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37\title{On the correctness of an assembler for the Intel MCS-51 microprocessor}
38\author{Jaap Boender \and Dominic P. Mulligan \and Claudio Sacerdoti Coen}
39\institute{Dipartimento di Scienze dell'Informazione, Universit\'a di Bologna}
40
41\bibliographystyle{splncs03}
42
43\begin{document}
44
45\maketitle
46
47\begin{abstract}
48We consider the formalisation of an assembler for the Intel MCS-51 8-bit microprocessor in the Matita proof assistant.
49This formalisation forms a major component of the EU-funded CerCo project, concering the construction and formalisation of a concrete complexity preserving compiler for a large subset of the C programming language.
50...
51\end{abstract}
52
53% ---------------------------------------------------------------------------- %
54% SECTION                                                                      %
55% ---------------------------------------------------------------------------- %
56\section{Introduction}
57\label{sect.introduction}
58
59We consider the formalisation of an assembler for the Intel MCS-51 8-bit microprocessor in the Matita proof assistant~\cite{asperti:user:2007}.
60This formalisation forms a major component of the EU-funded CerCo project~\cite{cerco:2011}, concering the construction and formalisation of a concrete complexity preserving compiler for a large subset of the C programming language.
61
62The MCS-51 dates from the early 1980s and commonly called the 8051/8052.
63Despite the microprocessor's age, derivatives are still widely manufactured by a number of semiconductor foundries.
64As a result the processor is widely used, especially in embedded systems development, where well-tested, cheap, predictable microprocessors find their niche.
65
66The MCS-51 has a relative paucity of features compared to its more modern brethren.
67In particular, the MCS-51 does not possess a cache or any instruction pipelining that would make predicting the concrete cost of executing a single instruction an involved process.
68Instead, each semiconductor foundry that produces an MCS-51 derivative is able to provide accurate timing information in clock cycles for each instruction in their derivative's instruction set.
69It is important to stress that this timing information, unlike in more sophisticated processors, is not an estimate, it is a definition.
70For the MCS-51, if a manufacturer states that a particular opcode takes three clock cycles to execute, then that opcode \emph{always} takes three clock cycles to execute.
71
72This predicability of timing information is especially attractive to the CerCo consortium.
73We are in the process of constructing a certified, concrete complexity compiler for a realistic processor, and not for building and formalising the worst case execution time (WCET) tools that would be necessary to achieve the same result with, for example, a modern ARM or PowerPC microprocessor.
74
75However, the MCS-51's paucity of features is a double edged sword.
76In particular, the MCS-51 features relatively miniscule memory spaces (including read-only code memory, stack and internal/external random access memory) by modern standards.
77As a result our compiler, to have any sort of hope of successfully compiling realistic C programs, ought to produce `tight' machine code.
78This is not simple.
79
80We here focus on a single issue in the MCS-51's instruction set: unconditional jumps.
81The MCS-51 features three conditional jump instructions: \texttt{LJMP} and \texttt{SJMP}---`long jump' and `short jump' respectively---and an 11-bit oddity of the MCS-51, \texttt{AJMP}, that the prototype CerCo compiler~\cite{cerco-report-code:2011} ignores for simplicity's sake.\footnote{Ignoring \texttt{AJMP} and its analogue \texttt{ACALL} is not idiosyncratic.  The Small Device C Compiler (SDCC)~\cite{sdcc:2011}, the leading open source C compiler for the MCS-51, also seemingly does not produce \texttt{AJMP} and \texttt{ACALL} instructions.  Their utility in a modern context remains unclear.}
82Each of these three instructions expects arguments in different sizes and behaves in different ways.
83\texttt{SJMP} may only perform a `local jump' whereas \texttt{LJMP} may jump to any address in the MCS-51's memory space.
84Consequently, the size of each opcode is different, and to squeeze as much code as possible into the MCS-51's limited code memory, the smallest possible opcode should be selected.
85
86The prototype CerCo C compiler does not attempt to select the smallest jump opcode in this manner, as this was thought to unneccessarily complicate the compilation chain.
