From ca5665c27691bbfa300976db30cafeeeebca493c Mon Sep 17 00:00:00 2001
From: Spencer Smith <smiths@mcmaster.ca>
Date: Sun, 18 Mar 2018 11:16:20 -0400
Subject: [PATCH] Addition of A3 Solution for Part 1

---
 Assignments/A3/A3Soln/A3P1_Spec.pdf | Bin 0 -> 118359 bytes
 Assignments/A3/A3Soln/A3P1_Spec.tex | 710 ++++++++++++++++++++++++++++
 2 files changed, 710 insertions(+)
 create mode 100644 Assignments/A3/A3Soln/A3P1_Spec.pdf
 create mode 100644 Assignments/A3/A3Soln/A3P1_Spec.tex

diff --git a/Assignments/A3/A3Soln/A3P1_Spec.pdf b/Assignments/A3/A3Soln/A3P1_Spec.pdf
new file mode 100644
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diff --git a/Assignments/A3/A3Soln/A3P1_Spec.tex b/Assignments/A3/A3Soln/A3P1_Spec.tex
new file mode 100644
index 00000000..6ef43a43
--- /dev/null
+++ b/Assignments/A3/A3Soln/A3P1_Spec.tex
@@ -0,0 +1,710 @@
+\documentclass[12pt]{article}
+
+\usepackage{graphicx}
+\usepackage{paralist}
+\usepackage{amsfonts}
+\usepackage{amsmath}
+\usepackage{hhline}
+\usepackage{booktabs}
+\usepackage{multirow}
+\usepackage{multicol}
+
+\oddsidemargin 0mm
+\evensidemargin 0mm
+\textwidth 160mm
+\textheight 200mm
+\renewcommand\baselinestretch{1.0}
+
+\pagestyle {plain}
+\pagenumbering{arabic}
+
+\newcounter{stepnum}
+
+%% Comments
+
+\usepackage{color}
+
+\newif\ifcomments\commentstrue
+
+\ifcomments
+\newcommand{\authornote}[3]{\textcolor{#1}{[#3 ---#2]}}
+\newcommand{\todo}[1]{\textcolor{red}{[TODO: #1]}}
+\else
+\newcommand{\authornote}[3]{}
+\newcommand{\todo}[1]{}
+\fi
+
+\newcommand{\wss}[1]{\authornote{blue}{SS}{#1}}
+
+\title{Assignment 3, Part 1, Specification}
+\author{SFWR ENG 2AA4}
+
+\begin {document}
+
+\maketitle
+
+The purpose of this software design exercise is to design and implement a
+portion of the specification for a Geographic Information System (GIS).  This
+document shows the complete specification, which will be the basis for your
+implementation and testing.  In this specification natural numbers
+($\mathbb{N}$) include zero ($0$).
+
+\wss{The parts that you need to fill in are marked by comments, like this one.
+  In several of the modules local functions are specified.  You can use these
+  local functions to complete the missing specifications.}
+
+\newpage
+
+\section* {Map Types Module}
+
+\subsection*{Module}
+
+MapTypes
+
+\subsection* {Uses}
+
+N/A
+
+\subsection* {Syntax}
+
+\subsubsection* {Exported Constants}
+
+None
+
+\subsubsection* {Exported Types}
+
+CompassT = \{N, S, E, W\}\\ 
+LanduseT = \{Recreational, Transport, Agricultural, Residential, Commercial\}\\
+RotateT = \{CW, CCW\}
+
+\subsubsection* {Exported Access Programs}
+
+None
+
+\subsection* {Semantics}
+
+\subsubsection* {State Variables}
+
+None
+
+\subsubsection* {State Invariant}
+
+None
+
+\newpage
+
+\section* {Point ADT Module}
+
+\subsection*{Template Module}
+
+PointT
+
+\subsection* {Uses}
+
+N/A
+
+\subsection* {Syntax}
+
+\subsubsection* {Exported Types}
+
+\wss{What should be written here?}PointT = ?
