diff -r 2c56b20032a7 -r 0679a84b11ad prio/PrioGDef.thy
--- a/prio/PrioGDef.thy	Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,483 +0,0 @@
-(*<*)
-theory PrioGDef
-imports Precedence_ord Moment
-begin
-(*>*)
-
-text {*
-  In this section, the formal model of Priority Inheritance is presented. 
-  The model is based on Paulson's inductive protocol verification method, where 
-  the state of the system is modelled as a list of events happened so far with the latest 
-  event put at the head. 
-
-  To define events, the identifiers of {\em threads},
-  {\em priority} and {\em critical resources } (abbreviated as @{text "cs"}) 
-  need to be represented. All three are represetned using standard 
-  Isabelle/HOL type @{typ "nat"}:
-*}
-
-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). *}
-
-text {*
-  \noindent
-  Every event in the system corresponds to a system call, the formats of which are
-  defined as follows:
-  *}
-
-datatype event = 
-  Create thread priority | -- {* Thread @{text "thread"} is created with priority @{text "priority"}. *}
-  Exit thread | -- {* Thread @{text "thread"} finishing its execution. *}
-  P thread cs | -- {* Thread @{text "thread"} requesting critical resource @{text "cs"}. *}
-  V thread cs | -- {* Thread @{text "thread"}  releasing critical resource @{text "cs"}. *}
-  Set thread priority -- {* Thread @{text "thread"} resets its priority to @{text "priority"}. *}
-
-text {* 
-\noindent
-  Resource Allocation Graph (RAG for short) is used extensively in our formal analysis. 
-  The following type @{text "node"} is used to represent nodes in RAG.
-  *}
-datatype node = 
-   Th "thread" | -- {* Node for thread. *}
-   Cs "cs" -- {* Node for critical resource. *}
-
-text {* 
-  In Paulson's inductive method, the states of system are represented as lists of events,
-  which is defined by the following type @{text "state"}:
-  *}
-type_synonym state = "event list"
-
-text {*
-  \noindent
-  The following function
-  @{text "threads"} is used to calculate the set of live threads (@{text "threads s"})
-  in state @{text "s"}.
-  *}
-fun threads :: "state \<Rightarrow> thread set"
-  where 
-  -- {* At the start of the system, the set of threads is empty: *}
-  "threads [] = {}" | 
-  -- {* New thread is added to the @{text "threads"}: *}
-  "threads (Create thread prio#s) = {thread} \<union> threads s" | 
-  -- {* Finished thread is removed: *}
-  "threads (Exit thread # s) = (threads s) - {thread}" | 
-  -- {* Other kind of events does not affect the value of @{text "threads"}: *}
-  "threads (e#s) = threads s" 
-text {* \noindent
-  Functions such as @{text "threads"}, which extract information out of system states, are called
-  {\em observing functions}. A series of observing functions will be defined in the sequel in order to 
-  model the protocol. 
-  Observing function @{text "original_priority"} calculates 
-  the {\em original priority} of thread @{text "th"} in state @{text "s"}, expressed as
-  : @{text "original_priority th s" }. The {\em original priority} is the priority 
-  assigned to a thread when it is created or when it is reset by system call 
-  @{text "Set thread priority"}.
-*}
-
-fun original_priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
-  where
-  -- {* @{text "0"} is assigned to threads which have never been created: *}
-  "original_priority thread [] = 0" |
-  "original_priority thread (Create thread' prio#s) = 
-     (if thread' = thread then prio else original_priority thread s)" |
-  "original_priority thread (Set thread' prio#s) = 
-     (if thread' = thread then prio else original_priority thread s)" |
-  "original_priority thread (e#s) = original_priority thread s"
-
-text {*
-  \noindent
-  In the following,
-  @{text "birthtime th s"} is the time when thread @{text "th"} is created, 
-  observed from state @{text "s"}.
-  The time in the system is measured by the number of events happened so far since the very beginning.
