theory Test 

imports Precedence_ord

begin

type_synonym thread = nat -- {* Type for thread identifiers. *}

type_synonym priority = nat  -- {* Type for priorities. *}

type_synonym cs = nat -- {* Type for critical sections (or critical resources). *}

datatype event = 

  Create thread priority 

| Exit thread 

| P thread cs 

| V thread cs 

| Set thread priority 

type_synonym state = "event list"

fun threads :: "state \<Rightarrow> thread set"

  where 

  "threads [] = {}" 

| "threads (Create th prio#s) = {th} \<union> threads s" 

| "threads (Exit th # s) = (threads s) - {th}" 

| "threads (_#s) = threads s" 

fun priority :: "thread \<Rightarrow> state \<Rightarrow> priority"

  where

  "priority th [] = 0" 

| "priority th (Create th' prio#s) = (if th' = th then prio else priority th s)" 

| "priority th (Set th' prio#s) = (if th' = th then prio else priority th s)" 

| "priority th (_#s) = priority th s"

fun last_set :: "thread \<Rightarrow> state \<Rightarrow> nat"

  where

  "last_set th [] = 0" 

| "last_set th ((Create th' prio)#s) = (if (th = th') then length s else last_set th s)" 

| "last_set th ((Set th' prio)#s) = (if (th = th') then length s else last_set th s)" 

| "last_set th (_#s) = last_set th s"

definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"

  where "preced th s \<equiv> Prc (priority th s) (last_set th s)"

definition

  "holds wq th cs \<equiv> th \<in> set (wq cs) \<and> th = hd (wq cs)"

definition

  "waits wq th cs \<equiv> th \<in> set (wq cs) \<and> th \<noteq> hd (wq cs)"

datatype node = 

  Th "thread" 

| Cs "cs" 

definition

  "RAG wq \<equiv> {(Th th, Cs cs) | th cs. waits wq th cs} \<union> {(Cs cs, Th th) | cs th. holds wq th cs}"

definition

  "dependants wq th \<equiv> {th' . (Th th', Th th) \<in> (RAG wq)^+}"

record schedule_state = 

  wq_fun :: "cs \<Rightarrow> thread list" 

  cprec_fun :: "thread \<Rightarrow> precedence" 

definition cpreced :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"

  where 

  "cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependants wq th})"

abbreviation

  "all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"

abbreviation 

  "initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"

abbreviation

  "release qs \<equiv> case qs of

             [] => [] 

fun schs :: "state \<Rightarrow> schedule_state"

  where 

  "schs [] = (| wq_fun = \<lambda> cs. [],  cprec_fun = (\<lambda>_. Prc 0 0) |)" 

| "schs (Create th prio # s) = 

       (let wq = wq_fun (schs s) in

          (|wq_fun = wq, cprec_fun = cpreced wq (Create th prio # s)|))"

|  "schs (Exit th # s) = 

       (let wq = wq_fun (schs s) in

          (|wq_fun = wq, cprec_fun = cpreced wq (Exit th # s)|))"

|  "schs (Set th prio # s) = 

       (let wq = wq_fun (schs s) in

          (|wq_fun = wq, cprec_fun = cpreced wq (Set th prio # s)|))"

|  "schs (P th cs # s) = 

       (let wq = wq_fun (schs s) in

        let new_wq = wq(cs := (wq cs @ [th])) in

          (|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|))"

|  "schs (V th cs # s) = 

       (let wq = wq_fun (schs s) in

        let new_wq = wq(cs := release (wq cs)) in

          (|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|))"

definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list" 

  where "wq s = wq_fun (schs s)"

definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"

  where "cp s \<equiv> cprec_fun (schs s)"

definition

  "holds2 s \<equiv> holds (wq_fun (schs s))"

definition

  "waits2 s \<equiv> waits (wq_fun (schs s))"

definition

  "RAG2 s \<equiv> RAG (wq_fun (schs s))"

definition

  "dependants2 s \<equiv> dependants (wq_fun (schs s))"

definition readys :: "state \<Rightarrow> thread set"

  where "readys s \<equiv> {th . th \<in> threads s \<and> (\<forall> cs. \<not> waits2 s th cs)}"

definition runing :: "state \<Rightarrow> thread set"

  where "runing s \<equiv> {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"

definition holding :: "state \<Rightarrow> thread \<Rightarrow> cs set"

  where "holding s th \<equiv> {cs . holds2 s th cs}"

lemma holding_RAG: 

