theory Correctness
imports PIPBasics
begin

text {* 
  The following two auxiliary lemmas are used to reason about @{term Max}.
*}
lemma image_Max_eqI: 
  assumes "finite B"
  and "b \<in> B"
  and "\<forall> x \<in> B. f x \<le> f b"
  shows "Max (f ` B) = f b"
  using assms
  using Max_eqI by blast 

lemma image_Max_subset:
  assumes "finite A"
  and "B \<subseteq> A"
  and "a \<in> B"
  and "Max (f ` A) = f a"
  shows "Max (f ` B) = f a"
proof(rule image_Max_eqI)
  show "finite B"
    using assms(1) assms(2) finite_subset by auto 
next
  show "a \<in> B" using assms by simp
next
  show "\<forall>x\<in>B. f x \<le> f a"
    by (metis Max_ge assms(1) assms(2) assms(4) 
            finite_imageI image_eqI subsetCE) 
qed

text {*
  The following locale @{text "highest_gen"} sets the basic context for our
  investigation: supposing thread @{text th} holds the highest @{term cp}-value
  in state @{text s}, which means the task for @{text th} is the 
  most urgent. We want to show that  
  @{text th} is treated correctly by PIP, which means
  @{text th} will not be blocked unreasonably by other less urgent
  threads. 
*}
locale highest_gen =
  fixes s th prio tm
  assumes vt_s: "vt s"
  and threads_s: "th \<in> threads s"
  and highest: "preced th s = Max ((cp s)`threads s)"
  -- {* The internal structure of @{term th}'s precedence is exposed:*}
  and preced_th: "preced th s = Prc prio tm" 

-- {* @{term s} is a valid trace, so it will inherit all results derived for
      a valid trace: *}
sublocale highest_gen < vat_s: valid_trace "s"
  by (unfold_locales, insert vt_s, simp)

context highest_gen
begin

text {*
  @{term tm} is the time when the precedence of @{term th} is set, so 
  @{term tm} must be a valid moment index into @{term s}.
*}
lemma lt_tm: "tm < length s"
  by (insert preced_tm_lt[OF threads_s preced_th], simp)

text {*
  Since @{term th} holds the highest precedence and @{text "cp"}
  is the highest precedence of all threads in the sub-tree of 
  @{text "th"} and @{text th} is among these threads, 
  its @{term cp} must equal to its precedence:
*}
lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R")
proof -
  have "?L \<le> ?R"
  by (unfold highest, rule Max_ge, 
        auto simp:threads_s finite_threads)
  moreover have "?R \<le> ?L"
    by (unfold vat_s.cp_rec, rule Max_ge, 
        auto simp:the_preced_def vat_s.fsbttRAGs.finite_children)
  ultimately show ?thesis by auto
qed

lemma highest_cp_preced: "cp s th = Max (the_preced s ` threads s)"
  using eq_cp_s_th highest max_cp_eq the_preced_def by presburger
  

lemma highest_preced_thread: "preced th s = Max (the_preced s ` threads s)"
  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)

lemma highest': "cp s th = Max (cp s ` threads s)"
  by (simp add: eq_cp_s_th highest)

end

locale extend_highest_gen = highest_gen + 
  fixes t 
  assumes vt_t: "vt (t@s)"
  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"

sublocale extend_highest_gen < vat_t: valid_trace "t@s"
  by (unfold_locales, insert vt_t, simp)

lemma step_back_vt_app: 
  assumes vt_ts: "vt (t@s)" 
  shows "vt s"
proof -
  from vt_ts show ?thesis
  proof(induct t)
    case Nil
    from Nil show ?case by auto
  next
    case (Cons e t)
    assume ih: " vt (t @ s) \<Longrightarrow> vt s"
      and vt_et: "vt ((e # t) @ s)"
    show ?case
    proof(rule ih)
      show "vt (t @ s)"
      proof(rule step_back_vt)
        from vt_et show "vt (e # t @ s)" by simp
      qed
    qed
  qed
qed

