--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/PIPDefs.thy Wed Jan 06 16:34:26 2016 +0000
@@ -0,0 +1,614 @@
+chapter {* Definitions *}
+(*<*)
+theory PIPDefs
+imports Precedence_ord Moment
+begin
+(*>*)
+
+text {*
+ In this section, the formal model of Priority Inheritance Protocol (PIP) 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.
+*}
+
+text {*
+ 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
+ The abstraction of Priority Inheritance Protocol (PIP) is set at the system call level.
+ Every system call is represented as an event. The format of events is defined
+ 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 {*
+ As mentioned earlier, 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
+ 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 {*
+ \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
+ The function @{text "threads"} defined above is one of
+ the so called {\em observation function}s which forms
+ the very basis of Paulson's inductive protocol verification method.
+ Each observation function {\em observes} one particular aspect (or attribute)
+ of the system. For example, the attribute observed by @{text "threads s"}
+ is the set of threads living in state @{text "s"}.
+ The protocol being modelled
+ The decision made the protocol being modelled is based on the {\em observation}s
+ returned by {\em observation function}s. Since {\observation function}s forms
+ the very basis on which Paulson's inductive method is based, there will be
+ a lot of such observation functions introduced in the following. In fact, any function
+ which takes event list as argument is a {\em observation function}.
+ *}
+
+text {* \noindent
+ Observation @{text "priority th s"} is
+ the {\em original priority} of thread @{text "th"} in state @{text "s"}.
+ The {\em original priority} is the priority
+ assigned to a thread when it is created or when it is reset by system call
+ (represented by event @{text "Set thread priority"}).
+*}
+
+fun priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
+ where
+ -- {* @{text "0"} is assigned to threads which have never been created: *}
+ "priority thread [] = 0" |
+ "priority thread (Create thread' prio#s) =
+ (if thread' = thread then prio else priority thread s)" |
+ "priority thread (Set thread' prio#s) =
+ (if thread' = thread then prio else priority thread s)" |
+ "priority thread (e#s) = priority thread s"
+
+text {*
+ \noindent
+ Observation @{text "last_set th s"} is the last time when the priority of thread @{text "th"} is set,
+ 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 last_set :: "thread \<Rightarrow> state \<Rightarrow> nat"
+ where
+ "last_set thread [] = 0" |
+ "last_set thread ((Create thread' prio)#s) =
+ (if (thread = thread') then length s else last_set thread s)" |
+ "last_set thread ((Set thread' prio)#s) =
+ (if (thread = thread') then length s else last_set thread s)" |
+ "last_set thread (_#s) = last_set 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 time} the priority is set.
+ 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 (priority thread s) (last_set thread s)"
+
+
+text {*
+ \noindent
+ A number of important notions in PIP are represented as the following functions,
+ defined in terms of the waiting queues of the system, where the waiting queues
+ , as a whole, is represented by the @{text "wq"} argument of every notion function.
+ The @{text "wq"} argument is itself a functions which maps every critical resource
+ @{text "cs"} to the list of threads which are holding or waiting for it.
+ The thread at the head of this list is designated as the thread which is current
+ holding the resrouce, which is slightly different from tradition where
+ all threads in the waiting queue are considered as waiting for the resource.
+ *}
+
+consts
+ holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+ waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+ RAG :: "'b \<Rightarrow> (node \<times> node) set"
+ dependants :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
+
+defs (overloaded)
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ This meaning of @{text "wq"} is reflected in the following definition of @{text "holding wq th cs"},
+ where @{text "holding wq th cs"} means thread @{text "th"} is holding the critical
+ resource @{text "cs"}. This decision is based on @{text "wq"}.
+ \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 "RAG wq"} generates RAG (a binary relations on @{text "node"})
+ out of waiting queues of the system (represented by the @{text "wq"} argument):
+ \end{minipage}
+ *}
+ cs_RAG_def:
+ "RAG (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 "dependants wq th"} represents the set of threads which are RAGing on
+ thread @{text "th"} in Resource Allocation Graph @{text "RAG wq"}.
