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[2088]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.}}
[2052]38\author{Dominic P. Mulligan \and Claudio Sacerdoti Coen}
[2089]39\institute{Dipartimento di Scienze dell'Informazione,\\ Universit\'a degli Studi di Bologna}
[2083]48We present a proof of correctness, in Matita, for an optimising assembler for the MCS-51 microcontroller.
[2327]50The efficient expansion of pseudoinstructions---namely jumps---into MCS-51 machine instructions is complex.
[2053]51We isolate the decision making over how jumps should be expanded from the expansion process itself as much as possible using `policies'.
52This makes the proof of correctness for the assembler significantly more straightforward.
[2053]54We observe that it is impossible for an optimising assembler to preserve the semantics of every assembly program.
55Assembly language programs can manipulate concrete addresses in arbitrary ways.
[2083]56Our proof strategy contains a tracking facility for `good addresses' and only programs that use good addresses have their semantics preserved under assembly.
[2087]57Our strategy offers increased flexibility over the traditional approach to proving the correctness of assemblers, wherein addresses in assembly are kept opaque and immutable.
[2053]58In particular, we may experiment with allowing the benign manipulation of addresses.
[2088]59\keywords{Verified software, CerCo (Certified Complexity), MCS-51 microcontroller, Matita proof assistant}
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68We consider the formalisation of an assembler for the Intel MCS-51 8-bit microprocessor in the Matita proof assistant~\cite{asperti:user:2007}.
[2053]69This 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.
71The MCS-51 dates from the early 1980s and is commonly called the 8051/8052.
[2083]72Despite the microprocessor's age, derivatives are still widely manufactured by a number of semiconductor foundries, with the processor being used especially in embedded systems development, where well-tested, cheap, predictable microprocessors find their niche.
[2083]74The 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.
75Yet---as with many things---what one hand giveth, the other taketh away, and the MCS-51's paucity of features---though an advantage in many respects---also quickly becomes a hindrance.
[2053]76In particular, the MCS-51 features a relatively minuscule series of memory spaces by modern standards.
[2052]77As a result our C compiler, to have any sort of hope of successfully compiling realistic programs for embedded devices, ought to produce `tight' machine code.
[2340]79In order to do this, we must solve the `branch displacement' problem---deciding how best to expand pseudojumps to labels in assembly language to machine code jumps. The branch displacement problem happears when pseudojumps can be expanded
80in different ways to real machine instructions, but the different expansions
81are not equivalent (e.g. do not have the same size or speed) and not always
82correct (e.g. correctness is only up to global constraints over the compiled
83code). For instance, some jump instructions (short jumps) are very small
84and fast, but they can only reach destinations within a
85certain distance from the current instruction. When the destinations are
86too far away, larger and slower long jumps must be used. The use of a long jump may
87augment the distance between another pseudojump and its target, forcing
[2343]88another long jump use, in a cascading effect. The job of the optimising
[2340]89compiler (assembler) is to individually expand every pseudo-instruction in such a way
90that all global constraints are satisfied and that the compiled program size
[2357]91is minimal in size and faster in time.
92This problem is known to be complex for most CISC architectures (see~\cite{hyde:branch:2006}).
[2358]94To free CerCo's C compiler from having to consider complications relating to branch displacement, we have chosen to implement an optimising assembler, whose input language the compiler will target.
[2053]95Labels, 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.
[2087]96We simplify the proof by assuming that all our assembly programs are pre-linked (i.e. we do not formalise a linker---this is left for future work).
[2340]98Another complication we have addressed is that of the cost model.
99CerCo imposes a cost model on C programs or, more specifically, on simple blocks of instructions.
[2344]100This 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.
[2052]101In short, we aim to obtain a very precise costing for a program by embracing the compilation process, not ignoring it.
[2340]102At the assembler level, this is reflected by our need to induce a cost
103model on the assembly code as a function of the assembly program and the
104strategy used to solve the branch displacement problem. In particular, the
[2343]105optimising compiler should also return a map that assigns a cost (in clock
[2340]106cycles) to every instruction in the source program. We expect the induced cost
107to be preserved by the compiler: we will prove that the compiled code
108tightly simulates the source code by taking exactly the predicted amount of
[2340]111Note that the temporal tightness of the simulation is a fundamental prerequisite
[2357]112of the correctness of the simulation because some functions of the MCS-51---timers and I/O---depend on the microprocessor's clock.
[2346]113If the pseudo- and concrete clock differ the result of an I/O operation may not be preserved.
[2340]115Branch displacement algorithms must have a deep knowledge of the way
116the rest of the assembler works in order to build globally correct solutions.
117Proving their correctness is quite a complex task (see, for instance,
118the compaion paper~\cite{boender:correctness:2012}).
