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37\title{On the correctness of an optimising assembler for the Intel MCS-51 microprocessor\thanks{The project CerCo acknowledges the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number: 243881.}}
38\author{Dominic P. Mulligan \and Claudio Sacerdoti Coen}
39\institute{Dipartimento di Scienze dell'Informazione,\\ Universit\'a degli Studi di Bologna}
40
41\bibliographystyle{splncs03}
42
43\begin{document}
44
45\maketitle
46
47\begin{abstract}
48We present a proof of correctness in Matita for an optimising assembler for the MCS-51 microcontroller.
49The efficient expansion of pseudoinstructions, namely jumps, into machine instructions is complex.
50We isolate the decision making over how jumps should be expanded from the expansion process itself as much as possible using `policies', making the proof of correctness for the assembler more straightforward.
51
52%We observe that it is impossible for an optimising assembler to preserve the semantics of every assembly program.
53%Assembly language programs can manipulate concrete addresses in arbitrary ways.
54Our proof strategy contains a tracking facility for `good addresses' and only programs that use good addresses have their semantics preserved under assembly, as we observe that it is impossible for an assembler to preserve the semantics of every assembly program.
55Our strategy offers increased flexibility over the traditional approach to proving the correctness of assemblers, wherein addresses in assembly are kept opaque and immutable.
56In particular, we may experiment with allowing the benign manipulation of addresses.
57\keywords{Verified software, CerCo (Certified Complexity), MCS-51 microcontroller, Matita proof assistant}
58\end{abstract}
59
60% ---------------------------------------------------------------------------- %
61% SECTION                                                                      %
62% ---------------------------------------------------------------------------- %
63\section{Introduction}
64\label{sect.introduction}
65
66We consider the formalisation of an assembler for the Intel MCS-51 8-bit microprocessor in the Matita proof assistant~\cite{asperti:user:2007}.
67This formalisation forms a major component of the EU-funded CerCo (`Certified Complexity') project~\cite{cerco:2011}, concerning the construction and formalisation of a concrete complexity preserving compiler for a large subset of the C programming language.
68
69The MCS-51 dates from the early 1980s and is commonly called the 8051/8052.
70Derivatives are still widely manufactured by a number of semiconductor foundries, with the processor being used especially in embedded systems.
71
72The MCS-51 has a relative paucity of features compared to its more modern brethren, with the lack of any caching or pipelining features meaning that timing of execution is predictable, making the MCS-51 very attractive for CerCo's ends.
73However, the MCS-51's paucity of features---though an advantage in many respects---also quickly becomes a hindrance, as the MCS-51 features a relatively minuscule series of memory spaces by modern standards.
74As a result our C compiler, to be able to successfully compile realistic programs for embedded devices, ought to produce `tight' machine code.
75
76To do this, we must solve the `branch displacement' problem---deciding how best to expand pseudojumps to labels in assembly language to machine code jumps.
77The branch displacement problem arises when pseudojumps can be expanded
78in different ways to real machine instructions, but the different expansions
79are not equivalent (e.g. differ in size or speed) and not always
80correct (e.g. correctness is only up to global constraints over the compiled
81code). For instance, some jump instructions (short jumps) are very small
82and fast, but they can only reach destinations within a
83certain distance from the current instruction. When the destinations are
84too far away, larger and slower long jumps must be used. The use of a long jump may
85augment the distance between another pseudojump and its target, forcing
86another long jump use, in a cascade. The job of the optimising
87compiler (assembler) is to individually expand every pseudo-instruction in such a way
88that all global constraints are satisfied and that the compiled program
89is minimal in size and faster in concrete time complexity.
90This problem is known to be computationally hard for most CISC architectures (see~\cite{hyde:branch:2006}).
91
92To simplify the CerCo C compiler we have chosen to implement an optimising assembler whose input language the compiler will target.
93Labels, 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 all feature in our assembly language.
94We further simplify by ignoring linking, assuming that all our assembly programs are pre-linked.
95
96Another complication we have addressed is that of the cost model.
97CerCo imposes a cost model on C programs or, more specifically, on simple blocks of instructions.
98This cost model is induced by the compilation process itself, and its non-compositional nature allows us to assign different costs to identical C statements depending on how they are compiled.
99In short, we aim to obtain a very precise costing for a program by embracing the compilation process, not ignoring it.
100At the assembler level, this is reflected by our need to induce a cost
101model on the assembly code as a function of the assembly program and the
102strategy used to solve the branch displacement problem. In particular, the
103optimising compiler should also return a map that assigns a cost (in clock
104cycles) to every instruction in the source program. We expect the induced cost
105to be preserved by the compiler: we will prove that the compiled code
106tightly simulates the source code by taking exactly the predicted amount of
107time.
108
109Note that the temporal tightness of the simulation is a fundamental prerequisite
110of the correctness of the simulation because some functions of the MCS-51---timers and I/O---depend on the microprocessor's clock.
111If the pseudo- and concrete clock differ the result of an I/O operation may not be preserved.
112
113Branch displacement algorithms must have a deep knowledge of the way
114the rest of the assembler works in order to build globally correct solutions.
115Proving their correctness is quite a complex task (see, for instance,
116the compaion paper~\cite{boender:correctness:2012}).
117Nevertheless, the correctness of the whole assembler only depends on the
118correctness of the branch displacement algorithm.
119Therefore, in the rest of the paper, we presuppose the
120existence of a correct policy, to be computed by a branch displacement
121algorithm if it exists. A policy is the decision over how
122any particular jump should be expanded; it is correct when the global
123constraints are satisfied.
124The assembler fails to assemble an assembly program if and only if a correct policy does not exist.
125This is stated in an elegant way in the dependent type of the assembler: the assembly function is total over a program, a policy and the proof that the policy is correct for that program.
126
127A final complication in the proof is due to the kind of semantics associated to pseudo-assembly programs.
128Should assembly programs be allowed to freely manipulate addresses?
129The traditional answer is `no': values stored in memory or registers are either
130concrete data or symbolic addresses. The latter can only be manipulated
131in very restricted ways and programs that do not do so are not assigned a semantics and cannot be reasoned about.
132All programs that have a semantics have it preserved by the assembler.
