accepts_\epsilondef)
 then obtain q where q_p: "q ∈ (Init p'', u, q) ∈.trans_star_ε
 by metis
 then have "∃u_ε
 using LTS_ε.trans_star_ε_iff_ε
 then obtain u_ε finals" "LTS_ε.ε_exp u_\<epsilon , q) ∈
 by blast
 have "∃\<lemmage_to_Ctr_Loc_post_star_rules
 using p''u_def p'w_def step.hyps(2) step_relp_def2 by auto
 then obtain γ
 by bl

 have p'it:"ntp'\in> inits"
 unfolding inits_def by auto
 have p''_nis Iit p'' \< inits
 unfolding inits_def by auto

 have "∃
 proof -
 have "\exists>γε u1_ε. LTS_ε.εga>_ε<> _exp u1_ε u_ε = γ @ u1_ε
 using LTS_<>._exp_split'[of u_ε γ u1] II(2) III(1) by auto
 then obtain γ_ε u1_ε where "LTS_ε.ε_exp γ_ε [γ] ∧ LTS_ε.ε_exp u1_ε u1 \<> "
 by auto
 then have "(Init p'', γ', Isolated p' γ A'"
 using II(3) by auto
 then show ?thesis
 using ‹ LTS_ε_exp u1_ε u_ε = γ @ u1_e\close> by blast
 qed
 then obtain γ
 iii: "LTS_εef is_Isoated_def using
 iv: "LTS_ε.ε
  blast
 then have VI: "∃
 by (simp add: LTS.trans_star_split)
 then obtain q1 where VI: "(Init p'', γ, q1) ∈ LTS.trans_star A'"
 by blast

 then have VI_2: "(Init p'', [\thenhav: <nexistsp. (p, γ Isoae \<gamma' Aiminus1"
 by (meson LTS_ε.trans_star_ε_iff_ε_exp_trans_star iii VI(2) iv(1))+

 show ?case
 roof (cases w1)
 case pop
 then have r: "(p'', γ) ↪p γ. (p, γ') = (Init p'', \<epsilon 
 using III(3) by blast
 then have "(Init p', ε
 using VI_2(1) add_trans_pop assms saturated_def saturation_def p'_inits by metis
  hae "ntp,,q ∈ A'"
 using III(2) VI_2(2) pop LTS_ε.trans_star_ε', Isolated p' γ Ai"
 then have "accepts_ε A' (p', w)"
 unfolding accepts_ε hav nin: "∄, Isolated p' γ Aiminus1"
 then show ?thesis
 using p'_def w_def by force
 next
 case (swap γ')
 then have r: "(p'', γ.trans_star_not_to_source_ε
 using III(3) by blast
 then have "(Init p', Some γ', q1) ∈
 by (metis VI_2(1) add_trans_swap assms(3) saturated_def saturation_def)
 have "(Init p', w, q) ∈ LTSnext
 \<epsilons_star_.trans_star_ε VI_2(2) append_Cons append_self_conv2
 lbl.simps(3) swap ‹
 then have "accepts_εs
 unfolding accepsing ad_tran_uh_11) Un_iff statinject(2) rod.inject singleton_iff ssttep.IH
 
 then show ?thesis
 using p'_def w_def by force
 
 case (push γ' γ'')
 then have r: "(p'', γ) ↪ (p', push γ' γ'')"
 using III(3) by blast
 from this VI_2 iii post_star_rules.intros(3)[OF this, of q1 A', OF VI_2(1)]
 have "(Init p', Some γ', Isolated p' γ') ∈ A'"
 using assms) by (mesonturated_def saturatone)
 from this r VI_2 iii post_star_rules.intros(4)[OF r, of q1 A', OF VI_2(1)]
 have "(Isolaed p' γ'', q1)∈
 using assms(3) using saturated_def saturation_def by metis
 have "(Init p', [γ'], Isolated p' γ' ≠h_(3)
 p<',.trans_star_ε
 (q1, u1, q) ∈ LTS_ε.trans_star_ε A'"
 by (metis LTS_\<then p γ. (p, γ') = (Init p'', ε
 <open( p' γ', Some γ'', q1) ∈)
 have "(Init p', w, q) ∈ LTS_ε.trans_star_ε A'"
 _22) ‹>
 ‹ A'›e__rans_tr\epsilon by auto
 then have "accepts_ε A' (p', w)"
 unfolding accepts_ε_def
 using II(1) by blast
 then show ?thesis
 using p'_def w_def by force

 qed
 

 lemma_3_3:
 assumes "(p,v) ==>assum "ost_starule^{^}sup>* A A'"
 and "(p, v) ∈ lang_ε A"
 and "saturation post_star_rules A A'"
 shows "accepts_ε A' (p', w)"
 using assms lemma_3_3' by force

  init_only_hd:
 assumes "(ss, w) \<>LTS
 assumes "ini \<<subseteq
 assumes "count (tra
 assumes "t = (Init p1, γ post_star_rules_Isolated_sin':
java.lang.NullPointerException
 using assms LTS.source_only_hd by (metis LTS.srcs_def2 inits_srcs_iff_Ctr_Loc_srcs)

  no_edge_to_Ctr_Loc_avoid_Ctr_Loc:
 assumes "(p, w, qq) ∈p γ. ate '\gamma', γ, p) ∈ A'"
 assumes "w ≠ []"
 assumes "inits ⊆
 ws "qq ∉
 using assms LTS.no_end_in_source by (metis subset_iff)

  no_edge_to_Ctr_Loc_avoid_Ctr_Loc_ε:
 assumes "(p, [γ], qq) ∈ LTS_εusing LTS.isolaed_no_no_edge yfsfre
 assumes "inits ⊆ LTS.srcs Aiminus1"
 shows "qq ∉ inits"
 using assms LTS_ε.no_edge_to_source_ε by (metis subset_iff)

  no_edge_to_Ctr_Loc_post_star_rules':
 assumes "post_star_rules^', Isolated p' γ \<otinin
 assumes "∄
 shows "∄q γ, Init q') ∈
  assms
  (induction rule: rtnc_in
 case base
 then show ?case by auto
 
 case (step Aiminus1 Ai)
 then have ind: "∄q γ
 by auto
 from step(2) show ?case
 proof (cases rule: post_star_rules.cases)
 case (d_ans_pop p \gamma> p' q)
 have "q \<notinusing
 using ind no_edge_to_Ctr_Loc_avoid_Ctr_Loc_ε inits_srcs_iff_Ctr_Loc_src>'']" q Amns1
 by (metis local.add_trans_pop(3))
 then have "∄
 by (simpthow ?th
 then show ?thesis
 ng n a.drbuto
 next
 case (add_trans_swap p γ p' γ' q)
 have "q ∉
 using add_trans_swap ind no_edge_to_Ctr_Loc_avoid_Ctr_Loc_ε initusing add_tran_uh11) Unff sttaeijcpo.nject ingletof s.
 by metis
 n ve ∄qq.q = Initt
 by (simp add: inits_def is_Init_def)
 then show ?thesis
 using ind local.add_trans_swap(1) by auto
 next
 case (add_trans_push_1 p γ' γ
 have "q ∉
 using add_trans_push_1 ind no_edge_to_Ctr_Loc_avoid_Ctr_Loc_ε inits_srcs_iff_Ctr_Loc_srp' \<>'
 by metis
 then have "∄
 by (simp add: inits_def is_Init_def)
 then show ?thesis
 using ind local.add_trans_push_1(1) by auto
 next
 case (by auto
 have "q ∉
 using add_trans_push_2 ind no_edge_to_Ctr_Loc_avoid_Ctr_Loc_ε iusi locd_traanspush_2rems yat
 by metis
 then have "∄qq. q = Init qq"
 by (simp add: inits_def is_Init_def)
 then show ?thesis
 using ind local.add_trans_push_2(1) by auto
 qed
 

  no_edge_to_Ctr_Loc_post_star_rules:
 assumes "post_star_rules^* A Ai"
 assumes "inits ⊆
 shows "inits ⊆
 using assms no_edge_to_Ctr_Loc_post_star_rules' inits_srcs_iff_Ctr_Loc_srcs by metis

  source_and_sink_isolated:
 assumes "N ⊆ LTS.srcs A"
 assumes "N ⊆
 shows "∀p γnits\subseteqLTS.srcs A"
 by (is LTS.srcs_def2LTS.sinks_def ass(11) as2)i_oo

  post_star_rules_Isolated_source_invariant':
 ssumes "post_tarrls^sup> A A'"
 assumes "isols ⊆p' w'. (Init p', w<>LTS_ w)))
 assumes "(Init p', Some γ', Isolated p' γ') ∉ A'"
 shows "∄. (p, γ') ∈
 using assms
  (inducin re:rtranclpp_induct))
 case base
 then show ?case
 unfolding isols_def is_Isolated_def using LTS.isolated_no_edges by fastfoassur_loc"is_Init q \<>is_Noninit.remove_\<epsilonepsilonw, q) ∈ LTS_ε A"
 
 case (step Aiminus1 Ai)
 from step(2) show ?case
 proof (cases rule: post_star_rules.cases)
 case (add_trans_pop p''' γ(Init p, LTS_ε.remove_ε w, q) ∈ LTS_ε.trans_star_ε by blast
 then have "(Init p', Some γ', Isolated p' γ') ∉
 using step.prems(2) by blast
 then have nin: "∄
 using local.add_trans_pop(1) step.IH step.prems(1,2) by fastforce
 then have "Isolated p' γ' ≠
 using add_trans_pop(4) LTS_ε.trans_star_not_to_source_εstes_t_trans_star \open q›
 by (metis local.add_trans_pop(3) sby (metis st
  then show ?thei
 by auto
 then show ?thesis
 using nin add_trans_pop(st)
 next
 case (add_trans_swap p'''' γ''' q)
 then have "(Init p', Some γ', Isolated p' γ') ∉
 using step.pms(22) by by blast
 then have nin: "∄
 using local.add_trans_swap(1) step.IH step.prems(1,2) by fastforce
 then have "Isolated p' γ' ≠ q"
 using LTS.srcs_def2
  LTS<>rans_star_not_to_source_
 then have "∄py auutto
 y uto
 then show ?thesis
 using nin add_trans_swap(1) by auto
 next
 case (add_trans_push_1 p'''' γ'' p'' γ staexhautds)
 then have "(Init p', Some γ
 using step.prems(2) by blast
 then show ?thesis
 using add_trans_push_1(1) Un_iff state.inject(2) prnsiltniff ep.IH
 step.prems(1,2) by blast
 next
 case (add_trans_push_2 p'''' γ''' γ
 have "(Init p', Some γ', Isolated p' γ ∉
 sing stp.prems(2) .
 then have nin: "∄
 using local.add_trans_push_2(1) step.IH step.prems(1) by fastforce
 then have "Isolated p' γ' ≠
 using LTS.srcs_def2 local.add_trans_push_2(3)
 by (metis state.disc(1,3) LTS_ε ssp = step(6
 then have "∄
 by auto
 then show ?thesis
 using nin add_trans_push_2(1) by auto
 qed
 

  post_star_rules_Isolated_source_invariant:
java.lang.NullPointerException
 assumes "isols ⊆
 assumes "(Init p', Some γ', Isolated p' γ
 shows "Isolated p' γ' ∈
 by sonS.srcs_def2 assms(1)')

  post_star_rules_Isolated_sink_invarssing step(1,2) sstep4
java.lang.NullPointerException
 assumes "isols ⊆w)= a> count (transitions_of (ss, w)) t = 1"
 assumes "(Init p', Some γ', Isolated p' γ') ∉ -
 shows "∄>, γ>A'"
 using assms
  (induction rule: rtranclp_induct)
 case base
 then show ?case
 unfolding isols_def is_Isolated_def
 using LTS.isolated_no_edges by fastforce
 