119Nevertheless, the correctness of the whole assembler only depends on the
120correctness of the branch displacement algorithm.
[2346]121Therefore, in the rest of the paper, we presuppose the
[2340]122existence of a correct policy, to be computed by a branch displacement
123algorithm if it exists. A policy is the decision over how
124any particular jump should be expanded; it is correct when the global
125constraints are satisfied.
[2346]126The assembler fails to assemble an assembly program if and only if a correct policy does not exist.
127This 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.
[2343]129The final complication in the proof of correctness of our optimising assembler
[2346]130is due to the kind of semantics associated to pseudo-assembly programs.
[2341]131Should assembly programs be allowed to freely manipulate addresses? The
132answer to the question deeply affects the proof of correctness.
133The traditional answer is no: values stored in memory or registers are either
134concrete data or symbolic addresses. The latters can be manipulated only
135in very restrictive ways and many programs that do not do so, for malign or
136benign reasons, are not assigned a semantics and cannot be reasoned about.
137All programs that have a semantics have it preserved by the compiler.
138Instead we took a different, novel approach: we allow programs to freely
140addresses non symbolically, but we only grant preservation of the semantics
141for those programs that do behave in a correct, anticipated way. At least
142in principle, this should allow some reasoning on the actual semantics of
143malign programs. In practice, we note how the alternative approach allows
144more code reusal between the semantics of assembly code and object code,
[2346]145with benefits on the size of the formalisation.
[2357]147The rest of this paper is a detailed description of our proof that is marginally still a work in progress.
[2052]148In Section~\ref{sect.matita} we provide a brief overview of the Matita proof assistant for the unfamiliar reader.
149In Section~\ref{sect.the.proof} we discuss the design and implementation of the proof proper.
150In Section~\ref{sect.conclusions} we conclude.
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158Matita is a proof assistant based on a variant of the Calculus of (Co)inductive Constructions~\cite{asperti:user:2007}.
[2358]159It features dependent types that we exploit in the formalisation.
160The (simplified) syntax of the statements and definitions in the paper should be self-explanatory.
[2052]161Pairs are denoted with angular brackets, $\langle-, -\rangle$.
163Matita features a liberal system of coercions.
164It is possible to define a uniform coercion $\lambda x.\langle x,?\rangle$ from every type $T$ to the dependent product $\Sigma x:T.P~x$.
165The coercion opens a proof obligation that asks the user to prove that $P$ holds for $x$.
166When 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
167 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 $x:A$.
168This 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.
[2346]169In 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.
[2349]171Throughout this paper we simplify the statements of lemmas and types of definitions in order to emphasise readability.
[2052]173% ---------------------------------------------------------------------------- %
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176\section{The proof}
[2345]179The aim of the section is to explain the main ideas and steps of the certified
[2347]180proof of correctness for an optimizing assembler for the MCS-51. The
[2346]181formalisation is available at~\url{}.
[2348]183In Section~\ref{subsect.machine.code.semantics} we sketch an operational semantics (a realistic and efficient emulator) for the MCS-51.
184We also introduce a syntax for decoded instructions that will be reused for the assembly language.
[2348]186In Section~\ref{subsect.assembly.code.semantics} we describe the assembly language and its operational semantics.
187The latter is parametric in the cost model that will be induced by the assembler.
188It reuses the semantics of the machine code on all `real' (i.e. non-pseudo-) instructions.
[2346]190Branch displacement policies are introduced in Section~\ref{subsect.the.assembler} where we also describe the assembler as a function over policies as previously described.
[2348]192The proof of correctness of the assembler consists in showing 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.
193The proof can be divided into two main lemmas.
194The first is correctness with respect to fetching, described in Section~\ref{}.
195It roughly states that one step of fetching at the assembly level that returns 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.
196The second lemma shows that $J_1,\ldots,J_n$ simulates $I$ but only if $I$ is well-behaved, i.e. it manipulates addresses in ways that are anticipated in the correctness proof.
197To keep track of well-behaved address manipulations, we couple the assembly status with a map that records where addresses are currently stored in memory or in the processor's accumulators.
[2356]198We then introduce a dynamic checking function that inspects the assembly status and this map to determine if the operation is well-behaved.
[2348]199An affirmative answer is the pre-condition of the lemma.
[2356]200The second lemma is detailed in Section~\ref{} 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.
[2066]202% ---------------------------------------------------------------------------- %
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[2354]206\subsection{Machine code and its semantics}
[2354]209We implemented a realistic and efficient emulator for the MCS-51 microprocessor.
210An MCS-51 program is just a sequence of bytes stored in the read-only code
211memory of the processor, represented as a compact trie of bytes addressed
212by the program counter.
213The \texttt{Status} of the emulator is described as
214a record that contains the microprocessor's program counter, registers, stack
215pointer, clock, special function registers, code memory, and so on.