87Instead, the compiler targets an assembly language, complete with pseudoinstructions including bespoke \texttt{Jmp} and \texttt{Call} instructions.
88Labels, conditional jumps to labels, a program preamble containing global data and a \texttt{MOV} instruction for moving this global data into the MCS-51's one 16-bit register also feature.
89This latter feature will ease any later consideration of separate compilation in the CerCo compiler.
90An assembler is used to expand pseudoinstructions into MCS-51 machine code.
91
92However, this assembly process is not trivial, for numerous reasons.
93For example, our conditional jumps to labels behave differently from their machine code counterparts.
94At the machine code level, conditional jumps may only jump to a relative offset, expressed in a byte, of the current program counter, limiting their range.
95However, at the assembly level, conditional jumps may jump to a label that appears anywhere in the program, significantly liberalising the use of conditional jumps and further simplifying the design of the CerCo compiler.
96
97Further, trying to na\"ively relate assembly programs with their machine code counterparts simply does not work.
98Machine code programs that fetch from code memory and programs that combine the program counter with constant shifts do not make sense at the assembly level.
99More generally, memory addresses can only be compared with other memory addresses.
100However, checking that memory addresses are only compared against each other at the assembly level is in fact undecidable.
101In short, the full preservation of the semantics of the two languages is impossible.
102
103Yet more complications are added by the peculiarities of the CerCo project itself.
104As mentioned, the CerCo consortium is in the business of constructing a verified compiler for the C programming language.
105However, unlike CompCert~\cite{compcert:2011,leroy:formal:2009,leroy:formally:2009}---which currently represents the state of the art for `industrial grade' verified compilers---CerCo considers not just the \emph{intensional correctness} of the compiler, but also its \emph{extensional correctness}.
106That is, CompCert focusses solely on the preservation of the \emph{meaning} of a program during the compilation process, guaranteeing that the program's meaning does not change as it is gradually transformed into assembly code.
107However in any realistic compiler (even the CompCert compiler!) there is no guarantee that the program's time properties are preserved during the compilation process; a compiler's `optimisations' could, in theory, even conspire to degrade the concrete complexity of certain classes of programs.
108CerCo aims to expand the current state of the art by producing a compiler where this temporal degradation is guaranteed not to happen.
109
110In order to achieve this CerCo imposes a cost model on programs, or more specifically, on simple blocks of instructions.
111This cost model is induced by the compilation process itself, and its non-compositional nature allows us to assign different costs to identical blocks of instructions depending on how they are compiled.
112In short, we aim to obtain a very precise costing for a program by embracing the compilation process, not ignoring it.
113This, however, complicates the proof of correctness for the compiler proper: for every translation pass from intermediate language to intermediate language, we must prove that not only has the meaning of a program been preserved, but also its complexity characteristics.
114This also applies for the translation from assembly language to machine code.
115
116How do we assign a cost to a pseudoinstruction?
117As mentioned, conditional jumps at the assembly level can jump to a label appearing anywhere in the program.
118However, at the machine code level, conditional jumps are limited to jumping `locally', using a measly byte offset.
119To translate a jump to a label, a single conditional jump pseudoinstruction may be translated into a block of three real instructions, as follows (here, \texttt{JZ} is `jump if accumulator is zero'):
120\begin{displaymath}
121\begin{array}{r@{\quad}l@{\;\;}l@{\qquad}c@{\qquad}l@{\;\;}l}
122       & \mathtt{JZ}  & label                      &                 & \mathtt{JZ}   & \text{size of \texttt{SJMP} instruction} \\
123       & \ldots       &                            & \text{translates to}   & \mathtt{SJMP} & \text{size of \texttt{LJMP} instruction} \\
124label: & \mathtt{MOV} & \mathtt{A}\;\;\mathtt{B}   & \Longrightarrow & \mathtt{LJMP} & \text{address of \textit{label}} \\
125       &              &                            &                 & \ldots        & \\
126       &              &                            &                 & \mathtt{MOV}  & \mathtt{A}\;\;\mathtt{B}
127\end{array}
128\end{displaymath}
129In the translation, if \texttt{JZ} fails, we fall through to the \texttt{SJMP} which jumps over the \texttt{LJMP}.