(* locale red_extend_highest_gen = extend_highest_gen +
   fixes i::nat
*)

(*
sublocale red_extend_highest_gen <   red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)"
  apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
  apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
  by (unfold highest_gen_def, auto dest:step_back_vt_app)
*)

context extend_highest_gen
begin

 lemma ind [consumes 0, case_names Nil Cons, induct type]:
  assumes 
    h0: "R []"
  and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e; 
                    extend_highest_gen s th prio tm t; 
                    extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
  shows "R t"
proof -
  from vt_t extend_highest_gen_axioms show ?thesis
  proof(induct t)
    from h0 show "R []" .
  next
    case (Cons e t')
    assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
      and vt_e: "vt ((e # t') @ s)"
      and et: "extend_highest_gen s th prio tm (e # t')"
    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
    from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto
    show ?case
    proof(rule h2 [OF vt_ts stp _ _ _ ])
      show "R t'"
      proof(rule ih)
        from et show ext': "extend_highest_gen s th prio tm t'"
          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
      next
        from vt_ts show "vt (t' @ s)" .
      qed
    next
      from et show "extend_highest_gen s th prio tm (e # t')" .
    next
      from et show ext': "extend_highest_gen s th prio tm t'"
          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
    qed
  qed
qed


lemma th_kept: "th \<in> threads (t @ s) \<and> 
                 preced th (t@s) = preced th s" (is "?Q t") 
proof -
  show ?thesis
  proof(induct rule:ind)
    case Nil
    from threads_s
    show ?case
      by auto
  next
    case (Cons e t)
    interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
    interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
    show ?case
    proof(cases e)
      case (Create thread prio)
      show ?thesis
      proof -
        from Cons and Create have "step (t@s) (Create thread prio)" by auto
        hence "th \<noteq> thread"
        proof(cases)
          case thread_create
          with Cons show ?thesis by auto
        qed
        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
          by (unfold Create, auto simp:preced_def)
        moreover note Cons
        ultimately show ?thesis
          by (auto simp:Create)
      qed
    next
      case (Exit thread)
      from h_e.exit_diff and Exit
      have neq_th: "thread \<noteq> th" by auto
      with Cons
      show ?thesis
        by (unfold Exit, auto simp:preced_def)
    next
      case (P thread cs)
      with Cons
      show ?thesis 
        by (auto simp:P preced_def)
    next
      case (V thread cs)
      with Cons
      show ?thesis 
        by (auto simp:V preced_def)
    next
      case (Set thread prio')
      show ?thesis
      proof -
        from h_e.set_diff_low and Set
        have "th \<noteq> thread" by auto
        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
          by (unfold Set, auto simp:preced_def)
        moreover note Cons
        ultimately show ?thesis
          by (auto simp:Set)
      qed
    qed
  qed
qed

text {*
  According to @{thm th_kept}, thread @{text "th"} has its living status
  and precedence kept along the way of @{text "t"}. The following lemma
  shows that this preserved precedence of @{text "th"} remains as the highest
  along the way of @{text "t"}.

  The proof goes by induction over @{text "t"} using the specialized
  induction rule @{thm ind}, followed by case analysis of each possible 
  operations of PIP. All cases follow the same pattern rendered by the 
  generalized introduction rule @{thm "image_Max_eqI"}. 

