semantics of a process \<open>P\<close> encompasses all possible behaviours of this process in the context of all
possible environments \<open>P \<lbrakk>S\<rbrakk> Env\<close> (where \<open>S\<close> is the set of \<open>atomic events\<close> both \<open>P\<close> and \<open>Env\<close> must
synchronize). This design objective has the consequence that two kinds of choice have to
be distinguished:
\<^enum> the \<^emph>\<open>external choice\<close>, written \<open>_\<box>_\<close>, which forces a process "to follow" whatever
the environment offers, and
\<^enum> the \<^emph>\<open>internal choice\<close>, written \<open>_\<sqinter>_\<close>, which imposes on the environment of a process
Generalizations of these two operators \<open>\<box>x\<in>A. P(x)\<close> and \<open>\<Sqinter>x\<in>A. P(x)\<close> allow for modeling the concepts
of \<^emph>\<open>input\<close> and \<^emph>\<open>output\<close>: Based on the prefix operator \<open>a\<rightarrow>P\<close> (event \<open>a\<close> happens, then the process
proceeds with \<open>P\<close>), receiving input is modeled by \<open>\<box>x\<in>A. x\<rightarrow>P(x)\<close> while sending output is represented
by \<open>\<Sqinter>x\<in>A. x\<rightarrow>P(x)\<close>. Setting choice in the center of the language semantics implies that
deadlock-freeness becomes a vital property for the well-formedness of a process, nearly as vital
as type-checking: Consider two events \<open>a\<close> and \<open>b\<close> not involved in a process \<open>P\<close>, then
\<open>(a\<rightarrow>P \<box> b\<rightarrow>P) \<lbrakk>{a,b}\<rbrakk> (a\<rightarrow>P \<sqinter> b\<rightarrow>P)\<close> is deadlock free provided \<open>P\<close> is, while
\<open>(a\<rightarrow>P \<sqinter> b\<rightarrow>P) \<lbrakk>{a,b}\<rbrakk> (a\<rightarrow>P \<sqinter> b\<rightarrow>P)\<close> deadlocks (both processes can make "ruthlessly" an opposite choice,
but are required to synchronize).
Verification of \<^csp> properties has been centered around the notion of \<^emph>\<open>process refinement orderings\<close>,
most notably \<open>_\<sqsubseteq>\<^sub>F\<^sub>D_\<close> and \<open>_\<sqsubseteq>_\<close>. The latter turns the denotational domain of \<^csp> into a Scott cpo
@{cite "scott:cpo:1972"}, which yields semantics for the fixed point operator \<open>\<mu>x. f(x)\<close> provided
that \<open>f\<close> is continuous with respect to \<open>_\<sqsubseteq>_\<close>. Since it is possible to express deadlock-freeness and
livelock-freeness as a refinement problem, the verification of properties has been reduced
traditionally to a model-checking problem for finite set of events \<open>A\<close>.
We are interested in verification techniques for arbitrary event sets \<open>A\<close> or arbitrarily
parameterized processes. Such processes can be used to model dense-timed processes, processes
with dynamic thread creation, and processes with unbounded thread-local variables and buffers.
However, this adds substantial complexity to the process theory: when it comes to study the
interplay of different denotational models, refinement-orderings, and side-conditions for
continuity, paper-and-pencil proofs easily reach their limits of precision.
Several attempts have been undertaken to develop a formal theory in an interactive proof system,
mostly in Isabelle/HOL @{cite "Camilleri91" and "tej.ea:corrected:1997" and "IsobeRoggenbach2010"
and "DBLP:journals/afp/Noce16"}.