216The value of the code memory is a parameter of the record since it is not
217changed during execution.
[2354]219The \texttt{Status} records is itself an instance of a more general
220datatype \texttt{PreStatus} that abstracts over the implementation of code
221memory in order to reuse the same datatype for the semantics of the assembly
222language in the next section.
224The execution of a single instruction is performed by the \texttt{execute\_1}
225function, parametric over the content \texttt{cm} of the code memory:
[2346]227definition execute_1: $\forall$cm. Status cm $\rightarrow$ Status cm
230The function \texttt{execute\_1} closely matches the fetch-decode-execute
231cycle of the MCS-51 hardware, as described by a Siemen's manufacturer's data sheet~\cite{siemens:2011}.
232Fetching and decoding are performed simultaneously:
233we 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.
234Decoded instructions are represented by the \texttt{instruction} data type
235which extends a data type of \texttt{preinstruction}s that will be reused
236for the assembly language.
238inductive preinstruction (A: Type[0]): Type[0] :=
[2083]239 | ADD: $\llbracket$acc_a$\rrbracket$ → $\llbracket$registr; direct; indirect; data$\rrbracket$ $\rightarrow$ preinstruction A
[2339]240 | DEC: $\llbracket$acc_a; registr; direct; indirect$\rrbracket$ $\rightarrow$ preinstruction A
[2083]241 | JB: $\llbracket$bit_addr$\rrbracket$ $\rightarrow$ A $\rightarrow$ preinstruction A
[2066]242 | ...
[2083]244inductive instruction: Type[0] :=
245 | LCALL: $\llbracket$addr16$\rrbracket$ $\rightarrow$ instruction
246 | AJMP: $\llbracket$addr11$\rrbracket$ $\rightarrow$ instruction
247 | RealInstruction: preinstruction $\llbracket$relative$\rrbracket$ $\rightarrow$ instruction.
248 | ...
[2354]250The MCS-51 has many operand modes, but an unorthogonal instruction set: every
251opcode is only enable for a finite subset of the possible operand modes.
252Here we exploit dependent types and an implicit coercion to synthesize
253the type of arguments of opcodes from a vector of names of operand modes.
254For example, \texttt{ACC} has two operands, the first one constrained to be
255the \texttt{A} accumulator, and the second one to be a disjoint union of
256register, direct, indirect and data operand modes.
[2354]258The parameterised type $A$ of \texttt{preinstruction} represents the addressing mode allowed for conditional jumps; in the \texttt{RealInstruction} constructor
259we constraint it to be a relative offset. A different instantiation will be
260used in the next Section for assembly programs.
[2066]262Once decoded, execution proceeds by a case analysis on the decoded instruction, following the operation of the hardware.
[2354]263For example, the \texttt{DEC} preinstruction (`decrement') is executed as follows:
[2083]265 | DEC addr $\Rightarrow$
266  let s := add_ticks1 s in
[2346]267  let $\langle$result, flags$\rangle$ := sub_8_with_carry (get_arg_8 s true addr)
[2083]268   (bitvector_of_nat 8 1) false in
[2346]269     set_arg_8 s addr result
[2354]272Here, \texttt{add\_ticks1} models the incrementing of the internal clock of the microprocessor; it is a parameter of the semantics of \texttt{preinstruction}s
273that is fixed in the semantics of \texttt{instruction}s according to the
274manufacturer datasheet.
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[2354]280\subsection{Assembly code and its semantics}
[2087]283An 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).
[2354]284All preinstructions are pseudoinstructions, but conditional jumps are now
285only allowed to use \texttt{Identifiers} (labels) as their target.
[2083]287inductive pseudo_instruction: Type[0] :=
288  | Instruction: preinstruction Identifier $\rightarrow$ pseudo_instruction
[2066]289    ...
[2083]290  | Jmp: Identifier $\rightarrow$ pseudo_instruction
291  | Call: Identifier $\rightarrow$ pseudo_instruction
292  | Mov: $\llbracket$dptr$\rrbracket$ $\rightarrow$ Identifier $\rightarrow$ pseudo_instruction.
294The 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.
[2354]295The object code calls and jumps that act on concrete memory addresses are ruled
[2359]296out of assembly programs not being included in the preinstructions (see previous
[2348]299Execution of pseudoinstructions is an endofunction on \texttt{PseudoStatus}.
[2354]300A \texttt{PseudoStatus} is an instance of \texttt{PreStatus} that differs
301from a \texttt{Status} only in the datatype used for code memory: a list
302of optionally labelled pseudoinstructions versus a trie of bytes.
303The \texttt{PreStatus} type is crucial for sharing the majority of the
304semantics of the two languages.