-*}
-fun birthtime :: "thread \<Rightarrow> state \<Rightarrow> nat"
-  where
-  "birthtime thread [] = 0" |
-  "birthtime thread ((Create thread' prio)#s) = 
-       (if (thread = thread') then length s else birthtime thread s)" |
-  "birthtime thread ((Set thread' prio)#s) = 
-       (if (thread = thread') then length s else birthtime thread s)" |
-  "birthtime thread (e#s) = birthtime thread s"
-
-text {*
-  \noindent 
-  The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of 
-  a thread is the combination of its {\em original priority} and {\em birth time}. The intention is
-  to discriminate threads with the same priority by giving threads whose priority
-  is assigned earlier higher precedences, becasue such threads are more urgent to finish. 
-  This explains the following definition:
-  *}
-definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
-  where "preced thread s \<equiv> Prc (original_priority thread s) (birthtime thread s)"
-
-
-text {*
-  \noindent
-  A number of important notions are defined here:
-  *}
-
-consts 
-  holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool" 
-  waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
-  depend :: "'b \<Rightarrow> (node \<times> node) set"
-  dependents :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
-
-text {*
-  \noindent
-  In the definition of the following several functions, it is supposed that
-  the waiting queue of every critical resource is given by a waiting queue 
-  function @{text "wq"}, which servers as arguments of these functions.
-  *}
-defs (overloaded) 
-  -- {* 
-  \begin{minipage}{0.9\textwidth}
-  We define that the thread which is at the head of waiting queue of resource @{text "cs"}
-  is holding the resource. This definition is slightly different from tradition where
-  all threads in the waiting queue are considered as waiting for the resource.
-  This notion is reflected in the definition of @{text "holding wq th cs"} as follows:
-  \end{minipage}
-  *}
-  cs_holding_def: 
-  "holding wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
-  -- {* 
-  \begin{minipage}{0.9\textwidth}
-  In accordance with the definition of @{text "holding wq th cs"}, 
-  a thread @{text "th"} is considered waiting for @{text "cs"} if 
-  it is in the {\em waiting queue} of critical resource @{text "cs"}, but not at the head.
-  This is reflected in the definition of @{text "waiting wq th cs"} as follows:
-  \end{minipage}
-  *}
-  cs_waiting_def: 
-  "waiting wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
-  -- {* 
-  \begin{minipage}{0.9\textwidth}
-  @{text "depend wq"} represents the Resource Allocation Graph of the system under the waiting 
-  queue function @{text "wq"}.
-  \end{minipage}
-  *}
-  cs_depend_def: 
-  "depend (wq::cs \<Rightarrow> thread list) \<equiv>
-      {(Th th, Cs cs) | th cs. waiting wq th cs} \<union> {(Cs cs, Th th) | cs th. holding wq th cs}"
-  -- {* 
-  \begin{minipage}{0.9\textwidth}
-  The following @{text "dependents wq th"} represents the set of threads which are depending on
-  thread @{text "th"} in Resource Allocation Graph @{text "depend wq"}:
-  \end{minipage}
-  *}
-  cs_dependents_def: 
-  "dependents (wq::cs \<Rightarrow> thread list) th \<equiv> {th' . (Th th', Th th) \<in> (depend wq)^+}"
-
-text {*
-  The data structure used by the operating system for scheduling is referred to as 
-  {\em schedule state}. It is represented as a record consisting of 
-  a function assigning waiting queue to resources and a function assigning precedence to 
-  threads:
-  *}
-record schedule_state = 
-    wq_fun :: "cs \<Rightarrow> thread list" -- {* The function assigning waiting queue. *}
-    cprec_fun :: "thread \<Rightarrow> precedence" -- {* The function assigning precedence. *}
-
-text {* \noindent 
-  The following
-  @{text "cpreced s th"} gives the {\em current precedence} of thread @{text "th"} under
-  state @{text "s"}. The definition of @{text "cpreced"} reflects the basic idea of 
-  Priority Inheritance that the {\em current precedence} of a thread is the precedence 
-  inherited from the maximum of all its dependents, i.e. the threads which are waiting 
-  directly or indirectly waiting for some resources from it. If no such thread exits, 
-  @{text "th"}'s {\em current precedence} equals its original precedence, i.e. 