+
+\subsubsection* {Exported Access Programs}
+
+\begin{tabular}{| l | l | l | l |}
+\hline
+\textbf{Routine name} & \textbf{In} & \textbf{Out} & \textbf{Exceptions}\\
+\hline
+PointT & $\mathbb{Z}$, $\mathbb{Z}$ & PointT & \\
+\hline
+x & ~ & $\mathbb{Z}$ & ~\\
+\hline
+y & ~ & $\mathbb{Z}$ & ~\\
+\hline
+translate & $\mathbb{Z}$, $\mathbb{Z}$ & PointT & ~\\
+\hline
+\end{tabular}
+
+\subsection* {Semantics}
+
+\subsubsection* {State Variables}
+
+$xc$: \wss{What is the type of the state variables?}$\mathbb{Z}$\\
+$yc$: \wss{What is the type of the state variables?}$\mathbb{Z}$
+
+\subsubsection* {State Invariant}
+
+None
+
+\subsubsection* {Assumptions}
+
+The constructor PointT is called for each object instance before any other
+access routine is called for that object.  The constructor cannot be called on
+an existing object.
+
+\subsubsection* {Access Routine Semantics}
+
+PointT($x, y$):
+\begin{itemize}
+\item transition: \wss{What should the state transition be for the constructor?}$xc, yc := x, y$
+\item output: $out := \mathit{self}$
+\item exception: None
+\end{itemize}
+
+\noindent x():
+\begin{itemize}
+\item output: $out := xc$
+\item exception: None
+\end{itemize}
+
+\noindent y():
+\begin{itemize}
+\item \wss{What should go here?}output: $out := yc$
+\item exception: None
+\end{itemize}
+
+\noindent translate($\Delta x$, $\Delta y$):
+\begin{itemize}
+\item \wss{What should go here?}output: $out := \mbox{PointT}(xc + \Delta x, yc + \Delta y)$
+\item exception: \wss{What should go here?}None
+\end{itemize}
+
+\newpage
+
+\section* {Line ADT Module}
+
+\subsection*{Template Module}
+
+LineT
+
+\subsection* {Uses}
+
+\wss{What should go here?}PointT, MapTypes
+
+\subsection* {Syntax}
+
+\subsubsection* {Exported Types}
+
+LineT = ?
+
+\subsubsection* {Exported Access Programs}
+
+\begin{tabular}{| l | l | l | l |}
+\hline
+\textbf{Routine name} & \textbf{In} & \textbf{Out} & \textbf{Exceptions}\\
+\hline
+LineT & PointT, CompassT, $\mathbb{N}$ & LineT & invalid\_argument\\
+\hline
+strt & ~ & PointT & ~\\
+\hline
+end & ~ & PointT & ~\\
+\hline
+orient & ~ & CompassT & ~\\
+\hline
+ len & ~ & $\mathbb{N}$ & ~\\
+\hline
+flip & ~ & LineT & ~\\
+\hline
+rotate & RotateT & LineT & ~\\
+\hline
+translate & $\mathbb{Z}$, $\mathbb{Z}$ & LineT  & ~\\
+\hline
+\end{tabular}
+
+\subsection* {Semantics}
+
+\subsubsection* {State Variables}
+
+$s$: PointT\\
+$o$: CompassT\\
+$L$: $\mathbb{N}$\\ 
+
+\subsubsection* {State Invariant}
+
+None
+
+\subsubsection* {Assumptions}
+
+The constructor LineT is called for each object instance before any other
+access routine is called for that object.  The constructor cannot be called on
+an existing object.
