doc: fix some typos

diff --git a/doc/implementation.tex b/doc/implementation.tex
index cec0874..758d27e 100644 (file)
--- a/doc/implementation.tex
+++ b/doc/implementation.tex
@@ -288,7 +288,7 @@ P = \{\, \vec x \to \vec y \mid
 \end{equation}
 and then the approximation computed in \eqref{eq:transitive:approx}
 is essentially the same as that of \shortciteN{Beletska2009}.
-If some of $\Delta_i$s are not singleton sets or if
+If some of the $\Delta_i$s are not singleton sets or if
 some of $\vec \delta_i$s are parametric, then we need
 to resort to further approximations.
 
@@ -988,7 +988,7 @@ $$
 instead of \eqref{eq:transitive:approx}.
 Typically, $D$ will be a strict superset of both $\domain R_i$
 and $\range R_i$.  We therefore need to check that domain
-and range of the transitive closure part of ${\cal C}(R_i,D)$,
+and range of the transitive closure are part of ${\cal C}(R_i,D)$,
 i.e., the part that results from the paths of positive length ($k \ge 1$),
 are equal to the domain and range of $R_i$.
 If not, then the incremental approach cannot be applied for
@@ -1017,7 +1017,7 @@ In particular, if we have
 \text{for each $j \ne i$ either }
 \domain R_j \subseteq D \text{ or } \domain R_j \cap \range R_i = \emptyset
 \end{equation}
-and, similarly, either
+and, similarly,
 \begin{equation}
 \label{eq:transitive:left}
 \text{for each $j \ne i$ either }