# source:src/ASM/CPP2012-policy/proof.tex@3341

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1\section{The proof}
2
3In this section, we present the correctness proof for the algorithm in more
4detail.  The main correctness statement is as follows (slightly simplified, here):
5
6
7\begin{align*}
8\mathtt{sigma\_policy\_specification} \equiv
9\lambda program.\lambda sigma.\lambda policy. \notag\\
10  sigma\ 0 = 0\ \wedge \notag\\
11        \mathbf{let}\ instr\_list \equiv code\ program\ \mathbf{in} \notag\\
12        \forall ppc.ppc < |instr\_list| \rightarrow \notag\\
13        \mathbf{let}\ pc \equiv sigma\ ppc\ \mathbf{in} \notag\\
14        \mathbf{let}\ instruction \equiv \mathtt{fetch\_pseudo\_instruction}\ instr\_list\ ppc\ \mathbf{in} \notag\\
15        \mathbf{let}\ next\_pc \equiv sigma\ (ppc+1)\ \mathbf{in}\notag\\
16                next\_pc = pc + \mathtt{instruction\_size}\ sigma\ policy\ ppc\ instruction\ \wedge\notag\\
17                (pc + \mathtt{instruction\_size}\ sigma\ policy\ ppc\ instruction < 2^{16}\ \vee\notag\\
18                (\forall ppc'.ppc' < |instr\_list| \rightarrow ppc < ppc' \rightarrow \notag\\
19          \mathbf{let}\ instruction' \equiv \mathtt{fetch\_pseudo\_instruction}\ instr\_list\ ppc'\ ppc\_ok'\ \mathbf{in} \notag\\
20                \mathtt{instruction\_size}\ sigma\ policy\ ppc'\ instruction' = 0)\ \wedge \notag\\
21        pc + \mathtt{instruction\_size}\ sigma\ policy\ ppc\ instruction = 2^{16})
22\end{align*}
23
24Informally, this means that when fetching a pseudo-instruction at $ppc$, the
25translation by $\sigma$ of $ppc+1$ is the same as $\sigma(ppc)$ plus the size
26of the instruction at $ppc$.  That is, an instruction is placed consecutively
27after the previous one, and there are no overlaps.
28
29Instructions are also stocked in
30order: the memory address of the instruction at $ppc$ should be smaller than
31the memory address of the instruction at $ppc+1$. There is one exeception to
32this rule: the instruction at the very end of the program, whose successor
33address can be zero (this is the case where the program size is exactly equal
34to the amount of memory).
35
36Finally, we enforce that the program starts at address 0, i.e. $\sigma(0) = 0$.
37
38Since our computation is a least fixed point computation, we must prove
39termination in order to prove correctness: if the algorithm is halted after
40a number of steps without reaching a fixed point, the solution is not
41guaranteed to be correct. More specifically, branch instructions might be
42encoded which do not coincide with the span between their location and their
43destination.
44
45Proof of termination rests on the fact that the encoding of branch
46instructions can only grow larger, which means that we must reach a fixed point
47after at most $2n$ iterations, with $n$ the number of branch instructions in
48the program. This worst case is reached if at every iteration, we change the
49encoding of exactly one branch instruction; since the encoding of any branch
50instructions can change first from short to absolute and then from absolute to
51long, there can be at most $2n$ changes.
52
53The proof has been carried out using the Russell'' style from~\cite{Sozeau2006}.
54We have proven some invariants of the {\sc f} function from the previous
55section; these invariants are then used to prove properties that hold for every
56iteration of the fixed point computation; and finally, we can prove some
57properties of the fixed point.
58
59\subsection{Fold invariants}
60
61These are the invariants that hold during the fold of {\sc f} over the program,
62and that will later on be used to prove the properties of the iteration.
63
64Note that during the fixed point computation, the $\sigma$ function is
65implemented as a trie for ease of access; computing $\sigma(x)$ is achieved by looking
66up the value of $x$ in the trie. Actually, during the fold, the value we
67pass along is a pair $\mathbb{N} \times \mathtt{ppc\_pc\_map}$. The first component
68is the number of bytes added to the program so far with respect to
69the previous iteration, and the second component, {\tt ppc\_pc\_map}, is a pair
70consisting of the current size of the program and our $\sigma$ function.
