Add some bitvector rules

This moves #31 forward!

spec/doc.tex

@@ -73,6 +73,7 @@
 \newcommand\lsymb[1]{\mathbf{#1}}
 \DeclareMathOperator*{\subst}{subst}
 \DeclareMathOperator*{\reify}{reify}
+\DeclareMathOperator*{\bvexplode}{bvexplode}
 \newcommand\groundbox[1]{\boxed{#1}}
 
 % Proofs
@@ -1688,6 +1689,42 @@ is functional congruence, and \proofRule{sko_forall} works like
 \end{AletheS}
 \end{example}
 
+\paragraph{Bitblasting}
+
+% Note: we could also use an equality wih bvexplode (maybe called bvpack then)
+% TODO: Add example
+Bitblasting\index{bitblasting} is the translation of
+bit-vector\index{bit-vector} functions to propositional formulas.  To express
+this in the Alethe proof rules, the format uses a family of helper predicates
+$\lsymb{bvexplode}$, one for each bit-vector sort $(\lsymb{BitVec}\;n)$.
+\[
+\lsymb{bvexplode} : (\lsymb{BitVec}\;n)\,\underbrace{\lsymb{Bool} \dots \lsymb{Bool}}_n\,\lsymb{Bool}.
+\]
+
+Intuitively, the predicate $\lsymb{bvexplode}$ is true if the boolean arguments
+correspond to the bits in the first argument.
+Let ${<}u_1, \dots, u_n{>}$ denote a bit-vector of sort $(\lsymb{BitVec}\;n)$
+where $u_i = \top$ if the bit at position $i$ is $1$, and $u_i = \bot$ otherwise.
+\[
+\lsymb{bvexplode}\; {<}u_1, \dots, u_n{>} \,v_1 \dots v_n = \top.
+\]
+iff $u_i = v_i$ for all $1 ≤ i ≤ n$.
+
+The addition of the $\lsymb{bvexplode}$ predicates is a conservative extension.
+They could be defined in terms of standard SMT-LIB functions.
+\begin{align*}
+\lsymb{bvexplode}\; {<}u_1, \dots, u_n{>} \,v_1 \dots v_n :=\;
+           &(\lsymb{extract}\;1\;1) ≈ (\lsymb{ite}\,v_1\,{<}\top{>}\,{<}\bot{>})\,\land\\
+           &(\lsymb{extract}\;2\;2) ≈ (\lsymb{ite}\,v_2\,{<}\top{>}\,{<}\bot{>})\,\land\\
+           &\cdots \\
+           &(\lsymb{extract}\;n\;n) ≈ (\lsymb{ite}\,v_n\,{<}\top{>}\,{<}\bot{>})
+\end{align*}
+
+To avoid name clashes with user defined functions, $\lsymb{bvexplode}$
+is written as \inlineAlethe{.bvexplode}.  The SMT-LIB standard specifies
+that simple symbols starting with ``\inlineAlethe{.}'' are reserved for
+solver generated functions.
+
 \section{The Alethe Rules}
 \label{apx:rules}
 \input{rule_list}