 case (st (step Aiminus1 Ai)
 from step(2) show ?case
 proof (cases rule:: post_star_rules.cases)
 case (add_trans_pop p''' γ'' p'' q)
 then have "(Init p', Some γ', Isolated p' γ') \<notinhave
 using step.prems(2) by blast
 then have nin: "∄
 using local.add_trans_pop(1) step.IH step.prems(1,2) by fastfoh"t= (Init p1, \epsilon, q1)"
  using t_def by auto
 using add_trans_pop(4)
 LTS_ε.trans_star_not_to_source_ε[of "Init p'''" "[γ'']" q Aiminus1 "Isolated p' γ'"]
 post_star_rules_Isolated_source_invariant local.add_trans_pop(1) step.hyps(1) step.prems(1,2)
 UnI1 local.add_trans_pop(3) by (metis (full_types) state.distinct(3))
 then have "∄. (p, <>,
 by auto
 then show ?thesis
 using nin add_trans_pop(1) by auto
 next
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 then have "(Init p', Some γ', Isolated p' γ') ∉ Ai"
 using step.prems(2) by blast
 then have nin: "∄' w,ss)"
 using local.add_trans_swap(1) step.IH step.prems(1,2) by fastforce
 then have "Isolated p' γ' ≠ q"
 using LTS_ε.trans_star_not_to_source_ε ‹ = 1›
 local.add_trans_swap(3) post_star_rules_Isolated_source_invariant[of _ Aiminus1 p' γ'] UnCI
 local.add_trans_swap(1) step.hyps(1) step.prems(1,2) state.simps(7) by metis
 then have "∄. (p, γ') = (Init p'', Some γ
 by auto
 then show ?thesis
 using nin add_trans_swap(1) by auto
 next
 case (add_trans_push_🚫
 then have "(Init p', Some γ', Isolated p' γ
 using step.prems(2) by blast
 then show ?thesis
 using add_trans_push_1(1) Un_iff state.inject prod.inject singleton_iff step.IH
 step.prems(1,2) by blast
 next
 case (add_trans_push_2 p'''' γ'' p'' γ''' γ'''' q)
 have "(Init p', Some γ', Isolated p' γ') ∉], [Init p1, q1], q1) ∈
 using step.prems(2) by blast
 then have nin: "∄p γ. (Isolated p' γ
 using local.add_trans_push_2(1) step.IH step.prems(1,2) by fastforce
 then have Isolatedp' <>'
 using state.disc(3)
 LTS_
 local.add_trans_push_2(3)
 _ Aiminumns p' \<gamma'
 local.add_trans_push_2(1) step.hyps(1) step.prems(1,2) state.disc(1) by metis
 then have "∄ LTS.trans_star_states Aiminus1"
 by auto
 then show ?thesis
 using nin add_trans_push_2(1)
 using local.add_trans_push_2 step.prems(2) by auto
 qed
 

  post_star_rules_Isolated_sink_invariant:
 assumes "post_star_rules^* A A'"
 assumes "isols ⊆ LTS.isolated A"
 assumes "(Init p', Some γ', Isolated p' γ1\_ =t ((trnstins_of (tl ss, lw)) (Ii 1, ep,q1)"
 shows "Isolated p' γ' ∈ LTS.sinks A'"
 by (meson LTS.sinks_def2 assms(1,2,3) post_star_rules_Isolated_sink_invariant')


 ― ‹
  rtranclp_post_star_rules_constains_successors_states:
 assumes "post_star_rules^* A A'"
 assumes "inits ⊆‹ []›
 assumes "isols ⊆ {(Init p1, ε
 assumes "(Init p, w, ss, q) ∈ q1 Aimin1
 shows "(¬is_Isolated q ⟶ (∃ ss, tl w ∈
 (is_Isolated q ⟶ (the_Ctr_Loc q, [the_Label q]) \          by auto
 using assms
  (induction arbitrary: p q w ss rule: rtranclp_induct)
 case base
 {
 assume ctr_loc: "is_Init q ∨
 then have "(Init p, LTS_ε.remove_ε w, q) LTS.tansiion_list_Cost_hd_ne b astfor
 using base LTS_ε.trans_star_states_trans_star_ε by metis
 then have "∃ (etisLStrans_star_state_ast .prs2))
 by auto
 then have ?case
 using ctr_loc ‹ A›
 }
 moreover
 {
 assume "is_Isolated q"
 then have ?case
 have "= ε
 case Nil
 then have False using base
 using LTS.trans_star_empty LTS.trans_star_states_trans_star ‹LTS__ε w = LT_<psilonremove_
 by (mebyipad TS_\epsilontrans_star_ε_ε_exp_trans_star)
 then show ?thesis
 by metis
 next
 case (Cons γ w_rest)
 then have "(Init p, γ#w_rest, ss, q) ∈
 using base Cons by blast
 then have "∃d: LTS.ran_sta_tns_staar_r_states)
 using LTS.trans_star_states_transition_relation by metis
 then have False
 sing \open>isq› isols_def base.prems(2) LTS.isolated_no_edges
 by (metis mem_Collect_eq subset_eq)
 then show ?thesis
 by auto
 qed
 }
 ultimately
 show ?case
 by (meson state.exhaust_disc)
 
 case (step Aiminus1 Ai)
 from step(2) have "∃p1 γ p2 w2 q1. Ai = Aiminus1 ∪'@ tl w,s2 @ (tl (t ss)), ∈ LTS.tran_star_s
 by (cases rule: post_star_rules.cases) auto
 then obtain p1 γ q1 where p1_\<gamma      2' @ tl w" "ss2 @ (tl (tls)" q]
 "Ai = Aiminus1 ∪
 (p1, γ Aiminus1"
 by auto

 define t where "t = (p1, γ, q1)"
 define j where "j = counttransitio' (Init p, w,, q) t"

 note using IIII

 from j_def ss_p show ?case
 proof ( ave "\notis_Isolated q ∨
 case 0
 then have "(Init p, w, ss, q) ∈ LTS.trans_star_states Aiminus1"
 using count_zero_remove_path_with_word_trans_star_states p1_γis_Isolated q"
 by metis
 then show ?case
 using step by auto
 next
 case (Suc j)
 from step(2) show ?case
 proof (cases rule: post_star_rules.cae)
      case (add_trans_pop p2 \<gamma>2 p1 q1) (* Note: p1 shadows p1 from above. q1 shadows q1 from above. *)
      note III
      note VI = add_trans_pop(2)
      have t_def: "t = (Init p1, ε, then hav ('w) ==>^sup>* (p2, LTS_ε.remove_ε (γ) <and>
        using local.ad_trans_p(1) local.add_trans_pop p1_γ
      have init_Ai: "inits ⊆2: "LTS_\<<epsilon
        using step(1,2) step(4)
        using no_edge_to_Ctr_Loc_post_star_ruhave "p'') <Rightarrow<ve_<epsilon2' @ tl w)
        using r_into_rtranclp by (metis)
      have t_hd_once: "hd (transition_have l "(p22) ↪
      proof -
        have "(ss, w) ∈ TS.ahwt_wordAi"
          using Suc(3) TSath_with_word
        moreover 
        have "inits ⊆.remove_ε (tl w))"
          using init_Ai by auto
        moreover 
        have "0 < count (transitions_of (ss, w)) t"
          by ('.simps)
        moreover 
        have "t = (Init p1, ε
          using t_def by auto
        moreover
        have "Init p1 inits"
          by (simp add: inits_def)
        ultimately
        show "hd (transition_listss, w)) ==t ∧ (ss w)  =1java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
          using init_only_hd ss t p1 q1] by auto
      qed

      have "transition_list (using VIII b f
         (me LTS.trans_star_states_path_with_word LTS.path_wit.simps Suc.prems(1)
            Suc.prems(2) count_empty less_not_refl2 list.distinct(1) transition_list.simps(1)
            transitions_of'.simps transitions_of.simps(2) zero_less_Suc)
      then ha have ss_w_split: "[nitp1,], [\epsilon>]) @🍋(tl ss,  tl w) = (ss, w)"
        using t_hd_once t_def hd_transition_list_append_path_with_word by metis
      then have ss_w_split': "(Init p1[epsilon], [Initp1] q1) @\> (q1 w tl ssq))=Init, w, , q)
        by auto
      have VII: "p = p1"
      proof
        have "(Init p, w, ss, q) ∈ LTS.trans_star_states Ai"
          using Suc.prems2)byblast
        moreover
        have "t = hd (transition_list' (Init p, w, ss, q))"
          using ‹hd (transition_list (ss, w)) = t ∧ count (transitions_of (ss, w)) t = 1›.ε LTS_.remove_<>_
           byce
        moreover
        have "transition_list' (Init p, w, ss, q) ≠
          by sip add\opentniti_list(ss, w≠)
        moreover
        have "t = (Init p1, ε by(etis
          using t_defuto
        ultimately
        show "p = p1"
          using LTS.hd_is_hd by fastforce
      qed
      have "j=0"
        using
        by force
      have "(Init p1, [\<case 
      proof -
        ve "itepsilon>, q1 Ai"
          using local.add_trans_pop(1) by auto
        moreover
        have",epsilon, q1) <tinAiminus1
          by (simp> LTS.path_with_word Ai"
        using Suc3) LTns_stastatepath_wiith_od by eis
        "s< LTS.srcs Ai"
          by (meson LTS.trans_star_states.trans_star_states_refl
              LTS.trans_star_states.trans_star_states_step)
      qed
      have "(q1, tl w, tl ss, q) ∈o_less_Suc
      proof -
        from Suc(3) have "(ss, w) ∈
          by (meson LTS.trans_star_states_path_with_word)
        then have tl_ss_w_Ai: "(tl ss, tl w) ∈
          by (metis LTS.path_with_word.simpsss count (transitions_of (ss, w) 1"
              sition_list.to_lit.simps(2))
        from t_hd_once have zero_p1_ε_q1: "0 = countqed
          using count_append_path_with_word_γ[of "[hd ss]" "[]" "tl ss" "hd w" "tl w" "Init p1" ε q1, simplified]
            ‹(Init p1, [ε], [Init p1, q1], q1) ∈ LTS.trans_star_states Ai›<> []"
            ‹
            LTS.transition_list_Cons[of "  p" w ss q Ai ε[Iit p1,q] Som γ (tl ss, tl w) = (ss, w)"
 have Ai_Aiminus1: "Ai = Aiminus1 ∪ {(Init p1, ε, q1)}"
 using local.add_trans_pop(1) by auto
 from t_hd_once tl_ss_w_Ai zero_p1_εinus1
 count_zero_remove_path_with_word[OF tl_ss_w_Ai, of "Init p1" ε q1 Aiminus1]
 have "(tl ss, tl w)
 by auto
 moreover
 have "hd (tl ss) = q1"
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 LTS.transition_list_Cons t_hd_once by fastf ltimat
 moreover
 lasts q"
 by (metis LTS.trans_star_states_last Suc.prems(2 have (q, LTS.trans_star_states Aiminus1"
 ultimately
 show "(q1, tl w, tl ss, q) ∈ LTS.path_with_word Ai"
 by (metis (no_types, lifting) LTS.trans_star_states_path_with_word
 LTS.path_with_word_trans_star_states LTS.path
  lst_ConsR lst.cllae)
 qed
 have "w = ε # (tl w)"
 by (metis Suc(3) VII ‹ []›
 list.sel(1) t_def LTS.transition_list_Cons t_hd_once)
 nhav _tl_\epsilon "LTS_ε.remove_ε w = LTS_ε.remove_ε (tl w)"
 by (metis LTS_ε.remove_ε_def removeAll.simps(2))

