#LyX 2.2 created this file. For more info see http://www.lyx.org/
-\lyxformat 504
+\lyxformat 508
\begin_document
\begin_header
\save_transient_properties true
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\begin_layout Columns
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Frame
\begin_inset Argument 4
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\begin_layout Columns
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\color inherit
is a
-\color inherit
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Frame
\begin_inset Argument 2
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout ExampleBlock
\begin_inset Argument 2
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout ExampleBlock
\begin_inset Argument 2
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Frame
\begin_inset Argument 4
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Frame
\begin_inset Argument 1
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Frame
\begin_inset Argument 4
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\begin_layout Examples
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\begin_layout Examples
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\begin_layout Examples
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\begin_layout Corollary
For directed graphs, we can solve
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_layout
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Fact
\begin_inset Formula $\Lang{reach}_{\operatorname{path}}$
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
such that
-\color inherit
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Frame
\begin_inset Argument 4
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
such that
-\color inherit
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Frame
\begin_inset Argument 4
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Frame
\begin_inset Argument 4
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Block
\begin_inset Argument 2
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\color inherit
gets as input
-\color inherit
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Frame
\begin_inset Argument 4
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
.
- \SpecialCharNoPassThru LyX
+ \SpecialChar LyX
has the FragileFrame layout for this.
\end_layout
\begin_layout FragileFrame
This is some program code:
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Block
\begin_inset Argument 2
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_deeper
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Frame
\begin_inset Argument 4
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_layout
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Theorem
For each
\end_layout
\begin_layout Standard
-
-\begin_inset Separator parbreak
+\begin_inset Separator plain
\end_inset
-\end_layout
+\end_layout
\begin_layout Theorem
For each
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\begin_layout Corollary
For undirected graphs, we can solve
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\end_inset
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper
\begin_layout Proof
-\begin_inset Separator parbreak
-\end_inset
-
\end_layout
\begin_deeper