133We take an alternative approach, allowing programs to freely
134manipulate addresses non-symbolically but only granting a preservation of semantics
135to those programs that act in `well-behaved' ways.
136In principle, this should allow some reasoning on the actual semantics of malign programs.
137In practice, we note how our approach facilitates more code reuse between the semantics of assembly code and object code.
138
139The rest of this paper is a detailed description of our proof that is marginally still a work in progress.
140\paragraph{Matita}
141Matita is a proof assistant based on a variant of the Calculus of (Co)inductive Constructions~\cite{asperti:user:2007}.
142It features dependent types that we exploit in the formalisation.
143The (simplified) syntax of the statements and definitions in the paper should be self-explanatory.
144Pairs are denoted with angular brackets, $\langle-, -\rangle$.
145
146Matita features a liberal system of coercions.
147It is possible to define a uniform coercion $\lambda x.\langle x,?\rangle$ from every type $T$ to the dependent product $\Sigma x:T.P~x$.
148The coercion opens a proof obligation that asks the user to prove that $P$ holds for $x$.
149When a coercion must be applied to a complex term (a $\lambda$-abstraction, a local definition, or a case analysis), the system automatically propagates the coercion to the sub-terms
150 For instance, to apply a coercion to force $\lambda x.M : A \to B$ to have type $\forall x:A.\Sigma y:B.P~x~y$, the system looks for a coercion from $M: B$ to $\Sigma y:B.P~x~y$ in a context augmented with $x:A$.
151This is significant when the coercion opens a proof obligation, as the user will be presented with multiple, but simpler proof obligations in the correct context.
152In this way, Matita supports the `Russell' proof methodology developed by Sozeau in~\cite{sozeau:subset:2006}, with an implementation that is lighter and more tightly integrated with the system than that of Coq.
153
154% ---------------------------------------------------------------------------- %
155% SECTION                                                                      %
156% ---------------------------------------------------------------------------- %
157\section{The proof}
158\label{sect.the.proof}
159
160Our aim here is to explain the main ideas and steps of the certified proof of correctness for an optimising assembler for the MCS-51.
161
162In Subsect.~\ref{subsect.machine.code.semantics} we sketch an operational semantics (a realistic and efficient emulator) for the MCS-51.
163We also introduce a syntax for decoded instructions that will be reused for the assembly language.
164
165In Subsect.~\ref{subsect.assembly.code.semantics} we describe the assembly language and its operational semantics.
166The latter is parametric in the cost model that will be induced by the assembler, reusing the semantics of the machine code on all `real' instructions.
167
168Branch displacement policies are introduced in Subsect.~\ref{subsect.the.assembler} where we also describe the assembler as a function over policies as previously described.
169
170To prove our assembler correct we show that the object code given in output, together with a cost model for the source program, simulates the source program executed using that cost model.
171The proof can be divided into two main lemmas.
172The first is correctness with respect to fetching, described in Subsect.~\ref{subsect.total.correctness.of.the.assembler}.
173Roughly it states that a step of fetching at the assembly level, returning the decoded instruction $I$, is simulated by $n$ steps of fetching at the object level that returns instructions $J_1,\ldots,J_n$, where $J_1,\ldots,J_n$ is, amongst the possible expansions of $I$, the one picked by the policy.
174The second lemma states that $J_1,\ldots,J_n$ simulates $I$ but only if $I$ is well-behaved, i.e. manipulates addresses in `good' ways.
175To keep track of well-behaved address manipulations we record where addresses are currently stored (in memory or an accumulator).
176We introduce a dynamic checking function that inspects this map to determine if the operation is well-behaved, with an affirmative answer being the pre-condition of the lemma.
177The second lemma is detailed in Subsect.~\ref{subsect.total.correctness.for.well.behaved.assembly.programs} where we also establish correctness of our assembler as a composition of the two lemmas: programs that are well-behaved when executed under the cost model induced by the compiler are correctly simulated by the compiled code.
178
179% ---------------------------------------------------------------------------- %
180% SECTION                                                                      %
181% ---------------------------------------------------------------------------- %
182
183\subsection{Machine code and its semantics}
184\label{subsect.machine.code.semantics}
185
186We implemented a realistic and efficient emulator for the MCS-51 microprocessor.
187An MCS-51 program is just a sequence of bytes stored in the read-only code
188memory of the processor, represented as a compact trie of bytes addressed
189by the program counter.
190The \texttt{Status} of the emulator is described as
191a record that contains the microprocessor's program counter, registers, stack
192pointer, clock, special function registers, code memory, and so on.
193The value of the code memory is a parameter of the record since it is not
194changed during execution.
195
196The \texttt{Status} records is itself an instance of a more general
197datatype \texttt{PreStatus} that abstracts over the implementation of code
198memory in order to reuse the same datatype for the semantics of the assembly
199language in the next section.
200
201The execution of a single instruction is performed by the \texttt{execute\_1}
202function, parametric over the content \texttt{cm} of the code memory:
203\begin{lstlisting}
204definition execute_1: $\forall$cm. Status cm $\rightarrow$ Status cm
205\end{lstlisting}
206
207The function \texttt{execute\_1} closely matches the fetch-decode-execute
208cycle of the MCS-51 hardware, as described by a Siemen's manufacturer's data sheet~\cite{siemens:2011}.
209Fetching and decoding are performed simultaneously:
210we first fetch, using the program counter, from code memory the first byte of the instruction to be executed, decoding the resulting opcode, fetching more bytes as is necessary to decode the arguments.
211Decoded instructions are represented by the \texttt{instruction} data type
212which extends a data type of \texttt{preinstruction}s that will be reused
213for the assembly language.
214\begin{lstlisting}
215inductive preinstruction (A: Type[0]): Type[0] :=
216 | ADD: $\llbracket$acc_a$\rrbracket$ → $\llbracket$registr; direct; indirect; data$\rrbracket$ $\rightarrow$ preinstruction A
217 | DEC: $\llbracket$acc_a; registr; direct; indirect$\rrbracket$ $\rightarrow$ preinstruction A
218 | JB: $\llbracket$bit_addr$\rrbracket$ $\rightarrow$ A $\rightarrow$ preinstruction A
219 | ...
