definition "wRAG (s::state) = {(Th th, Cs cs) | th cs. waiting s th cs}"

definition "hRAG (s::state) =  {(Cs cs, Th th) | th cs. holding s th cs}"

definition "tRAG s = wRAG s O hRAG s"

lemma RAG_split: "RAG s = (wRAG s \<union> hRAG s)"

  by (unfold s_RAG_abv wRAG_def hRAG_def s_waiting_abv 

             s_holding_abv cs_RAG_def, auto)

lemma tRAG_alt_def: 

  "tRAG s = {(Th th1, Th th2) | th1 th2. 

                  \<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"

 by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)

lemma tRAG_mono:

  assumes "RAG s' \<subseteq> RAG s"

  shows "tRAG s' \<subseteq> tRAG s"

  using assms 

  by (unfold tRAG_alt_def, auto)

lemma holding_next_thI:

  assumes "holding s th cs"

  and "length (wq s cs) > 1"

  obtains th' where "next_th s th cs th'"

proof -

  from assms(1)[folded eq_holding, unfolded cs_holding_def]

  have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)" .

  then obtain rest where h1: "wq s cs = th#rest" 

    by (cases "wq s cs", auto)

  with assms(2) have h2: "rest \<noteq> []" by auto

  let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"

  have "next_th s th cs ?th'" using  h1(1) h2 

    by (unfold next_th_def, auto)

  from that[OF this] show ?thesis .

lemma RAG_tRAG_transfer:

  assumes "vt s'"

  assumes "RAG s = RAG s' \<union> {(Th th, Cs cs)}"

  and "(Cs cs, Th th'') \<in> RAG s'"

  shows "tRAG s = tRAG s' \<union> {(Th th, Th th'')}" (is "?L = ?R")

proof -

  interpret rtree: rtree "RAG s'"

  proof

  show "single_valued (RAG s')"

  apply (intro_locales)

    by (unfold single_valued_def, 

        auto intro:unique_RAG[OF assms(1)])

  show "acyclic (RAG s')"

     by (rule acyclic_RAG[OF assms(1)])

  qed

  { fix n1 n2

    assume "(n1, n2) \<in> ?L"

    from this[unfolded tRAG_alt_def]

    obtain th1 th2 cs' where 

      h: "n1 = Th th1" "n2 = Th th2" 

         "(Th th1, Cs cs') \<in> RAG s"

         "(Cs cs', Th th2) \<in> RAG s" by auto

    from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \<in> RAG s'" by auto

    from h(3) and assms(2) 

    have "(Th th1, Cs cs') = (Th th, Cs cs) \<or> 

          (Th th1, Cs cs') \<in> RAG s'" by auto

    hence "(n1, n2) \<in> ?R"

    proof

      assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"

      hence eq_th1: "th1 = th" by simp

      moreover have "th2 = th''"

      proof -

        from h1 have "cs' = cs" by simp

        from assms(3) cs_in[unfolded this] rtree.sgv

        show ?thesis

          by (unfold single_valued_def, auto)

      qed

      ultimately show ?thesis using h(1,2) by auto

    next

      assume "(Th th1, Cs cs') \<in> RAG s'"

      with cs_in have "(Th th1, Th th2) \<in> tRAG s'"

        by (unfold tRAG_alt_def, auto)

      from this[folded h(1, 2)] show ?thesis by auto

    qed

  } moreover {

    fix n1 n2

    assume "(n1, n2) \<in> ?R"

    hence "(n1, n2) \<in>tRAG s' \<or> (n1, n2) = (Th th, Th th'')" by auto

    hence "(n1, n2) \<in> ?L" 

    proof

      assume "(n1, n2) \<in> tRAG s'"

      moreover have "... \<subseteq> ?L"

      proof(rule tRAG_mono)

        show "RAG s' \<subseteq> RAG s" by (unfold assms(2), auto)

      qed

      ultimately show ?thesis by auto

    next

      assume eq_n: "(n1, n2) = (Th th, Th th'')"