130Naturally, if \textit{label} is close enough, a conditional jump pseudoinstruction is mapped directly to a conditional jump machine instruction; the above translation only applies if \textit{label} is not sufficiently local.
131This leaves the problem, addressed below, of calculating whether a label is indeed `close enough' for the simpler translation to be used.
132
133Crucially, the above translation demonstrates the difficulty in predicting how many clock cycles a pseudoinstruction will take to execute.
134A conditional jump may be mapped to a single machine instruction or a block of three.
135Perhaps more insidious, the number of cycles needed to execute the instructions in the two branches of a translated conditional jump may be different.
136Depending on the particular MCS-51 derivative at hand, an \texttt{SJMP} could in theory take a different number of clock cycles to execute than an \texttt{LJMP}.
137These issues must also be dealt with in order to prove that the translation pass preserves the concrete complexity of the code.
138
139The question remains: how do we decide whether to expand a jump into an \texttt{SJMP} or an \texttt{LJMP}?
140To understand why this problem is not trivial, consider the following snippet of assembly code:
141\begin{displaymath}
142\text{dpm: finish me}
143\end{displaymath}
144
145As our example shows, given an occurence $l$ of an \texttt{LJMP} instruction, it may be possible to shrink $l$ to an occurence of an \texttt{SJMP} providing we can shrink any \texttt{LJMP}s that exist between $l$ and its target location.
146However, shrinking these \texttt{LJMP}s may in turn depend on shrinking $l$ to an \texttt{SJMP}, as it is perfectly possible to jump backwards.
147In short, unless we can somehow break this loop of circularity, and similar knotty configurations of jumps, we are stuck with a suboptimal solution to the expanding jumps problem.
148
149How we went about resolving this problem affected the shape of our proof of correctness for the whole assembler in a rather profound way.
150We first attempted to synthesize a solution bottom up: starting with no solution, we gradually refine a solution using the same functions that implement the jump expansion.
151Using this technique, solutions can fail to exist, and the proof quickly descends into a diabolical quagmire.
152
153Abandoning this attempt, we instead split the `policy'---the decision over how any particular jump should be expanded---from the implementation that actually expands assembly programs into machine code.
154Assuming the existence of a correct policy, we proved the implementation of the assembler correct.
155Further, we proved that the assembler fails to assemble an assembly program if and only if a correct policy does not exist.
156Policies do not exist in only a limited number of circumstances: namely, if a pseudoinstruction attempts to jump to a label that does not exist, or the program is too large to fit in code memory.
157The first case would constitute a serious compiler error, and hopefully certifying the rest of the compiler would rule this possibility out.
158The second case is unavoidable---certified compiler or not, trying to load a huge program into a small code memory will break \emph{something}.
159
160The rest of this paper is a detailed description of this proof.
161
162% ---------------------------------------------------------------------------- %
163% SECTION                                                                      %
164% ---------------------------------------------------------------------------- %
165\subsection{Overview of the paper}
166\label{subsect.overview.of.the.paper}
167In Section~\ref{sect.matita} we provide a brief overview of the Matita proof assistant for the unfamiliar reader.
168In Section~\ref{sect.the.proof} we discuss the design and implementation of the proof proper.
169In Section~\ref{sect.conclusions} we conclude.
170
171% ---------------------------------------------------------------------------- %
172% SECTION                                                                      %
173% ---------------------------------------------------------------------------- %
174\section{Matita}
175\label{sect.matita}
176
177Matita is a proof assistant based on the (Co)inductive Calculus of Constructions~\cite{asperti:user:2007}.
178For those familiar with Coq, Matita's syntax and mode of operation should be entirely familiar.
179We take time here to explain one of Matita's syntactic idiosyncracies, however.