[2066]306Emulation for pseudoinstructions is handled by \texttt{execute\_1\_pseudo\_instruction}:
[2058]308definition execute_1_pseudo_instruction:
[2359]309 $\forall$cm. $\forall$costing:($\forall$ppc: Word. ppc < $\mid$snd cm$\mid$ $\rightarrow$ nat $\times$ nat).
310  $\forall$s:PseudoStatus cm. program_counter s < $\mid$snd cm$\mid$ $\rightarrow$ PseudoStatus cm
[2354]312The type of \texttt{execute\_1\_pseudo\_instruction} is more involved than
313that of \texttt{execute\_1}. The first difference is that execution is only
314defined when the program counter points to a valid instruction, i.e.
315it is smaller than the length $\mid$\texttt{snd cm}$\mid$ of the program.
316The second difference is the abstraction over the cost model, abbreviated
317here as \emph{costing}.
318The 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
319to represent different costs for the `true branch' and the `false branch'.
320In the next Section we will see how the optimizing
321assembler induces the only costing that is preserved by compilation.
322Obviously the induced costing is determined by the branch displacement policy
323that decides how to expand every pseudojump to a label into concrete
[2066]326Execution proceeds by first fetching from pseudo-code memory using the program counter---treated as an index into the pseudoinstruction list.
[2339]327This index is always guaranteed to be within the bounds of the pseudoinstruction list due to the dependent type placed on the function.
[2066]328No decoding is required.
329We 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.
330For all newly introduced pseudoinstructions, we simply translate labels to concrete addresses before behaving as a `real' instruction.
332In contrast to other approaches, we do not perform any kind of symbolic execution, wherein data is the disjoint union of bytes and addresses, with addresses kept opaque and immutable.
333Labels are immediately translated to concrete addresses, and registers and memory locations only ever contain bytes, never labels.
334As 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.
[2359]335The only limitation introduced by this approach is that the size of
336assembly programs is bounded by $2^16$.
[2066]337This will be further discussed in Subsection~\ref{}.
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343\subsection{The assembler}
346Conceptually the assembler works in two passes.
[2359]347The first pass expands every pseudoinstruction into a list of machine code instructions using the function \texttt{expand\_pseudo\_instruction}.
[2066]348The second pass encodes as a list of bytes the expanded instruction list by mapping the function \texttt{assembly1} across the list, and then flattening.
[2359]349The program obtained as a list of bytes is ready to be loaded in code memory
350for execution.
353\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$\mathtt{[\ldots0110\ldots]}$}
[2092]355The most complex of the two passes is the first, which expands pseudoinstructions and must perform the task of branch displacement~\cite{hyde:branch:2006}.
[2083]356The function \texttt{assembly\_1\_pseudo\_instruction} used in the body of the paper is essentially the composition of the two passes.
[2327]358The branch displacement problem refers to the task of expanding pseudojumps into their concrete counterparts, preferably as efficiently as possible.
[2066]359For 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}.
360Each 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.
361Consequently, 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.
363Similarly, a conditional pseudojump must be translated potentially into a configuration of machine code instructions, depending on the distance to the jump's target.
364For 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'):
368       & \mathtt{JZ}  & \mathtt{label}                      &                 & \mathtt{JZ}   & \text{size of \texttt{SJMP} instruction} \\
369       & \ldots       &                            & \text{translates to}   & \mathtt{SJMP} & \text{size of \texttt{LJMP} instruction} \\
370\mathtt{label:} & \mathtt{MOV} & \mathtt{A}\;\;\mathtt{B}   & \Longrightarrow & \mathtt{LJMP} & \text{address of \textit{label}} \\
371       &              &                            &                 & \ldots        & \\
372       &              &                            &                 & \mathtt{MOV}  & \mathtt{A}\;\;\mathtt{B}
375Here, if \texttt{JZ} fails, we fall through to the \texttt{SJMP} which jumps over the \texttt{LJMP}.
376Naturally, 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.
378In order to implement branch displacement it is impossible to really make the \texttt{expand\_pseudo\_instruction} function completely independent of the encoding function.
379This is due to branch displacement requiring the distance in bytes of the target of the jump.
380Moreover 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.
381Proving 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.
382For 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'.
385definition expand_pseudo_instruction:
386 $\forall$lookup_labels: Identifier $\rightarrow$ Word.
[2349]387 $\forall$policy.
[2066]388 $\forall$ppc: Word.
389 $\forall$lookup_datalabels: Identifier $\rightarrow$ Word.
390 $\forall$pi: pseudo_instruction.
391  list instruction := ...
393Here, 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.
394The input \texttt{pi} is the pseudoinstruction to be expanded and is found at address \texttt{ppc} in the assembly program.
[2349]395The function takes \texttt{policy} as an input.
396In reality, this is a pair of functions, but for the purposes of this paper we simplify.
397The \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)}.
[2066]398Of 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.