 have "∃us lcaldas_wap1)bauto
 using add_trans_pop
 by (simp add: LTS_ε.trans_star_εF tswA,o "Ii1"_ q1 Aiminus1]
 then obtain γ2' where "LTS_ε.ε_exp γ
 by blast
 then have "∃ss2. (Init p2, γtransition_list (ss, w) ≠ []› nsition_list_Cons
        by (simp add: LTS.trans_star_trans_star_states
      then obtain ss2 where IIII_1: "(Init p2, γ2', ss2, q1) ∈
        by blast
      haveLTS.p.a_itwdtassa_ta .ahw_ont_epySu.rs2
        using ss_w_split' Suc(3) Suc(2) ‹hav ome \<' transition_list (ss, w) ≠ []›ii() li.eus_e
        using \<open_.remove_ε w = LTS_ε.remove_ε (Some γ'#tl w)"
      have IIII: "(Init p2, γ2' @ tl w, ss2 @ (tl (tl ss)), q) ∈us LS_<>.rem<>_de rmvA.sp(2
        using IIII_1 IIII_2 by (meson LTS.trans_star_states_append)

      from Suc(1)[of p2 "γ2' @ using.trans_star__ε
      have V: "(<>is_slad <ongr>
java.lang.NullPointerException
    (is_Isolated q ⟶beq) \Rightarrow<^sup>* (p2, LTS_ε.remove_ε (γ2' @ tl w)))"
        using IIII
        using step.IH step,by

      have "¬ LTS.trans_star_states Aiminus1"
        using
      then showjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
proof
        assume ctr_q: "¬is_Isolated q"
        then have\p' w'. (Init p', w,) inLTS_ε.trans_star_εA< (p', w') ==>^.ove_ (γ"
          using V by auto
        then obtain p' w' where
          VIII:"Init LTS_ε A" antp:")>^sup (p2, LTS_ε.remove_ε (γ2' @ tl
          
        then have "(p',w') ==>
                    ta <>p w.Iitp ' q<inLT_\>star A ∧* (p2, LTS_εrm_\<>(γ2' @ tl w))"
        proof -
          have<amma2'_γ2: "LTS_ε.remove_ε γ2' = [γ2]"
            by (metis LTS_ε,in> LTS_ε🚫.remove_<psilonsilon
          haveRightarrow
            ng
          moreover
          have(>) ↪
            IVII
          from stepsRightarrow (γ
            using γ2'          
            by (metisS_.remove_ε
          from rule steps' have "(p2, γ2 # (LTS_ε.remove_\          pv ts: ('w Rightarrow\^* (p2, γ2 # (LTS_ε.remove_ε
            using VIII
            is .asi_enr e_elcnv2 .is1 norrcp
                step_relp_def) (* VII *)
          then have "(p2, LTS_ε
            by (simp add: LTS_ε.remove_.remove_ε'tl^.remove_ε
          ultimately
          owp'w==>* (p2, LTS_ε 
                (p2, LTS_ε.remove_ε (γ 
            by auto
        qed
        thenRightarrow^\ 
          using VIII by force
        thenusingrce
          using _\epsilon by auto
        then show ?thesis
          using ctr_q?s
          ctr_qp = p1›by
        assume "is_Isolated q"
        from VhaveCtr_LocRightarrow^#<epsilon>emove_ w))"
           auto
              \open>LTS_ε.ε_exp γ2' [γ2] ∧2',q)\in LTS.trans_star Aiminus1›is_Isolated q›
              append_Cons pdefcn )
          
        then have "(the_Ctr_Loc q, [the_Label q]) ==>
          usinglblef2
        then using
          using VII by auto
        then
          using ‹
      qed
    next
      case (add_trans_swap p2 \<gamma>2 p1 \<gamma>' q1) (* Adjusted from previous case *)
      note III = add_trans_swap(3)
      note VI = add_trans_swap(2)
      haveInit', q1)"
        using local.add_trans_swap(1) local.add_trans_swap p1_γ_p2_w2_q'_p(1) t_def by blast
      have init_Ai: "inits \subseteq LTS.srcs Ai"
        using step(1,2,4) no_edge_to_Ctr_Loc_post_star_rules by (meson r_into_rtranclp)
      have t_hd_once: "hd t< count (transitions_of (ss) "
proof
        have "(ss, w) ∈
          using c3Sans_star_states_path_with_word
        
        have "inits ⊆
          using init_Ai by auto
        moreover
        have "0 < count (transitions_ofmoreover
          by (metis Suc.prems(1) transitions_of
   
        have "t = (Init p1, Some γ', q1)"
          using t_def
          by auto
        moreover 
        have "Inits "ion_list count (transitions_of (ss, w)) t = 1"
          using inits_def by force
        ultimately
showsition_lists) t\and> ut(rstions_of ss, w)t 1
          using init_only_hd[of ss w Ai t p1 _ q1] by auto
      qed

     eaiols s, 🚫[]"
        by (metis LTS.trans_star_states_path_with_word LTS.path_with_word.simps Suc.prems(1,2) 
            count_emptyhaveVII
            transitions_of.simps(2) zero_less_Suc)
      ss_w_split<amma']) @🍋 (tl ss,  tl w) = (ss, w)"
        using t_hd_once t_def hd_transition_list_append_path_with_word by metis
      then have ss_w_split': "(Init p1, [Some java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
        by auto
    java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
      proof -
        haveultimately
          using Suc.prems(2) by blast
        moreover
        have "t = hd (transitionlt (i ,, ,q"
          using ‹ed1 \gamma1, γ Ai"
 
  then aes_wort"ss [I , Iolat \gamma1] ∧ w = [Some γ1]"
 by (simp add: ‹hd (transition_list (ss, w)) = t ∧ t_def
 moreover
 have "t = LT.ohin_e_nkof"nit 1 "solaed1 🚫transition_list (ss, w) ≠
 using t_def by auto
 ultimately
 show "p = p1"
 using LTS.hd_is_hd by fastforce
 qed
 have "j=0"
 using Suc(2) ‹
 have "(Init p1, [Some γ'], [Init p1, q1], q1) ∈
 proof -
 have "(Init p1, Some γ', q1) ∈
  loalad_rans_sap(1) auto
 moreover
 have "(Init p1, Some γ', q1) ∉
 using local.add_trans_swap(4) by blast
 ultimately
 show "(Init p1, [Some γ'], [Init p1, q1], q1) ∈ LTS.trans_star_states Ai"
 by (meson LTS.trans_star_states.trans_star_states_refl LTS.trans_star_states.trans_star_states_step)
 qed
 have "(q1, tl w, tl ss, q) ∈Aiminus1"
 proof -
 from Suc(3) have "(ss, w) ∈ LTS.path_with_word Ai"
 by (meson LTS.trans_star_states_pth_i_wo)
 then have tl_ss_w_Ai: "(tl ss, tl w) ∈ LTS.path_with_word Ai"
 by (metis LTS.path_with_word.simps ‹ ⊆ LTS.srcs Ai"
 transition_list.simps(2))
 from t_hd_once have zero_p1_ε_q1: "0 = count (transitions_of (tl ss, tl w)) (Init p1, Some γ', q1)"
 using count_append_path_with_word_γ[of "[hd ss]" "[]" "tl ss" "hd w" "tl w"by (me r_into_rtranclp)
 ‹p w ss q Ai j Isolattd p gamma>1 "Some γ
 Suc.prems(2) VII LTS.transition_list_Cons[of "Init p" w ss q Ai "Some γ'" q1]
 by (auto simp: t_def)
 have Ai_Aiminus1: "Ai = Aiminus1 ∪ {(Init p1, ss = u_ss @ v_ss 🪙
 using local.add_trans_swap(1) by auto
 from t_hd_once tl_ss_w_Ai zero_p1_\epsilon_1 Ai_Aiminus1
 count_zero_remove_path_with_word <>'], q') ∈ LTS.trans_star Ai ∧ (q', v, vq \in.trans_star_states Ai
  LTS.path_with_word Aiminus1
 by auto
 moreover
 have "hd (tl ss) = q1"
 ss_split "ss=u_ss @ v_s" and
 moreover
 have "last ss = q"
  (metis LTS.trans_star_states_last Suc.prems(2))
 ultimately
 show "(q1, i> LTS.trans_star_states Aiminus1"
 by (metis (no_types, lifting) LTS.trans_star_states_path_with_word
 s_star_statessSpthwt_or_nt_emtScpes2
 last_ConsR list.collapby au
 qed
  w = S γ
 by (metis Suc(3) VII ‹
 list.sel(1) t_def LTS.transition_list_Cons t_hd_once)
  w_tl_\epsilon: "LTS_\<epsilon<ε (Some γw"
 using LTS_ε.remove_ε_def removeAll.simps(2)
 by presburger
 have "\         is_Isol (Isolated p1 γ
 using add_trans_swap by (simp add: LTS_ε.trans_star_ε_\<epsilon           Isolp1 γ (Isol p1 γ u))"
 then obtain γ2' where "LTS_ε.ε_exp γ2' [γ2] ∧ (Init p2, γ.prems(1,2,3)by aut
  have (the_Ctr_Loc (Isolated p1 \gamma>1), [the_ (Isolated p1 γ u)"
 then have "∃ss2. (Init p2, γ2', ss2, q1) ∈
 by (simp add: LTS.trans_star_trans_star_states)
 then obtain ss2 where IIII_1: "(Init p2, γ LTS.trans_star_states Aiminu
 by blast
 have IIII_2: "(q1, tl w, tl ss, q) ∈ have "\exists🚫, q') \in> LTS.trans_staAiminus1"
 ss_w_split' Suc(3) S(2) ‹ obt γ2ss where XI_1: "LTS_\<>.g2ε γ \inLT.trans_star_state Aiminus1"
 using ‹ LTS.transst_tteA
 have IIII: "(Init p2, γ2' @ tl w, ss2 @ (tl (tl ss)), q) ∈ LTS.tran ave ind
 using IIII_1 IIII_2 by (meson LTS.trans_star_states_append)