180The use of `$\mathtt{?}$' or `$\mathtt{\ldots}$' in an argument position denotes a type or types to be inferred automatically by unification respectively.
181The use of `$\mathtt{?}$' in the body of a definition, lemma or axiom denotes an incomplete term that is to be closed, by hand, using tactics.
182
183% ---------------------------------------------------------------------------- %
184% SECTION                                                                      %
185% ---------------------------------------------------------------------------- %
186\section{The proof}
187\label{sect.the.proof}
188
189\subsection{The assembler and semantics of machine code}
190\label{subsect.the.assembler.and.semantics.of.machine.code}
191
192The formalisation in Matita of the semantics of MCS-51 machine code is described in~\cite{mulligan:executable:2011}.
193We merely describe enough here to understand the rest of the proof.
194
195At heart, the MCS-51 emulator centres around a \texttt{Status} record, describing the current state of the microprocessor.
196This record contains fields corresponding to the microprocessor's program counter, special function registers, and so on.
197At the machine code level, code memory is implemented as a trie of bytes, addressed by the program counter.
198Machine code programs are loaded into \texttt{Status} using the \texttt{load\_code\_memory} function.
199
200We may execut a single step of a machine code program using the \texttt{execute\_1} function, which returns an updated \texttt{Status}:
201\begin{lstlisting}
202definition execute_1: Status $\rightarrow$ Status := $\ldots$
203\end{lstlisting}
204The function \texttt{execute} allows one to execute an arbitrary, but fixed (due to Matita's normalisation requirement!) number of steps of a program.
205
206Naturally, assembly programs have analogues.
207The counterpart of the \texttt{Status} record is \texttt{PseudoStatus}.
208Instead of code memory being implemented as tries of bytes, code memory is here implemented as lists of pseudoinstructions, and program counters are merely indices into this list.
209Our analogue of \texttt{execute\_1} is \texttt{execute\_1\_pseudo\_instruction}:
210\begin{lstlisting}
211definition execute_1_pseudo_instruction: (Word $\rightarrow$ nat $\times$ nat) $\rightarrow$
212                                         PseudoStatus $\rightarrow$ PseudoStatus := $\ldots$
213\end{lstlisting}
214Notice, here, that the emulation function for pseudoprograms takes an additional argument.
215This is a function that maps program counters (for the pseudoprogram) to pairs of natural numbers representing the number of clock ticks that the pseudoinstruction needs to execute, post expansion.
216We call this function a \emph{costing}, and note that the costing is induced by the particular strategy we use to expand pseudoinstructions.
217If we change how we expand conditional jumps to labels, for instance, then the costing needs to change, hence \texttt{execute\_1\_pseudo\_instruction}'s parametricity in the costing.
218
219The costing returns \emph{pairs} of natural numbers because, in the case of expanding conditional jumps to labels, the expansion of the `true branch' and `false branch' may differ in the number of clock ticks needed for execution.
220This timing information is used inside \texttt{execute\_1\_pseudo\_instruction} to update the clock of the \texttt{PseudoStatus}.
221During the proof of correctness of the assembler we relate the clocks of \texttt{Status}es and \texttt{PseudoStatus}es.
222
223The assembler, mapping programs consisting of lists of pseudoinstructions to lists of bytes, operates in a mostly straightforward manner.
224To a degree of approximation, the assembler on an assembly program, consisting of $n$ pseudoinstructions $\mathtt{P_i}$ for $1 \leq i \leq n$, works as in the following diagram:
225\begin{displaymath}
226[\mathtt{P_1}, \ldots \mathtt{P_n}] \xrightarrow{\mathtt{flatten}\left(\mathtt{P_i} \xrightarrow{\mbox{\fontsize{7}{9}\selectfont$\mathtt{expand}$}} \mathtt{[I_1^i, \ldots I^q_i]} \xrightarrow{\mbox{\fontsize{7}{9}\selectfont$\mathtt{assembly1}^*$}} \mathtt{[0110]}\right)^{*}} \mathtt{[010101]}
227\end{displaymath}
228Here $\mathtt{I^i_j}$ for $1 \leq j \leq q$ are the $q$ machine code instructions obtained by expanding, with \texttt{expand\_pseudo\_instruction}, a single pseudoinstruction.