  The very essence is to show that precedences, no matter whether they 
  are newly introduced or modified, are always lower than the one held 
  by @{term "th"}, which by @{thm th_kept} is preserved along the way.
*}
lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s"
proof(induct rule:ind)
  case Nil
  from highest_preced_thread
  show ?case by simp
next
  case (Cons e t)
    interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
    interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
  show ?case
  proof(cases e)
    case (Create thread prio')
    show ?thesis (is "Max (?f ` ?A) = ?t")
    proof -
      -- {* The following is the common pattern of each branch of the case analysis. *}
      -- {* The major part is to show that @{text "th"} holds the highest precedence: *}
      have "Max (?f ` ?A) = ?f th"
      proof(rule image_Max_eqI)
        show "finite ?A" using h_e.finite_threads by auto 
      next
        show "th \<in> ?A" using h_e.th_kept by auto 
      next
        show "\<forall>x\<in>?A. ?f x \<le> ?f th"
        proof 
          fix x
          assume "x \<in> ?A"
          hence "x = thread \<or> x \<in> threads (t@s)" by (auto simp:Create)
          thus "?f x \<le> ?f th"
          proof
            assume "x = thread"
            thus ?thesis 
              apply (simp add:Create the_preced_def preced_def, fold preced_def)
              using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 
              preced_th by force
          next
            assume h: "x \<in> threads (t @ s)"
            from Cons(2)[unfolded Create] 
            have "x \<noteq> thread" using h by (cases, auto)
            hence "?f x = the_preced (t@s) x" 
              by (simp add:Create the_preced_def preced_def)
            hence "?f x \<le> Max (the_preced (t@s) ` threads (t@s))"
              by (simp add: h_t.finite_threads h)
            also have "... = ?f th"
              by (metis Cons.hyps(5) h_e.th_kept the_preced_def) 
            finally show ?thesis .
          qed
        qed
      qed
     -- {* The minor part is to show that the precedence of @{text "th"} 
           equals to preserved one, given by the foregoing lemma @{thm th_kept} *}
      also have "... = ?t" using h_e.th_kept the_preced_def by auto
      -- {* Then it follows trivially that the precedence preserved
            for @{term "th"} remains the maximum of all living threads along the way. *}
      finally show ?thesis .
    qed 
  next 
    case (Exit thread)
    show ?thesis (is "Max (?f ` ?A) = ?t")
    proof -
      have "Max (?f ` ?A) = ?f th"
      proof(rule image_Max_eqI)
        show "finite ?A" using h_e.finite_threads by auto 
      next
        show "th \<in> ?A" using h_e.th_kept by auto 
      next
        show "\<forall>x\<in>?A. ?f x \<le> ?f th"
        proof 
          fix x
          assume "x \<in> ?A"
          hence "x \<in> threads (t@s)" by (simp add: Exit) 
          hence "?f x \<le> Max (?f ` threads (t@s))" 
            by (simp add: h_t.finite_threads) 
          also have "... \<le> ?f th" 
            apply (simp add:Exit the_preced_def preced_def, fold preced_def)
            using Cons.hyps(5) h_t.th_kept the_preced_def by auto
          finally show "?f x \<le> ?f th" .
        qed
      qed
      also have "... = ?t" using h_e.th_kept the_preced_def by auto
      finally show ?thesis .
    qed 
  next
    case (P thread cs)
    with Cons
    show ?thesis by (auto simp:preced_def the_preced_def)
  next
    case (V thread cs)
    with Cons
    show ?thesis by (auto simp:preced_def the_preced_def)
  next 
    case (Set thread prio')
    show ?thesis (is "Max (?f ` ?A) = ?t")
    proof -
      have "Max (?f ` ?A) = ?f th"
      proof(rule image_Max_eqI)
        show "finite ?A" using h_e.finite_threads by auto 
      next
        show "th \<in> ?A" using h_e.th_kept by auto 
      next
        show "\<forall>x\<in>?A. ?f x \<le> ?f th"
        proof 
          fix x
          assume h: "x \<in> ?A"
          show "?f x \<le> ?f th"
          proof(cases "x = thread")
            case True
            moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th"
            proof -
              have "the_preced (t @ s) th = Prc prio tm"  
                using h_t.th_kept preced_th by (simp add:the_preced_def)
              moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto
              ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def)
            qed
            ultimately show ?thesis
              by (unfold Set, simp add:the_preced_def preced_def)
          next
            case False
            then have "?f x  = the_preced (t@s) x"
              by (simp add:the_preced_def preced_def Set)
            also have "... \<le> Max (the_preced (t@s) ` threads (t@s))"
              using Set h h_t.finite_threads by auto 
            also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def) 
            finally show ?thesis .
          qed
        qed
      qed
      also have "... = ?t" using h_e.th_kept the_preced_def by auto
      finally show ?thesis .
    qed 
  qed
qed

lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))"
  by (insert th_kept max_kept, auto)

text {*
  The reason behind the following lemma is that:
  Since @{term "cp"} is defined as the maximum precedence 
  of those threads contained in the sub-tree of node @{term "Th th"} 
  in @{term "RAG (t@s)"}, and all these threads are living threads, and 
  @{term "th"} is also among them, the maximum precedence of 
  them all must be the one for @{text "th"}.
*}
lemma th_cp_max_preced: 
  "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R") 
proof -
  let ?f = "the_preced (t@s)"
  have "?L = ?f th"
  proof(unfold cp_alt_def, rule image_Max_eqI)
    show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
    proof -
      have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} = 
            the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and>
                            (\<exists> th'. n = Th th')}"
      by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps)
      moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree) 
      ultimately show ?thesis by simp
    qed
  next
    show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
      by (auto simp:subtree_def)
  next
    show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}.
               the_preced (t @ s) x \<le> the_preced (t @ s) th"
    proof
      fix th'
      assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
      hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto
      moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}"
        by (meson subtree_Field)
      ultimately have "Th th' \<in> ..." by auto
      hence "th' \<in> threads (t@s)" 
      proof
        assume "Th th' \<in> {Th th}"
        thus ?thesis using th_kept by auto 
      next
        assume "Th th' \<in> Field (RAG (t @ s))"
        thus ?thesis using vat_t.not_in_thread_isolated by blast 
      qed
      thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th"
        by (metis Max_ge finite_imageI finite_threads image_eqI 
               max_kept th_kept the_preced_def)
    qed
  qed
  also have "... = ?R" by (simp add: max_preced the_preced_def) 
  finally show ?thesis .
qed

lemma th_cp_max[simp]: "Max (cp (t@s) ` threads (t@s)) = cp (t@s) th"
  using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger

lemma [simp]: "Max (cp (t@s) ` threads (t@s)) = Max (the_preced (t@s) ` threads (t@s))"
  by (simp add: th_cp_max_preced)
  
lemma [simp]: "Max (the_preced (t@s) ` threads (t@s)) = the_preced (t@s) th"
  using max_kept th_kept the_preced_def by auto

lemma [simp]: "the_preced (t@s) th = preced th (t@s)"
  using the_preced_def by auto

lemma [simp]: "preced th (t@s) = preced th s"
  by (simp add: th_kept)

lemma [simp]: "cp s th = preced th s"
  by (simp add: eq_cp_s_th)

lemma th_cp_preced [simp]: "cp (t@s) th = preced th s"
  by (fold max_kept, unfold th_cp_max_preced, simp)

lemma preced_less:
  assumes th'_in: "th' \<in> threads s"
  and neq_th': "th' \<noteq> th"
  shows "preced th' s < preced th s"
  using assms
by (metis Max.coboundedI finite_imageI highest not_le order.trans 
    preced_linorder rev_image_eqI threads_s vat_s.finite_threads 
    vat_s.le_cp)

section {* The `blocking thread` *}

text {* 
  The purpose of PIP is to ensure that the most 
  urgent thread @{term th} is not blocked unreasonably. 
  Therefore, a clear picture of the blocking thread is essential 
  to assure people that the purpose is fulfilled. 
  
  In this section, we are going to derive a series of lemmas 
  with finally give rise to a picture of the blocking thread. 

  By `blocking thread`, we mean a thread in running state but 
  different from thread @{term th}.
*}

text {*
  The following lemmas shows that the @{term cp}-value 
  of the blocking thread @{text th'} equals to the highest
  precedence in the whole system.
*}
lemma runing_preced_inversion:
  assumes runing': "th' \<in> runing (t@s)"
  shows "cp (t@s) th' = preced th s" (is "?L = ?R")
proof -
  have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms
      by (unfold runing_def, auto)
  also have "\<dots> = ?R"
      by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) 
  finally show ?thesis .
qed

text {*

  The following lemma shows how the counters for @{term "P"} and
  @{term "V"} operations relate to the running threads in the states
  @{term s} and @{term "t @ s"}.  The lemma shows that if a thread's
  @{term "P"}-count equals its @{term "V"}-count (which means it no
  longer has any resource in its possession), it cannot be a running
  thread.