220
221inductive instruction: Type[0] :=
222 | LCALL: $\llbracket$addr16$\rrbracket$ $\rightarrow$ instruction
223 | AJMP: $\llbracket$addr11$\rrbracket$ $\rightarrow$ instruction
224 | RealInstruction: preinstruction $\llbracket$relative$\rrbracket$ $\rightarrow$ instruction.
225 | ...
226\end{lstlisting}
227The MCS-51 has many operand modes, but an unorthogonal instruction set: every
228opcode is only enable for a finite subset of the possible operand modes.
229Here we exploit dependent types and an implicit coercion to synthesize
230the type of arguments of opcodes from a vector of names of operand modes.
231For example, \texttt{ACC} has two operands, the first one constrained to be
232the \texttt{A} accumulator, and the second one to be a disjoint union of
233register, direct, indirect and data operand modes.
234
235The parameterised type $A$ of \texttt{preinstruction} represents the addressing mode allowed for conditional jumps; in the \texttt{RealInstruction} constructor
236we constraint it to be a relative offset.
237A different instantiation will be used in the next section for assembly programs.
238
239Once decoded, execution proceeds by a case analysis on the decoded instruction, following the operation of the hardware.
240For example, the \texttt{DEC} preinstruction (`decrement') is executed as follows:
241\begin{lstlisting}
242 | DEC addr $\Rightarrow$
243  let s := add_ticks1 s in
244  let $\langle$result, flags$\rangle$ := sub_8_with_carry (get_arg_8 s true addr)
245   (bitvector_of_nat 8 1) false in
246     set_arg_8 s addr result
247\end{lstlisting}
248
249Here, \texttt{add\_ticks1} models the incrementing of the internal clock of the microprocessor; it is a parameter of the semantics of \texttt{preinstruction}s
250that is fixed in the semantics of \texttt{instruction}s according to the
251manufacturer datasheet.
252
253% ---------------------------------------------------------------------------- %
254% SECTION                                                                      %
255% ---------------------------------------------------------------------------- %
256
257\subsection{Assembly code and its semantics}
258\label{subsect.assembly.code.semantics}
259
260An assembly program is a list of potentially labelled pseudoinstructions, bundled with a preamble consisting of a list of symbolic names for locations in data memory (i.e. global variables).
261All preinstructions are pseudoinstructions, but conditional jumps are now
262only allowed to use \texttt{Identifiers} (labels) as their target.
263\begin{lstlisting}
264inductive pseudo_instruction: Type[0] :=
265  | Instruction: preinstruction Identifier $\rightarrow$ pseudo_instruction
266    ...
267  | Jmp: Identifier $\rightarrow$ pseudo_instruction
268  | Call: Identifier $\rightarrow$ pseudo_instruction
269  | Mov: $\llbracket$dptr$\rrbracket$ $\rightarrow$ Identifier $\rightarrow$ pseudo_instruction.
270\end{lstlisting}
271The pseudoinstructions \texttt{Jmp}, \texttt{Call} and \texttt{Mov} are generalisations of machine code unconditional jumps, calls and move instructions respectively, all of whom act on labels, as opposed to concrete memory addresses.
272The object code calls and jumps that act on concrete memory addresses are ruled
273out of assembly programs not being included in the preinstructions (see previous
274Section).
275
276Execution of pseudoinstructions is an endofunction on \texttt{PseudoStatus}.
277A \texttt{PseudoStatus} is an instance of \texttt{PreStatus} that differs
278from a \texttt{Status} only in the datatype used for code memory: a list
279of optionally labelled pseudoinstructions versus a trie of bytes.
280The \texttt{PreStatus} type is crucial for sharing the majority of the
281semantics of the two languages.
282
283Emulation for pseudoinstructions is handled by \texttt{execute\_1\_pseudo\_instruction}:
284\begin{lstlisting}
285definition execute_1_pseudo_instruction:
286 $\forall$cm. $\forall$costing:($\forall$ppc: Word. ppc < $\mid$snd cm$\mid$ $\rightarrow$ nat $\times$ nat).
287  $\forall$s:PseudoStatus cm. program_counter s < $\mid$snd cm$\mid$ $\rightarrow$ PseudoStatus cm
288\end{lstlisting}
289The type of \texttt{execute\_1\_pseudo\_instruction} is more involved than
290that of \texttt{execute\_1}. The first difference is that execution is only
291defined when the program counter points to a valid instruction, i.e.
292it is smaller than the length $\mid$\texttt{snd cm}$\mid$ of the program.
293The second difference is the abstraction over the cost model, abbreviated
294here as \emph{costing}.
295The costing is a function that maps valid program counters to pairs of natural numbers representing the number of clock ticks used by the pseudoinstructions stored at those program counters. For conditional jumps the two numbers differ
296to represent different costs for the `true branch' and the `false branch'.
297In the next section we will see how the optimising
298assembler induces the only costing (induced by the branch displacement policy deciding how to expand pseudojumps) that is preserved by compilation.
299
300Execution proceeds by first fetching from pseudo-code memory using the program counter---treated as an index into the pseudoinstruction list.
301This index is always guaranteed to be within the bounds of the pseudoinstruction list due to the dependent type placed on the function.
302No decoding is required.
303We then proceed by case analysis over the pseudoinstruction, reusing the code for object code for all instructions present in the MCS-51's instruction set.
304For all newly introduced pseudoinstructions, we simply translate labels to concrete addresses before behaving as a `real' instruction.
305
306We do not perform any kind of symbolic execution, wherein data is the disjoint union of bytes and addresses, with addresses kept opaque and immutable.
307Labels are immediately translated to concrete addresses, and registers and memory locations only ever contain bytes, never labels.
308As a consequence, we allow the programmer to mangle, change and generally adjust addresses as they want, under the proviso that the translation process may not be able to preserve the semantics of programs that do this.
309The only limitation introduced by this approach is that the size of
310assembly programs is bounded by $2^16$.
311This will be further discussed in Subsect.~\ref{subsect.total.correctness.for.well.behaved.assembly.programs}.
