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Add some bitvector rules

Merged Hans-Jörg requested to merge devel/bb2024 into master
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@@ -1532,58 +1532,58 @@ form and the reordering of equalities.
\begin{RuleDescription}{bitblast_extract}
\begin{AletheX}
$i$. & \ctxsep & $((\_\ \mathrm{extract}\ j\ i)\ x) \simeq (\mathrm{bbterm}\ x[i]\ \ldots\ x[j])$ & (\currule) \\
$i$. & \ctxsep & $((\lsymb{extract}\ j\ i)\ x) (\lsymb{bbT}\ \varphi_i\ \ldots\ \varphi_j)$ & (\currule) \\
\end{AletheX}
Each term $x[i]$ corresponds to whether the $i$-th bit of $x$ is true or not, which
will be represented via an application of the operator ``bit\_of'', i.e.,
$((\_\ bit\_of\ i)\ x)$, which has a Boolean return type.
%
The ``bbterm'' operator takes $n$ Booleans and yields a bit-vector of size $n$
where the least significant bit is 1 if the first argument 1 is true, 0
otherwise, and so on.
%
\noindent
where the formulas $\varphi_k$ are $(\lsymb{bitOf}_k\ x)$ for $i \leq k \leq j$.
Alternatively, the rule may also be phrased as a ``short-circuiting'' of the
above when $x$ is a ``bbterm'' application:
above when $x$ is a $\lsymb{bbT}$ application:
\medskip
\begin{AletheX}
$i$. & \ctxsep & $((\_\ \mathrm{extract}\ j\ i)\ (\mathrm{bbterm}\ x_0\ \ldots\
x_i\ \ldots \ x_j\ \ldots\ x_n)) \simeq (\mathrm{bbterm}\ x_i\ \ldots\ x_j)$ & (\currule) \\
$i$. & \ctxsep & $((\lsymb{extract}\ j\ i)\ (\lsymb{bbT}\ x_0\ \ldots\
x_i\ \ldots \ x_j\ \ldots\ x_n)) (\lsymb{bbT}\ x_i\ \ldots\ x_j)$ & (\currule) \\
\end{AletheX}
\noindent
This alternative is based on the validity of the equality
$$(\mathrm{bbterm}\ x_0\ \ldots\ x_i\ \ldots \ x_j\ \ldots\ x_n)[i]\simeq x_i$$
for any bit-vector $x$ of size $n+1$, where $0\leq i\leq n$.
\[
\lsymb{bitOf}_k\ (\lsymb{bbT}\ x_0\ \ldots\ x_i\ \ldots \ x_j\ \ldots\ x_n) ≈ x_k
\]
for any bit-vector $x$ of size $n+1$, where $0\leq k\leq n$.
\end{RuleDescription}
\begin{RuleDescription}{bitblast_ult}
\begin{AletheX}
$i$. & \ctxsep & $(\mathrm{bvult}\ x\ y) \simeq res_{n-1}$ & (\currule) \\
$i$. & \ctxsep & $(\lsymb{bvult}\ x\ y) \mathrm{res}_{n-1}$ & (\currule) \\
\end{AletheX}
in which both $x$ and $y$ must have the same type $(\_\ \mathrm{BitVec}\ n)$ and, for
$i\geq 0$:
in which both $x$ and $y$ must have the same type $(\lsymb{BitVec}\ n)$ and, for
$i\geq 0$
\[
\begin{array}{lcl}
\mathrm{res}_0&=&\neg x[0] \wedge y[0]\\
\mathrm{res}_{i+1}&=&((x[i+1]\simeq y[i+1])\wedge
\mathrm{res}_i)\vee (\neg x[i+1]\wedge y[i+1])
\mathrm{res}_0&=&\neg (\lsymb{bitOf}_0\ x) \wedge (\lsymb{bitOf}_0\ y)\\
\mathrm{res}_{i+1}&=&(((\lsymb{bitOf}_{i+1}\ x)(\lsymb{bitOf}_{i+1}\ y))\wedge \mathrm{res}_i)\vee
(\neg (\lsymb{bitOf}_{i+1}\ x) \wedge (\lsymb{bitOf}_{i+1}\ y))
\end{array}
\]
\noindent
Alternatively, the rule may also be phrased as a ``short-circuiting'' of the
above when $x$ and $y$ are ``bbterm'' applications. So given that
above when $x$ and $y$ are ``$\lsymb{bbT}$'' applications. So given that
\[
\begin{array}{lcl}
x&=&(\mathrm{bbterm}\ x_0\ \ldots\ x_i\ \ldots \ x_j\ \ldots\ x_n)\\
y&=&(\mathrm{bbterm}\ y_0\ \ldots\ y_i\ \ldots \ y_j\ \ldots\ y_n)\\
x&=&(\lsymb{bbT}\ x_0\ \ldots\ x_i\ \ldots \ x_j\ \ldots\ x_n)\\