+
+\subsubsection* {Access Routine Semantics}
+
+LineT($st, ornt, l$):
+\begin{itemize}
+\item transition: $s, o, L := st, ornt, l$
+\item output: $out := \mathit{self}$
+\item exception: \wss{Write the spec for an exception when the length of the line is 0}$exc := (L = 0 \Rightarrow \mathrm{invalid\_argument})$
+\end{itemize}
+
+\noindent strt():
+\begin{itemize}
+\item output: $out := \mbox{PointT}(s.x(), s.y())$
+\item exception: None
+\end{itemize}
+
+\noindent end():
+\begin{itemize}
+% \item output:
+%   $out := (o = \mbox{N} \Rightarrow \mbox{PointT}(s.x, s.y + (L-1)) | o = \mbox{S} \Rightarrow
+%   \mbox{PointT}(s.x, s.y - (L-1)) | o = \mbox{W} \Rightarrow \mbox{PointT}(s.x - (L-1), s.y) |
+%   o = \mbox{E} \Rightarrow \mbox{PointT}(s.x + (L-1), s.y))$
+\item output: \wss{Write the spec for returning the end point of the
+    line.}$out := \mbox{PointT}((o = \mbox{W} \Rightarrow s.x - (L-1) | o =
+  \mbox{E} \Rightarrow s.x + (L-1) | \mbox{True} \Rightarrow s.x), (o = \mbox{N}
+  \Rightarrow s.y + (L-1) | o = \mbox{S} \Rightarrow s.y - (L-1) | \mbox{True}
+  \Rightarrow s.y))$
+
+\item exception: None
+\end{itemize}
+
+\noindent orient():
+\begin{itemize}
+\item output: $out := o$
+\item exception: None
+\end{itemize}
+
+\noindent len():
+\begin{itemize}
+\item output: $out := L$
+\item exception: None
+\end{itemize}
+
+\noindent flip():
+\begin{itemize}
+\item output: \wss{Write the spec for returning a new line that is the mirror
+    image of the current line.  That is, the start point and length of the new
+    line will remain the same, but the orientation will be changed by 180
+    degrees}$out := \mbox{LineT}(s, (o = \mbox{N} \Rightarrow \mbox{S} | o =
+  \mbox{S} \Rightarrow \mbox{N} | o = \mbox{W} \Rightarrow \mbox{E} | o =
+  \mbox{E} \Rightarrow \mbox{W}), L)$
+\item exception: None
+\end{itemize}
+
+\noindent rotate(r):
+\begin{itemize}
+\item output: 
+\begin{tabular}{|p{2.cm}|p{2.3cm}|l|}
+\hhline{~|~|-|}
+\multicolumn{1}{r}{} & \multicolumn{1}{r|}{} & \multicolumn{1}{l|}{$out :=$}\\
+\hhline{|-|-|-|}
+$r = \mbox{CW}$ & $o = \mbox{N}$ & \wss{?}$\mbox{LineT}(s, E, L)$\\
+\hhline{|~|-|-|}
+~ & $o = \mbox{S}$ & \wss{?}$\mbox{LineT}(s, W, L)$\\
+\hhline{|~|-|-|}
+~ & $o = \mbox{W}$ & \wss{?}$\mbox{LineT}(s, N, L)$\\
+\hhline{|~|-|-|}
+~ & $o = \mbox{E}$ & \wss{?}$\mbox{LineT}(s, S, L)$\\
+\hhline{-|-|-|}
+$r = \mbox{CCW}$ & $o = \mbox{N}$ & \wss{?}$\mbox{LineT}(s, W, L)$\\
+\hhline{|~|-|-|}
+~ & $o = \mbox{S}$ & \wss{?}$\mbox{LineT}(s, E, L)$\\
+\hhline{|~|-|-|}
+~ & $o = \mbox{W}$ & \wss{?}$\mbox{LineT}(s, S, L)$\\
+\hhline{|~|-|-|}
+~ & $o = \mbox{E}$ & \wss{?}$\mbox{LineT}(s, N, L)$\\
+\hhline{-|-|-|}
+\end{tabular}
+
+\item exception: None
+\end{itemize}
+
+\noindent translate($\Delta x$, $\Delta y$):
+\begin{itemize}
+\item output: \wss{Add the missing spec}$out := \mbox{LineT}(s.\mbox{translate}(\Delta x, \Delta y), o, L)$
+\item exception: None
+\end{itemize}
+
+\newpage
+
+\section* {Path ADT Module}
+
+\subsection*{Template Module}
+
+PathT
+
+\subsection* {Uses}
+
+PointT, LineT, MapTypes
+
+\subsection* {Syntax}
+
+\subsubsection* {Exported Types}
+
+PathT = ?
+
+\subsubsection* {Exported Access Programs}
+
+\begin{tabular}{| l | l | l | l |}
+\hline
+\textbf{Routine name} & \textbf{In} & \textbf{Out} & \textbf{Exceptions}\\
+\hline
+PathT & PointT, CompassT, $\mathbb{N}$ & PathT & \\ %LineT constructor for invalid\_argument
+\hline
+append & CompassT, $\mathbb{N}$ & & invalid\_argument\\ 
+\hline
+strt & ~ & PointT & ~\\
+\hline
+end & ~ & PointT & ~\\
+\hline
+line & $\mathbb{N}$ & LineT & outside\_bounds\\
+\hline
+size & ~ & $\mathbb{N}$ & \\
+\hline
+ len & ~ & $\mathbb{N}$ & ~\\
+\hline
+translate & $\mathbb{Z}$, $\mathbb{Z}$ & PathT  & ~\\
+\hline
+\end{tabular}
+
+\subsection* {Semantics}
+
+\subsubsection* {State Variables}
+
+$s$: sequence of LineT
+
+\subsubsection* {State Invariant}
+
+None
+
+\subsubsection* {Assumptions}
+
+\begin{itemize}
+\item The constructor PathT is called for each object instance before any other
+access routine is called for that object.  The constructor cannot be called on
+an existing object.