312
313% ---------------------------------------------------------------------------- %
314% SECTION                                                                      %
315% ---------------------------------------------------------------------------- %
316
317\subsection{The assembler}
318\label{subsect.the.assembler}
319
320Conceptually the assembler works in two passes.
321The first pass expands every pseudoinstruction into a list of machine code instructions using the function \texttt{expand\_pseudo\_instruction}.
322The second pass encodes as a list of bytes the expanded instruction list by mapping the function \texttt{assembly1} across the list, and then flattening.
323The program obtained as a list of bytes is ready to be loaded in code memory
324for execution.
325\begin{displaymath}
326\hspace{-0.5cm}
327\mbox{\fontsize{7}{9}\selectfont$[\mathtt{P_1}, \ldots \mathtt{P_n}]$} \underset{\mbox{\fontsize{7}{9}\selectfont$\mathtt{assembly}$}}{\xrightarrow{\left(P_i \underset{\mbox{\fontsize{7}{9}\selectfont$\mathtt{assembly\_1\_pseudo\_instruction}$}}{\xrightarrow{\mathtt{P_i} \xrightarrow{\mbox{\fontsize{7}{9}\selectfont$\mathtt{expand\_pseudo\_instruction}$}} \mathtt{[I^1_i, \ldots I^q_i]} \xrightarrow{\mbox{\fontsize{7}{9}\selectfont$\mathtt{~~~~~~~~assembly1^{*}~~~~~~~~}$}} \mathtt{[0110]}}} \mathtt{[0110]}\right)^{*}}} \mbox{\fontsize{7}{9}\selectfont$\mathtt{[\ldots0110\ldots]}$}
328\end{displaymath}
329The most complex of the two passes is the first, which expands pseudoinstructions and must perform the task of branch displacement~\cite{hyde:branch:2006}.
330The function \texttt{assembly\_1\_pseudo\_instruction} used in the body of the paper is essentially the composition of the two passes.
331
332The branch displacement problem refers to the task of expanding pseudojumps into their concrete counterparts, preferably as efficiently as possible.
333For instance, the MCS-51 features three unconditional jump instructions: \texttt{LJMP} and \texttt{SJMP}---`long jump' and `short jump' respectively---and an 11-bit oddity of the MCS-51, \texttt{AJMP}.
334Each of these three instructions expects arguments in different sizes and behaves in markedly different ways: \texttt{SJMP} may only perform a `local jump'; \texttt{LJMP} may jump to any address in the MCS-51's memory space and \texttt{AJMP} may jump to any address in the current memory page.
335Consequently, 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 that will suffice should be selected.
336
337Similarly, a conditional pseudojump must be translated potentially into a configuration of machine code instructions, depending on the distance to the jump's target.
338For example, to 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'):
339{\small{
340\begin{displaymath}
341\begin{array}{r@{\quad}l@{\;\;}l@{\qquad}c@{\qquad}l@{\;\;}l}
342       & \mathtt{JZ}  & \mathtt{label}                      &                 & \mathtt{JZ}   & \text{size of \texttt{SJMP} instruction} \\
343       & \ldots       &                            & \text{translates to}   & \mathtt{SJMP} & \text{size of \texttt{LJMP} instruction} \\
344\mathtt{label:} & \mathtt{MOV} & \mathtt{A}\;\;\mathtt{B}   & \Longrightarrow & \mathtt{LJMP} & \text{address of \textit{label}} \\
345       &              &                            &                 & \ldots        & \\
346       &              &                            &                 & \mathtt{MOV}  & \mathtt{A}\;\;\mathtt{B}
347\end{array}
348\end{displaymath}}}
349Here, if \texttt{JZ} fails, we fall through to the \texttt{SJMP} which jumps over the \texttt{LJMP}.
350Naturally, if \texttt{label} is `close enough', a conditional jump pseudoinstruction is mapped directly to a conditional jump machine instruction; the above translation only applies if \texttt{label} is not sufficiently local.
351
352In order to implement branch displacement it is impossible to really make the \texttt{expand\_pseudo\_instruction} function completely independent of the encoding function.
353This is due to branch displacement requiring the distance in bytes of the target of the jump.
354Moreover the standard solutions for solving the branch displacement problem find their solutions iteratively, by either starting from a solution where all jumps are long, and shrinking them when possible, or starting from a state where all jumps are short and increasing their length as needed.
355Proving the correctness of such algorithms is already quite involved and the correctness of the assembler as a whole does not depend on the `quality' of the solution found to a branch displacement problem.
356For this reason, we try to isolate the computation of a branch displacement problem from the proof of correctness for the assembler by parameterising our \texttt{expand\_pseudo\_instruction} by a `policy'.
357
358\begin{lstlisting}
359definition expand_pseudo_instruction:
360 $\forall$lookup_labels: Identifier $\rightarrow$ Word.
361 $\forall$policy.
362 $\forall$ppc: Word.
363 $\forall$lookup_datalabels: Identifier $\rightarrow$ Word.
364 $\forall$pi: pseudo_instruction.
365  list instruction := ...
366\end{lstlisting}
367Here, the functions \texttt{lookup\_labels} and \texttt{lookup\_datalabels} are the functions that map labels and datalabels to program counters respectively, both of them used in the semantics of assembly.
368The input \texttt{pi} is the pseudoinstruction to be expanded and is found at address \texttt{ppc} in the assembly program.
369The function takes \texttt{policy} as an input.
370In reality, this is a pair of functions, but for the purposes of this paper we simplify.
371The \texttt{policy} maps pseudo-program counters to program counters: the encoding of the expansion of the pseudoinstruction found at address \texttt{a} in the assembly code should be placed into code memory at address \texttt{policy(a)}.
372Of course this is possible only if the policy is correct, which means that the encoding of consecutive assembly instructions must be consecutive in code memory.
373\begin{displaymath}
374\texttt{policy}(\texttt{ppc} + 1) = \texttt{pc} + \texttt{current\_instruction\_size}
375\end{displaymath}
376Here, \texttt{current\_instruction\_size} is the size in bytes of the encoding of the expanded pseudoinstruction found at \texttt{ppc}.