71
72\begin{align*}
73\mathtt{out\_of\_program\_none} \equiv \lambda prefix.\lambda sigma. \notag\\
74\forall i.i < 2^{16} \rightarrow (i > |prefix| \leftrightarrow
75        \mathtt{lookup\_opt}\ i\ sigma = \mathtt{None})
76\end{align*}
77
78This invariant states that any pseudo-address not yet examined is not
79present in the lookup trie.
80
81\begin{align*}
82\mathtt{not\_jump\_default} \equiv \lambda prefix.\lambda sigma.\notag\\
83\forall i.i < |prefix| \rightarrow\notag\\
84        \neg\mathtt{is\_jump}\ (\mathtt{nth}\ i\ prefix) \rightarrow\notag\\
85        \mathtt{lookup}\ i\ sigma = \mathtt{short\_jump}
86\end{align*}
87
88This invariant states that when we try to look up the jump length of a
89pseudo-address where there is no branch instruction, we will get the default
90value, a short jump.
91
92\begin{align*}
93        \mathtt{jump\_increase} \equiv \lambda pc.\lambda op\lambda p.
94        \forall i.i < |prefix| \rightarrow\notag\\
95        \mathbf{let}\ oj \equiv \mathtt{lookup}\ i\ op\ \mathbf{in} \notag\\
96        \mathbf{let}\ j \equiv \mathtt{lookup}\ i\ p\ \mathbf{in} \notag\\
97                \mathtt{jmpleq}\ oj\ j
98\end{align*}
99
100This invariant states that between iterations (with $op$ being the previous
101iteration, and $p$ the current one), jump lengths either remain equal or
102increase. It is needed for proving termination.
103
104\begin{align*}
105\mathtt{sigma\_compact\_unsafe} \equiv \lambda prefix.\lambda sigma.\notag\\
106\forall n.n < |prefix| \rightarrow\notag\\
107\mathbf{match}\ \mathtt{lookup\_opt}\ n\ sigma\ \mathbf{with}\notag\\
108\mathtt{None} \Rightarrow \mathrm{False} \notag\\
109\mathtt{Some}\ \langle pc, j \rangle \Rightarrow
110        \mathbf{match}\ \mathtt{lookup\_opt}\ (n+1)\ sigma\ \mathbf{with}\notag\\
111        \mathtt{None} \Rightarrow \mathrm{False} \notag\\
112        \mathtt{Some}\ \langle pc_1, j_1 \rangle \Rightarrow
113                pc_1 = pc + \mathtt{instruction\_size\_jmplen}\ j\ (\mathtt{nth}\ n\ prefix)
114\end{align*}
115
116This is a temporary formulation of the main property\\
117({\tt sigma\_policy\_specification}); its main difference
118from the final version is that it uses {\tt instruction\_size\_jmplen} to
119compute the instruction size. This function uses $j$ to compute the span
120of branch instructions  (i.e. it uses the $\sigma$ function under construction),
121instead of looking at the distance between source and destination. This is
122because $\sigma$ is still under construction; later on we will prove that after
123the final iteration, {\tt sigma\_compact\_unsafe} is equivalent to the main
124property.
125
126\begin{align*}
127\mathtt{sigma\_safe} \equiv \lambda prefix.\lambda labels.\lambda old\_sigma.\lambda sigma\notag\\
128\forall i.i < |prefix| \rightarrow \notag\\
129\mathbf{let}\ instr \equiv \mathtt{nth}\ i\ prefix\ \mathbf{in}\notag\\
130\forall dest\_label.\mathtt{is\_jump\_to\_instr}\ dest\_label \rightarrow\notag\\
131\mathbf{let}\ paddr \equiv \mathtt{lookup}\ labels\ dest\_label\ \mathbf{in} \notag\\
132\mathbf{let}\ \langle j, src, dest \rangle \equiv\notag\\
134  \mathbf{let}\ \langle \_, j \rangle \equiv \mathtt{lookup}\ i\ sigma\ \mathbf{in} \notag\\
135        \mathbf{let}\ \langle pc\_plus\_jl, \_ \rangle \equiv \mathtt{lookup}\ (i+1)\ sigma\ \mathbf{in}\notag\\
137        \langle j, pc\_plus\_jl, addr \rangle\notag\\
138\mathbf{else} \\
139  \mathbf{let}\ \langle \_, j \rangle \equiv \mathtt{lookup}\ i\ sigma\ \mathbf{in} \notag\\
140        \mathbf{let}\ \langle pc\_plus\_jl, \_ \rangle \equiv \mathtt{lookup}\ (i+1)\ old\_sigma\ \mathbf{in}\notag\\
142        \langle j, pc\_plus\_jl, addr \rangle\ \mathbf{in}\notag\\
143        \mathbf{match}\ j\ \mathbf{with} \notag\\
144        \mathrm{short\_jump} \Rightarrow \mathtt{short\_jump\_valid}\ src\ dest\notag\\
145        \mathrm{absolute\_jump} \Rightarrow \mathtt{absolute\_jump\_valid}\ src\ dest\notag\\
146        \mathrm{long\_jump} \Rightarrow \mathrm{True}
147\end{align*}
148
149This is a more direct safety property: it states that branch instructions are
150encoded properly, and that no wrong branch instructions are chosen.