  "holding s th = {cs . (Cs cs, Th th) \<in> RAG2 s}"

unfolding holding_def

unfolding holds2_def

unfolding RAG2_def RAG_def

unfolding holds_def waits_def

by auto

inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"

  where

  step_Create: "\<lbrakk>th \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create th prio)" 

| step_Exit: "\<lbrakk>th \<in> runing s; holding s th = {}\<rbrakk> \<Longrightarrow> step s (Exit th)" 

| step_P: "\<lbrakk>th \<in> runing s;  (Cs cs, Th th)  \<notin> (RAG2 s)^+\<rbrakk> \<Longrightarrow> step s (P th cs)" 

| step_V: "\<lbrakk>th \<in> runing s;  holds2 s th cs\<rbrakk> \<Longrightarrow> step s (V th cs)" 

| step_Set: "\<lbrakk>th \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set th prio)"

inductive vt :: "state \<Rightarrow> bool"

  where

  vt_nil[intro]: "vt []" 

| vt_cons[intro]: "\<lbrakk>vt s; step s e\<rbrakk> \<Longrightarrow> vt (e#s)"

lemma runing_ready: 

  shows "runing s \<subseteq> readys s"

  unfolding runing_def readys_def

  by auto 

lemma readys_threads:

  shows "readys s \<subseteq> threads s"

  unfolding readys_def

  by auto

lemma wq_distinct_step: 

  assumes "step s e" "distinct (wq s cs)" 

  shows "distinct (wq (e # s) cs)"

using assms

apply(induct)

apply(auto simp add: wq_def Let_def)

apply(auto simp add: wq_def Let_def RAG2_def RAG_def holds_def runing_def waits2_def waits_def readys_def)[1]

apply(auto split: list.split)

apply(rule someI2)

apply(auto)

done

lemma wq_distinct: 

  assumes "vt s" 

  shows "distinct (wq s cs)"

using assms

apply(induct)

apply(simp add: wq_def)

apply(simp add: wq_distinct_step)

done

lemma RAG_set_unchanged[simp]: 

  shows "RAG2 (Set th prio # s) = RAG2 s"

unfolding RAG2_def

by (simp add: Let_def)

lemma RAG_create_unchanged[simp]: 

  "RAG2 (Create th prio # s) = RAG2 s"

unfolding RAG2_def

by (simp add: Let_def)

lemma RAG_exit_unchanged[simp]: 

  shows "RAG2 (Exit th # s) = RAG2 s"

unfolding RAG2_def

by (simp add: Let_def)

lemma RAG_p1:

  assumes "wq s cs = []"

lemma RAG_p2:

lemma waiting_unique:

  assumes "vt s"

  and "waits2 s th cs1"

  and "waits2 s th cs2"

  shows "cs1 = cs2"

sorry

lemma acyclic_RAG_step: 

  assumes vt: "vt s"

  and     stp: "step s e"

  and     ac: "acyclic (RAG2 s)"

  shows "acyclic (RAG2 (e # s))"

using stp vt ac

proof (induct)

  case (step_P th s cs)

  have ac: "acyclic (RAG2 s)" by fact

  have ds: "(Cs cs, Th th) \<notin> (RAG2 s)\<^sup>+" by fact

  { assume a: "wq s cs = []" -- "case waiting queue is empty"

    have "(Th th, Cs cs) \<notin> (RAG2 s)\<^sup>*"  

    proof

      assume "(Th th, Cs cs) \<in> (RAG2 s)\<^sup>*"

      then have "(Th th, Cs cs) \<in> (RAG2 s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)

      then obtain x where "(x, Cs cs) \<in> RAG2 s" using tranclD2 by metis 

      with a show False by (auto simp: RAG2_def RAG_def wq_def waits_def)

    qed

    with acyclic_insert ac

    have "acyclic (RAG2 s \<union> {(Cs cs, Th th)})" by simp

  }

  moreover

  { assume a: "wq s cs \<noteq> []" -- "case waiting queue is not empty"

    from ds have "(Cs cs, Th th) \<notin> (RAG2 s)\<^sup>*" by (metis node.distinct(1) rtranclD)

    with acyclic_insert ac 

    have "acyclic (RAG2 s \<union> {(Th th, Cs cs)})" by auto

  }

  ultimately show "acyclic (RAG2 (P th cs # s))" by metis

next    

  case (step_V th s cs) 

  have vt: "vt s" by fact

  have ac: "acyclic (RAG2 s)" by fact

  have rn: "th \<in> runing s" by fact

  have hd: "holds2 s th cs" by fact

qed (simp_all)