377Note that the entanglement we hinted at is only partially solved in this way: the assembler code can ignore the implementation details of the algorithm that finds a policy;
378however, the algorithm that finds a policy must know the exact behaviour of the assembly program because it needs to predict the way the assembly will expand and encode pseudoinstructions, once fed with a policy.
379A companion submission to this one~\cite{boender:correctness:2012} certifies an algorithm that finds branch displacement policies for the assembler described in this paper.
380
381The \texttt{expand\_pseudo\_instruction} function uses the \texttt{policy} map to determine the size of jump required when expanding pseudojumps, computing the jump size by examining the size of the differences between program counters.
382For instance, if at address \texttt{ppc} in the assembly program we found \texttt{Jmp l} such that \texttt{lookup\_labels l = a}, if the offset \texttt{d = policy(a) - policy(ppc + 1)} is such that \texttt{d} $< 128$ then \texttt{Jmp l} is normally translated to the best local solution, the short jump \texttt{SJMP d}.
383A global best solution to the branch displacement problem, however, is not always made of locally best solutions.
384Therefore, in some circumstances, it is necessary to force the assembler to expand jumps into larger ones.
385This is achieved by another boolean-valued function such that if the function applied to \texttt{ppc} returns true then a \texttt{Jmp l} at address \texttt{ppc} is always translated to a long jump.
386An essentially identical mechanism exists for call instructions.
387
388% ---------------------------------------------------------------------------- %
389% SECTION                                                                      %
390% ---------------------------------------------------------------------------- %
391\subsection{Correctness of the assembler with respect to fetching}
392\label{subsect.total.correctness.of.the.assembler}
393Using our policies, we now work toward proving the correctness of the assembler.
394Correctness means that the assembly process never fails when provided with a correct policy and that the process does not change the semantics of a certain class of well-behaved assembly programs.
395
396The aim of this section is to prove the following informal statement: when we fetch an assembly pseudoinstruction \texttt{I} at address \texttt{ppc}, then we can fetch the expanded pseudoinstruction(s) \texttt{[J1, \ldots, Jn] = fetch\_pseudo\_instruction \ldots\ I\ ppc} from \texttt{policy ppc} in the code memory obtained by loading the assembled object code.
397This constitutes the first major step in the proof of correctness of the assembler, the next one being the simulation of \texttt{I} by \texttt{[J1, \ldots, Jn]} (see Subsect.~\ref{subsect.total.correctness.for.well.behaved.assembly.programs}).
398
399The \texttt{assembly} function is given a Russell type (slightly simplified here):
400\begin{lstlisting}
401definition assembly:
402  $\forall$program: pseudo_assembly_program.
403  $\forall$policy.
404    $\Sigma$assembled: list Byte $\times$ (BitVectorTrie costlabel 16).
405      policy is correct for program $\rightarrow$
406      $\mid$program$\mid$ < $2^{16}$ $\rightarrow$ $\mid$fst assembled$\mid$ < $2^{16}$ $\wedge$
407      (policy ($\mid$program$\mid$) = $\mid$fst assembled$\mid$ $\vee$
408      (policy ($\mid$program$\mid$) = 0 $\wedge$ $\mid$fst assembled$\mid$ = $2^{16}$)) $\wedge$
409      $\forall$ppc: pseudo_program_counter. ppc < $2^{16}$ $\rightarrow$
410        let pseudo_instr := fetch from program at ppc in
411        let assembled_i := assemble pseudo_instr in
412          $\mid$assembled_i$\mid$ $\leq$ $2^{16}$ $\wedge$
413            $\forall$n: nat. n < $\mid$assembled_i$\mid$ $\rightarrow$ $\exists$k: nat.
414              nth assembled_i n = nth assembled (policy ppc + k).
415\end{lstlisting}
416In plain words, the type of \texttt{assembly} states the following.
417Suppose we are given a policy that is correct for the program we are assembling.
418Then we return a list of assembled bytes, complete with a map from program counters to cost labels, such that the following properties hold for the list of bytes.
419Under the condition that the policy is `correct' for the program and the program is fully addressable by a 16-bit word, the assembled list is also fully addressable by a 16-bit word, the policy maps the last program counter that can address the program to the last instruction of the assemble pseudoinstruction or overflows, and if we fetch from the pseudo-program counter \texttt{ppc} we get a pseudoinstruction \texttt{pi} and a new pseudo-program counter \texttt{ppc}.
420Further, assembling the pseudoinstruction \texttt{pseudo\_instr} results in a list of bytes, \texttt{assembled\_i}.
421Then, indexing into this list with any natural number \texttt{n} less than the length of \texttt{assembled\_i} gives the same result as indexing into \texttt{assembled} with \texttt{policy ppc} (the program counter pointing to the start of the expansion in \texttt{assembled}) plus \texttt{k}.
422
423Essentially the lemma above states that the \texttt{assembly} function correctly expands pseudoinstructions, and that the expanded instruction reside consecutively in memory.
424This result is lifted from lists of bytes into a result on tries of bytes (i.e. code memories), using an additional lemma: \texttt{assembly\_ok}.
425
426Lemma \texttt{fetch\_assembly} establishes that the \texttt{fetch} and \texttt{assembly1} functions interact correctly.
427The \texttt{fetch} function, as its name implies, fetches the instruction indexed by the program counter in the code memory, while \texttt{assembly1} maps a single instruction to its byte encoding:
428\begin{lstlisting}
429lemma fetch_assembly:
430 $\forall$pc: Word.
431 $\forall$i: instruction.
432 $\forall$code_memory: BitVectorTrie Byte 16.
433 $\forall$assembled: list Byte.
434  assembled = assemble i $\rightarrow$
435  let len := $\mid$assembled$\mid$ in
436  let pc_plus_len := pc + len in
437   encoding_check pc pc_plus_len assembled $\rightarrow$
438   let $\langle$instr, pc', ticks$\rangle$ := fetch pc in
439    instr = i $\wedge$ ticks = (ticks_of_instruction instr) $\wedge$ pc' = pc_plus_len.
440\end{lstlisting}
441We read \texttt{fetch\_assembly} as follows.
442Given an instruction, \texttt{i}, we first assemble the instruction to obtain \texttt{assembled}, checking that the assembled instruction was stored in code memory correctly.