      from assms(2, 3) have "(Cs cs, Th th'') \<in> RAG s" by auto

      moreover have "(Th th, Cs cs) \<in> RAG s" using assms(2) by auto

      ultimately show ?thesis 

        by (unfold eq_n tRAG_alt_def, auto)

    qed

  } ultimately show ?thesis by auto

qed

lemma readys_root:

  assumes "vt s"

  and "th \<in> readys s"

  shows "root (RAG s) (Th th)"

proof -

  { fix x

    assume "x \<in> ancestors (RAG s) (Th th)"

    hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)

    from tranclD[OF this]

    obtain z where "(Th th, z) \<in> RAG s" by auto

    with assms(2) have False

         apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)

         by (fold wq_def, blast)

  } thus ?thesis by (unfold root_def, auto)

qed

lemma readys_in_no_subtree:

  assumes "vt s"

  and "th \<in> readys s"

  and "th' \<noteq> th"

  shows "Th th \<notin> subtree (RAG s) (Th th')" 

proof

   assume "Th th \<in> subtree (RAG s) (Th th')"

   thus False

   proof(cases rule:subtreeE)

      case 1

      with assms show ?thesis by auto

   next

      case 2

      with readys_root[OF assms(1,2)]

      show ?thesis by (auto simp:root_def)

   qed

qed

lemma image_id:

  assumes "\<And> x. x \<in> A \<Longrightarrow> f x = x"

  shows "f ` A = A"

  using assms by (auto simp:image_def)

fun the_thread :: "node \<Rightarrow> thread" where

   "the_thread (Th th) = th"

definition "cp_gen s x =

                  Max ((the_preced s \<circ> the_thread) ` subtree (tRAG s) x)"

lemma cp_gen_alt_def:

  "cp_gen s = (Max \<circ> (\<lambda>x. (the_preced s \<circ> the_thread) ` subtree (tRAG s) x))"

    by (auto simp:cp_gen_def)

lemma tRAG_nodeE:

  assumes "(n1, n2) \<in> tRAG s"

  obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"

  using assms

  by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def)

lemma subtree_nodeE:

  assumes "n \<in> subtree (tRAG s) (Th th)"

  obtains th1 where "n = Th th1"

proof -

  show ?thesis

  proof(rule subtreeE[OF assms])

    assume "n = Th th"

    from that[OF this] show ?thesis .

  next

    assume "Th th \<in> ancestors (tRAG s) n"

    hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)

    hence "\<exists> th1. n = Th th1"

    proof(induct)

      case (base y)

      from tRAG_nodeE[OF this] show ?case by metis

    next

      case (step y z)

      thus ?case by auto

    qed

    with that show ?thesis by auto

  qed

qed

lemma threads_set_eq: 

   "the_thread ` (subtree (tRAG s) (Th th)) = 

                  {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")

proof -

  { fix th'

    assume "th' \<in> ?L"

    then obtain n where h: "th' = the_thread n" "n \<in>  subtree (tRAG s) (Th th)" by auto

    from subtree_nodeE[OF this(2)]

    obtain th1 where "n = Th th1" by auto

    with h have "Th th' \<in>  subtree (tRAG s) (Th th)" by auto

    hence "Th th' \<in>  subtree (RAG s) (Th th)"

    proof(cases rule:subtreeE[consumes 1])

      case 1

      thus ?thesis by (auto simp:subtree_def)

    next

      case 2

      hence "(Th th', Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)

      thus ?thesis

      proof(induct)

        case (step y z)

        from this(2)[unfolded tRAG_alt_def]

        obtain u where 

         h: "(y, u) \<in> RAG s" "(u, z) \<in> RAG s"

          by auto

        hence "y \<in> subtree (RAG s) z" by (auto simp:subtree_def)

        with step(3)

        show ?case by (auto simp:subtree_def)

      next

        case (base y)

        from this[unfolded tRAG_alt_def]

        show ?case by (auto simp:subtree_def)

      qed

    qed

    hence "th' \<in> ?R" by auto

  } moreover {

    fix th'