This paper is based on @{cite "tej.ea:corrected:1997"}, which has been the most comprehensive
attempt to formalize denotational \<^csp> semantics covering a part of Bill Roscoe's Book
@{cite "roscoe:csp:1998"}. Our contributions are as follows:
% @{cartouche [display] \<open>C \<Longrightarrow> A \<sqsubseteq>\<^sub>F\<^sub>D A' \<Longrightarrow> B \<sqsubseteq>\<^sub>F\<^sub>D B' \<Longrightarrow> A \<lbrakk>S\<rbrakk> B \<sqsubseteq>\<^sub>F\<^sub>D A' \<lbrakk>S\<rbrakk> B'\<close>}
% \end{minipage}
% \end{center}
% are of particular interest since they allow to avoid the costly automata-product construction
% of model-checkers and to separate infinite sub-systems from finite (model-checkable) ones; however,
% their side-conditions \<open>C\<close> are particularly tricky to work out. Decomposition rules may pave the
% way for future tool combinations for model-checkers such as FDR4~@{cite "fdr4"} or
% PAT~@{cite "SunLDP09"} based on proof certifications.*)
text\<open>These two processes \<open>P\<^sub>d\<^sub>e\<^sub>t\<close> and \<open>P\<^sub>n\<^sub>d\<^sub>e\<^sub>t\<close> cannot be distinguished by using
the trace semantics: \<open>\<T>(P\<^sub>d\<^sub>e\<^sub>t) = \<T>(P\<^sub>n\<^sub>d\<^sub>e\<^sub>t) = {[],[a],[b]}\<close>. To resolve this problem, Brookes @{cite "BrookesHR84"}
proposed the failures model, where communication traces were augmented with the
constraint information for further communication that is represented negatively as a refusal set.
A failure \<open>(t, X)\<close> is a pair of a trace \<open>t\<close> and a set of events \<open>X\<close> that a process can refuse if
any of the events in \<open>X\<close> were offered to him by the environment after performing the trace \<open>t\<close>.
The semantic function \<open>\<F>\<close> in the failures model maps a process to a set of refusals.
Let \<open>\<Sigma>\<close> be the set of events. Then, \<open>{([],\<Sigma>)} \<subseteq> \<F> Stop\<close> as the process \<open>Stop\<close> refuses all events.
For Example 1, we have \<open>{([],\<Sigma>\{a,b}),([a],\<Sigma>),([b],\<Sigma>)} \<subseteq> \<F> P\<^sub>d\<^sub>e\<^sub>t\<close>, while
\<open>{([],\<Sigma>\{a}),([],\<Sigma>\{b}),([a],\<Sigma>),([b],\<Sigma>)} \<subseteq> \<F> P\<^sub>n\<^sub>d\<^sub>e\<^sub>t\<close> (the \<open>_\<subseteq>_\<close> refers to the fact that
the refusals must be downward closed; we show only the maximal refusal sets here).
Thus, internal and external choice, also called \<^emph>\<open>nondeterministic\<close> and \<^emph>\<open>deterministic\<close>
choice, can be distinguished in the failures semantics.
However, it turns out that the failures model suffers from another deficiency with respect to
the phenomenon called infinite internal chatter or \<^emph>\<open>divergence\<close>.\<close>
text\<open> First, we need a slight revision of the concept
of \<^emph>\<open>trace\<close>: if \<open>\<Sigma>\<close> is the type of the atomic events (represented by a type variable), then
we need to extend this type by a special event \<open>\<surd>\<close> (called "tick") signaling termination.
Thus, traces have the type \<open>(\<Sigma>+\<surd>)\<^sup>*\<close>, written \<open>\<Sigma>\<^sup>\<surd>\<^sup>*\<close>; since \<open>\<surd>\<close> may only occur at the end of a trace,
we need to define a predicate \<open>front\<^sub>-tickFree t\<close> that requires from traces that \<open>\<surd>\<close> can only occur
at the end.
Second, in the traditional literature, the semantic domain is implicitly described by 9 "axioms"
over the three semantic functions \<open>\<T>\<close>, \<open>\<F>\<close> and \<open>\<D>\<close>.
Therefore third, we define a pre-type for processes \<open>\<Sigma> process\<^sub>0\<close> by \<open> \<P>(\<Sigma>\<^sup>\<surd>\<^sup>* \<times> \<P>(\<Sigma>\<^sup>\<surd>)) \<times> \<P>(\<Sigma>\<^sup>\<surd>)\<close>.
Forth, we turn our wishlist of "axioms" above into the definition of a predicate \<open>is_process P\<close>
of type \<open>\<Sigma> process\<^sub>0 \<Rightarrow> bool\<close> deciding if its conditions are fulfilled. Since \<open>P\<close> is a pre-process,
we replace \<open>\<F>\<close> by \<open>fst\<close> and \<open>\<D>\<close> by \<open>snd\<close> (the HOL projections into a pair).
but this can be constructed in a straight-forward manner. Suitable definitions for
\<open>\<T>\<close>, \<open>\<F>\<close> and \<open>\<D>\<close> lifting \<open>fst\<close> and \<open>snd\<close> on the new \<open>'\<alpha> process\<close>-type allows to derive
text\<open> Now, the operators of \<^csp> \<open>Skip\<close>, \<open>Stop\<close>, \<open>_\<sqinter>_\<close>, \<open>_\<box>_\<close>, \<open>_\<rightarrow>_\<close>,\<open>_\<lbrakk>_\<rbrakk>_\<close> etc.