+%\item Appends will not be made while iterating through the list of lines.
+\end{itemize}
+
+\subsubsection* {Access Routine Semantics}
+
+PathT($st, ornt, l$):
+\begin{itemize}
+\item transition: \wss{What is the spec to add the first element to the sequence
+  of LineT?}$s[0] := \mbox{LineT}(st, ornt, l)$
+\item output: $out := \mathit{self}$
+\item exception: None
+\end{itemize}
+
+\noindent append($ornt, l$):
+\begin{itemize}
+\item transition: \wss{What is the missing specification?  The appended line
+    starts at a point adjacent to the end point of the previous line in the
+    direction $ornt$.  The lines are not allowed to
+    overlap.}$s := s || \langle \mbox{LineT}(\mbox{adjPt}(ornt), ornt, l\rangle$
+\item exception: \wss{What is the specification for the exception?  An exception
+    should be generated if the introduced line overlaps with any of the previous
+    points in the existing path.}
+\begin{multline*}
+  exc :=\\
+  (\mbox{pointsInLine}(\mbox{LineT}(st, ornt, l)) \cap (\cup
+  (l: \mbox{LineT} | l \in s : \mbox{pointsInLine}(l))) \neq \emptyset \Rightarrow \mbox{invalid\_argument})
+\end{multline*}
+\end{itemize}
+
+\noindent strt():
+\begin{itemize}
+\item output: \wss{What is the missing spec?}$out := s[0].\mbox{strt}$
+\item exception: None
+\end{itemize}
+
+\noindent end():
+\begin{itemize}
+\item output: \wss{What is the missing spec?}$out := s[|s|-1].\mbox{end}$
+\item exception: None
+\end{itemize}
+
+\noindent line($i$):
+\begin{itemize}
+\item output: \wss{Returns the ith line in the sequence.  What is the missing spec?}$out := \mbox{LineT}(s[i].\mbox{strt}(), s[i].\mbox{orient}(), s[i].\mbox{len}())$
+\item exception: \wss{Generate the exception if the index is not in the sequence.}$exc := (i \geq |s| \Rightarrow \mbox{outside\_bounds})$
+\end{itemize}
+
+\noindent size:
+\begin{itemize}
+\item output: \wss{Output the number of lines in the path.}$out := |s|$
+\item exception: None
+\end{itemize}
+
+\noindent len:
+\begin{itemize}
+\item output: \wss{Output the total number of points (grid cells) on the path,
+    including the beginning and end points (cells).}$out := +(l: \mbox{LineT} | i \in s: l.\mbox{len})$
+\item exception: None
+\end{itemize}
+
+\noindent translate($\Delta x$, $\Delta y$):
+\begin{itemize}
+\item output: Create a new PathT object with state variable $s'$ such that:
+$$\forall(i: \mathbb{N} | i \in [0..|s|-1] : s'[i] = s[i].\mbox{translate}(\Delta x, \Delta y))$$
+\item exception: None
+\end{itemize}
+
+\subsection*{Local Functions}
+
+pointsInLine: $\mbox{LineT} \rightarrow \mbox{(set of PointT)}$\\
+
+\noindent pointsInLine ($l$) 
+\begin{multline*}
+\equiv \{ i: \mathbb{N} | i \in [0
+  .. (l.\mbox{len} - 1)] : l.\mbox{strt}.\mbox{translate}(\wss{Complete the spec.}\\
+(l.\mbox{orient}=\mbox{W} \Rightarrow -i |
+  l.\mbox{orient}=\mbox{E} \Rightarrow i | \mbox{True} \Rightarrow 0), (l.\mbox{orient}=\mbox{N} \Rightarrow i |
+  l.\mbox{orient}=\mbox{S} \Rightarrow -i | \mbox{True} \Rightarrow 0) ) \}
+\end{multline*}
+
+\noindent adjPt: $\mbox{CompassT} \rightarrow \mbox{PointT}$\\