443Fetching from code memory, we obtain a tuple consisting of the instruction, new program counter, and the number of ticks this instruction will take to execute.
444We finally check that the fetched instruction is the same instruction that we began with, and the number of ticks this instruction will take to execute is the same as the result returned by a lookup function, \texttt{ticks\_of\_instruction}, devoted to tracking this information.
445Or, in plainer words, assembling and then immediately fetching again gets you back to where you started.
446
447Lemma \texttt{fetch\_assembly\_pseudo} is obtained by composition of \texttt{expand\_pseudo\_instruction} and \texttt{assembly\_1\_pseudoinstruction}:
448\begin{lstlisting}
449lemma fetch_assembly_pseudo:
450 $\forall$program: pseudo_assembly_program.
451 $\forall$policy.
452 $\forall$ppc.
453 $\forall$code_memory.
454 let $\langle$preamble, instr_list$\rangle$ := program in
455 let pi := $\pi_1$ (fetch_pseudo_instruction instr_list ppc) in
456 let pc := policy ppc in
457 let instrs := expand_pseudo_instructio policy ppc pi in
458 let $\langle$l, a$\rangle$ := assembly_1_pseudoinstruction policy ppc pi in
459 let pc_plus_len := pc + l in
460  encoding_check code_memory pc pc_plus_len a $\rightarrow$
461   fetch_many code_memory pc_plus_len pc instructions.
462\end{lstlisting}
463Here, \texttt{l} is the number of machine code instructions the pseudoinstruction at hand has been expanded into.
464We assemble a single pseudoinstruction with \texttt{assembly\_1\_pseudoinstruction}, which internally calls \texttt{expand\_pseudo\_instruction}.
465The function \texttt{fetch\_many} fetches multiple machine code instructions from code memory and performs some routine checks.
466
467Intuitively, Lemma \texttt{fetch\_assembly\_pseudo} can be read as follows.
468Suppose we expand the pseudoinstruction at \texttt{ppc} with the policy, obtaining the list of machine code instructions \texttt{instrs}.
469Suppose we also assemble the pseudoinstruction at \texttt{ppc} to obtain \texttt{a}, a list of bytes.
470Then, we check with \texttt{fetch\_many} that the number of machine instructions that were fetched matches the number of instruction that \texttt{expand\_pseudo\_instruction} expanded.
471
472The final lemma in this series is \texttt{fetch\_assembly\_pseudo2} that combines the Lemma \texttt{fetch\_assembly\_pseudo} with the correctness of the functions that load object code into the processor's memory:
473\begin{lstlisting}
474lemma fetch_assembly_pseudo2:
475 $\forall$program.
476 $\mid$snd program$\mid$ $\leq$ $2^{16}$ $\rightarrow$
477 $\forall$policy.
478 policy is correct for program $\rightarrow$
479 $\forall$ppc. ppc < $\mid$snd program$\mid$ $\rightarrow$
480  let $\langle$labels, costs$\rangle$ := create_label_cost_map program in
481  let $\langle$assembled, costs'$\rangle$ := $\pi_1$ (assembly program policy) in
482  let cmem := load_code_memory assembled in
483  let $\langle$pi, newppc$\rangle$ := fetch_pseudo_instruction program ppc in
484  let instructions := expand_pseudo_instruction policy ppc pi in
485    fetch_many cmem (policy newppc) (policy ppc) instructions.
486\end{lstlisting}
487
488Here we use $\pi_1$ to project the existential witness from the Russell-typed function \texttt{assembly}.
489We read \texttt{fetch\_assembly\_pseudo2} as follows.
490Suppose we are given an assembly program which can be addressed by a 16-bit word and a policy that is correct for this program.
491Suppose we are able to successfully assemble an assembly program using \texttt{assembly} and produce a code memory, \texttt{cmem}.
492Then, fetching a pseudoinstruction from the pseudo-code memory at address \texttt{ppc} corresponds to fetching a sequence of instructions from the real code memory using \texttt{policy} to expand pseudoinstructions.
493The fetched sequence corresponds to the expansion, according to the policy, of the pseudoinstruction.
494
495At first, the lemma appears to immediately imply the correctness of the assembler, but this property is \emph{not} strong enough to establish that the semantics of an assembly program has been preserved by the assembly process since it does not establish the correspondence between the semantics of a pseudoinstruction and that of its expansion.
496In particular, the two semantics differ on instructions that \emph{could} directly manipulate program addresses.
497
498% ---------------------------------------------------------------------------- %
499% SECTION                                                                      %
500% ---------------------------------------------------------------------------- %
501\subsection{Correctness for `well-behaved' assembly programs}
502\label{subsect.total.correctness.for.well.behaved.assembly.programs}
503
504The traditional approach to verifying the correctness of an assembler is to treat memory addresses as opaque structures that cannot be modified.
505Memory is represented as a map from opaque addresses to the disjoint union of data and opaque addresses---addresses are kept opaque to prevent their possible `semantics breaking' manipulation by assembly programs:
506\begin{displaymath}
507\mathtt{Mem} : \mathtt{Addr} \rightarrow \mathtt{Bytes} + \mathtt{Addr} \qquad \llbracket - \rrbracket : \mathtt{Instr} \rightarrow \mathtt{Mem} \rightarrow \mathtt{option\ Mem}
508\end{displaymath}
509The semantics of a pseudoinstruction, $\llbracket - \rrbracket$, is given as a possibly failing function from pseudoinstructions and memory spaces to new memory spaces.
510The semantic function proceeds by case analysis over the operands of a given instruction, failing if either operand is an opaque address, or otherwise succeeding, updating memory.
511\begin{gather*}
512\llbracket \mathtt{ADD\ @A1\ @A2} \rrbracket^\mathtt{M} = \begin{cases}
513                                                              \mathtt{Byte\ b1},\ \mathtt{Byte\ b2} & \rightarrow \mathtt{Some}(\mathtt{M}\ \text{with}\ \mathtt{b1} + \mathtt{b2}) \\
514                                                              -,\ \mathtt{Addr\ a} & \rightarrow \mathtt{None} \\
515                                                              \mathtt{Addr\ a},\ - & \rightarrow \mathtt{None}
516                                                            \end{cases}
517\end{gather*}
518In this paper we take a different approach, tracing memory locations (and accumulators) that contain memory addresses.