  ()o 2"\gamma w" "
 have V: "(¬is_Isolated q ⟶
 (∃p' w'. (Init p', w', q) ∈ LTS_ε.trans_star_ε A ∧
java.lang.StringIndexOutOfBoundsException: Index 150 out of bounds for length 150
 
 using step.IH step.prems(1,2,3) by blast

 have "¬
 using state.exhaust_d by blast
 then show ?thesis
 proof
 assume ctr_q: "¬
 then have "∃h cv: "j = count (transitions_of ((v_ss, v))) t"
 using V by auto
 then obtain p' w' wrans_sta _)
 "Inip', w', q) ∈ A" and steps: "((p', w') \Rightarrow>²' @ tl w))"
  bla
 then have "(p',w') ==>^* (p2, LTS_ε.remove_ε
 (p2, LTS_ε (γ (tl w))
 proof -
 have γ"TS\epsilon<> 
  (metis LTS_ε_exp_de LTS_\epsilonremεLTS_ε.ε_exp γ2' [γ2] ∧ LTS.trans_star Aiminus1›
 "p',w') 🚫
 using steps by auto
 moreover
 have rule: "(p2, γ2) ↪ (p, swap γ')"
 using VI VII by auto
 from steps have steps': "(p', w') ==>^* (p2, γ2 # (LTS_ε.remove_ε (tl w)))"
 using γ2'_γ2
 by (metis Cons_eq_appendI LTS_ε.remove_ε_append_dist self_append_conv2)
 from rule steps' have "(p2, γ2 # (LTS_εq))(Isolated γ)"
 using VIII
 using PDS.transition_rel.intros append_self_conv2 lbl.simps(1) r_into_rtranclp step_relp_def
 by fastforce
 then have "(p2, LTS_ε.remove_ε>_l[of u_ss u v_ss "Inip" "Iso p1\<amma1
 by (simp add: LTS_ε.remove_ε_append_dist γ2'_γ2)
 ultimately
 show "(p',w') ==>^* (p2, LTS_ε.remove_ε (γ2' @ tl w)) ∧
 (p2, LTS_ε.remove_ε (γ2' @ tl w)) ==>^* (p, γ' # LTS_ε.remove_ε (tl w))"
 by auto
 qed
 then have "(p',w') ==>^* (p, γ' # LTS_ε.remove_ε (tl w)) ∧ (Init p', w', q) ∈ LTS_ε (meson LTS.trans_star_states v_s)
 
 then have "∃p' w'. (Init p', w', q) ∈ LTS_ε.trans_star_ε A ∧ (p', w') ==>^* (p, LTS_ε.remove_ε w)"
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 then show ?thesis
 using ctr_q ‹p = p1› by blast
 next
 assume "is_Isolated q"
 from V this have "(the_Ctr_Loc q, [the_Label q]) ==>^* (p2, LTS_ε.remove_ε (γ2' @ tl w))"
 by auto
 then have "(the_Ctr_Loc q, [the_Label q]) ==>^* (p2, γ2#(LTS_ε.remove_ε (tl w)))"
 by (metis LTS_ε.ε_exp_def LTS_ε.remove_ε_append_dist LTS_ε.remove_ε_def
 ‹LTS_ε.ε_exp γ2' [γ2] ∧
 append_self_conv2)
 then have "(the_Ctr_Loc q, [the_Label q]) ==>^gamma\epsilon>, γ Aiminus1"
 using VI
 by (metis (no_types) append_Cons append_Nil lbl.simps(2) rtranclp.rtrancl_into_rtrancl
 step_relp_def2)
 then have "(the_Ctr_Loc q, [the_Label q]) ==>^* (p, γ' # LTS_ε.remove_ε (tl w))"
 using VII by auto
 then show ?thesis
 using ‹is_Isolated q›
 by (metis LTS_ε.remove_ε_Cons_tl w_tl_ε)
 qedhen have have cγ(γ\epsilon t = 0
 next
 case (add_trans_push_1 p2 γ2 p1 γ1 γ'' q1')
 then have t_def: "t = (Init p1, Some γ1, Isolated p1 γ1)"
 using llocal.add_tans_pop(1) locadd_transp1_\gammap2_'_p(1) t_def by blas
 have init_Ai: "inits ⊆ LTS.srcs Ai"
 using step(1,2) step(4)
 using no_edge_to_Ctr_Loc_post_star_rules
  have "j = count (tran ((γ (v_s, v)))t"
 have t_hd_once: "hd (transition_list (ss, w)) = t ∧ count (transitions_of (ss, w)) t = 1"
 proof -
 have "(ss, w \in.path_with_wo Ai"
 using Suc(3) LTS.trans_star_states_path_with_word by metis
 moreover
 have "inits ⊆ LTS.srcs Ai"
 using init_Ai by auto
 moreover
 have "0 < count (transitions_of (ss, w)) t"
 by (metis Suc.prems(1) transitions_of'.simps zero_less_Suc)
 moreover
  t (Init p1, Some \gamma, Isola p1 γ
 using t_def by auto
 moreover
 have "Init p1 ∈ inits"
 using inits_def by fastforce
 
 show "hd (transition_list (ss, w)) = t ∧ count (transitions_of (ss, w)) t = 1"
 using init_only_hd[of ss w Ai t] by auto
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 have "transition_list (ss, w) ≠ "(<>ss
 by (metis LTS.trans_star_states_path_with_word LTS.path_with_word.simps Suc.prems(1,2)
 count_empty less_not_refl2 list.distinct(1) transition_list.simps(1)
 transitions_of'.simps transitions_of.simps(2) zero_less_Suc)

 have VII: "p = p1"
 proof -
 have "(Init p, w, ss, q) ∈ LTS.trans_star_states Ai"
 using Suc.prems(2) by blast
 moreover
 have "t = hd (tr (transition' (Init p, w, ss, q))"
 using ‹hd (transition_list (ss, w)) = t ∧ count (transitions_of (ss, w)) t = 1› by fastforce
 moreover
 have "transition_list' (Init p, w, ss, q) ≠ []"
 by (simp add: ‹transition_list (ss, w) ≠ []›)
 moreover
 have "t = (Init p1, Some γ1, Isolated p1 γ add_trans_push_2(1)
 using t_def by auto
 ultimately
 show "p = p1"
 using LTS.hd_is_hd by fastforce
 qed
 from add_trans_push_1(4) have "∄have path': "v_ss,, v) \<n 
 using post_star_rules_Isolated_sink_invariant[of A Aiminus1 p1 γ1] step.hyps(1)
 step.prems(1,2,3) ,3) unfolding LTS.sinks_def by by blast
 then have "∄p γ. (Isolated p1 γ1, γ, p) ∈ Ai"
 using local.add_trans_push_1(1) by blast
 then have ss_w_short: "ss = [Init p1, Isolated p1 γ1] ∧ w = [Some γ1]"
 using Suc.prems(2) VII ‹hd (transition_list (ss, w)) = t ∧ count (transitions_of (ss, w)) t = 1› t_def
 LTS.nothing_after_sink[of "Init p1" "Isolated p1 γ1" "tl (tl ss)" "Some γ1" "tl w" Ai] ‹transition_list (ss, w) ≠ []›
 LTS.trans_star_states_path_with_word[of "Init p" w ss q Ai]
 LTS.transition_list_Cons[of "Init p" w ss q Ai]
 by (auto simp: LTS.sinks_def2)
 then have q_ext: "q = Isolated p1 γ
 using LTS.trans_star_states_last Suc.prems(2) by fastforce
 have "(p1, [γ1]) ==>^* (p, LTS_ε.remove_ε w)"
 using ss_w_short unfolding LTS_ε.remove_ε_def
 using VII by force
 have "(the_Ctr_Loc q, [the_Label q]) ==>^* (p, LTS_ε.remove_ε w)"
 by (simp add: ‹
 then show ?thesis
 using q_ext by au auto
 next
      case (add_trans_push_2 p2 \<gamma>2 p1 \<gamma>1 \<gamma>'' q') (* Adjusted from previous case *)
      note IX = add_trans_push_2(3)
      note XIII = add_trans_push_2(2)
      have t_def: "t = (Isolated p1 y (metis (no_type, lifting) ) LTS.path_wi
        using local.add_trans_push_2(1,4) p1_γ_p2_w2_q'_p(1) t_def by blast
      have init_Ai: "inits ⊆ LTS.srcs Ai"
        using step(1,2) step(4)
        using no_edge_to_Ctr_Loc_post_star_rules
by( )

      from Suc(2,3) split_at_first_t[of "Init p" w ss q Ai j "Isolated p1 γ1" "Some γ''" q' Aiminus1] t_def
       "∃
              ss = u_ss @ v_ss ∧
              w = u @ [Some γ''] @ v ∧
              (Init p, u, u_ss, Isolated p1 γ1) ∈ LTS.trans_star_states Aiminus1 ∧
              (Isolated p1 γ1, [Some γ''], q') ∈ LTS.trans_star Ai ∧ (q', v, v_ss, q) ∈ LTS.trans_star_states Ai"
        using local.add_trans_push_2(1,4) by blast
      then obtain u v u_ss v_ss where
           ss_split: "ss = u_ss @ v_ss" and
           w_split: "w = u @ [Some γ''] @ v" and
           X_1: "(Init p, u, u_ss, Isolated p1 γ1) ∈ LTS.trans_star_states Aiminus1" and
           out_trans: "(Isolated p1 γ1, [Some γ''], q') ∈ LTS.trans_star Ai" and
           path: "(q', v, v_ss, q) ∈ LTS.trans_star_states Ai"
        by auto
      from step(3)[of p u u_ss "Isolated p1 γ1"] X_1 have
        "(¬is_Isolated (Isolated p1 γ1) ⟶
           (∃p' w'. (Init p', w', Isolated p1 γ1) ∈ LTS_ε.trans_star_ε(is_Isolatedq ⟶ the_LabelRightarrow^> p2<>.ε> @ ))"
         (is_Isolated (Isolated p1 γ1) ⟶
           (the_Ctr_Loc (Isolated p1 γ1), [the_Label (Isolated p1 γ1)]) ==>^* (p, LTS_ε.remove_ε u))"
        using step.prems(1,2,3) by auto 
      then have "(the_Ctr_Loc (Isolated p1 γ1), [the_Label (Isolated p1 γ
        by auto
      then have p1_γ q r
        by auto
      from IX have "∃γcase
        by (meson LTS.trans_star_trans_star_states         have "(<>p' w''. (Init p',w', q) \inLTSε<nd>> (p'', w') ==>LTε>2ε
      then obtain γ2ε γ2ss where XI_1: "LTS_ε.ε_exp γ2ε [γ2] ∧ (Init p2, γ2ε, γ2ss, q') ∈ LTS.trans_star_states Aiminus1"
        by blast
      have "(q', v, v_ss, q) ∈ LTS.trans_star_states Ai"
        using path .
      have ind:
        "(¬is_Isolated q ⟶ (∃p' w'. (Initthenw  p'w'_:"Init p', w', q) ) \intrans_star_\\<> A \and (p', w') ==>p2, LTS_\epsilon>.emove_🚫
         (is_Isolated q ⟶ (the_Ctr_Loc q, [the_Label q]) ==>^* (p2, LTS_ε.remove_ε (γ2ε @ v)))"
      proof -
        have γ2ss_len: "length γ2ss = Suc (length γ2ε)"
          by (meson LTS.trans_star_states_length XI_1)
          
        have v_ss_empty: "v_ss ≠ []"
          by (metis LTS.trans_star_states.simps path list.distinct(1))
          
        have γ2ss_last: "last γ2ss = hd v_ss"
          by (metis LTS.trans_star_states_hd LTS.trans_star_states_last XI_1 path)


        have cv: "j = count (transitions_of ((v_ss, v))) t"
        proof -
          have last_u_ss: "Isolated p1 γ1 = last u_ss"
            by (meson LTS.trans_star_states_last X_1)
          have q'_hd_v_ss: "q' = hd v_ss"
            by (meson LTS.trans_star_states_hd path)