151
152Note that we compute the distance using the memory address of the instruction
153plus its size: this follows the behaviour of the MCS-51 microprocessor, which
154increases the program counter directly after fetching, and only then executes
155the branch instruction (by changing the program counter again).
156
157\begin{lstlisting}
158\fst (bvt_lookup $\ldots$ (bitvector_of_nat ? 0) (\snd policy)
159 $\langle$0,short_jump$\rangle$) = 0)
160\fst policy = \fst (bvt_lookup $\ldots$
161 (bitvector_of_nat ? (|prefix|)) (\snd policy) $\langle$0,short_jump$\rangle$)
162\end{lstlisting}
163
164These two properties give the values of $\sigma$ for the start and end of the
165program; $\sigma(0) = 0$ and $\sigma(n)$, where $n$ is the number of
166instructions up until now, is equal to the maximum memory address so far.
167
168\begin{lstlisting}
169(added = 0 $\rightarrow$ policy_pc_equal prefix old_sigma policy))
170(policy_jump_equal prefix old_sigma policy $\rightarrow$ added = 0))
171\end{lstlisting}
172
173And finally, two properties that deal with what happens when the previous
174iteration does not change with respect to the current one. $added$ is a
175variable that keeps track of the number of bytes we have added to the program
176size by changing the encoding of branch instructions. If $added$ is 0, the program
177has not changed and vice versa.
178
179We need to use two different formulations, because the fact that $added$ is 0
180does not guarantee that no branch instructions have changed.  For instance,
181it is possible that we have replaced a short jump with an absolute jump, which
182does not change the size of the branch instruction.
183
184Therefore {\tt policy\_pc\_equal} states that $old\_sigma_1(x) = sigma_1(x)$,
185whereas {\tt policy\_jump\_equal} states that $old\_sigma_2(x) = sigma_2(x)$.
186This formulation is sufficient to prove termination and compactness.
187
188Proving these invariants is simple, usually by induction on the prefix length.
189
190\subsection{Iteration invariants}
191
192These are invariants that hold after the completion of an iteration. The main
193difference between these invariants and the fold invariants is that after the
194completion of the fold, we check whether the program size does not supersede
19564 Kb, the maximum memory size the MCS-51 can address.
196
197The type of an iteration therefore becomes an option type: {\tt None} in case
198the program becomes larger than 64 Kb, or $\mathtt{Some}\ \sigma$
199otherwise. We also no longer use a natural number to pass along the number of
200bytes added to the program size, but a boolean that indicates whether we have
201changed something during the iteration or not.
202
203If an iteration returns {\tt None}, we have the following invariant:
204
205\clearpage
206\begin{lstlisting}
207definition nec_plus_ultra :=
208 $\lambda$program:list labelled_instruction.$\lambda$p:ppc_pc_map.
209 ¬($\forall$i.i < |program| $\rightarrow$
210  is_jump (\snd (nth i ? program $\langle$None ?, Comment []$\rangle$)) $\rightarrow$
211  \snd (bvt_lookup $\ldots$ (bitvector_of_nat 16 i) (\snd p) $\langle$0,short_jump$\rangle$) =
212   long_jump).
213\end{lstlisting}
214
215This invariant is applied to $old\_sigma$; if our program becomes too large
216for memory, the previous iteration cannot have every branch instruction encoded
217as a long jump. This is needed later in the proof of termination.
218
219If the iteration returns $\mathtt{Some}\ \sigma$, the invariants
220{\tt out\_of\_program\_none},\\
221{\tt not\_jump\_default}, {\tt jump\_increase},
222and the two invariants that deal with $\sigma(0)$ and $\sigma(n)$ are
223retained without change.