y&=&(\lsymb{bbT}\ y_0\ \ldots\ y_i\ \ldots \ y_j\ \ldots\ y_n)\\
\end{array}
\]
then ``res'' can be defined, for $i \geq 0$, as
then ``$\mathrm{res}$'' can be defined, for $i \geq 0$, as
\[
\begin{array}{lcl}
\mathrm{res}_0&=&\neg x_0 \wedge y_0\\
\mathrm{res}_{i+1}&=&((x_{i+1}\simeq y_{i+1})\wedge
\mathrm{res}_{i+1}&=&((x_{i+1} y_{i+1})\wedge
\mathrm{res}_i)\vee (\neg x_{i+1}\wedge y_{i+1})
\end{array}
\]
@@ -1592,37 +1592,45 @@ then ``res'' can be defined, for $i \geq 0$, as
\begin{RuleDescription}{bitblast_add}
\begin{AletheX}
$i$. & \ctxsep & $(\mathrm{bvadd}\ x\ y) \simeq (\mathrm{bbterm}\ (x[0] \oplus y[0])\oplus\mathrm{carry}_0\ \ldots\ (x[n-1]
\oplus y[n-1])\oplus\mathrm{carry}_{n-1})$ & (\currule) \\
$i$. & \ctxsep & $(\lsymb{bvadd}\ x\ y) ≈ A_1$ & (\currule) \\
\end{AletheX}
in which both $x$ and $y$ must have the same type $(\_\ \mathrm{BitVec}\ n)$ and, for
$i\geq 0$:
\[
\begin{array}{lcl}
\mathrm{carry}_0&=&\bot\\
\mathrm{carry}_{i+1}&=&(x[i]\wedge y[i])\vee((x[i]\oplus y[i])\wedge \mathrm{carry}_i)
\end{array}
\]
in which both $x$ and $y$ must have the same type $(\lsymb{BitVec}\ n)$.
The term ``$A_1$'' is
\begin{align*}
(\lsymb{bbT}\;& (((\lsymb{bitOf}_{0}\ x) \,\lsymb{xor}\,(\lsymb{bitOf}_{0}\ y))\,\lsymb{xor}\,\mathrm{carry}_0) \\
& (((\lsymb{bitOf}_{1}\ x) \,\lsymb{xor}\,(\lsymb{bitOf}_{1}\ y))\,\lsymb{xor}\,\mathrm{carry}_1) \\
& \ldots \\
& (((\lsymb{bitOf}_{n-1}\ x) \,\lsymb{xor}\, (\lsymb{bitOf}_{n-1}\ y))\,\lsymb{xor}\,\mathrm{carry}_{n-1}))
\end{align*}
and for $i\geq 0$
\[
\begin{array}{lcl}
\mathrm{carry}_0&=&\bot\\
\mathrm{carry}_{i+1}&=&((\lsymb{bitOf}_{i}\ x) \wedge (\lsymb{bitOf}_{i}\ y))\vee(((\lsymb{bitOf}_{i}\ x)\,\lsymb{xor}\, (\lsymb{bitOf}_{i}\ y))\wedge \mathrm{carry}_i)
\end{array}
\]
\noindent
Alternatively, the rule may also be phrased as a ``short-circuiting'' of the
above when $x$ and $y$ are ``bbterm'' applications. So given that
above when $x$ and $y$ are ``$\lsymb{bbT}$'' applications. So given that
\[
\begin{array}{lcl}
x&=&(\mathrm{bbterm}\ x_0\ \ldots\ x_i\ \ldots \ x_j\ \ldots\ x_n)\\
y&=&(\mathrm{bbterm}\ y_0\ \ldots\ y_i\ \ldots \ y_j\ \ldots\ y_n)\\
x&=&(\lsymb{bbT}\ x_0\ \ldots\ x_i\ \ldots \ x_j\ \ldots\ x_n)\\
y&=&(\lsymb{bbT}\ y_0\ \ldots\ y_i\ \ldots \ y_j\ \ldots\ y_n)\\
\end{array}
\]
then the rule can be alternatively phrased as
\begin{AletheX}
$i$. & \ctxsep & $(\mathrm{bvadd}\ x\ y) \simeq (\mathrm{bbterm}\ (x_0 \oplus y_0)\oplus\mathrm{carry}_0\ \ldots\ (x_{n-1}
\oplus y_{n-1})\oplus\mathrm{carry}_{n-1})$ & (\currule) \\
$i$. & \ctxsep & $(\mathrm{bvadd}\ x\ y) ≈ A_2$ & (\currule) \\
\end{AletheX}
with ``carry'' being defined, for $i \geq 0$, as
with $A_2 := (\lsymb{bbT}\ (x_0 \,\lsymb{xor}\, y_0)\,\lsymb{xor}\,\mathrm{carry}_0\ \ldots\ (x_{n-1}
\,\lsymb{xor}\, y_{n-1})\,\lsymb{xor}\,\mathrm{carry}_{n-1})$ and
``$\mathrm{carry}$'' being defined, for $i \geq 0$, as
\[
\begin{array}{lcl}
\mathrm{carry}_0&=&\bot\\
\mathrm{carry}_{i+1}&=&(x_i\wedge y_i)\vee((x_i\oplus y_i)\wedge \mathrm{carry}_i)
\mathrm{carry}_{i+1}&=&(x_i\wedge y_i)\vee((x_i\,\lsymb{xor}\, y_i)\wedge \mathrm{carry}_i)
\end{array}
\]
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