[2349]400\texttt{policy}(\texttt{ppc} + 1) = \texttt{pc} + \texttt{current\_instruction\_size}
402Here, \texttt{current\_instruction\_size} is the size in bytes of the encoding of the expanded pseudoinstruction found at \texttt{ppc}.
[2083]403Note 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;
404however, 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.
[2087]405A companion submission to this one~\cite{boender:correctness:2012} certifies an algorithm that finds branch displacement policies for the assembler described in this paper.
[2349]407The \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.
408For 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}.
[2083]409A global best solution to the branch displacement problem, however, is not always made of locally best solutions.
410Therefore, in some circumstances, it is necessary to force the assembler to expand jumps into larger ones.
[2349]411This 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.
[2083]412An essentially identical mechanism exists for call instructions.
[2052]414% ---------------------------------------------------------------------------- %
415% SECTION                                                                      %
416% ---------------------------------------------------------------------------- %
417\subsection{Correctness of the assembler with respect to fetching}
[2355]419Using our policies, we now work toward proving the correctness of the assembler.
[2356]420Correctness 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.
[2351]422The 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.
[2083]423This 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 Subsection~\ref{}).
[2083]425The \texttt{assembly} function is given a Russell type (slightly simplified here):
[2083]427definition assembly:
[2339]428  $\forall$program: pseudo_assembly_program.
429  $\forall$policy.
430    $\Sigma$assembled: list Byte $\times$ (BitVectorTrie costlabel 16).
431      policy is correct for program $\rightarrow$
[2342]432      $\mid$program$\mid$ < $2^{16}$ $\rightarrow$ $\mid$fst assembled$\mid$ < $2^{16}$ $\wedge$
433      (policy ($\mid$program$\mid$) = $\mid$fst assembled$\mid$ $\vee$
434      (policy ($\mid$program$\mid$) = 0 $\wedge$ $\mid$fst assembled$\mid$ = $2^{16}$)) $\wedge$
435      $\forall$ppc: pseudo_program_counter. ppc < $2^{16}$ $\rightarrow$
[2339]436        let pseudo_instr := fetch from program at ppc in
437        let assembled_i := assemble pseudo_instr in
438          $\mid$assembled_i$\mid$ $\leq$ $2^{16}$ $\wedge$
[2342]439            $\forall$n: nat. n < $\mid$assembled_i$\mid$ $\rightarrow$ $\exists$k: nat.
[2339]440              nth assembled_i n = nth assembled (policy ppc + k).
[2327]442In plain words, the type of \texttt{assembly} states the following.
[2349]443Suppose we are given a policy that is correct for the program we are assembling.
444Then 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.
445Under 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}.
446Further, assembling the pseudoinstruction \texttt{pseudo\_instr} results in a list of bytes, \texttt{assembled\_i}.
447Then, 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}.
[2087]449Essentially the lemma above states that the \texttt{assembly} function correctly expands pseudoinstructions, and that the expanded instruction reside consecutively in memory.
[2083]450This 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}.
[2058]452Lemma \texttt{fetch\_assembly} establishes that the \texttt{fetch} and \texttt{assembly1} functions interact correctly.
[2052]453The \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:
[2058]455lemma fetch_assembly:
456 $\forall$pc: Word.
457 $\forall$i: instruction.
458 $\forall$code_memory: BitVectorTrie Byte 16.
459 $\forall$assembled: list Byte.
[2342]460  assembled = assemble i $\rightarrow$
461  let len := $\mid$assembled$\mid$ in
462  let pc_plus_len := pc + len in
463   encoding_check pc pc_plus_len assembled $\rightarrow$
464   let $\langle$instr, pc', ticks$\rangle$ := fetch pc in
[2083]465    instr = i $\wedge$ ticks = (ticks_of_instruction instr) $\wedge$ pc' = pc_plus_len.
[2358]467We read \texttt{fetch\_assembly} as follows.
[2052]468Given an instruction, \texttt{i}, we first assemble the instruction to obtain \texttt{assembled}, checking that the assembled instruction was stored in code memory correctly.
[2058]469Fetching 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.
470We 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.
[2052]471Or, in plainer words, assembling and then immediately fetching again gets you back to where you started.
[2349]473Lemma \texttt{fetch\_assembly\_pseudo} is obtained by composition of \texttt{expand\_pseudo\_instruction} and \texttt{assembly\_1\_pseudoinstruction}:
475lemma fetch_assembly_pseudo:
[2058]476 $\forall$program: pseudo_assembly_program.
[2342]477 $\forall$policy.
[2058]478 $\forall$ppc.
479 $\forall$code_memory.