+ Here, "RAGing" means waiting directly or indirectly on the critical resource.
+ \end{minipage}
+ *}
+ cs_dependants_def:
+ "dependants (wq::cs \<Rightarrow> thread list) th \<equiv> {th' . (Th th', Th th) \<in> (RAG wq)^+}"
+
+
+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 dependants, 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> dependants wq th)))"
+
+text {*
+ Notice that the current precedence (@{text "cpreced"}) of one thread @{text "th"} can be boosted
+ (becoming larger than its own precedence) by those threads in
+ the @{text "dependants wq th"}-set. If one thread get boosted, we say
+ it inherits the priority (or, more precisely, the precedence) of
+ its dependants. This is how the word "Inheritance" in
+ Priority Inheritance Protocol comes.
+*}
+
+(*<*)
+lemma
+ cpreced_def2:
+ "cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependants wq th})"
+ unfolding cpreced_def image_def
+ apply(rule eq_reflection)
+ apply(rule_tac f="Max" in arg_cong)
+ by (auto)
+(*>*)
+
+
+text {* \noindent
+ Assuming @{text "qs"} be the waiting queue of a critical resource,
+ the following abbreviation "release qs" is the waiting queue after the thread
+ holding the resource (which is thread at the head of @{text "qs"}) released
+ the resource:
+*}
+abbreviation
+ "release qs \<equiv> case qs of
+ [] => []
+ | (_#qs') => (SOME q. distinct q \<and> set q = set qs')"
+text {* \noindent
+ It can be seen from the definition that the thread at the head of @{text "qs"} is removed
+ from the return value, and the value @{term "q"} is an reordering of @{text "qs'"}, the
+ tail of @{text "qs"}. Through this reordering, one of the waiting threads (those in @{text "qs'"} }
+ is chosen nondeterministically to be the head of the new queue @{text "q"}.
+ Therefore, this thread is the one who takes over the resource. This is a little better different
+ from common sense that the thread who comes the earliest should take over.
+ The intention of this definition is to show that the choice of which thread to take over the
+ release resource does not affect the correctness of the PIP protocol.
+*}
+
+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
+ (to be used as the @{text "wq"} argument in @{text "holding"}, @{text "waiting"}
+ and @{text "RAG"}, etc) 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 two abbreviations (@{text "all_unlocked"} and @{text "initial_cprec"})
+ are used to set the initial values of the @{text "wq_fun"} @{text "cprec_fun"} fields
+ respectively of the @{text "schedule_state"} record by the following function @{text "sch"},
+ which is used to calculate the system's {\em schedule state}.
+
+ Since there is no thread at the very beginning to make request, all critical resources
+ are free (or unlocked). This status is represented by the abbreviation
+ @{text "all_unlocked"}.
+ *}
+abbreviation
+ "all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"
+
+
+text {* \noindent
+ The initial current precedence for a thread can be anything, because there is no thread then.
+ We simply assume every thread has precedence @{text "Prc 0 0"}.
+ *}
+
+abbreviation
+ "initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"
+
+
+text {* \noindent
+ The following function @{text "schs"} is used to calculate the system's schedule state @{text "schs s"}
+ out of the current system state @{text "s"}. It is the central function to model Priority Inheritance:
+ *}
+fun schs :: "state \<Rightarrow> schedule_state"
+ where
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ Setting the initial value of the @{text "schedule_state"} record (see the explanations above).
+ \end{minipage}
+ *}
+ "schs [] = (| wq_fun = all_unlocked, cprec_fun = initial_cprec |)" |
+
+ -- {*
+ \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 RAGency 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}
+
+
+ The calculation of @{text "cprec_fun"} depends on the value of @{text "wq_fun"}.
+ Therefore, in the following cases, @{text "wq_fun"} is always calculated first, in
+ the name of @{text "wq"} (if @{text "wq_fun"} is not changed
+ by the happening event) or @{text "new_wq"} (if the value of @{text "wq_fun"} is changed).
+ \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)|))"
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ Different from the forth coming cases, the @{text "wq_fun"} field of the schedule state
+ is changed. So, the new value is calculated first, in the name of @{text "new_wq"}.