224
225Instead of using {\tt sigma\_compact\_unsafe}, we can now use the proper
226invariant:
227
228\begin{lstlisting}
229definition sigma_compact :=
230 $\lambda$program:list labelled_instruction.$\lambda$labels:label_map.$\lambda$sigma:ppc_pc_map.
231 $\forall$n.n < |program| $\rightarrow$
232  match bvt_lookup_opt $\ldots$ (bitvector_of_nat ? n) (\snd sigma) with
233  [ None $\Rightarrow$ False
234  | Some x $\Rightarrow$ let $\langle$pc,j$\rangle$ := x in
235    match bvt_lookup_opt $\ldots$ (bitvector_of_nat ? (S n)) (\snd sigma) with
236    [ None $\Rightarrow$ False
237    | Some x1 $\Rightarrow$ let $\langle$pc1,j1$\rangle$ := x1 in
238      pc1 = pc + instruction_size
239       ($\lambda$id.bitvector_of_nat ? (lookup_def ?? labels id 0))
240       ($\lambda$ppc.bitvector_of_nat ?
241        (\fst (bvt_lookup $\ldots$ ppc (\snd sigma) $\langle$0,short_jump$\rangle$)))
242       ($\lambda$ppc.jmpeqb long_jump (\snd (bvt_lookup $\ldots$ ppc
243        (\snd sigma) $\langle$0,short_jump$\rangle$))) (bitvector_of_nat ? n)
244       (\snd (nth n ? program $\langle$None ?, Comment []$\rangle$))
245    ]
246  ].
247\end{lstlisting}
248
249This is almost the same invariant as ${\tt sigma\_compact\_unsafe}$, but differs in that it
250computes the sizes of branch instructions by looking at the distance between
251position and destination using $\sigma$.
252
253In actual use, the invariant is qualified: $\sigma$ is compact if there have
254been no changes (i.e. the boolean passed along is {\tt true}). This is to
255reflect the fact that we are doing a least fixed point computation: the result
256is only correct when we have reached the fixed point.
257
258There is another, trivial, invariant if the iteration returns
259$\mathtt{Some}\ \sigma$:
260
261\begin{lstlisting}
262\fst p < 2^16
263\end{lstlisting}
264
265The invariants that are taken directly from the fold invariants are trivial to
266prove.
267
268The proof of {\tt nec\_plus\_ultra} works as follows: if we return {\tt None},
269then the program size must be greater than 64 Kb. However, since the
270previous iteration did not return {\tt None} (because otherwise we would
271terminate immediately), the program size in the previous iteration must have
272been smaller than 64 Kb.
273
274Suppose that all the branch instructions in the previous iteration are
275encoded as long jumps. This means that all branch instructions in this
276iteration are long jumps as well, and therefore that both iterations are equal
277in the encoding of their branch instructions. Per the invariant, this means that
278$added = 0$, and therefore that all addresses in both iterations are equal.
279But if all addresses are equal, the program sizes must be equal too, which
280means that the program size in the current iteration must be smaller than
28164 Kb. This contradicts the earlier hypothesis, hence not all branch
282instructions in the previous iteration are encoded as long jumps.
283
284The proof of {\tt sigma\_compact} follows from {\tt sigma\_compact\_unsafe} and
285the fact that we have reached a fixed point, i.e. the previous iteration and
286the current iteration are the same. This means that the results of
287{\tt instruction\_size\_jmplen} and {\tt instruction\_size} are the same.
288
289\subsection{Final properties}
290
291These are the invariants that hold after $2n$ iterations, where $n$ is the
292program size (we use the program size for convenience; we could also use the
293number of branch instructions, but this is more complex). Here, we only
294need {\tt out\_of\_program\_none}, {\tt sigma\_compact} and the fact that
295$\sigma(0) = 0$.
296
297Termination can now be proved using the fact that there is a $k \leq 2n$, with
298$n$ the length of the program, such that iteration $k$ is equal to iteration
299$k+1$. There are two possibilities: either there is a $k < 2n$ such that this
300property holds, or every iteration up to $2n$ is different. In the latter case,
301since the only changes between the iterations can be from shorter jumps to
302longer jumps, in iteration $2n$ every branch instruction must be encoded as
303a long jump. In this case, iteration $2n$ is equal to iteration $2n+1$ and the
304fixpoint is reached.
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