    assume "th' \<in> ?R"

    hence "(Th th', Th th) \<in> (RAG s)^*" by (auto simp:subtree_def)

    from star_rpath[OF this]

    obtain xs where "rpath (RAG s) (Th th') xs (Th th)" by auto

    hence "Th th' \<in> subtree (tRAG s) (Th th)"

    proof(induct xs arbitrary:th' th rule:length_induct)

      case (1 xs th' th)

      show ?case

      proof(cases xs)

        case Nil

          from rpath_nilE[OF 1(2)[unfolded this]]

          have "th' = th" by auto

          thus ?thesis by (auto simp:subtree_def)

      next

        case (Cons x1 xs1) note Cons1 = Cons

        show ?thesis

        proof(cases "xs1")

          case Nil

            from 1(2)[unfolded Cons[unfolded this]]

            have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .

            hence "(Th th', x1) \<in> (RAG s)" by (cases, simp)

            then obtain cs where "x1 = Cs cs" 

              by (unfold s_RAG_def, auto)

              find_theorems rpath "_ = _@[_]"

            from rpath_nnl_lastE[OF rp[unfolded this]]

            show ?thesis by auto

        next

          case (Cons x2 xs2)

          from 1(2)[unfolded Cons1[unfolded this]]

          have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .

          from rpath_edges_on[OF this]

          have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .

          have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"

            by (simp add: edges_on_unfold)

          with eds have rg1: "(Th th', x1) \<in> RAG s" by auto

          then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)

          have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"

            by (simp add: edges_on_unfold)

          from this eds

          have rg2: "(x1, x2) \<in> RAG s" by auto

          from this[unfolded eq_x1] 

          obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)

          from rp have "rpath (RAG s) x2 xs2 (Th th)"

           by  (elim rpath_ConsE, simp)

          from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .

          from 1(1)[rule_format, OF _ this, unfolded Cons1 Cons]

          have "Th th1 \<in> subtree (tRAG s) (Th th)" by simp

          moreover have "(Th th', Th th1) \<in> (tRAG s)^*"

          proof -

            from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]

            show ?thesis by (unfold RAG_split tRAG_def wRAG_def hRAG_def, auto)

          qed

          ultimately show ?thesis by (auto simp:subtree_def)

        qed

      qed

    qed

    from imageI[OF this, of the_thread]

    have "th' \<in> ?L" by simp

  } ultimately show ?thesis by auto

qed

lemma cp_alt_def1: 

  "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"

proof -

  have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =

       ((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"

       by auto

  thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)

qed

lemma cp_gen_def_cond: 

  assumes "x = Th th"

  shows "cp s th = cp_gen s (Th th)"

by (unfold cp_alt_def1 cp_gen_def, simp)

lemma cp_gen_over_set:

  assumes "\<forall> x \<in> A. \<exists> th. x = Th th"

  shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"

proof(rule f_image_eq)

  fix a

  assume "a \<in> A"

  from assms[rule_format, OF this]

  obtain th where eq_a: "a = Th th" by auto

  show "cp_gen s a = (cp s \<circ> the_thread) a"

    by  (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)

qed

locale valid_trace = 

  fixes s

  assumes vt : "vt s"

context valid_trace

begin

lemma wf_RAG: "wf (RAG s)"

proof(rule finite_acyclic_wf)

  from finite_RAG[OF vt] show "finite (RAG s)" .

next

  from acyclic_RAG[OF vt] show "acyclic (RAG s)" .

qed

end

context valid_trace

begin

lemma sgv_wRAG: "single_valued (wRAG s)"

  using waiting_unique[OF vt] 

  by (unfold single_valued_def wRAG_def, auto)

lemma sgv_hRAG: "single_valued (hRAG s)"

  using holding_unique 

  by (unfold single_valued_def hRAG_def, auto)

lemma sgv_tRAG: "single_valued (tRAG s)"

  by (unfold tRAG_def, rule single_valued_relcomp, 

              insert sgv_wRAG sgv_hRAG, auto)

lemma acyclic_tRAG: "acyclic (tRAG s)"

proof(unfold tRAG_def, rule acyclic_compose)

  show "acyclic (RAG s)" using acyclic_RAG[OF vt] .