229Each machine code instruction $\mathtt{I^i_j}$ is then assembled, using the \texttt{assembly1} function, into a list of bytes.
230This process is iterated for each pseudoinstruction, before the lists are flattened into a single bit list representation of the original assembly program.
231
232By inspecting the above diagram, it would appear that the best way to proceed with a proof that the assembly process does not change the semantics of an assembly program is via a decomposition of the problem into two subproblems.
233Namely, we first expand any and all pseudoinstructions into lists of machine instructions, and provide a proof that this process does not change our program's semantics.
234Finally, we assemble all machine code instructions into machine code---lists of bytes---and prove once more that this process does not have an adverse effect on a program's semantics.
235By composition, we then have that the whole assembly process is semantics preserving.
236
237This is a tempting approach to the proof, but ultimately the wrong approach.
238In particular, it is important that we track how the program counter indexing into the assembly program, and the machine's program counter evolve, so that we can relate them.
239Expanding pseudoinstructions requires that the machine's program counter be incremented by $n$ steps, for $1 \leq n$, for every increment of the assembly program's program counter.
240Keeping track of the correspondence between the two program counters quickly becomes unfeasible using a compositional approach, and hence the proof must be monolithic.
241
242% ---------------------------------------------------------------------------- %
243% SECTION                                                                      %
244% ---------------------------------------------------------------------------- %
245\subsection{Policies}
246\label{subsect.policies}
247
248Policies exist to dictate how conditional and unconditional jumps at the assembly level should be expanded into machine code instructions.
249Using policies, we are able to completely decouple the decision over how jumps are expanded with the act of expansion, simplifying our proofs.
250As mentioned, the MCS-51 instruction set includes three different jump instructions: \texttt{SJMP}, \texttt{AJMP} and \texttt{LJMP}; call these `short', `medium' and `long' jumps, respectively:
251\begin{lstlisting}
252inductive jump_length: Type[0] :=
253  | short_jump: jump_length
254  | medium_jump: jump_length
255  | long_jump: jump_length.
256\end{lstlisting}
257A \texttt{jump\_expansion\_policy} is a map from \texttt{Word}s to \texttt{jump\_length}s, implemented as a trie.
258Intuitively, a policy maps positions in a program (indexed using program counters implemented as \texttt{Word}s) to a particular variety of jump.
259\begin{lstlisting}
260definition jump_expansion_policy := BitVectorTrie jump_length 16.
261\end{lstlisting}
262Next, we require a series of `sigma' functions.
263These functions map assembly program counters to their machine code counterparts, establishing the correspondence between `positions' in an assembly program and `positions' in a machine code program.
264At the heart of this process is \texttt{sigma0} which traverses an assembly program building maps from program counter to program counter.
265This function fails if and only if an internal call to \texttt{assembly\_1\_pseudoinstruction} fails:
266\begin{lstlisting}
267definition sigma0: pseudo_assembly_program
268  $\rightarrow$ option (nat $\times$ (nat $\times$ (BitVectorTrie Word 16))) := $\ldots$
269\end{lstlisting}
270We eventually lift this to functions from program counters to program counters:
271\begin{lstlisting}
272definition sigma_safe:
273  pseudo_assembly_program $\rightarrow$ option (Word $\rightarrow$ Word) := $\ldots$
274\end{lstlisting}
275Now, it's possible to define what a `good policy' is i.e. one that does not cause \texttt{sigma\_safe} to fail.
276As mentioned, \texttt{sigma\_safe} can only fail if an assembly program fails to be assembled:
277\begin{lstlisting}
278definition policy_ok := $\lambda$p. sigma_safe p $\neq$ None $\ldots$.