480 let $\langle$preamble, instr_list$\rangle$ := program in
481 let pi := $\pi_1$ (fetch_pseudo_instruction instr_list ppc) in
[2342]482 let pc := policy ppc in
[2349]483 let instrs := expand_pseudo_instructio policy ppc pi in
484 let $\langle$l, a$\rangle$ := assembly_1_pseudoinstruction policy ppc pi in
[2342]485 let pc_plus_len := pc + l in
[2058]486  encoding_check code_memory pc pc_plus_len a $\rightarrow$
487   fetch_many code_memory pc_plus_len pc instructions.
[2058]489Here, \texttt{l} is the number of machine code instructions the pseudoinstruction at hand has been expanded into.
[2087]490We assemble a single pseudoinstruction with \texttt{assembly\_1\_pseudoinstruction}, which internally calls \texttt{expand\_pseudo\_instruction}.
[2052]491The function \texttt{fetch\_many} fetches multiple machine code instructions from code memory and performs some routine checks.
493Intuitively, Lemma \texttt{fetch\_assembly\_pseudo} can be read as follows.
[2349]494Suppose we expand the pseudoinstruction at \texttt{ppc} with the policy, obtaining the list of machine code instructions \texttt{instrs}.
[2058]495Suppose we also assemble the pseudoinstruction at \texttt{ppc} to obtain \texttt{a}, a list of bytes.
[2052]496Then, 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.
[2349]498The 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:
500lemma fetch_assembly_pseudo2:
[2058]501 $\forall$program.
[2342]502 $\mid$snd program$\mid$ $\leq$ $2^{16}$ $\rightarrow$
[2058]503 $\forall$policy.
[2342]504 policy is correct for program $\rightarrow$
505 $\forall$ppc. ppc < $\mid$snd program$\mid$ $\rightarrow$
[2349]506  let $\langle$labels, costs$\rangle$ := create_label_cost_map program in
[2351]507  let $\langle$assembled, costs'$\rangle$ := $\pi_1$ (assembly program policy) in
[2058]508  let cmem := load_code_memory assembled in
[2349]509  let $\langle$pi, newppc$\rangle$ := fetch_pseudo_instruction program ppc in
[2342]510  let instructions := expand_pseudo_instruction policy ppc pi in
511    fetch_many cmem (policy newppc) (policy ppc) instructions.
[2346]514Here we use $\pi_1$ to project the existential witness from the Russell-typed function \texttt{assembly}.
[2052]516We read \texttt{fetch\_assembly\_pseudo2} as follows.
[2349]517Suppose we are given an assembly program which can be addressed by a 16-bit word and a policy that is correct for this program.
[2058]518Suppose we are able to successfully assemble an assembly program using \texttt{assembly} and produce a code memory, \texttt{cmem}.
[2349]519Then, 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.
520The fetched sequence corresponds to the expansion, according to the policy, of the pseudoinstruction.
[2358]522At 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.
[2052]523In particular, the two semantics differ on instructions that \emph{could} directly manipulate program addresses.
525% ---------------------------------------------------------------------------- %
526% SECTION                                                                      %
527% ---------------------------------------------------------------------------- %
[2356]528\subsection{Correctness for `well-behaved' assembly programs}
[2058]531The traditional approach to verifying the correctness of an assembler is to treat memory addresses as opaque structures that cannot be modified.
[2083]532Memory 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:
534\mathtt{Mem} : \mathtt{Addr} \rightarrow \mathtt{Bytes} + \mathtt{Addr} \qquad \llbracket - \rrbracket : \mathtt{Instr} \rightarrow \mathtt{Mem} \rightarrow \mathtt{option\ Mem}
536The semantics of a pseudoinstruction, $\llbracket - \rrbracket$, is given as a possibly failing function from pseudoinstructions and memory spaces to new memory spaces.
537The 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.
539\llbracket \mathtt{ADD\ @A1\ @A2} \rrbracket^\mathtt{M} = \begin{cases}
540                                                              \mathtt{Byte\ b1},\ \mathtt{Byte\ b2} & \rightarrow \mathtt{Some}(\mathtt{M}\ \text{with}\ \mathtt{b1} + \mathtt{b2}) \\
541                                                              -,\ \mathtt{Addr\ a} & \rightarrow \mathtt{None} \\
542                                                              \mathtt{Addr\ a},\ - & \rightarrow \mathtt{None}
543                                                            \end{cases}
545In contrast, in this paper we take a different approach.
546We trace memory locations (and, potentially, registers) that contain memory addresses.
[2087]547We then 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.
[2058]549We believe that this approach is more flexible when compared to the traditional approach, as in principle it allows us to introduce some permitted \emph{benign} manipulations of addresses that the traditional approach, using opaque addresses, cannot handle, therefore expanding the set of input programs that can be assembled correctly.
[2327]550This approach, of using real addresses coupled with a weak, dynamic typing system sitting atop of memory, is similar to one taken by Tuch \emph{et al}~\cite{tuch:types:2007}, for reasoning about low-level C code.