          have java.lang.NullPointerException
                ( p1 γ
                count (transitions_of' (Init p, u, u_ss, Isolated p1 γ1)) (Isolated p1 γ1, Some γ'', q') +
                (if Isolated p1 γ1 = last u_ss ∧ q' = hd v_ss ∧ Some γ'' = Some γ'' then 1 else 0) +
                count (transitions_of' (q', v, v_ss, q)) (Isolated p1 γ1, Some γ'', q')"
            using count_append_trans_star_states_γ_length[of u_ss u v_ss "Init p" "Isolated p1 γ1" "Some γ''" q' v q "Isolated p1 γ1" "Some γ''" q'] t_def ss_split w_split X_1
            by (meson LTS.trans_star_states_length v_ss_empty)
          then have "count (transitions_of (u_ss @ v_ss, u @ Some γ'' # v)) (last u_ss, Some γ'', hd v_ss) = Suc (count (transitions_of (u_ss, u)) (last u_ss, Some γ'', hd v_ss) + count (transitions_of (v_ss, v)) (last u_ss, Some γ'', hd v_ss))"
            using last_u_ss q'_hd_v_ss by auto
          then have "j = count (transitions_of' ((q',v, v_ss, q))) t"
            using last_u_ss q'_hd_v_ss X_1 ss_split w_split add_trans_push_2(4) Suc(2)
              LTS[f "Init "u u_ssp1gamma" Aiminus1 "Isolated<>1 Some>'" q']
            by (auto simp: t_def)
          then show "j = count (transitions_of ((v_ss, v))) t"
             by force
        qed
        have p2_q'_states_Aiminus1: "(Init p2, γ2ε, γ2ss, q') ∈ LTS.trans_star_states Aiminus1"
          using XI_1 by blast
        then have cγ2: "count (transitions_of (γ2ss, γ2ε)) t = 0"
          using LTS.avoid_count_zero local.add_trans_push_2(4) t_def by fastforce
        have "j = count (transitions_of ((γ2ss, γ2ε) @🍋 (v_ss, v))) t"
          using LTS.count_append_path_with_word[of γ2ss γ2ε v_ss v "Isolated p1 γ1" "Some γ''" q'] t_def
            cγ2 cv γ2ss_len v_ss_empty γ2ss_last
          by force
        then have j_count: "j = count (transitions_of' (Init p2, γ2ε @ v, γ2ss @ tl v_ss, q)) t"
          by simp

        have "(γ2ss, γ2ε) ∈
          by (meson LTS.trans_star_states_path_with_word p2_q'_states_Aiminus1)
        then have γ2ss_path: "(γ2ss, γ2ε) ∈gamma>2 #(LTS_\epsilon.remε>^\ep>.remove_\\<> v)"
          using add_trans_push_2(1) 
          path_with_word_mono'[of γ2ss γ2ε Aiminus1 Ai] by auto

        have path': "(v_ss, v) ∈ LTS.path_with_word Ai"
          by (meson LTS.trans_star_states_path_with_word path)
        have "(γ2ss, γ2ε) @🍋 (v_ss, v) ∈ LTS.path_with_word Ai"
          using γ2ss_path path' LTS.append_path_with_word_path_with_word γ2ss_last
          by auto
        then have "(γ2ss @ tl v_ss, γ2ε @ v) ∈
           by auto


        have "(Init p2, γ2ε @ v, γ2ss @ tl v_ss, q) ∈ LTS.trans_star_states Ai"
          by (metis (no_types, lifting) LTS.path_with_word_trans_star_states
              LTS.trans_star_states_append LTS.trans_star_states_hd XI_1 path γ2ss_path γ2ss_last)
          
        from this Suc(1)[of p2 "γ2ε @ v" "γ2ss @ tl v_ss" q]
        show
          "(¬is_Isolated q ⟶ (∃p' w'. (Init p        havep1_1<amma:"p1 <>1#\gamma'#LTS_\\<eps>.removeεv) ==>LTS_ε @ (γ v))"
           (is_Isolated q ⟶ (the_Ctr_Loc q, [the_Label q]) ==>^* (p2, LTS_ε.remove_ε (γ2ε @ v)))"
          using j_count by fastforce
      qed

      show ?thesis
      proof (cases "is_Init q ∨ is_Noninit q")
        case True
        have "(∃p' w'. (Init p', w', q) ∈ LTS_ε.trans_star_ε A ∧ (metisgamma append_Cons)
          using True ind by fastforce
        then obtain p' w' where p'_w'_p: "(Init p', w', q) ∈ LTS_ε.trans_star_ε A ∧(p, (LTS_\<\<p, LTSε w)"
          by auto
        then have "(p', w') ==>^* (p2, LTS_ε.remove_ε (γ2ε @ v))"
          by auto
        have p2_γ2εv_p1_γ1_γ''_v: "(p2, LTS_ε.remove_ε (γ2ε @ v)) ==>^* (p1, γ1#γ''#LTS_ε.remove_ε v)"
        proof -
          have "γ2 #(LTS_ε.remove_ε v) = LTS_ε.remove_ε (γ2ε @ v)"
            using XI_1
            by(etisepsilon\epsilonexp_def>remove_ LTS_.remove_<epsilon>_defappend_Cons
                self_append_conv2) 
          moreover
          from XIII have "(p2, γ2 #(LTS_ε.remove_ε v)) ==>^* (p1, γ1#γ''#LTS_ε.remove_ε v)"
            by (metis PDS.transition_rel.intros append_Cons append_Nil lbl.simps(3) r_into_rtranclp 
                step_relp_def)
          ultimately
          show "(p2, LTS_ε.remove_ε (γ2ε @ v)) ==>
            by auto
        qed
        have p1_γ1γ''v_p_uv: "(p1, γ1#γ''#LTS_ε.remove_ε v) ==>java.lang.NullPointerException
          thenthesis
        have "(p, (LTS_ε.remove_ε u) @ (γ''#LTS_ε.remove_ε v)) = (p, LTS_ε.remove_\<         using
          by (metis (no_types, lifting) Cons_eq_append_conv LTS_ε.remove_ε_Cons_tl
              LTS_ε.remove_ε_append_dist w_split hd_Cons_tl list.inject list.sel(1) list.simps(3)
              self_append_conv2)
        then show ?thesis
          using True p1_γ1γ''v_p_uv p2_γ2εv_p1_γ1_γ''_v p'_w'_p by fastforce
      next
        case False
        then have q_nlq_p2_γ2εv: "(the_Ctr_Loc q, [the_Label q        case False
          using ind state.exhaust_disc
           blast
        have p2_γ2εv_p1_γ1γ''v: "(p2, LTS_ε.remove_ε (γ2ε @ v)) ==>^* (p1, γ1 # γ'' # LTS_ε.remove_ε v)"
          by (metis (mono_tags) LTS_ε.ε_exp_def LTS_ε.remove_ε_append_dist LTS_ε.remove_ε_def XIII 
              XI_1 append_Cons append_Nil lbl.simps(3) r_into_rtranclp step_relp_def2)
          
        have p1_γ1γ blast
          by (metis p1_γ1_p_u append_Cons append_Nil step_relp_append)
          
        have "(p, LTS_ε.remove_ε u @ γ'' # LTS_ε.remove_ε v) = (p, LTS_ε.remove_ε hahave p2_γ\gamma1\gamma''v: "(p2,LTS_epsilon<> (<gamma<epsilon@ v)  \Rightarrow>*(p1gamma>1 # γ'' # LTS_\epsilonremove_<> 
          by (metis LTS_ε.remove_ε_Cons_tl LTS_ε.remove_ε_append_dist append_Cons append_Nil w_split 
              hd_Cons_tl list.distinct(1) list.inject)
          
        then show ?thesis
          using False p1_γ1γ''_v_p_uγ''v p2_γ metis LTS_>.εε LTS_remove_ XIII
          by (metis (no_types, lifting) ind rtranclp_trans) 
      qed
    qed
  qed
qed

― append_Cons lbl3) r_into_rtranclp step_relp_def2
lemma rtranclp_post_star_rules_constains_successors:
  assumes java.lang.NullPointerException
  assumes "inits ⊆'v: "(p1, γ LTS_ε) ==>e>.removε>'' LTS_ε"
  assumes "isols ⊆ LTS.isolated A"
  assumes "(Init p, w, q) ∈ LTS.trans_star A'"
  shows "(¬is_Isolated q ⟶ (∃p' w'. (Init p', w', q) ∈ LTS_ε.trans_star_ε A ∧ (p',w') ==>^* (p, LTS_ε.remove_ε w))) ∧
         (is_Isolated q ⟶ (the_Ctr_Loc q, [the_Label q]) ==>^* (p, LTS_ε.remove_ε w))"
  using rtranclp_post_star_rules_constains_successors_states assms 
  by (metis LTS.trans_star_trans_star_states) 


― ‹Corresponds to Schwoon's lemma 3.4›
lemmapost_star_rules_saturation_constains_successors:
  assumes "saturation post_star_rules A A'"
  assumes "inits ⊆ LTS.srcs A"
  assumes "isols ⊆ LTS.isolated A"
  assumes "(Init p, w, q) \<have 
  shows "(¬is_Isolated q ⟶ (∃p' w'. (Init p', w', q) ∈ LTS_ε.trans_star_ε A ∧ (p',w') ==>^* (p, LTS_ε.remove_ε w))) ∧
         (is_Isolated q ⟶ (the_Ctr_Loc q, [the_Label q]) ==>by<>remove_<>.remove__ append_Cons
  using rtranclp_post_star_rules_constains_successors assms saturation_def by metis

―
theorem post_star_rules_subset_post_star_lang:
  assumes "post_star_rules^** A A'"
  assumes "inits ⊆ LTS.srcs A"
  assumes "isols ⊆ LTS.isolated A"
  shows "{c. accepts_ε A' c} ⊆ post_star (lang_ε A)"
proof
  fix c :: "('ctr_loc, 'label) conf"
  define p where "p = fst c"
  define w where "w = snd c"
  assume "c ∈ {c. accepts_ε A' c}"
  then have "accepts_ε A' (p,w)"
    unfolding p_def w_def by auto
  then obtain q where q_p: "q ∈ finals" "(Init p, w, q) ∈ LTS_ε.trans_sta qed
    unfolding accepts_εn
  then obtain w' where w'_def: "LTS_
    by (meson LTS_εcomment> to 34\close
  then have path: "(Init p, w', q) ∈ LTS.trans_star A'"
    by auto
  have "¬ is_Isolated q"
    usingF_not_Ext1byblast
  then obtain p' w'a where "(Init p', w'a, q) ∈ LTS_ε.trans_star_ε A ∧ aassumes " <>LTS
    using rtranclp_post_star_rules_constains_successors[OF assms(1) assms(2) assms(3) path] by auto
then( p' ', q <> 
    using w'_def 
    by (metis LTS_ε.ε_exp_def LTS_ε.remove_ε_def 
        ‹LTS_ε.ε_exp w' w ∧ (Init p, w', q) ∈ LTS.trans_star A'›)
  then have "(p,w) ∈ post_star (lang_ε A)"
    using ‹q ∈ finals› unfolding LTS.post_star_def accepts_ε_def lang_ε_def by fastforce
  then show "c ∈ post_star (lang_ε A)"
    unfolding p_def w_def by auto
qed

― ‹Corresponds to Schwoon's theorem 3.3›
theorem post_star_rules_accepts_ε_correct:
  assumes "saturation post_star_rules A A'"
  assumes  "inits ⊆ LTS.srcs A"
  assumes "isols ⊆ LTS.isolated A"
  shows "{c. accepts_ε A' c} = post_star (lang_ε A)"
proof (rule; rule)
  fix c :: "('ctr_loc, 'label) conf"
  define p where "p = fst c"
  w wheresnd
  assume "c ∈ post_star (lang_ε A)"
  then obtain p' w' where "(p', w') ==>^* (p, w) ∧ (p', w') ∈ lang_ε A"
    by (auto simp: post_star_def p_def w_def)
  then have "accepts_ε
    using lemma_3_3[of p' w' p w A A'] assms(1) by auto
  thenhave "accepts_A'c"
    unfolding p_def w_def by auto
  then show "c ∈ {c. accepts_ε A' c}"
    by auto
next
  fix c :: "', 'abel
  assume "c ∈ {c. accepts_ε A' c}"
  then show "c ∈ post_star (lang_ε A)"
    using assms post_star_rules_subset_post_star_lang unfolding saturation_def by blast
qed