-  @{text "preced th s"}.
-  *}
-definition cpreced :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"
-  where "cpreced wq s = (\<lambda> th. Max ((\<lambda> th. preced th s) ` ({th} \<union> dependents wq th)))"
-
-(*<*)
-lemma 
-  cpreced_def2:
-  "cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependents wq th})"
-  unfolding cpreced_def image_def
-  apply(rule eq_reflection)
-  apply(rule_tac f="Max" in arg_cong)
-  by (auto)
-(*>*)
-
-abbreviation
-  "all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"
-
-abbreviation 
-  "initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"
- 
-abbreviation
-  "release qs \<equiv> case qs of
-             [] => [] 
-          |  (_#qs) => (SOME q. distinct q \<and> set q = set qs)"
-
-text {* \noindent
-  The following function @{text "schs"} is used to calculate the schedule state @{text "schs s"}.
-  It is the key function to model Priority Inheritance:
-  *}
-fun schs :: "state \<Rightarrow> schedule_state"
-  where 
-  "schs [] = (| wq_fun = \<lambda> cs. [],  cprec_fun = (\<lambda>_. Prc 0 0) |)" |
-
-  -- {*
-  \begin{minipage}{0.9\textwidth}
-  \begin{enumerate}
-  \item @{text "ps"} is the schedule state of last moment.
-  \item @{text "pwq"} is the waiting queue function of last moment.
-  \item @{text "pcp"} is the precedence function of last moment (NOT USED). 
-  \item @{text "nwq"} is the new waiting queue function. It is calculated using a @{text "case"} statement:
-  \begin{enumerate}
-      \item If the happening event is @{text "P thread cs"}, @{text "thread"} is added to 
-            the end of @{text "cs"}'s waiting queue.
-      \item If the happening event is @{text "V thread cs"} and @{text "s"} is a legal state,
-            @{text "th'"} must equal to @{text "thread"}, 
-            because @{text "thread"} is the one currently holding @{text "cs"}. 
-            The case @{text "[] \<Longrightarrow> []"} may never be executed in a legal state.
-            the @{text "(SOME q. distinct q \<and> set q = set qs)"} is used to choose arbitrarily one 
-            thread in waiting to take over the released resource @{text "cs"}. In our representation,
-            this amounts to rearrange elements in waiting queue, so that one of them is put at the head.
-      \item For other happening event, the schedule state just does not change.
-  \end{enumerate}
-  \item @{text "ncp"} is new precedence function, it is calculated from the newly updated waiting queue 
-        function. The dependency of precedence function on waiting queue function is the reason to 
-        put them in the same record so that they can evolve together.
-  \end{enumerate}
-  \end{minipage}
-     *}
-   "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)|))"
-
-lemma cpreced_initial: 
-  "cpreced (\<lambda> cs. []) [] = (\<lambda>_. (Prc 0 0))"
-apply(simp add: cpreced_def)
-apply(simp add: cs_dependents_def cs_depend_def cs_waiting_def cs_holding_def)
-apply(simp add: preced_def)
-done
-
-lemma sch_old_def:
-  "schs (e#s) = (let ps = schs s in 
-                  let pwq = wq_fun ps in 
-                  let nwq = case e of
-                             P th cs \<Rightarrow>  pwq(cs:=(pwq cs @ [th])) |
-                             V th cs \<Rightarrow> let nq = case (pwq cs) of
-                                                      [] \<Rightarrow> [] | 
-                                                      (_#qs) \<Rightarrow> (SOME q. distinct q \<and> set q = set qs)
-                                            in pwq(cs:=nq)                 |
-                              _ \<Rightarrow> pwq
-                  in let ncp = cpreced nwq (e#s) in 
-                     \<lparr>wq_fun = nwq, cprec_fun = ncp\<rparr>
-                 )"
-apply(cases e)
-apply(simp_all)
-done
-
-
-text {* 
-  \noindent
-  The following @{text "wq"} is a shorthand for @{text "wq_fun"}. 