next

  show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto

next

  show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto

qed

lemma sgv_RAG: "single_valued (RAG s)"

  using unique_RAG[OF vt] by (auto simp:single_valued_def)

lemma rtree_RAG: "rtree (RAG s)"

  using sgv_RAG acyclic_RAG[OF vt]

  by (unfold rtree_def rtree_axioms_def sgv_def, auto)

end

sublocale valid_trace < rtree_s: rtree "tRAG s"

proof(unfold_locales)

  from sgv_tRAG show "single_valued (tRAG s)" .

next

  from acyclic_tRAG show "acyclic (tRAG s)" .

qed

sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"

proof -

  show "fsubtree (RAG s)"

  proof(intro_locales)

    show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG[OF vt]] .

  next

    show "fsubtree_axioms (RAG s)"

    proof(unfold fsubtree_axioms_def)

    find_theorems wf RAG

      from wf_RAG show "wf (RAG s)" .

    qed

  qed

qed

sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"

proof -

  have "fsubtree (tRAG s)"

  proof -

    have "fbranch (tRAG s)"

    proof(unfold tRAG_def, rule fbranch_compose)

        show "fbranch (wRAG s)"

        proof(rule finite_fbranchI)

           from finite_RAG[OF vt] show "finite (wRAG s)"

           by (unfold RAG_split, auto)

        qed

    next

        show "fbranch (hRAG s)"

        proof(rule finite_fbranchI)

           from finite_RAG[OF vt] 

           show "finite (hRAG s)" by (unfold RAG_split, auto)

        qed

    qed

    moreover have "wf (tRAG s)"

    proof(rule wf_subset)

      show "wf (RAG s O RAG s)" using wf_RAG

        by (fold wf_comp_self, simp)

    next

      show "tRAG s \<subseteq> (RAG s O RAG s)"

        by (unfold tRAG_alt_def, auto)

    qed

    ultimately show ?thesis

      by (unfold fsubtree_def fsubtree_axioms_def,auto)

  qed

  from this[folded tRAG_def] show "fsubtree (tRAG s)" .

qed

lemma Max_UNION: 

  assumes "finite A"

  and "A \<noteq> {}"

  and "\<forall> M \<in> f ` A. finite M"

  and "\<forall> M \<in> f ` A. M \<noteq> {}"

  shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")

  using assms[simp]

proof -

  have "?L = Max (\<Union>(f ` A))"

    by (fold Union_image_eq, simp)

  also have "... = ?R"

    by (subst Max_Union, simp+)

  finally show ?thesis .

qed

lemma max_Max_eq:

  assumes "finite A"

    and "A \<noteq> {}"

    and "x = y"

  shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")

proof -

  have "?R = Max (insert y A)" by simp

  also from assms have "... = ?L"

      by (subst Max.insert, simp+)

  finally show ?thesis by simp

qed

context valid_trace

begin

(* ddd *)

lemma cp_gen_rec:

  assumes "x = Th th"

  shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"

proof(cases "children (tRAG s) x = {}")

  case True

  show ?thesis

    by (unfold True cp_gen_def subtree_children, simp add:assms)

next

  case False

  hence [simp]: "children (tRAG s) x \<noteq> {}" by auto

  note fsbttRAGs.finite_subtree[simp]

  have [simp]: "finite (children (tRAG s) x)"

     by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], 

            rule children_subtree)

  { fix r x

    have "subtree r x \<noteq> {}" by (auto simp:subtree_def)

  } note this[simp]

  have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"

  proof -

    from False obtain q where "q \<in> children (tRAG s) x" by blast

    moreover have "subtree (tRAG s) q \<noteq> {}" by simp

    ultimately show ?thesis by blast

  qed

  have h: "Max ((the_preced s \<circ> the_thread) `

                ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =

        Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"

                     (is "?L = ?R")

  proof -