279\end{lstlisting}
280Finally, we obtain \texttt{sigma}, a map from program counters to program counters, which is guranteed not to fail as we internally provide a that
281\begin{lstlisting}
282definition sigma: pseudo_assembly_program $\rightarrow$ Word $\rightarrow$ Word := $\ldots$
283\end{lstlisting}
284
285% ---------------------------------------------------------------------------- %
286% SECTION                                                                      %
287% ---------------------------------------------------------------------------- %
288\subsection{Total correctness of the assembler}
289\label{subsect.total.correctness.of.the.assembler}
290
291Using our policies, we now work toward proving the total correctness of the assembler.
292By total correctness, we mean that the assembly process does not change the semantics of an assembly program.
293Naturally, this necessitates keeping some sort of correspondence between program counters at the assembly level, and program counters at the machine code level.
294For this, we use the \texttt{sigma} machinery defined at the end of Subsection~\ref{subsect.policies}.
295
296CSC: no options using policy
297\begin{lstlisting}
298lemma fetch_assembly_pseudo2: $\forall$program, assembled, costs, labels.
299  Some $\ldots$ $\langle$labels,costs$\rangle$ = build_maps program $\rightarrow$
300  Some $\ldots$ $\langle$assembled,costs$\rangle$ = assembly program $\rightarrow$ $\forall$ppc.
301  let code_memory := load_code_memory assembled in
302  let preamble := $\pi_1$ program in
303  let data_labels := construct_datalabels preamble in
304  let lk_labels :=
305    λx. sigma program (address_of_word_labels_code_mem ($\pi_2$ program) x) in
306  let lk_dlabels := λx. lookup ? ? x data_labels (zero ?) in
307  let expansion := jump_expansion ppc program in
308  let $\langle$pi,newppc$\rangle$ := fetch_pseudo_instruction ($\pi_2$ program) ppc in
309  let ppc' := sigma program ppc in
310  let newppc' := sigma program newppc in
311  let instructions' :=
312    expand_pseudo_instruction lk_labels lk_dlabels ppc' expansion pi in
313  let fetched := fetch_many code_memory newppc' ppc' instructions in
314    $\exists$instructions. Some ? instructions = instructions' $\wedge$ fetched.
315\end{lstlisting}
316
317We expand pseudoinstructions using the function \texttt{expand\_pseudo\_instruction}.
318This function accepts a `policy decision'---an element of type \texttt{jump\_length}---that is used when expanding a \texttt{Call}, \texttt{Jmp} or conditional jump to a label into the correct machine instruction.
319This \texttt{policy\_decision} is asssumed to originate from a policy as defined in Subsection~\ref{subsect.policies}.
320\begin{lstlisting}
321definition expand_pseudo_instruction:
322  ∀lookup_labels, lookup_datalabels, pc, policy_decision.
323    pseudo_instruction $\rightarrow$ option list instruction := $\ldots$
324\end{lstlisting}
325Under the assumption that a correct policy exists, \texttt{expand\_pseudo\_instruction} should never fail, and therefore the option type may be dispensed with.
326This is because the only failure conditions for \texttt{expand\_pseudo\_instruction} result from trying to expand a pseudoinstruction into an `impossible' combination of machine code instructions.
327For instance, if the policy decision dictates that we should expand a \texttt{Call} pseudoinstruction into a `short jump', then we fail, as the MCS-51's instruction set only features instructions \texttt{ACALL} and \texttt{LCALL}.
328
329\begin{lstlisting}
330axiom assembly_ok: ∀program,assembled,costs,labels.
331  Some $\ldots$ $\langle$labels, costs$\rangle$ = build_maps program $\rightarrow$
332  Some $\ldots$ $\langle$assembled, costs$\rangle$ = assembly program $\rightarrow$
333  let code_memory := load_code_memory assembled in
334  let preamble := $\pi_1$ program in
335  let datalabels := construct_datalabels preamble in
336  let lk_labels :=
337    $\lambda$x. sigma program (address_of_word_labels_code_mem ($\pi_2$ program) x) in
338  let lk_dlabels := $\lambda$x. lookup ? ? x datalabels (zero ?) in
339   ∀ppc,len,assembledi.