519We prove that only those assembly programs that use addresses in `safe' ways have their semantics preserved by the assembly process---a sort of dynamic type system sitting atop memory.
520In principle this approach allows us to introduce some permitted \emph{benign} manipulations of addresses that the traditional approach cannot handle, therefore expanding the set of input programs that can be assembled correctly.
521This approach seems similar to one taken by Tuch \emph{et al}~\cite{tuch:types:2007} for reasoning about low-level C code.
522
523Our analogue of the semantic function above is merely a wrapper around the function that implements the semantics of machine code, paired with a function that keeps track of addresses.
524The semantics of pseudo- and machine code are then essentially shared.
525The only thing that changes at the assembly level is the presence of the new tracking function.
526
527However, with this approach we must detect (at run time) programs that manipulate addresses in well-behaved ways, according to some approximation of well-behavedness.
528We use an \texttt{internal\_pseudo\_address\_map} to trace addresses of code memory addresses in internal RAM:
529\begin{lstlisting}
530definition address_entry := upper_lower $\times$ Byte.
531
532definition internal_pseudo_address_map :=
533  (BitVectorTrie address_entry 7) $\times$ (BitVectorTrie address_entry 7)
534    $\times$ (option address_entry).
535\end{lstlisting}
536Here, \texttt{upper\_lower} is a type isomorphic to the booleans.
537The implementation of \texttt{internal\_pseudo\_address\_map} is complicated by some peculiarities of the MCS-51's instruction set.
538Note here that all addresses are 16 bit words, but are stored (and manipulated) as 8 bit bytes.
539All \texttt{MOV} instructions in the MCS-51 must use the accumulator \texttt{A} as an intermediary, moving a byte at a time.
540The third component of \texttt{internal\_pseudo\_address\_map} therefore states whether the accumulator currently holds a piece of an address, and if so, whether it is the upper or lower byte of the address (using the \texttt{upper\_lower} flag) complete with the corresponding source address in full.
541The first and second components, on the other hand, performs a similar task for the higher and lower external RAM.
542Again, we use our \texttt{upper\_lower} flag to describe whether a byte is the upper or lower component of a 16-bit address.
543
544The \texttt{low\_internal\_ram\_of\_pseudo\_low\_internal\_ram} function converts the lower internal RAM of a \texttt{PseudoStatus} into the lower internal RAM of a \texttt{Status}.
545A similar function exists for high internal RAM.
546Note that both RAM segments are indexed using addresses 7-bits long:
547\begin{lstlisting}
548definition low_internal_ram_of_pseudo_low_internal_ram:
549 internal_pseudo_address_map $\rightarrow$ policy $\rightarrow$ BitVectorTrie Byte 7
550  $\rightarrow$ BitVectorTrie Byte 7.
551\end{lstlisting}
552
553Next, we are able to translate \texttt{PseudoStatus} records into \texttt{Status} records using \texttt{status\_of\_pseudo\_status}.
554Translating a \texttt{PseudoStatus}'s code memory requires we expand pseudoinstructions and then assemble to obtain a trie of bytes.
555This never fails, provided that our policy is correct:
556\begin{lstlisting}
557definition status_of_pseudo_status:
558 internal_pseudo_address_map $\rightarrow$ $\forall$pap. $\forall$ps: PseudoStatus pap.
559 $\forall$policy. Status (code_memory_of_pseudo_assembly_program pap policy)
560\end{lstlisting}
561
562The \texttt{next\_internal\_pseudo\_address\_map} function is responsible for run time monitoring of the behaviour of assembly programs, in order to detect well-behaved ones.
563It returns a map that traces memory addresses in internal RAM after execution of the next pseudoinstruction, failing when the instruction tampers with memory addresses in unanticipated (but potentially correct) ways.
564It thus decides the membership of a strict subset of the set of well-behaved programs.
565\begin{lstlisting}
566definition next_internal_pseudo_address_map: internal_pseudo_address_map $\rightarrow$
567 $\forall$cm. (Identifier $\rightarrow$ PseudoStatus cm $\rightarrow$ Word) $\rightarrow$ $\forall$s: PseudoStatus cm.
568   program_counter s < $2^{16}$ $\rightarrow$ option internal_pseudo_address_map
569\end{lstlisting}
570If we wished to allow `benign manipulations' of addresses, it would be this function that needs to be changed.
571Note we once again use dependent types to ensure that program counters are properly within bounds.
572The third argument is a function that resolves the concrete address of a label.
573
574The function \texttt{ticks\_of0} computes how long---in clock cycles---a pseudoinstruction will take to execute when expanded in accordance with a given policy.
575The function returns a pair of natural numbers, needed for recording the execution times of each branch of a conditional jump.
576\begin{lstlisting}
577definition ticks_of0:
578 pseudo_assembly_program $\rightarrow$ (Identifier $\rightarrow$ Word) $\rightarrow$ $\forall$policy. Word $\rightarrow$
579   pseudo_instruction $\rightarrow$ nat $\times$ nat
580\end{lstlisting}
581An additional function, \texttt{ticks\_of}, is merely a wrapper around this function.
582
583Finally, we are able to state and prove our main theorem, relating the execution of a single assembly instruction and the execution of (possibly) many machine code instructions, as long as we are able to track memory addresses properly:
584\begin{lstlisting}
585theorem main_thm:
586 $\forall$M, M': internal_pseudo_address_map.
587 $\forall$program: pseudo_assembly_program.
588 $\forall$program_in_bounds: $\mid$program$\mid$ $\leq$ $2^{16}$.
589 let maps := create_label_cost_map program in
590 let addr_of := ... in
591 program is well labelled $\rightarrow$
592 $\forall$policy. policy is correct for program.
593 $\forall$ps: PseudoStatus program. ps < $\mid$program$\mid$.
594  next_internal_pseudo_address_map M program ... = Some M' $\rightarrow$
595   $\exists$n. execute n (status_of_pseudo_status M ps policy) =
596    status_of_pseudo_status M'
597      (execute_1_pseudo_instruction program
598       (ticks_of program ($\lambda$id. addr_of id ps) policy) ps) policy.