  The proof is by contraction with the assumption @{text "th' \<noteq> th"}.
  The key is the use of @{thm eq_pv_dependants} to derive the
  emptiness of @{text th'}s @{term dependants}-set from the balance of
  its @{term P} and @{term V} counts.  From this, it can be shown
  @{text th'}s @{term cp}-value equals to its own precedence.

  On the other hand, since @{text th'} is running, by @{thm
  runing_preced_inversion}, its @{term cp}-value equals to the
  precedence of @{term th}.

  Combining the above two resukts we have that @{text th'} and @{term
  th} have the same precedence. By uniqueness of precedences, we have
  @{text "th' = th"}, which is in contradiction with the assumption
  @{text "th' \<noteq> th"}.

*} 
                      
lemma eq_pv_blocked: (* ddd *)
  assumes neq_th': "th' \<noteq> th"
  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
  shows "th' \<notin> runing (t@s)"
proof
  assume otherwise: "th' \<in> runing (t@s)"
  show False
  proof -
    have th'_in: "th' \<in> threads (t@s)"
        using otherwise readys_threads runing_def by auto 
    have "th' = th"
    proof(rule preced_unique)
      -- {* The proof goes like this: 
            it is first shown that the @{term preced}-value of @{term th'} 
            equals to that of @{term th}, then by uniqueness 
            of @{term preced}-values (given by lemma @{thm preced_unique}), 
            @{term th'} equals to @{term th}: *}
      show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
      proof -
        -- {* Since the counts of @{term th'} are balanced, the subtree
              of it contains only itself, so, its @{term cp}-value
              equals its @{term preced}-value: *}
        have "?L = cp (t@s) th'"
         by (unfold cp_eq_cpreced cpreced_def eq_dependants vat_t.eq_pv_dependants[OF eq_pv], simp)
        -- {* Since @{term "th'"} is running, by @{thm runing_preced_inversion},
              its @{term cp}-value equals @{term "preced th s"}, 
              which equals to @{term "?R"} by simplification: *}
        also have "... = ?R" 
        thm runing_preced_inversion
            using runing_preced_inversion[OF otherwise] by simp
        finally show ?thesis .
      qed
    qed (auto simp: th'_in th_kept)
    with `th' \<noteq> th` show ?thesis by simp
 qed
qed