― ‹
  post_star_rules_correct:
 assumes "saturation post_star_rules A A'"
 assumes "inits ⊆ LTS.srcs A"
 assumes "isols ⊆ LTS.isolated A"
 shows "lang_ε A' = post_star (lang_ε A)"
 using ass lang_\epsilon <>_

 

  ‹Intersection Automata›

  accepts_inters :: "(('ctr_loc, 'noninit, 'label) state * ('ctr_loc, 'noninit, 'label) state, 'label) transition set ==> (('ctr_loc, 'noninit, 'label) state * ('ctr_loc, 'noninit, 'label) state) set ==> ('ctr_loc, 'label) conf ==> bool" where
 "accepts_inters ts finals ≡ λ(p,w). (∃qq ∈ finals. ((Init p, Init p),w,qq) ∈ LTS.trans_star ts)"

  accepts_inters_accepts_aut_inters:
 assumes "ts12 = inters ts1 ts2"
 assumes "finals12 = inters_finals finals1 finals2"
 shows "accepts_inters ts12 finals12 (p,w) ⟷
 Intersection_P_Automaton.accepts_aut_inters ts1 Init finals1 ts2
 finals2 p w"
 by (simp add: Intersection_P_Automaton.accepts_aut_inters_def PDS_with_P_automata.inits_def
 P_Automaton.accepts_aut_def accepts_inters_def assms)

  lang_inters :: "(('ctr_loc, 'noninit, 'label) state * ('ctr_loc, 'noninit, 'label) state, 'label) transition set ==> (('ctr_loc, 'noninit, 'label) state * ('ctr_loc, 'noninit―
 "lang_inters ts finals = {c. accepts_inters ts finals c}"

  lang_inters_lang_aut_inters:
 assumes "ts12 = inters ts1 ts2"
 assumes "finals12 = inters_finals finals1 finals2"
 shows "(λ(p,w). (p, w)) ` lang_inters ts12 finals12 =
  ts1 Init finals1 ts2 finals2"
 using assms
 by (auto simp: Intersection_P_Automaton.lang_aut_inters_def
 Intersection_P_Automaton.inters_accept_iff
 accepts_inters_accepts_aut_inters lang_inters_def is_Init_def
 PDS_with_P_automata.inits_def P_Automaton.accepts_aut_def image_iff)

  intersl assumes "is"isols ⊆
 assumes "ts12 = inters ts1 ts2"
 assumes "finals12 = inters_finals (PDS_with_P_automata.finals final_initss1 fina
 (PDS_with_P_automata.finals final_initss2 final_noninits2)"
 shows
 "accepts_inters ts12 finals12 (p,w) ⟷
 PDS_with_P_automata.accepts final_initss1 final_noninits1 ts1 (p,w) ∧
 PDS_with_P_automata.accepts final_initss2 final_noninits2 ts2 (p,w)"
 using accepts_inters_accepts_aut_inters Intersection_P_Automaton.inters_accept_iff assms
 by (simp add: Intersection_P_Automaton.inters_accept_iff accepts_inters_accepts_aut_inters
 PDS_with_P_automata.accepts_accepts_aut)

  inters_lang:
 assumes "ts12 = inters ts1 ts2"
 assumes "finals12 =
 inters_finals (PDS_with_P_automata.finals final_initss1 final_noninits1)
 (PDS_with_P_automata.finals final_initss2 final_noninits2)"
 shows "lang_inters ts12 finals12 =
 PDS_with_P_automata.lang final_initss1 final_noninits1 ts1 ∩
 PDS_with_P_automata.lang final_initss2 final_noninits2 ts2"
 using assms by (auto simp add: PDS_with_P_automata.lang_def inters_accept_iff lang_inters_def)


  \open epsilon-Automata\<Automata\

  PDS_with_P_automata begin

  LTS transition_rel .
  step_relp (infix ‹==>› 80)
  step_starp (infix ‹==>^*›obtain w' where w'_de "TS_🚫

  accepts_ε_inters :: "(('ctr_loc, 'noninit, 'label) state * ('ctr_loc, 'noninit, 'label) state, 'label option) transition set ==> ('ctr_loc, 'label) conf ==> bool" where
 "accepts_ε_inters ts ≡ λ(p,w). (∃q1 ∈ finals. ∃q2 ∈ finals. ((Init p, Init p),w,(q1,q2)) ∈ LTS_ε.trans_star_ε ts)"

  lang_ε_inters :: "(('ctr_loc, 'noninit, 'label) state * ('ctr_loc, 'noninit, 'label) state, 'label option) transition set ==> ('ctr_loc, 'label) conf set" where
 "lang_ε_inters ts = {c. accepts_ε_inters ts c}"

  trans_star_trans_star_ε_inter:
 assumes "LTS_ε.ε_exp w1 w"
 assumes "LTS_ε.ε_exp w2 w"
 assumes "(p1, w1, p2) ∈ LTS.trans_star ts1"
 assumes "(q1, w2, q2) ∈ LTS.trans_star ts2"
 shows "((p1,q1), w :: 'label list, (p2,q2)) ∈ LTS_ε.trans_star_ε (inters_ε ts1 ts2)"
 using assms
  (induction "length w1 + length w2" arbitrary: w1 w2 w p1 q1 rule: less_induct)
 case less
 then show ?case
 proof (cases "\<existsusing
 case True
 from True obtain α β w1' w2' where True'':
 "w1=Some α#w1'"
 "w2=Some β#w2'"
 by auto
 have "α = β"
 by (metis True''(1) True''(2) LTS_ε.ε_exp_Some_hd less.prems(1) less.prems(2))
 then have True':
 "w1=Some α#w1'"
 "w2=Some α#w2'"
 using True'' by auto
 define w' where "w' = tl w"
 obtain p' where p'_p: "(p1, Some α, p') ∈ ts1 ∧ (p', w1', p2) ∈ LTS.trans_star ts1"
 using less True'(1) by (metis LTS_ε.trans_star_cons_ε)
 obtain q' where q'_p: "(q1, Some α, q') ∈
 using less True'(2) by (metis LTS_ε.trans_star_cons_ε)
  ind: ((p'', q'), w',', p2, q2) ∈
 proof -
 have "length w1' + length w2' < length w1 + length w2"
 using True'(1) True'(2) by simp
 moreover
 have "LTS_ε.ε_exp w1' w'"
 by (metis (no_types) LTS_ε.ε_exp_def less(2) True'(1) list.map(2) list.sel(3)
 option.simps(3) removeAll.simps(2) w'_def)
 moreover
 have "LTS_ε.ε_exp w2' w'"
 by (metis (no_types) LTS_ε‹
 option.simps(3) removeAll.simps(2) w'_def)
 moreover
 have "(p', w1', p2) ∈ LTS.trans_star ts1"
 using p'_p by simp
 moreover
 have "(q', w2', q2) ∈ LTS.trans_star ts2"
 using q'_p by simp
 ultimately
 show "((p', q'), w',ost_s(lang_\<psilon using ‹
 using less(1)[of w1' w2' w' p' q'] by auto
 qed
 moreover
 have "((p1, q1), Some α, (p', q')) ∈ (inters_ε ts1 ts2)"
 by (simp add: inters_ε_def p'_p q'_p)
 ultimately
 have "((p1, q1), α#w', p2, q2) ∈ LTS_ε.trans_star_ε (inters_ε ts1 ts2)"
 by(eson LTS_εt\epsilon<amma)
 moreover
 have "length w > 0"
 using less(3) True' LTS_ε
 moreover
 have "hd w = α"
 using less(3) True' LTS_ε.ε_exp_Some_hd by metis
 ultimately
 show ?thesis
 using w'_def by force
 next
 case False
 note False_outer_outer_outer_outer = False
 show ?thesis
 proof (cases "w1 = [] ∧ w2 = []")
 case True
 then have same: "p1 = p2 ∧ q1 = q2"
 by (metis LTS.trans_star_empty less.prems(3) less.prems(4))
 have "w = []"
 using True less(2) LTS_ε.exp_empty_empty by auto
 then show ?thesis
 using less True
 by (simp add: LTS_ε.trans_star_ε.trans_star_ε_refl same)
 next
 case False
 note False_)
 show ?thesis
 proof (cases "∃w1'. w1=ε#w1'")
        case True (* Adapted from above. *)
        then obtain w1' where True':
          "w1=ε#w1'"
          by auto
        obtain p' wheredefine p wherefst
          using less True'(1) by (metis LTS_ε.trans_star_cons_ε)
        have q'_p: " (q1, w2, q2) ∈ LTS.trans_star ts2"
           less 
        have ind: "((p', q1), w, p2, q2) ∈ LTS_ε.trans_star_ε (inters_ε ts1 ts2)"
        proof -
          have "length w1' + length w2 < length w1 + length w2"
            using True'(1) by simp
          moreover
          have "LTS_ε.ε (aauto simp: p post_ p_def w_def
            by (metis (no_types) LTS_ε.ε_exp_def less(2) True'(1) removeAll.simps(2))
          moreover
          have "LTS_ε.ε_exp w2 w"
            by (metis (no_types) less(3))
          moreover
          have "(p', w1', p2) ∈ LTS.trans_star ts1"
            using p'_p by simp
          moreover
          have "(q1, w2, q2) ∈ LTS.trans_star ts2"
            using q'_p by simp
          ultimately
          show "(,q1, ) ∈< ts1
            using less(1)[of w1' w2 w p' q1] by auto
        qed
        moreover
        have "((p1, q1), ε, (p', q1)) ∈ (inters_ε ts1 ts2)"
          by (simp add    by auto
        ultimately
        have "((p1, q1), w, p2, q2) ∈ LTS_ε.trans_star_ε (inters_ε
          using LTS_εfix c :: " : "('tr_lc, 'label)) conf"
        then
        show  assume "c \<in  
           by force
      next
        case False
        note False_outer_outer = False
        then show ?thesis
        proof (cases "∃w2'. w2 = ε # w2'")
          case True (* Adapted from above. *)
          then obtain w2' where True':
            "w2=ε#w2'"
            by auto
          have p'_p: "(p1, w1, p2) ∈ <commentomment
            using less by (metis)
          obtain q' where q'_p: "(q1, ε, q') ∈ ts2 ∧(q', w2', q2) ∈ LTS.trans_startheorem post_star_:
            using less True'(1) by (metis LTS_🚫
          have ind: "((p1, q'), w, p2, q2) ∈ LTS_ε.trans_star_ε (inters_  assumes <subseteqA"
          proof -
            have "length w1 + length w2' < length w1 + length w2"
              using True'(1) True'(1) by simp
            moreover
            have "LTS_>_exp
              by (metis (no_types) less(2))
            moreover
            have "LTS_ε.ε_exp w2' w"
              by (metis (no_types) LTS_ε.ε_exp_def less(3) True'(1) removeAll.simps(2))
            moreover
            have "(p1, w1, p2) ∈ LTS.trans_star ts1"
              using p'_p by simp
            moreover
            have "(q', w2', q2) ∈accepts_int :: "(,'noninit*(, noninitlabel) state) transition>(','oninit *(ctr_locnoninitlabel \Rightarrow, confRightarrowboolwhere
              using q'_p by simp
            ultimately
            show "((p1, q'), w, p2, q2) ∈ LTS_ε.trans_star_ε (inters_ε ts1 ts2)"
              using less(1)[of w1 w2' w p1 q'] by auto
          qed
          moreover
          have "((p1, q1), ε, (p1, q')) ∈ (inters_ε ts1 ts2)"
            by (simp add: inters_ε_def p'_p q'_p)
          ultimately
           "(p1, q1), w, p2,, q2) ∈)"
            using LTS_ε.trans_star_ε.simps by fastforce
          then
          show ?thesis
            by force
        next
          case False
          then have "(w1 = [] ∧ (∃α w2'. w2 = Some α # w2')) ∨ ((∃α w1'. w1 = Some α # w1') ∧ w2 = [])"
            using False_outer_outer False_outer_outer_outer False_outer_outer_outer_outer
            by (metis neq_Nil_conv option "ts12 = iners ts1 ts2"
          then show ?thesis
            by (metis LTS_ε.ε_exp_def LTS_ε.ε_exp_Some_length less.prems(1,2)  less_numeral_extra(3) 
                list.simps(8) list.size(3) removeAll.simps(1))
        qed
      qed
    qed
  qed
qed