+ \end{minipage}
+ *}
+| "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_dependants_def cs_RAG_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 "RAG"} and
+ @{text "dependants"} still have the
+ same meaning, but redefined so that they no longer RAG 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_RAG_abv:
+ "RAG (s::state) \<equiv> RAG (wq_fun (schs s))"
+ s_dependants_abv:
+ "dependants (s::state) \<equiv> dependants (wq_fun (schs s))"
+
+
+text {*
+ The following lemma can be proved easily, and the meaning is obvious.
+ *}
+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_RAG_def:
+ "RAG (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_RAG_abv wq_def cs_RAG_def)
+
+lemma
+ s_dependants_def:
+ "dependants (s::state) th \<equiv> {th' . (Th th', Th th) \<in> (RAG (wq s))^+}"
+ by (auto simp:s_dependants_abv wq_def cs_dependants_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
+ Notice that the definition of @{text "running"} reflects the preemptive scheduling strategy,
+ because, if the @{text "running"}-thread (the one in @{text "runing"} set)
+ lowered its precedence by resetting its own priority to a lower
+ one, it will lose its status of being the max in @{text "ready"}-set and be superseded.
+*}
+
+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> RAG s}"
+unfolding holdents_def
+unfolding s_RAG_def
+unfolding s_holding_abv
+unfolding wq_def
+by (simp)
+
+text {* \noindent
+ Observation @{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
+ According to the convention of Paulson's inductive method,
+ the decision made by a protocol that event @{text "e"} is eligible to happen next under state @{text "s"}
+ is expressed as @{text "step s e"}. The predicate @{text "step"} is inductively defined as
+ follows (notice how the decision is based on the {\em observation function}s
+ defined above, and also notice how a complicated protocol is modeled by a few simple
+ observations, and how such a kind of simplicity gives rise to improved trust on
+ faithfulness):
+ *}
+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> (RAG 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)" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ A thread can adjust its own priority as long as it is current running.
+ With the resetting of one thread's priority, its precedence may change.
+ If this change lowered the precedence, according to the definition of @{text "running"}
+ function,
+ \end{minipage}
+ *}
+ thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
+
+text {*
+ In Paulson's inductive method, every protocol is defined by such a @{text "step"}
+ predicate. For instance, the predicate @{text "step"} given above
+ defines the PIP protocol. So, it can also be called "PIP".
+*}
+
+abbreviation
+ "PIP \<equiv> step"
+
+
+text {* \noindent
+ For any protocol defined by a @{text "step"} predicate,
+ the fact that @{text "s"} is a legal state in
+ the protocol is 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 of the protocol defined by predicate @{text "step"},
+ and event @{text "e"} is allowed to happen under state @{text "s"} by the protocol
+ predicate @{text "step"}, then @{text "e#s"} is a new legal state rendered by the
+ happening of @{text "e"}:
+ \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 specific protocol specified by a @{text "step"}-predicate to get the set of
+ legal states of that particular protocol.
+ *}
+
+text {*
+ The following are two very basic properties of @{text "vt"}.
+*}
+
+lemma step_back_vt: "vt (e#s) \<Longrightarrow> vt s"
+ by(ind_cases "vt (e#s)", simp)
+
+lemma step_back_step: "vt (e#s) \<Longrightarrow> step s e"
+ by(ind_cases "vt (e#s)", simp)
+
+text {* \noindent
+ The following two auxiliary 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.
+ Notice how this definition is backed up by the @{text "release"} function and its use
+ in the @{text "V"}-branch of @{text "schs"} function. This @{text "next_th"} function
+ is not needed for the execution of PIP. It is introduced as an auxiliary function
+ to state lemmas. The correctness of this definition will be confirmed by
+ lemmas @{text "step_v_hold_inv"}, @{text " step_v_wait_inv"},
+ @{text "step_v_get_hold"} and @{text "step_v_not_wait"}.
+ *}
+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 aux 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 observation @{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 observation @{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
+(*>*)
+