-  *}
-definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list" 
-  where "wq s = wq_fun (schs s)"
-
-text {* \noindent 
-  The following @{text "cp"} is a shorthand for @{text "cprec_fun"}. 
-  *}
-definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
-  where "cp s \<equiv> cprec_fun (schs s)"
-
-text {* \noindent
-  Functions @{text "holding"}, @{text "waiting"}, @{text "depend"} and 
-  @{text "dependents"} still have the 
-  same meaning, but redefined so that they no longer depend on the 
-  fictitious {\em waiting queue function}
-  @{text "wq"}, but on system state @{text "s"}.
-  *}
-defs (overloaded) 
-  s_holding_abv: 
-  "holding (s::state) \<equiv> holding (wq_fun (schs s))"
-  s_waiting_abv: 
-  "waiting (s::state) \<equiv> waiting (wq_fun (schs s))"
-  s_depend_abv: 
-  "depend (s::state) \<equiv> depend (wq_fun (schs s))"
-  s_dependents_abv: 
-  "dependents (s::state) \<equiv> dependents (wq_fun (schs s))"
-
-
-text {* 
-  The following lemma can be proved easily:
-  *}
-lemma
-  s_holding_def: 
-  "holding (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th = hd (wq_fun (schs s) cs))"
-  by (auto simp:s_holding_abv wq_def cs_holding_def)
-
-lemma s_waiting_def: 
-  "waiting (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th \<noteq> hd (wq_fun (schs s) cs))"
-  by (auto simp:s_waiting_abv wq_def cs_waiting_def)
-
-lemma s_depend_def: 
-  "depend (s::state) =
-    {(Th th, Cs cs) | th cs. waiting (wq s) th cs} \<union> {(Cs cs, Th th) | cs th. holding (wq s) th cs}"
-  by (auto simp:s_depend_abv wq_def cs_depend_def)
-
-lemma
-  s_dependents_def: 
-  "dependents (s::state) th \<equiv> {th' . (Th th', Th th) \<in> (depend (wq s))^+}"
-  by (auto simp:s_dependents_abv wq_def cs_dependents_def)
-
-text {*
-  The following function @{text "readys"} calculates the set of ready threads. A thread is {\em ready} 
-  for running if it is a live thread and it is not waiting for any critical resource.
-  *}
-definition readys :: "state \<Rightarrow> thread set"
-  where "readys s \<equiv> {th . th \<in> threads s \<and> (\<forall> cs. \<not> waiting s th cs)}"
-
-text {* \noindent
-  The following function @{text "runing"} calculates the set of running thread, which is the ready 
-  thread with the highest precedence. 
-  *}
-definition runing :: "state \<Rightarrow> thread set"
-  where "runing s \<equiv> {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
-
-text {* \noindent
-  The following function @{text "holdents s th"} returns the set of resources held by thread 
-  @{text "th"} in state @{text "s"}.