[2358]552Our 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.
553The semantics of pseudo- and machine code are then essentially shared.
[2087]554The only thing that changes at the assembly level is the presence of the new tracking function.
[2356]556However, with this approach we must detect (at run time) programs that manipulate addresses in well-behaved ways, according to some approximation of well-behavedness.
[2060]557We use an \texttt{internal\_pseudo\_address\_map} to trace addresses of code memory addresses in internal RAM:
[2342]559definition address_entry := upper_lower $\times$ Byte.
[2060]561definition internal_pseudo_address_map :=
[2342]562  (BitVectorTrie address_entry 7) $\times$ (BitVectorTrie address_entry 7)
563    $\times$ (option address_entry).
[2350]565Here, \texttt{upper\_lower} is a type isomorphic to the booleans denoting whether a byte value is the upper or lower byte of some 16-bit address.
[2060]567The implementation of \texttt{internal\_pseudo\_address\_map} is complicated by some peculiarities of the MCS-51's instruction set.
568Note here that all addresses are 16 bit words, but are stored (and manipulated) as 8 bit bytes.
569All \texttt{MOV} instructions in the MCS-51 must use the accumulator \texttt{A} as an intermediary, moving a byte at a time.
[2350]570The 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.
571The first and second components, on the other hand, performs a similar task for the higher and lower external RAM.
572Again, we use our \texttt{upper\_lower} flag to describe whether a byte is the upper or lower component of a 16-bit address.
574The \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}.
[2087]575A similar function exists for high internal RAM.
[2350]576Note that both RAM segments are indexed using addresses 7-bits long:
[2342]578definition low_internal_ram_of_pseudo_low_internal_ram:
579 internal_pseudo_address_map $\rightarrow$ policy $\rightarrow$ BitVectorTrie Byte 7
580  $\rightarrow$ BitVectorTrie Byte 7.
583Next, we are able to translate \texttt{PseudoStatus} records into \texttt{Status} records using \texttt{status\_of\_pseudo\_status}.
584Translating a \texttt{PseudoStatus}'s code memory requires we expand pseudoinstructions and then assemble to obtain a trie of bytes.
[2063]585This never fails, provided that our policy is correct:
[2058]587definition status_of_pseudo_status:
588 internal_pseudo_address_map $\rightarrow$ $\forall$pap. $\forall$ps: PseudoStatus pap.
[2351]589 $\forall$policy. Status (code_memory_of_pseudo_assembly_program pap policy)
[2356]592The \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.
[2052]593It 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.
[2356]594It thus decides the membership of a strict subset of the set of well-behaved programs.
[2342]596definition next_internal_pseudo_address_map: internal_pseudo_address_map $\rightarrow$
597 $\forall$cm. (Identifier $\rightarrow$ PseudoStatus cm $\rightarrow$ Word) $\rightarrow$ $\forall$s: PseudoStatus cm.
598   program_counter s < $2^{16}$ $\rightarrow$ option internal_pseudo_address_map
[2350]600If we wished to allow `benign manipulations' of addresses, it would be this function that needs to be changed.
601Note we once again use dependent types to ensure that program counters are properly within bounds.
602The third argument is a function that resolves the concrete address of a label.
[2083]604The function \texttt{ticks\_of0} computes how long---in clock cycles---a pseudoinstruction will take to execute when expanded in accordance with a given policy.
[2052]605The function returns a pair of natural numbers, needed for recording the execution times of each branch of a conditional jump.
[2083]607definition ticks_of0:
[2342]608 pseudo_assembly_program $\rightarrow$ (Identifier $\rightarrow$ Word) $\rightarrow$ $\forall$policy. Word $\rightarrow$
609   pseudo_instruction $\rightarrow$ nat $\times$ nat
[2083]611An additional function, \texttt{ticks\_of}, is merely a wrapper around this function.
613Finally, we are able to state and prove our main theorem.
[2058]614This relates 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:
616theorem main_thm:
[2058]617 $\forall$M, M': internal_pseudo_address_map.
618 $\forall$program: pseudo_assembly_program.
[2342]619 $\forall$program_in_bounds: $\mid$program$\mid$ $\leq$ $2^{16}$.
620 let maps := create_label_cost_map program in
621 let addr_of := ... in
622 program is well labelled $\rightarrow$
623 $\forall$policy. policy is correct for program.
624 $\forall$ps: PseudoStatus program. ps < $\mid$program$\mid$.
625  next_internal_pseudo_address_map M program ... = Some M' $\rightarrow$
626   $\exists$n. execute n (status_of_pseudo_status M ps policy) =
627    status_of_pseudo_status M'
628      (execute_1_pseudo_instruction program
629       (ticks_of program ($\lambda$id. addr_of id ps) policy) ps) policy.