340    let $\langle$pi, newppc$\rangle$ := fetch_pseudo_instruction ($\pi_2$ program) ppc in
341    let assembly' := assembly_1_pseudoinstruction program ppc
342      (sigma program ppc) lk_labels lk_dlabels pi in
343    let newpc := (sigma program ppc) + len in
344    let echeck :=
345      encoding_check code_memory (sigma program ppc) slen assembledi in
346     Some $\ldots$ $\langle$len, assembledi$\rangle$ = assembly' $\rightarrow$
347      echeck $\wedge$ sigma program newppc = newpc.
348\end{lstlisting}
349
350Lemma \texttt{fetch\_assembly\_pseudo} establishes the correctness of expanding and then assembling pseudoinstructions:
351\begin{lstlisting}
352lemma fetch_assembly_pseudo: $\forall$program, ppc, lk_labels, lk_dlabels.
353  $\forall$pi, code_memory, len, assembled, instructions, pc.
354  let jexp := jump_expansion ppc program in
355  let exp :=
356    expand_pseudo_instruction lk_labels lk_dlabels pc jexp pi
357  let ass :=
358    assembly_1_pseudoinstruction program ppc pc lk_labels lk_dlabels pi in
359  Some ? instructions = exp $\rightarrow$
360    Some $\ldots$ $\langle$len, assembled$\rangle$ = ass $\rightarrow$
361      encoding_check code_memory pc (pc + len) assembled $\rightarrow$
362        fetch_many code_memory (pc + len) pc instructions.
363\end{lstlisting}
364Here, \texttt{len} is the number of machine code instructions the pseudoinstruction at hand has been expanded into, \texttt{encoding\_check} is a recursive function that checks for any possible corruption of the code memory, resulting from expanding the pseudoinstruction.
365The function \texttt{fetch\_many} fetches multiple machine code instructions from code memory and performs some routine checks.
366
367Intuitively, lemma \texttt{fetch\_assembly\_pseudo} can be read as follows.
368Suppose our policy \texttt{jump\_expansion} dictates that the pseudoinstruction indexed by the pseudo program counter \texttt{ppc} in assembly program \texttt{program} gives us the policy decision \texttt{jexp}.
369Further, suppose we expand the pseudoinstruction at \texttt{ppc} with the policy decision \texttt{jexp}, obtaining an (optional) list of machine code instructions \texttt{exp}.
370Suppose we also
371
372\begin{lstlisting}
373theorem fetch_assembly: $\forall$pc, i, cmem, assembled.
374  assembled = assembly1 i $\rightarrow$
375  let len := length $\ldots$ assembled in
376    encoding_check cmem pc (pc + len) assembled $\rightarrow$
377    let fetched := fetch code_memory (bitvector_of_nat $\ldots$ pc) in
378    let $\langle$instr_pc, ticks$\rangle$ := fetched in
379    let $\langle$instr, pc'$\rangle$ := instr_pc in
380      (eq_instruction instr i $\wedge$
381       eqb ticks (ticks_of_instruction instr) $\wedge$
382       eq_bv $\ldots$ pc' (pc + len)) = true.
383\end{lstlisting}
384
385An \texttt{internal\_pseudo\_address\_map} positions in the memory of a \texttt{PseudoStatus} with a physical memory address.
386\begin{lstlisting}
387definition internal_pseudo_address_map := list (BitVector 8).
388\end{lstlisting}
389We use \texttt{internal\_pseudo\_address\_map}s to convert the lower internal RAM of a \texttt{PseudoStatus} into the lower internal RAM of a \texttt{Status}.
390Notice, the MCS-51's internal RAM is addressed with a 7-bit `byte'.
391% dpm: ugly English, fix
392The whole of the internal RAM space is addressed with bytes: the first bit is used to distinguish between the programmer addressing low and high internal memory.
393\begin{lstlisting}
394axiom low_internal_ram_of_pseudo_low_internal_ram:
395  internal_pseudo_address_map $\rightarrow$ BitVectorTrie Byte 7 $\rightarrow$ BitVectorTrie Byte 7.