-  *}
-definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
-  where "holdents s th \<equiv> {cs . holding s th cs}"
-
-lemma holdents_test: 
-  "holdents s th = {cs . (Cs cs, Th th) \<in> depend s}"
-unfolding holdents_def
-unfolding s_depend_def
-unfolding s_holding_abv
-unfolding wq_def
-by (simp)
-
-text {* \noindent
-  @{text "cntCS s th"} returns the number of resources held by thread @{text "th"} in
-  state @{text "s"}:
-  *}
-definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
-  where "cntCS s th = card (holdents s th)"
-
-text {* \noindent
-  The fact that event @{text "e"} is eligible to happen next in state @{text "s"} 
-  is expressed as @{text "step s e"}. The predicate @{text "step"} is inductively defined as 
-  follows:
-  *}
-inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
-  where
-  -- {* 
-  A thread can be created if it is not a live thread:
-  *}
-  thread_create: "\<lbrakk>thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
-  -- {*
-  A thread can exit if it no longer hold any resource:
-  *}
-  thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
-  -- {*
-  \begin{minipage}{0.9\textwidth}
-  A thread can request for an critical resource @{text "cs"}, if it is running and 
-  the request does not form a loop in the current RAG. The latter condition 
-  is set up to avoid deadlock. The condition also reflects our assumption all threads are 
-  carefully programmed so that deadlock can not happen:
-  \end{minipage}
-  *}
-  thread_P: "\<lbrakk>thread \<in> runing s;  (Cs cs, Th thread)  \<notin> (depend s)^+\<rbrakk> \<Longrightarrow> 
-                                                                step s (P thread cs)" |
-  -- {*
-  \begin{minipage}{0.9\textwidth}
-  A thread can release a critical resource @{text "cs"} 
-  if it is running and holding that resource:
-  \end{minipage}
-  *}
-  thread_V: "\<lbrakk>thread \<in> runing s;  holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
-  -- {*
-  A thread can adjust its own priority as long as it is current running:
-  *}  
-  thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
-
-text {* \noindent
-  With predicate @{text "step"}, the fact that @{text "s"} is a legal state in 
-  Priority Inheritance protocol can be expressed as: @{text "vt step s"}, where
-  the predicate @{text "vt"} can be defined as the following:
-  *}
-inductive vt :: "state \<Rightarrow> bool"
-  where
-  -- {* Empty list @{text "[]"} is a legal state in any protocol:*}
-  vt_nil[intro]: "vt []" |
-  -- {* 
-  \begin{minipage}{0.9\textwidth}
-  If @{text "s"} a legal state, and event @{text "e"} is eligible to happen
-  in state @{text "s"}, then @{text "e#s"} is a legal state as well:
-  \end{minipage}
-  *}
-  vt_cons[intro]: "\<lbrakk>vt s; step s e\<rbrakk> \<Longrightarrow> vt (e#s)"
-
-text {*  \noindent
-  It is easy to see that the definition of @{text "vt"} is generic. It can be applied to 
-  any step predicate to get the set of legal states.
-  *}
-
-text {* \noindent
-  The following two functions @{text "the_cs"} and @{text "the_th"} are used to extract
-  critical resource and thread respectively out of RAG nodes.
-  *}
-fun the_cs :: "node \<Rightarrow> cs"
-  where "the_cs (Cs cs) = cs"
-
-fun the_th :: "node \<Rightarrow> thread"
-  where "the_th (Th th) = th"
-
-text {* \noindent
-  The following predicate @{text "next_th"} describe the next thread to 
-  take over when a critical resource is released. In @{text "next_th s th cs t"}, 
-  @{text "th"} is the thread to release, @{text "t"} is the one to take over.
-  *}
-definition next_th:: "state \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> thread \<Rightarrow> bool"
-  where "next_th s th cs t = (\<exists> rest. wq s cs = th#rest \<and> rest \<noteq> [] \<and> 
-                                                t = hd (SOME q. distinct q \<and> set q = set rest))"
-
-text {* \noindent
-  The function @{text "count Q l"} is used to count the occurrence of situation @{text "Q"}
-  in list @{text "l"}:
-  *}
-definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
-  where "count Q l = length (filter Q l)"
-
-text {* \noindent
-  The following @{text "cntP s"} returns the number of operation @{text "P"} happened 
-  before reaching state @{text "s"}.
-  *}
-definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
-  where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
-
-text {* \noindent
-  The following @{text "cntV s"} returns the number of operation @{text "V"} happened 
-  before reaching state @{text "s"}.
-  *}
-definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
-  where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
-(*<*)
-
-end
-(*>*)
-