for internal choice, external choice, prefix and parallel composition, can
be defined indirectly on the process-type. For example, for the simple case of the internal choice,
we construct it such that \<open>_\<sqinter>_\<close> has type \<open>'\<alpha> process \<Rightarrow> '\<alpha> process \<Rightarrow> '\<alpha> process\<close> and
such that its projection laws satisfy the properties \<open>\<F> (P \<sqinter> Q) = \<F> P \<union> \<F> Q\<close> and
\<open>\<D> (P \<sqinter> Q) = \<D> P \<union> \<D> Q\<close> required from @{cite "roscoe:csp:1998"}.
This boils down to a proof that an equivalent definition on the pre-process type \<open>\<Sigma> process\<^sub>0\<close>
maintains \<open>is_process\<close>, \<^ie> this predicate remains invariant on the elements of the semantic domain.
For example, we define \<open>_\<sqinter>_\<close> on the pre-process type as follows:
\<^item> \<^theory_text>\<open>definition "P \<sqinter> Q \<equiv> Abs_process(\<F> P \<union> \<F> Q , \<D> P \<union> \<D> Q)"\<close>
\<open>Abs_process\<close> are the representation and abstraction morphisms resulting from the
type definition linking \<open>'\<alpha> process\<close> isomorphically to \<open>'\<alpha> process\<^sub>0\<close>. Proving the above properties
for \<open>\<F> (P \<sqinter> Q)\<close> and \<open>\<D> (P \<sqinter> Q)\<close> requires a proof that \<open>(\<F> P \<union> \<F> Q , \<D> P \<union> \<D> Q)\<close>
satisfies the 9 "axioms", which is fairly simple in this case.
The definitional presentation of the \<^csp> process operators according to @{cite "roscoe:csp:1998"}
of recursive equations is considered as keystone. Its prerequisite is a complete partial ordering
\<open>_\<sqsubseteq>_\<close>. The natural candidate \<open>_\<sqsubseteq>\<^sub>\<F>\<^sub>\<D>_\<close> is unfortunately not complete for infinite \<open>\<Sigma>\<close> for the
generalized deterministic choice, and thus for the building block of the read-operations.
Roscoe and Brooks @{cite "Roscoe1992AnAO"} finally proposed another ordering, called the
\<^emph>\<open>process ordering\<close>, and restricted the generalized deterministic choice in a particular way such
that completeness could at least be assured for read-operations. This more complex ordering
is based on the concept \<^emph>\<open>refusals after\<close> a trace \<open>s\<close> and defined by \<open>\<R> P s \<equiv> {X | (s, X) \<in> \<F> P}\<close>.\<close>
text\<open>While the original work @{cite "tej.ea:corrected:1997"} was based on an own --- and different ---
fixed-point theory, we decided to base HOL-\<^csp> 2 on HOLCF (initiated by @{cite "muller.ea:holcf:1999"}
and substantially extended in @{cite "huffman.ea:axiomatic:2005"}).
HOLCF is based on parametric polymorphism with type classes. A type class is actually a
constraint on a type variable by respecting certain syntactic and semantics
requirements. For example, a type class of partial ordering, denoted by \<open>\<alpha>::po\<close>, is restricted to
all types \<open>\<alpha>\<close> possessing a relation \<open>\<le>:\<alpha>\<times>\<alpha>\<rightarrow>bool\<close> that is reflexive, anti-symmetric, and transitive.
Isabelle possesses a construct that allows to establish, that the type \<open>nat\<close> belongs to this class,
with the consequence that all lemmas derived abstractly on \<open>\<alpha>::po\<close> are in particular applicable on
\<open>nat\<close>. The type class of \<open>po\<close> can be extended to the class of complete partial ordering \<open>cpo\<close>.
A \<open>po\<close> is said to be complete if all non-empty directed sets have a least upper bound (\<open>lub\<close>).
Finally the class of \<open>pcpo\<close> (Pointed cpo) is a \<open>cpo\<close> ordering that has a least element,
denoted by \<open>\<bottom>\<close>. For \<open>pcpo\<close> ordering, two crucial notions for continuity (\<open>cont\<close>) and fixed-point operator
(\<open>\<mu>X. f(X)\<close>) are defined in the usual way. A function from one \<open>cpo\<close> to another one is said
to be continuous if it distributes over the \<open>lub\<close> of all directed sets (or chains).