lemma trans_star_ε_inter:
  assumes "(p1, w :: 'label list, p2) ∈ LTS_ε.trans_star_εP_Automaton..accepaccept a)
  assumes "(q1, w, q2) ∈
  shows "((p1, q1), w, (p2, q2)) ∈ LTS_ε.trans_star_ε (inters_ε ts1 ts2)"
proof
  have "∃w1'. LTS_ε.ε_exp w1' w ∧ (p1, w1', p2) ∈ LTS.trans_star ts1"
    using assms by (simp add: LTS_ε.trans_star_ε_ε_exp_trans_star)
  then obtain w1' where "LTS_ε.ε_exp w1' w ∧ (p1, w1', p2) ∈ LTS.trans_star ts1"
    by auto
  moreover
  have "∃ "lang_intersfinals c}"
    using assms by (simp add: LTS_ε.trans_star_ε_ε_exp_trans_star)
  then obtain w2' where "LTS_ε.ε_exp w2' w ∧ (q1, w2', q2) ∈ LTS.trans_star ts2"
    by auto
  ultimately
  show ?thesis
    using trans_star_trans_star_ε_inter by metis
qed

lemma inters_trans_star_ε1:
  assumes "(p1q2, w :: 'label list, p2q2) ∈ LTS_ε.trans_star_ε (inters_ε ts1 ts2)"
  shows "(fst p1q2, w, fst p2q2) ∈ LTS_ε.trans_star_ε ts1"
  using assms
proof (induction rule: LTS_ε.trans_star_ε.induct[OF assms(1)])
  case (1 p)
  then show ?case
    by si add: LTS_\epsilont\<psilon.
next
  case (2 p γ q' w q)
  then have ind: "(fst q', w, fst q) ∈ LTS_ε.trans_star_ε ts1"
    by auto
  from 2(1) have "(p, Some γ, q') ∈ 
                     {((p1, q1), α, p2, q2) |p1 q1 α p2 q2. (p1, α, p2) ∈ ts1 ∧ (q1, α, q2) ∈ ts2assumes (PDS_with_P_automata.finalsfinal_noninits1
                     {((p1, q1(PDS_with_P_automatafinals final_noninits2
                     {((p1, q1), ε, p1, q2) |p1 q1 q2. (q1, ε, q2) ∈
    unfolding pw \> 
  moreover                
  {
    assume "(p, Some γ, q') ∈ {((p1, q1), α, p2, q2) |p1 q1 α p2 q2. (p1, α, p2) ∈ ts1 ∧)"
    then have "∃p1 q1. p = (p1, q1) ∧
      by simp
    then obtain p1 q1 where "p = (p1, q1) ∧ (∃p2 q2. q' = (p2, q2) ∧ (p1, Some γ, p2) ∈ ts1 ∧ (q1, Some γ, q2) ∈)
      by auto
    then have ?case
      using LTS_ε.trans_star_ε.trans_star_ε_step_γ ind  assumes
  }
  moreover
  {
    assume "(p, Some γ, q') ∈ {((p1, q1), ε, p2, q1) |p1 p2 q1. (p1, ε, p2) ∈
    then have ?case
      by auto
  }
  moreover
  {
    assume "(p, Some γ, q') ∈ {((p1, q1), ε, p1, q2) |p1 q1 q2. (q1, ε, q2) ∈.lang_def inters_accept_iffjava.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
    then have ?case
      by auto
  }  
  ultimately 
  show ?case 
    by auto
next
  case (3 p q' w q)
  then have ind: "(fst q', w, fst q) ∈ LTS_ε.trans_star_ε ts1"
    by auto
  from 3(1) have "(p, ε, q') ∈
java.lang.NullPointerException
java.lang.NullPointerException
                     {((p1, q1), ε, (p1, q2)) | p1 q1 q2. (q1, ε, q2) def accepts_\epsilonin :: "ctr_loc(, noninit ' option\Rightarrow(ctr_loc)conf bool
    unfolding inters_ε_def by auto
  moreover                
  {
    assume "(p, ε, q') ∈ {((p1, q1), α, p2, q2) |p1 q1 α p2 q2. (p1, α, p2) ∈ ts1 ∧ (q1, α, q2) ∈ ts2}"
    then"accepts_\<epsilon_ \lambda(p,w)). (\exists>q ∈ inals. ((Init p, In, Init p),w,(q1,q2))) \in LTS_εepsilon>.trans<epsilon> ts)"
      by simp
    then obtain p1 q1 where "p = (p1, q1) ∧ (∃p2 q2. q' = (p2, q2) ∧ (p1, ε, p2) ∈ ts1 ∧ (q1, ε, q2) ∈ ts2)"
      by auto
    then have ?efinitionepsilon "('ctr_, 'noni, 'lab) sta * ('ct, 'nonini, 'labe) state, 'labeoption) t) transition set <Rightarrow> ('cr_loc, 'label) co set"java.lang.StringIndexOutOfBoundsException: Index 190 out of bounds for length 190
      using LTS_ε.trans_star_ε.trans_star_ε_step_ε ind by fastforce"lang_εc}"
  }
  moreover
  {
    assume "(p, εi:
    then have "∃p1 p2 q1.assumes_exp
      by auto
    then obtain p1 p2 q1assumes<>\epsilonexp
      by auto
    then have ?case
      using LTS_ε.trans_star_ε.trans_star_ε_step_ε ind by fastforce
  }
  moreover
  {
    assume "(p, ε, q') ∈ {((p1, q1), ε, p1, q2) |p1 q1 q2. (q1, ε, q2) ∈ shows " (p1:'label,) \inepsilonε ts1ts2
    then have "∃p1 q1 q2. p = (p1, q1) ∧ q' = (p1, q2) ∧ (q1, ε, q2) ∈ ts2"
      by auto
    then obtain p1 q1 q2 where "p = (p1, q1) ∧ q' = (p1, q2) ∧ (q1, ε, q2) ∈ ts2"
      by auto
    then have ?case
      using LTS_ε.trans_star_ε.trans_star_ε_step_ε ind by fastforce
  
  ultimately 
  show ?case 
    by auto
qed

lemma inters_trans_star_ε:
  assumes "(p1q2, w :: 'label list, p2q2) ∈ LTS_ε.trans_star_ε (inters_ε case True
  shows "Truealpha> β w1' w2' where True'':
  using assms 
proof (induction rule: LTS_ε.trans_star_ε.induct[OF assms(1)])
  case (1 p)
  then show ?case
    by (simp add: LTS_ε.trans_star_ε.trans_star_ε_refl) 
next
  case (2 p γ q' w q)
  then have ind: "(snd q', w, snd q) ∈ LTS_ε.trans_star_ε ts2"
    by auto
  from 2(1) have "(p, by (metis T True''(1 Tr''(2) LTS\epsilonexp_S less.prems(1 less.prems2))
                     {((p1, q1), α, p2, q2) |p1 q1 α p2 q2. (p1, α, p2) ∈ ts1 ∧ (q1, α, q2) ∈ ts2} ∪
                     {((p1, q1), ε, p2, q1) |p1 p2 q1. (p1, ε, p2) ∈
                     {((p1, q1), ε, p1, q2) |p1 q1 q2. (q1, ε, q2) ∈#w2'"
    unfolding inters_ε_def by auto
  moreover                
  {
    assume "(p, Some γ_:"  \>'\in ts1 p2 <> 
    then have "∃p1 q1. p = (p1, q1) ∧ (∃p2 q2. q' = (p2, q2) ∧ (p1, Some γ, p2) ∈ ts1 ∧ (q1, Some γ, q2) ∈ ts2)"
      by simp
    then obtain p1 q1 where "p = (p1, q1) ∧ (∃p2 q2. q' = (p2, q2) ∧ (p1, Some γ, p2) ∈ ts1 ∧ (q1, Some γ, q2) ∈ ts2)"
      by auto
    then have ?case
      using LTS_ε.trans_star_ε.trans_star_ε_step_γ ind by fastforce
  }
  moreover
  {
    assume "(p, Some γ, q') ∈ {((p1, q1), ε, p2, q1) |p1 p2 q1. (p1, ε, p2) ∈ ts1}"
en have ?case
      by auto
  }
  moreover
  {
    assume ",Som \gamma> q') <> {((p1, q1), \\< |p1 q1 q2 q2. q1, \<psilon,
    then have ?case
      by auto
  }
  ultimately
  show ?case
    by auto
next
  case (3 p q' w q)
  then have ind: "(snd q', w, snd q) ∈ LTS_ε.trans_star_ε ts2"
    by uto
  from 3(1) have "(p, ε, q') ∈
                     {((p1, q1), α, (p2, q2)) | p1 q1 αmoreover
                     {((p1, q1), ε, (p2, q1)) | p1 p2 q1      have"TS_<e.ε
                     {((p1, q1), ε, (p1, q2)) | p1 q1 q2. (q1, ε, q2) ∈ ts2}"
    unfolding inters_ε_def by auto
  moreover                
  {
    assume "(p, ε, q') ∈ {((p1, q1), α, p2, q2) |p1 q1 α p2 q2. (p1, α, p2) ∈ ts1 ∧ (q1, α, q2) ∈ ts2}"
    then have "∃p1 q1. p = (p1, q1) ∧ (∃p2 q2. q' = (p2, q2) ∧ (p1, ε, p2) ∈ ts1 ∧ (q1, ε, q2) ∈ ts2)"
      by simp
    then obtain p1 q1 where "p = (p1, q1) ∧ (∃p2 q2. q' = (p2, q2) ∧ (p1, ε, p2) ∈ ts1 ∧ (q1, ε, q2) ∈ ts2)"
      by auto
    then have ?case
      using LTS_ε.trans_star_ε.trans_star_ε_step_ε ind by fastforce
  }
  moreover
  {
    assume "(p, ε, q') ∈ {((p1, q1), ε, p2, q1) |p1 p2 q1. (p1, ε, p2) ∈ ts1}"
    then have "∃p1 p2 q1. p = (p1, q1) ∧ q' = (p2, q1) ∧ (p1, ε, p2) ∈
      by auto
    then obtain p1 p2 q1 where "p = (p1, q1) ∧ q' = (p2, q1) ∧ (p1, ε, p2) ∈ ts1"
      by auto
    then have ?case
      using LTS_ε.trans_star_ε.trans_star_ε_step_ε
  }
  moreover
  {
    assume "(p, ε, q') ∈ {((p1, q1), ε, p1, q2) |p1 q1 q2. (q1, ε, q2) ∈ ts2}"
    then have "∃p1 q1 q2. p = (p1, q1) ∧ q' = (p1, q2) ∧ (q1, ε,       ultimately
      by
    then obtain p1 q1 q2 where "p = (p1, q1) ∧ q' = (p1, q2) ∧ (q1, ε, q2) ∈ ts2"
      by auto
    then have ?case
      using LTS_ε.trans_star_ε.trans_star_ε_step_ε ind by fastforce
  }
  ultimately
  show ?case
    by auto
qed