396\end{lstlisting}
397A similar axiom exists for high internal RAM.
398
399Next, we are able to translate \texttt{PseudoStatus} records into \texttt{Status} records using \texttt{status\_of\_pseudo\_status}.
400Translating a \texttt{PseudoStatus}'s code memory requires we expand pseudoinstructions and then assemble to obtain a trie of bytes.
401This can fail, as mentioned, in a limited number of situations, related to improper use of labels in an assembly program.
402However, it is possible to `tighten' the type of \texttt{status\_of\_pseudo\_status}, removing the option type, by using the fact that if any `good policy' exists, assembly will never fail.
403\begin{lstlisting}
404definition status_of_pseudo_status:
405 internal_pseudo_address_map → PseudoStatus → option Status
406\end{lstlisting}
407After fetching an assembly instruction we must update any \texttt{internal\_pseudo\hyp{}\_address\_map}s that may be laying around.
408This is done with the following function:
409\begin{lstlisting}
410definition next_internal_pseudo_address_map: internal_pseudo_address_map
411  $\rightarrow$ PseudoStatus $\rightarrow$ option internal_pseudo_address_map
412\end{lstlisting}
413Finally, we are able to state and prove our main theorem.
414This relates the execution of a single assembly instruction and the execution of (possibly) many machine code instructions.
415That is, the assembly process preserves the semantics of an assembly program, as it is translated into machine code:
416\begin{lstlisting}
417theorem main_thm:
418  ∀M,M',ps,s,s''.
419    next_internal_pseudo_address_map M ps = Some $\ldots$ M' $\rightarrow$
420      status_of_pseudo_status M ps = Some $\ldots$ s $\rightarrow$
421        status_of_pseudo_status M'
422          (execute_1_pseudo_instruction
423            (ticks_of (code_memory $\ldots$ ps)) ps) = Some $\ldots$ s'' $\rightarrow$
424              $\exists$n. execute n s = s''.
425\end{lstlisting}
426The statement can be given an intuitive reading as follows.
427Suppose our \texttt{PseudoStatus} $ps$ can be successfully converted into a \texttt{Status} $s$.
428Suppose further that, after executing a single assembly instruction and converting the resulting \texttt{PseudoStatus} into a \texttt{Status}, we obtain $s''$, being careful to track the number of ticks executed.
429Then, there exists some number $n$, so that executing $n$ machine code instructions in \texttt{Status} $s$ gives us \texttt{Status} $s''$.
430Theorem \texttt{main\_thm} establishes the correctness of the assembly process.
431
432% ---------------------------------------------------------------------------- %
433% SECTION                                                                      %
434% ---------------------------------------------------------------------------- %
435\section{Conclusions}
436\label{sect.conclusions}
437
438We have proved the total correctness of an assembler for MCS-51 assembly language.
439In particular, our assembly language featured labels, arbitrary conditional and unconditional jumps to labels, global data and instructions for moving this data into the MCS-51's single 16-bit register.
440
441\subsection{Use of dependent types and Russell}
442\label{subsect.use.of.dependent.types.and.Russell}
443
444Our formalisation makes sparing use of dependent types.
445In certain datastructures, such as tries and vectors, they are used to guarantee invariants.
446However, we have currently shyed away from making extensive use of dependent types and inductive predicates in the proof of correctness for the assembler itself.
447This is because complex dependent types and inductive predicates tend not to co\"operate particularly well with tactics such as inversion.
448
449However, there are certain cases where the use of dependent types is unavoidable.
450For instance, when proving that the \texttt{build\_maps} function is correct, a function that collates the cost and data labels of an assembly program into map datastructures.
451In cases such as these we make use of Matita's implementation of Russell~\cite{}.
452
453\subsection{Related work}
454\label{subsect.related.work}
455
456\bibliography{cpp-2011.bib}
457
458\end{document}\renewcommand{\verb}{\lstinline}
459\def\lstlanguagefiles{lst-grafite.tex}
460\lstset{language=Grafite}
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