One key result of the fixed-point theory is the proof of the fixed-point theorem:
@{cartouche [display, indent=5] \<open>cont f \<Longrightarrow> adm P \<Longrightarrow> P \<bottom> \<Longrightarrow> (\<And>X. P X \<Longrightarrow> P(f X)) \<Longrightarrow> P(\<mu>X. f X)\<close>}
Fixed-point induction requires a small side-calculus for establishing the admissibility
of a predicate; basically, predicates are admissible if they are valid for any least upper bound
of a chain \<open>x\<^sub>1 \<sqsubseteq> x\<^sub>2 \<sqsubseteq> x\<^sub>3 ... \<close> provided that \<open>\<forall>i. P(x\<^sub>i)\<close>. It turns out that \<open>_\<sqsubseteq>_\<close> and \<open>_\<sqsubseteq>\<^sub>F\<^sub>D_\<close> as
well as all other refinement orderings that we introduce in this paper are admissible.
Fixed-point inductions are the main proof weapon in verifications,
together with monotonicities and the \<^csp> laws. Denotational arguments can be hidden as they are not
In our framework, we implemented the pcpo process refinement together with the five refinement
orderings introduced in @{technical "orderings"}. To enable fixed-point induction, we first have
the admissibility of the refinements.
@{cartouche [display, indent=7] \<open>cont u \<Longrightarrow> mono v \<Longrightarrow> adm(\<lambda>x. u x \<sqsubseteq>\<^sub>\<FF> v x) where \<FF>\<in>{\<T>,\<F>,\<D>,\<T>\<D>,\<F>\<D>}\<close>}
\<^item> External choice is not monotonic only under \<open>\<sqsubseteq>\<^sub>\<F>\<close>, with the following monotonicities proved:
@{cartouche [display,indent=5]
\<open>P \<sqsubseteq>\<^sub>\<FF> P' \<Longrightarrow> Q \<sqsubseteq>\<^sub>\<FF> Q' \<Longrightarrow> (P \<box> Q) \<sqsubseteq>\<^sub>\<FF> (P' \<box> Q') where \<FF>\<in>{\<T>,\<D>,\<T>\<D>,\<F>\<D>}\<close>}
\<^item> Sequence operator is not monotonic under \<open>\<sqsubseteq>\<^sub>\<F>\<close>, \<open>\<sqsubseteq>\<^sub>\<D>\<close> or \<open>\<sqsubseteq>\<^sub>\<T>\<close>:
%All refinements are right-side monotonic but \<open>\<sqsubseteq>\<^sub>\<F>\<close>, \<open>\<sqsubseteq>\<^sub>\<D>\<close> and \<open>\<sqsubseteq>\<^sub>\<T>\<close> are not left-side monotonic,
%which can be explained by
%the interdependence relationship of failure and divergence projections for the first component.
%We thus proved:
\<^item> Hiding operator is not monotonic under \<open>\<sqsubseteq>\<^sub>\<D>\<close>:
@{cartouche [display,indent=5] \<open>P \<sqsubseteq>\<^sub>\<FF> Q \<Longrightarrow> P \ A \<sqsubseteq>\<^sub>\<FF> Q \ A where \<FF>\<in>{\<T>,\<F>,\<T>\<D>,\<F>\<D>}\<close>}
%Intuitively, for the divergence refinement of the hiding operator, there may be
%some trace \<open>s\<in>\<T> Q\<close> and \<open>s\<notin>\<T> P\<close> such that it becomes divergent in \<open>Q \ A\<close> but
%not in \<open>P \ A\<close>.
%when the condition in the corresponding projection laws is satisfied, which makes it is not monotonic.
\<^item> Parallel composition is not monotonic under \<open>\<sqsubseteq>\<^sub>\<F>\<close>, \<open>\<sqsubseteq>\<^sub>\<D>\<close> or \<open>\<sqsubseteq>\<^sub>\<T>\<close>:
@{cartouche [display,indent=5] \<open>P \<sqsubseteq>\<^sub>\<FF> P' \<Longrightarrow> Q \<sqsubseteq>\<^sub>\<FF> Q' \<Longrightarrow> (P \<lbrakk>A\<rbrakk> Q) \<sqsubseteq>\<^sub>\<FF> (P' \<lbrakk>A\<rbrakk> Q') where \<FF>\<in>{\<T>\<D>,\<F>\<D>}\<close>}
%The failure and divergence projections of this operator are also interdependent, similar to the
%sequence operator.