631The 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}.
[2358]632We 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.
633Theorem \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.
635% ---------------------------------------------------------------------------- %
636% SECTION                                                                      %
637% ---------------------------------------------------------------------------- %
[2355]641We are proving the correctness of an assembler for MCS-51 assembly language.
[2358]642Our 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.
[2052]643Expanding 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.
[2358]645The formalisation is a component of CerCo which aims to produce a verified concrete complexity preserving compiler for a large subset of the C language.
646The 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.
648It is interesting to compare our work to an `industrial grade' assembler for the MCS-51: SDCC~\cite{sdcc:2011}.
649SDCC is the only open source C compiler that targets the MCS-51 instruction set.
650It 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.
651Note that this policy is the only possible policy \emph{in theory} that can preserve the semantics of an assembly program during the assembly process.
652However, this comes at the expense of assembler completeness: the generated program may be too large to fit into code memory.
653In this respect, there is a trade-off between the completeness of the assembler and the efficiency of the assembled program.
[2087]654The definition and proof of a terminating, correct jump expansion policy is described in a companion publication to this one~\cite{boender:correctness:2012}.
[2095]656Aside from their application in verified compiler projects such as CerCo, CompCert~\cite{leroy:formally:2009} and CompCertTSO~\cite{sevcik:relaxed-memory:2011}, verified assemblers such as ours could also be applied to the verification of operating system kernels.
[2352]657Of particular note is the verified seL4 kernel~\cite{klein:sel4:2009}.
[2052]658This verification explicitly assumes the existence of, amongst other things, a trustworthy assembler and compiler.
[2095]660Note that both CompCert, CompCertTSO and the seL4 formalisation assume the existence of `trustworthy' assemblers.
661For instance, the CompCert C compiler's backend stops at the PowerPC assembly language, in the default backend.
662The observation that an optimising assembler cannot preserve the semantics of every assembly program may have important consequences for these projects (though in the case of CompCertTSO, targetting a multiprocessor, what exactly constitutes the subset of `good programs' may not be entirely clear).
[2052]663If CompCert chooses to assume the existence of an optimising assembler, then care should be made to ensure that any assembly program produced by the CompCert compiler falls into the subset of programs that have a hope of having their semantics preserved by an optimising assembler.
[2060]665Our formalisation exploits dependent types in different ways and for multiple purposes.
666The first purpose is to reduce potential errors in the formalisation of the microprocessor.
[2358]667Dependent types are used to constrain the size of bitvectors and tries that represent memory quantities and memory areas respectively.
[2327]668They 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.
[2060]669Polymorphic 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.
[2357]671The second purpose is to single out sources of incompleteness.
[2060]672By 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.
[2087]673This also allowed to simplify the initial proof by dropping lemmas establishing that one function fails if and only if some previous function does so.
[2357]675Finally, 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}.
676Not 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.
677It would not be natural to see the proof that fetch and assembly commute as the specification of one of the two functions.
[2358]678\paragraph{Related work}
[2052]679% piton
[2355]680We are not the first to consider the correctness of an assembler for a non-trivial assembly language.
[2358]681The 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}.
683% jinja
[2352]684Klein and Nipkow consider a Java-like programming language, Jinja~\cite{klein:machine:2006}.
[2052]685They provide a compiler, virtual machine and operational semantics for the programming language and virtual machine, and prove that their compiler is semantics and type preserving.
687We believe some other verified assemblers exist in the literature.
688However, what sets our work apart from that above is our attempt to optimise the machine code generated by our assembler.
[2083]689This complicates any formalisation effort, as an attempt at the best possible selection of machine instructions must be made, especially important on a device such as the MCS-51 with a minuscule code memory.
[2052]690Further, care must be taken to ensure that the time properties of an assembly program are not modified by the assembly process lest we affect the semantics of any program employing the MCS-51's I/O facilities.
691This is only possible by inducing a cost model on the source code from the optimisation strategy and input program.
692This will be a \emph{leit motif} of CerCo.
694Our source files are available at~\url{}.
695We assumed several properties of `library functions', e.g. modular arithmetic and datastructure manipulation.
696We 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.
697We believe that the proof strategy is sound and that we will be able to close all axioms, up to minor bugs that should have local fixes that do not affect the global proof strategy.
[2358]699The development, including the definition of the executable semantics of the MCS-51, is spread across 29 files, with around 18,500 lines of Matita source.
700The bulk of the proof is contained in a series of files, \texttt{}, \texttt{} and \texttt{} consisting of approximately 4500 lines of Matita source.
[2060]701Numerous 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.
[2358]702The 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.
703Many lines of code are required to describe the complex semantics of the processor, but for the shared cases the proof of preservation of the semantics is essentially trivial.
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