text {*
  The following lemma is the extrapolation of @{thm eq_pv_blocked}.
  It says if a thread, different from @{term th}, 
  does not hold any resource at the very beginning,
  it will keep hand-emptied in the future @{term "t@s"}.
*}
lemma eq_pv_persist: (* ddd *)
  assumes neq_th': "th' \<noteq> th"
  and eq_pv: "cntP s th' = cntV s th'"
  shows "cntP (t@s) th' = cntV (t@s) th'"
proof(induction rule:ind) -- {* The proof goes by induction. *}
  -- {* The nontrivial case is for the @{term Cons}: *}
  case (Cons e t)
  -- {* All results derived so far hold for both @{term s} and @{term "t@s"}: *}
  interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp
  interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp
  interpret vat_es: valid_trace_e "t@s" e using Cons(1,2) by (unfold_locales, auto)
  show ?case
  proof -
    -- {* It can be proved that @{term cntP}-value of @{term th'} does not change
          by the happening of event @{term e}: *}
    have "cntP ((e#t)@s) th' = cntP (t@s) th'"
    proof(rule ccontr) -- {* Proof by contradiction. *}
      -- {* Suppose @{term cntP}-value of @{term th'} is changed by @{term e}: *}
      assume otherwise: "cntP ((e # t) @ s) th' \<noteq> cntP (t @ s) th'"
      -- {* Then the actor of @{term e} must be @{term th'} and @{term e}
            must be a @{term P}-event: *}
      hence "isP e" "actor e = th'" by (auto simp:cntP_diff_inv) 
      with vat_es.actor_inv
      -- {* According to @{thm vat_es.actor_inv}, @{term th'} must be running at 
            the moment @{term "t@s"}: *}
      have "th' \<in> runing (t@s)" by (cases e, auto)
      -- {* However, an application of @{thm eq_pv_blocked} to induction hypothesis
            shows @{term th'} can not be running at moment  @{term "t@s"}: *}
      moreover have "th' \<notin> runing (t@s)" 
               using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .
      -- {* Contradiction is finally derived: *}
      ultimately show False by simp
    qed
    -- {* It can also be proved that @{term cntV}-value of @{term th'} does not change
          by the happening of event @{term e}: *}
    -- {* The proof follows exactly the same pattern as the case for @{term cntP}-value: *}
    moreover have "cntV ((e#t)@s) th' = cntV (t@s) th'"
    proof(rule ccontr) -- {* Proof by contradiction. *}
      assume otherwise: "cntV ((e # t) @ s) th' \<noteq> cntV (t @ s) th'"
      hence "isV e" "actor e = th'" by (auto simp:cntV_diff_inv) 
      with vat_es.actor_inv
      have "th' \<in> runing (t@s)" by (cases e, auto)
      moreover have "th' \<notin> runing (t@s)"
          using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .
      ultimately show False by simp
    qed
    -- {* Finally, it can be shown that the @{term cntP} and @{term cntV} 
          value for @{term th'} are still in balance, so @{term th'} 
          is still hand-emptied after the execution of event @{term e}: *}
    ultimately show ?thesis using Cons(5) by metis
  qed
qed (auto simp:eq_pv)

text {*
  By combining @{thm  eq_pv_blocked} and @{thm eq_pv_persist},
  it can be derived easily that @{term th'} can not be running in the future:
*}
lemma eq_pv_blocked_persist:
  assumes neq_th': "th' \<noteq> th"
  and eq_pv: "cntP s th' = cntV s th'"
  shows "th' \<notin> runing (t@s)"
  using assms
  by (simp add: eq_pv_blocked eq_pv_persist) 

text {*
  The following lemma shows the blocking thread @{term th'}
  must hold some resource in the very beginning. 
*}
lemma runing_cntP_cntV_inv: (* ddd *)
  assumes is_runing: "th' \<in> runing (t@s)"
  and neq_th': "th' \<noteq> th"
  shows "cntP s th' > cntV s th'"
  using assms
proof -
  -- {* First, it can be shown that the number of @{term P} and
        @{term V} operations can not be equal for thred @{term th'} *}
  have "cntP s th' \<noteq> cntV s th'"
  proof
     -- {* The proof goes by contradiction, suppose otherwise: *}
    assume otherwise: "cntP s th' = cntV s th'"
    -- {* By applying @{thm  eq_pv_blocked_persist} to this: *}
    from eq_pv_blocked_persist[OF neq_th' otherwise] 
    -- {* we have that @{term th'} can not be running at moment @{term "t@s"}: *}
    have "th' \<notin> runing (t@s)" .
    -- {* This is obvious in contradiction with assumption @{thm is_runing}  *}
    thus False using is_runing by simp
  qed
  -- {* However, the number of @{term V} is always less or equal to @{term P}: *}
  moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
  -- {* Thesis is finally derived by combining the these two results: *}
  ultimately show ?thesis by auto
qed


text {*
  The following lemmas shows the blocking thread @{text th'} must be live 
  at the very beginning, i.e. the moment (or state) @{term s}. 