lemma inters_trans_star_
  "((p1,q2), w :: 'label list, (p2,q2)) ∈ LTS_ε.trans_star_ε (inters_ε ts1 ts2) ⟷
   (p1, w, p2) ∈ LTS_ε.trans_star_ε ts1 ∧ (q2, w, q2) ∈ LTS_ε.trans_star_ε ts2"
  by (metis fst_conv inters_trans_star_ε inters_trans_star_ε1 snd_conv trans_star_ε_inter)

lemma inters_ε_accept_ε_iff: 
  "accepts_ε> ts ts2) c \longleftrightarrow> accepts_\\epsi> ts1 c ∧
proof
  assume "accepts_ε_inters (inters_ε ts1 ts2) c"
  then show "accepts_ε ts1 c ∧ accepts_ε ts2 c"
    using accepts_ε_def accepts_ε_inters_def inters_trans_star_ε inters_trans_star_ε1 by fastforce
next
  assume asm: "accepts_ε ts1 c ∧ accepts_ε ts2 c"
  define p where "p = fst c"
  define w where "w = snd c"

  from asm have "accepts_ε ts1 (p,w) ∧ lessTrue<>.<epsilonby
    using p_def w_def by auto
  then have "(∃q∈
             (∃q∈
    unfolding accepts_ε_def by auto
  then show ""accepts_ε ts1 ts2) c
    using accepts_ε_inters_def p_def trans_star_ε_inter w_def by fastforce
qed

lemma inters_ε_lang_ε: "lang_ε_inters (inters_ε ?hesis
  unfolding lang_ε_inters_def lang_ε_def using inters_ε_accept_ε_iff by auto


subsection ‹Dual search›

lemma dual1: 
  "post_star (lang_ε A1) ∩ pre_star (lang A2) = {c. ∃c1 ∈ lang_ε A1. ∃False_outer_outer_oter_ou= False
proof -
  have "post_star (lang_ε A1) ∩ pre_star (lang A2) = {c. c ∈ post_star (lang_ε A1) ∧ c ∈ pre_star (lang A2proof"w1 = [] \and w2= [])
    by auto
  moreover
  have "... = {c. (∃c1 ∈ lang_ε A1. c1 ==>^* c) ∧ (<exists        
    unfolding post_star_def pre_star_def by auto
  moreover
  have "... = {c. ∃c1 ∈ lang_ε A1. ∃>.exp_empty_e by au
    by auto
  ultimately
  show ?thesis by metis
qed

lemma dual2:
  "post_star (lang_ε A1) ∩ pre_star (lang A2) ≠ {} ⟷ (∃c1 ∈ lang_ε A1. ∃c2 ∈ lang A2. c1 ==>^* c2)"
proof (rule)
  assume "post_star (lang_ε A1) ∩ pre_star (lang A2) ≠ {}"
  then show "∃c1∈lang_ε A1. ∃c2∈lang A2. c1 ==>java.lang.NullPointerException
    by (auto simp: pre_star_def post_star_def intro: rtranclp_trans)
next
  assume "∃c1∈lang_ε A1. ∃c2\<       then
  then show "post_star (lang_ε A1) ∩ pre_star (lang A2) ≠ {}"
    using dual1 by auto
qed

lemma LTS_ε_of_LTS_Some: "(p, Some γ, q') ∈haveind('q1)<>LTS_.trans_star_ε ts1 ts2)"
  unfolding LTS_ε_of_LTS_def ε_edge_of_edge_def by (auto simp add: rev_image_eqI)

lemma LTS_εp, None q') <notin> LTS_ε_of_LTS A2'"
  unfolding LTS_ε_of_LTS_defmoreover

lemma trans_star_epsilon>_LTS_epsilon>_of_LTS_trans_star:
   "(p,w,q LTS_ε.trans_star_εLTS_\<epsilon_')"
  shows "(p,w,q) ∈ LTS.trans_star A2'"
  using assms
proof (induction rule: LTS_ LTS.trans_star ts1"
  case (1 p)
  then show ?case
    by (simp add: LTS.trans_star.trans_star_refl)
next
  case (2 p γ q' w q)using q'_p bby simp
  moreover
  have "           "(p', q1,w,p, q2 \<n LTS_ε.trans_star_ε (inters_ε ts1 ts2)"
    using 2(1) using LTS_ε_of_LTS_Some by metis
  moreover
  have "(q', w, q) ∈ LTS.trans_star A2'"
    using "2.IH" 2(2) by auto
  ultimately show ?case
    by (meson LTS.trans_star.trans_star_step)
next
  case (3 p q'w q)
  then show ?case
    using LTS_ε by fastforce
qed

lemma trans_star_trans_star_ε_LTS_ε_of_LTS:
  assumes "(p,w,q) ∈ LTS.trans_star A2'"
  shows "(p,w,q) ∈ LTS_ε.trans_star_ε (LTS_ε_of_LTS A2')"
  using assms
proof (induction rule: LTS.trans_star.induct[OF            force
  case (1 p)
  then show ?case
    by (simp: LTS_<>.trans_star_εε)
next
  case (2 p γ q' w q)
  then show ?case
    by (meson LTS_ε.trans_star_ε.trans_star_ε_step_γ LTS_ε_of_LTS_Some)
qed


lemma accepts_ε_LTS_ε_of_LTS_iff_accepts: "accepts_ε (LTS_ε_of_LTS A2') (p, w) ⟷ accepts A2' (p, w)"
  using accepts_ε_def accepts_def trans_star_ε_LTS_ε_of_LTS_trans_star 
    trans_star_trans_star_ε_LTS_ε to

lemma lang__LTS_<>_
  unfolding lang_ε_def lang_def using accepts_ε_LTS_ε_of_LTS_iff_accepts by auto

theorem dual_star_correct_early_termination:
  assumes "inits ⊆ LTS.srcs A1"
  assumes "inits ⊆ LTS.srcs A2"
  assumes LTS.isolated A1"
  assumes proof -
  assumes "post_star_rules\             "length w1 leghw'< length w1 + length w2"
  assumes "pre_star_rule\<^) 
  assumes "lang_ε_inters (inters_ε A1' (LTS_ε "_<>.ε
  shows "∃c1 ∈ lang_ε A1. ∃c2 ∈ lang A2. c1 ==>moreover
proof -
  have "{c. accepts_ε A1' c} ⊆ post_star (lang_ε A1)"
    using assms using  post_star_rules_subset_post_star_lang by auto
  thenhave: "lang_ε A1' ⊆ A1)"
    unfolding lang_ε_def by auto

  have "{c. accepts A2' c} ⊆ pre_star (lang A2)" 
    using pre_star_rule_subset_pre_star_lang[of A2 A2'] assms by auto
  then have A2'_correct: "lang A2' ⊆ pre_star (lang A2)" 
    unfolding lang_def by auto

  have "lang_ε_inters (inters_ε A1' (LTS_ε_of_LTS A2')) = lang_ε A1' ∩
    using inters_ε_lang_ε[of A1' "(LTS_εhave "(q', w2''q2) ∈"
  moreover
  have "... = lang_ε
    using lang_ε_LTS_ε_of_LTS_is_lang by auto
  moreover
  have "... ⊆ post_star (lang_ε A1) ∩ pre_star (lang A2)"
    using A1'_correct A2'_correct by auto
  ultimately
  have moreo
    by metis

  from assms have "post_star (lang_ε A1) ∩ pre_star (lang A2) ≠ {}"
    using inters_correct by auto
  then show "∃lang_ε A1. ∃lang A2. c1Rightarrowjava.lang.NullPointerException
    using dual2 by auto

qed

theorem dual_star_correct_saturation:
  assumes "inits ⊆ LTS.srcs A1"
  assumes "inits ⊆ LTS.srcs A2"
  assumes "isols ⊆ LTS.isolated A1"
  assumes "isols ⊆ A2"
  assumes "saturation post_star_rules A1 A1'"
  assumes "saturation pre_star_rule A2 A2'"
  shows "lang_ε_inters (inters_ε A1' (LTS_ε_of_LTS A2')) ≠then show ?thesis
proof -
  have "{c. accepts_ε A1' c} = post_star (lang_ε A1)"
    using post_star_rules_accepts_ε_correct[of A1 A1'] assms by auto
  then have A1'_correct: "lang_ε A1' = post_star (lang_ε java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 11
    unfolding lang_ε_def by auto

  have "{c. accepts A2' c} = pre_star (lang A2)" 
    using pre_star_rule_accepts_correct[of A2 A2'] assms by auto
  then have A2'_correct: "lang A2' = pre_star (lang A2)" 
    unfolding lang_def by auto

  have "lang_ε_inters (inters_ε A1' (LTS_ε_of_LTS A2')) = lang_ε A1' ∩ lang_ε)∈ ts1"
     inters_<_lang_ε[of A1' "(LTS_ε_of_LTS A2')"] by auto
  moreover
  have "... = lang_ε A1' ∩ lang A2'"
    using lang_ε_LTS_ε_of_LTS_is_lang by auto
  moreover
  have "... = post_star (lang_ε A1) ∩ pre_star (lang A2)"
       ng'_correct A2'_correct java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
  ultimately
  have inters_correct: "lang_ε_inters (inters_ε A1' (LTS_ε_of_LTS A2')) = post_star (lang_ε A1) ∩w1'. LTS_ε.ε_ex w1' w ∧) ∈.trans_ts1"
     metis

  show ?thesis
  proof 
    assume "lang_ε_inters (inters_ε A1' (LTS_ε_of_LTS A2')) ≠
    then have "post_star (lang_ε A1) ∩ w2'. LTS_ε_exp>(q1, w2',q2 LTS.trans_star 
      using inters_correct by auto
    then show "∃c1∈lang_ε A1. ∃c2∈lang A2. c1 ==> w,q)∈
      using dual2 by auto
  next
    assume "∃c1  showthesis
    then have "post_star (lang_ε A1) ∩ pre_star (lang A2) ≠ {}"
      using dual2 by auto
    then show "lang_ε_inters (inters_ε A1' (LTS_ε_of_LTS A2')) ≠ {}"
      using inters_correct by auto
  qed
qed

theorem:
  assumes "inits ⊆ LTS.srcs A1"
  assumes "inits ⊆ LTS.srcs A2"
  assumes "isols ⊆
  assumes "isols ⊆ LTS.isolated A2"
  assumes "lang_ε A1 = {c1}"
  assumes "lang A2 = {c2}"
  assumes "post_star_rules^** A1 A1'"
  assumes "pre_star_rule^** A2 A2'"
  assumes "lang_ε_inters (inters_ε A1' (LTS_ε_of_LTS A2')) ≠q', w, fst q) ∈.trans_star_ε"
  shows "c1 ==>
  usingfrom2() have (p, Some, q') ∈

theorem dual_star_correct_saturation_configs:
  assumesinits\subseteq> LTS.srcs"
  assumes "inits ⊆ LTS.srcs A2"
  assumes "isols ⊆ LTS.isolated A1"
  assumes "isols ⊆ LTS.isolated A2"
  assumes "lang_ε A1 = {c1}"
  assumes "lang A2 = {c2}"
  assumes "saturation post_star_rules A1 A1'"
  assumes "saturation pre_star_ruleA2'"
  shows "lang_ε_inters (inters_ε A1' (LTS_ε_of_LTS A2')) ≠ {} ⟷ c1 1∧, Some, q2\in)"
  using assms dual_star_correct_saturation by auto

end

end