%Hence, this operator is not monotonic with \<open>\<sqsubseteq>\<^sub>\<F>\<close>, \<open>\<sqsubseteq>\<^sub>\<D>\<close> and \<open>\<sqsubseteq>\<^sub>\<T>\<close>, but monotonic when their
(* Besides the monotonicity results on the above \<^csp> operators,
we have also proved that for other \<^csp> operators, such as multi-prefix and non-deterministic choice,
they are all monotonic with these five refinement orderings. Such theoretical results provide significant indicators
for semantics choices when considering specification decomposition.
We want to emphasize that this is the first work on such substantial
analysis in a formal way, as far as we know.
%In the literature, these processes are defined in a way that does not distinguish the special event \<open>tick\<close>. To be consistent with the idea that ticks should be distinguished on the semantic level, besides the above
three processes,
one can directly prove 3 since for both \<open>CHAOS\<close> and \<open>DF\<close>,
the version with \<open>SKIP\<close> is constructed exactly in the same way from that without \<open>SKIP\<close>.
And 4 is obtained based on the projection laws of internal choice \<open>\<sqinter>\<close>.
Finally, for 5, the difference between \<open>DF\<close> and \<open>RUN\<close> is that the former applies internal choice
while the latter with external choice. From the projection laws of both operators,
the failure set of \<open>RUN\<close> has more constraints, thus being a subset of that of \<open>DF\<close>,
when the divergence set is empty, which is true for both processes.
We now present reference processes that exhibit basic behaviors, introduced in
fundamental \<^csp> works @{cite "Roscoe:UCS:2010"}. The process \<open>RUN A\<close> always
accepts events from \<open>A\<close> offered by the environment. The process \<open>CHAOS A\<close> can always choose to
accept or reject any event of \<open>A\<close>. The process \<open>DF A\<close> is the most non-deterministic deadlock-free
process on \<open>A\<close>, \<^ie>, it can never refuse all events of \<open>A\<close>.
To handle termination better, we added two new processes \<open>CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P\<close> and \<open>DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P\<close>.
%Note that we do not redefine \<open>RUN\<close> with \<open>SKIP\<close> because this process is supposed to never terminate,
Definition*[X4]\<open>\<open>CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<equiv> \<mu> X. (SKIP \<sqinter> STOP \<sqinter> (\<box> x \<in> A \<rightarrow> X))\<close>\<close>
Definition*[X5]\<open>\<open>DF A \<equiv> \<mu> X. (\<sqinter> x \<in> A \<rightarrow> X)\<close> \<close>
Definition*[X6]\<open>\<open>DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<equiv> \<mu> X. ((\<sqinter> x \<in> A \<rightarrow> X) \<sqinter> SKIP)\<close> \<close>
Last, regarding trace refinement, for any process P,
its set of traces \<open>\<T> P\<close> is a subset of \<open>\<T> (CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV)\<close> and of \<open>\<T> (DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV)\<close> as well.
%As we already proved that \<open>CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P\<close> covers all failures,
%we can immediately infer that it also covers all traces.
%The \<open>DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P\<close> case requires a longer denotational proof.
text\<open>\<^noindent> A process \<open>P\<close> is deadlock-free if and only if after any trace \<open>s\<close> without \<open>\<surd>\<close>, the union of \<open>\<surd>\<close>
and all events of \<open>P\<close> can never be a refusal set associated to \<open>s\<close>, which means that \<open>P\<close> cannot
\<^enum> \<open>livelock\<^sub>-free P \<longleftrightarrow> \<PP> UNIV \<sqsubseteq>\<^sub>\<D> P where \<PP> \<in> \<R>\<P>\<close>
\<^enum> @{cartouche [display]\<open>livelock\<^sub>-free P \<longleftrightarrow> DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV \<sqsubseteq>\<^sub>\<T>\<^sub>\<D> P
to prove that \<open>SYSTEM\<close> refines \<open>COPY\<close> using the \<open>pcpo\<close> process ordering \<open>\<sqsubseteq>\<close> that implies all other
\<open>base: \<bottom> \<sqsubseteq> Q\<close> and \<open>1-ind: X \<sqsubseteq> Q \<Longrightarrow> (_ \<rightarrow> X) \<sqsubseteq> Q\<close>. Now, if unfolding the fixed-point process \<open>Q\<close>
reveals two steps, the second goal becomes
\<open>X \<sqsubseteq> Q \<Longrightarrow> _ \<rightarrow> X \<sqsubseteq> _ \<rightarrow> _ \<rightarrow> Q\<close>. Unfortunately, this way, it becomes improvable
using monotonicities rules.