  The proof is a  simple combination of the results above:
*}
lemma runing_threads_inv: 
  assumes runing': "th' \<in> runing (t@s)"
  and neq_th': "th' \<noteq> th"
  shows "th' \<in> threads s"
proof(rule ccontr) -- {* Proof by contradiction: *}
  assume otherwise: "th' \<notin> threads s" 
  have "th' \<notin> runing (t @ s)"
  proof -
    from vat_s.cnp_cnv_eq[OF otherwise]
    have "cntP s th' = cntV s th'" .
    from eq_pv_blocked_persist[OF neq_th' this]
    show ?thesis .
  qed
  with runing' show False by simp
qed

text {*
  The following lemma summarizes several foregoing 
  lemmas to give an overall picture of the blocking thread @{text "th'"}:
*}
lemma runing_inversion: (* ddd, one of the main lemmas to present *)
  assumes runing': "th' \<in> runing (t@s)"
  and neq_th: "th' \<noteq> th"
  shows "th' \<in> threads s"
  and    "\<not>detached s th'"
  and    "cp (t@s) th' = preced th s"
proof -
  from runing_threads_inv[OF assms]
  show "th' \<in> threads s" .
next
  from runing_cntP_cntV_inv[OF runing' neq_th]
  show "\<not>detached s th'" using vat_s.detached_eq by simp
next
  from runing_preced_inversion[OF runing']
  show "cp (t@s) th' = preced th s" .
qed

section {* The existence of `blocking thread` *}

text {* 
  Suppose @{term th} is not running, it is first shown that
  there is a path in RAG leading from node @{term th} to another thread @{text "th'"} 
  in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}).

  Now, since @{term readys}-set is non-empty, there must be
  one in it which holds the highest @{term cp}-value, which, by definition, 
  is the @{term runing}-thread. However, we are going to show more: this running thread
  is exactly @{term "th'"}.
     *}
lemma th_blockedE: (* ddd, the other main lemma to be presented: *)
  assumes "th \<notin> runing (t@s)"
  obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
                    "th' \<in> runing (t@s)"
proof -
  -- {* According to @{thm vat_t.th_chain_to_ready}, either 
        @{term "th"} is in @{term "readys"} or there is path leading from it to 
        one thread in @{term "readys"}. *}
  have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)" 
    using th_kept vat_t.th_chain_to_ready by auto
  -- {* However, @{term th} can not be in @{term readys}, because otherwise, since 
       @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *}
  moreover have "th \<notin> readys (t@s)" 
    using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto 
  -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in 
        term @{term readys}: *}
  ultimately obtain th' where th'_in: "th' \<in> readys (t@s)"
                          and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto
  -- {* We are going to show that this @{term th'} is running. *}
  have "th' \<in> runing (t@s)"
  proof -
    -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *}
    have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R")
    proof -
      have "?L =  Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))"
        by (unfold cp_alt_def1, simp)
      also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)"
      proof(rule image_Max_subset)
        show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads)
      next
        show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)"
          by (metis Range.intros dp trancl_range vat_t.rg_RAG_threads vat_t.subtree_tRAG_thread) 
      next
        show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp
                    by (unfold tRAG_subtree_eq, auto simp:subtree_def)
      next
        show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) =
                      (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _")
        proof -
          have "?L = the_preced (t @ s) `  threads (t @ s)" 
                     by (unfold image_comp, rule image_cong, auto)
          thus ?thesis using max_preced the_preced_def by auto
        qed
      qed
      also have "... = ?R"
        using th_cp_max th_cp_preced th_kept 
              the_preced_def vat_t.max_cp_readys_threads by auto
      finally show ?thesis .
    qed 
    -- {* Now, since @{term th'} holds the highest @{term cp} 
          and we have already show it is in @{term readys},
          it is @{term runing} by definition. *}
    with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def) 
  qed
  -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *}
  moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)" 
    using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def)
  ultimately show ?thesis using that by metis
qed

text {*
  Now it is easy to see there is always a thread to run by case analysis
  on whether thread @{term th} is running: if the answer is Yes, the 
  the running thread is obviously @{term th} itself; otherwise, the running
  thread is the @{text th'} given by lemma @{thm th_blockedE}.
*}
lemma live: "runing (t@s) \<noteq> {}"
proof(cases "th \<in> runing (t@s)") 
  case True thus ?thesis by auto
next
  case False
  thus ?thesis using th_blockedE by auto
qed

end
end