599\end{lstlisting}
600The statement is standard for forward simulation, but restricted to \texttt{PseudoStatuses} \texttt{ps} whose next instruction to be executed is well-behaved with respect to the \texttt{internal\_pseudo\_address\_map} \texttt{M}.
601We explicitly require proof that the policy is correct, the program is well-labelled (i.e. no repeated labels, etc.) and the pseudo-program counter is in the program's bounds.
602Theorem \texttt{main\_thm} establishes the correctness of the assembly process and can be lifted to the forward simulation of an arbitrary number of well-behaved steps on the assembly program.
603
604% ---------------------------------------------------------------------------- %
605% SECTION                                                                      %
606% ---------------------------------------------------------------------------- %
607\section{Conclusions}
608\label{sect.conclusions}
609
610We are proving the correctness of an assembler for MCS-51 assembly language.
611Our assembly language features 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.
612Expanding these pseudoinstructions into machine code instructions is not trivial, and the proof that the assembly process is `correct', in that the semantics of a subset of assembly programs are not changed is complex.
613
614The formalisation is a component of CerCo which aims to produce a verified concrete complexity preserving compiler for a large subset of the C language.
615The verified assembler, complete with the underlying formalisation of the semantics of MCS-51 machine code, will form the bedrock layer upon which the rest of CerCo will build its verified compiler platform.
616
617We may compare our work to an `industrial grade' assembler for the MCS-51: SDCC~\cite{sdcc:2011}, the only open source C compiler that targets the MCS-51 instruction set.
618It appears that all pseudojumps in SDCC assembly are expanded to \texttt{LJMP} instructions, the worst possible jump expansion policy from an efficiency point of view.
619Note that this policy is the only possible policy \emph{in theory} that can preserve the semantics of an assembly program during the assembly process, coming at the expense of assembler completeness as the generated program may be too large for code memory, there being a trade-off between the completeness of the assembler and the efficiency of the assembled program.
620The definition and proof of a terminating, correct jump expansion policy is described elsewhere~\cite{boender:correctness:2012}.
621
622Verified assemblers could also be applied to the verification of operating system kernels and other formalised compilers.
623For instance the verified seL4 kernel~\cite{klein:sel4:2009}, CompCert~\cite{leroy:formally:2009} and CompCertTSO~\cite{sevcik:relaxed-memory:2011} all explicitly assume the existence of trustworthy assemblers.
624The fact that an optimising assembler cannot preserve the semantics of all assembly programs may have consequences for these projects.
625
626Our formalisation exploits dependent types in different ways and for multiple purposes.
627The first purpose is to reduce potential errors in the formalisation of the microprocessor.
628Dependent types are used to constrain the size of bitvectors and tries that represent memory quantities and memory areas respectively.
629They are also used to simulate polymorphic variants in Matita, in order to provide precise typings to various functions expecting only a subset of all possible addressing modes that the MCS-51 offers.
630Polymorphic variants nicely capture the absolutely unorthogonal instruction set of the MCS-51 where every opcode must accept its own subset of the 11 addressing mode of the processor.
631
632The second purpose is to single out sources of incompleteness.
633By abstracting our functions over the dependent type of correct policies, we were able to manifest the fact that the compiler never refuses to compile a program where a correct policy exists.
634This also allowed to simplify the initial proof by dropping lemmas establishing that one function fails if and only if some previous function does so.
635
636Finally, dependent types, together with Matita's liberal system of coercions, allow us to simulate almost entirely in user space the proof methodology `Russell' of Sozeau~\cite{sozeau:subset:2006}.
637Not every proof has been carried out in this way: we only used this style to prove that a function satisfies a specification that only involves that function in a significant way.
638It would not be natural to see the proof that fetch and assembly commute as the specification of one of the two functions.
639%\paragraph{Related work}
640
641% piton
642We are not the first to consider the correctness of an assembler for a non-trivial assembly language.
643The most impressive piece of work in this domain is Piton~\cite{moore:piton:1996}, a stack of verified components, written and verified in ACL2, ranging from a proprietary FM9001 microprocessor verified at the gate level, to assemblers and compilers for two high-level languages---Lisp and $\mu$Gypsy~\cite{moore:grand:2005}.
644% jinja
645Klein and Nipkow also provide a compiler, virtual machine and operational semantics for the Jinja~\cite{klein:machine:2006} language and prove that their compiler is semantics and type preserving.
646
647Though other verified assemblers exist what sets our work apart from that above is our attempt to optimise the generated machine code.
648This complicates a formalisation as an attempt at the best possible selection of machine instructions must be made---especially important on devices with limited code memory.
649Care must be taken to ensure that the time properties of an assembly program are not modified by assembly lest we affect the semantics of any program employing the MCS-51's I/O facilities.
650This is only possible by inducing a cost model on the source code from the optimisation strategy and input program.
651%\paragraph{Resources}
652
653Our source files are available at~\url{http://cerco.cs.unibo.it}.
654We assumed several properties of `library functions', e.g. modular arithmetic and datastructure manipulation.
655We axiomatised various small functions needed to complete the main theorems, as well as some `routine' proof obligations of the theorems themselves, in focussing on the main meat of the theorems.
656We believe that the proof strategy is sound and that all axioms can be closed, up to minor bugs that should have local fixes that do not affect the global proof strategy.
657
658The complete development is spread across 29 files with around 20,000 lines of Matita source.
659Relavent files are: \texttt{AssemblyProof.ma}, \texttt{AssemblyProofSplit.ma} and \texttt{AssemblyProofSplitSplit.ma}, consisting of approximately 4500 lines of Matita source.
660Numerous other lines of proofs are spread all over the development because of dependent types and the Russell proof style, which does not allow one to separate the code from the proofs.
661The low ratio between source lines and the number of lines of proof is unusual, but justified by the fact that the pseudo-assembly and the assembly language share most constructs and large swathes of the semantics are shared.
662
663\bibliography{cpp-2012-asm.bib}
664
665\end{document}\renewcommand{\verb}{\lstinline}
666\def\lstlanguagefiles{lst-grafite.tex}
667\lstset{language=Grafite}
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