We need here a two-step induction of the form \<open>base0: \<bottom> \<sqsubseteq> Q\<close>, \<open>base1: _ \<rightarrow> \<bottom> \<sqsubseteq> Q\<close> and
\<open>2-ind: X \<sqsubseteq> Q \<Longrightarrow> _ \<rightarrow> _ \<rightarrow> X \<sqsubseteq> _ \<rightarrow> _ \<rightarrow> Q\<close> to have a sufficiently powerful induction scheme.
For this reason, we derived a number of alternative induction schemes (which are not available
in the HOLCF library), which are also relevant for our final Dining Philophers example.
These are essentially adaptions of k-induction schemes applied to domain-theoretic
setting (so: requiring \<open>f\<close> continuous and \<open>P\<close> admissible; these preconditions are
skipped here):
\<^item> @{cartouche [display]\<open>... \<Longrightarrow> \<forall>i<k. P (f\<^sup>i \<bottom>) \<Longrightarrow> (\<forall>X. (\<forall>i<k. P (f\<^sup>i X)) \<longrightarrow> P (f\<^sup>k X))
\<Longrightarrow> P (\<mu>X. f X)\<close>}
\<^item> \<open>... \<Longrightarrow> \<forall>i<k. P (f\<^sup>i \<bottom>) \<Longrightarrow> (\<forall>X. P X \<longrightarrow> P (f\<^sup>k X)) \<Longrightarrow> P (\<mu>X. f X)\<close>
\<^item> @{cartouche [display]\<open>... \<Longrightarrow> P \<bottom> \<bottom> \<Longrightarrow> (\<forall>X Y. P X Y \<Longrightarrow> P (f X) (g Y))
\<Longrightarrow> P (\<mu>X. f X) (\<mu>X. g X)\<close>}
\<^noindent> This form does not help in cases like in \<open>P \<lbrakk>\<emptyset>\<rbrakk> Q \<sqsubseteq> S\<close> with the interleaving operator on the
Here, \<open>(f X \<lbrakk>\<emptyset>\<rbrakk> g Y)\<close> does not reduce to the \<open>(X \<lbrakk>\<emptyset>\<rbrakk> Y)\<close> term but to two terms \<open>(f X \<lbrakk>\<emptyset>\<rbrakk> Y)\<close> and
\<open>(X \<lbrakk>\<emptyset>\<rbrakk> g Y)\<close>.
To handle these cases, we developed an advanced parallel induction scheme and we proved its
\<open>definition trans\<^sub>f i \<sigma> \<equiv> if \<sigma> = 0 then {picks i i, picks (i+1)%N i}
else if \<sigma> = 1 then {putsdown i i}
else if \<sigma> = 2 then {putsdown (i+1)%N i}
else {}
definition upd\<^sub>f i \<sigma> e \<equiv> if e = (picks i i) then 1
else if e = (picks (i+1)%N) i then 2
else 0
definition FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i \<equiv> P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>trans\<^sub>f i, upd\<^sub>f i\<rbrakk> \<close>}
To validate our choice for the states, transition function \<open>trans\<^sub>f\<close> and update function \<open>upd\<^sub>f\<close>,
we prove that they are equivalent to the original process components: \<open>FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i = FORK i\<close>.
The anti-symmetry of refinement breaks this down to the two refinement proofs \<open>FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i \<sqsubseteq> FORK i\<close>
and \<open>FORK i \<sqsubseteq> FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i\<close>, which are similar to the CopyBuffer example shown in
@{technical "illustration"}. Note, again, that this fairly automatic induct-simplify-proof just
involves reasoning on the derived algebraic rules, not any reasoning on the level of the
denotational semantics.
%Second we prove that the normal form process is equivalent to the original fork process
%by proving refinements in both directions. We note here that the first refinement \<open>FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i \<sqsubseteq> FORK i\<close>
%requires a two steps induction as unfolding the original fixed-point process brings two steps
%\<open>FORK i = picks \<rightarrow> putsdown \<rightarrow> FORK i\<close>. After that we apply the same method
%to get the philosopher process under a normal form.