+adjPt$(ornt) \equiv$
+
+\medskip
+
+\begin{tabular}{|l|l|}
+\hline
+$ornt = \mbox{N}$ & $s[|s|-1].\mbox{end}.\mbox{translate}\wss{?}(0, 1)$\\
+\hline
+$ornt = \mbox{S}$ & $s[|s|-1].\mbox{end}.\mbox{translate}\wss{?}(0, -1)$\\
+\hline
+$ornt = \mbox{W}$ & $s[|s|-1].\mbox{end}.\mbox{translate}\wss{?}(-1, 0)$\\
+\hline
+$ornt = \mbox{E}$ & $s[|s|-1].\mbox{end}.\mbox{translate}\wss{?}(1, 0)$\\
+\hline
+\end{tabular}
+
+\newpage
+
+\section* {Generic Seq2D Module}
+
+\subsection* {Generic Template Module}
+
+Seq2D(T)
+
+\subsection* {Uses}
+
+MapTypes, PointT, LineT, PathT
+
+\subsection* {Syntax}
+
+\subsubsection* {Exported Types}
+
+Seq2D(T) = ?
+
+\subsubsection* {Exported Constants}
+
+None
+
+\subsubsection* {Exported Access Programs}
+
+\begin{tabular}{| l | l | l | p{5cm} |}
+\hline
+\textbf{Routine name} & \textbf{In} & \textbf{Out} & \textbf{Exceptions}\\
+\hline
+Seq2D & seq of (seq of T), $\mathbb{R}$ & Seq2D & invalid\_argument\\
+\hline
+set & PointT, T & ~ & outside\_bounds\\
+\hline
+get & PointT & T & outside\_bounds\\
+\hline
+getNumRow & ~ & $\mathbb{N}$ & \\
+\hline
+getNumCol & ~ & $\mathbb{N}$ & \\
+\hline
+getScale & ~ & $\mathbb{R}$ & \\
+\hline
+count & T & $\mathbb{N}$ & \\
+\hline
+count & LineT, T & $\mathbb{N}$ & invalid\_argument\\
+\hline
+count & PathT, T & $\mathbb{N}$ & invalid\_argument\\
+\hline
+length & PathT & $\mathbb{R}$ & invalid\_argument\\
+\hline
+connected & PointT, PointT & $\mathbb{B}$ & invalid\_argument\\
+\hline
+\end{tabular}
+
+\subsection* {Semantics}
+
+\subsubsection* {State Variables}
+
+$s$: seq of (seq of T)\\
+scale: $\mathbb{R}$\\
+nRow: $\mathbb{N}$\\
+nCol: $\mathbb{N}$
+
+\subsubsection* {State Invariant}
+
+None
+
+\subsubsection* {Assumptions}
+
+\begin{itemize}
+\item The Seq2D(T) constructor is called for each object instance before any
+other access routine is called for that object.  The constructor can only be
+called once.
+\item Assume that the input to the constructor is a sequence of rows, where each
+  row is a sequence of elements of type T.  The number of columns (number of
+  elements) in each row is assumed to be equal. That is each row
+  of the grid has the same number of entries.  $s[i][j]$ means the ith row and
+  the jth column.  The 0th row is at the bottom of the map and the 0th column
+  is at the leftmost side of the map.
+\end{itemize}
+
+\subsubsection* {Access Routine Semantics}
+
+Seq2D($S$, scl):
+\begin{itemize}
+\item transition: \wss{Fill in the transition.}$s, \mbox{scale}, \mbox{nCol}, \mbox{nRow} := S,
+  \mbox{scl}, |S[0]|, |S|$
+\item output: $\mathit{out} := \mathit{self}$
+\item exception: \wss{Fill in the exception.  One should be generated if the
+    scale is less than zero, or the input sequence is empty, or the number of
+    columns is zero in the first row, or the number of columns in any row is
+    different from the number of columns in the first row.}$exc := (\mbox{scale} \leq 0 \lor |S| = 0 \lor |S[0]| = 0 \Rightarrow \mbox{invalid\_argument}|\\
+  \lnot\forall(l : \mbox{seq of T}|l \in S : |l| = |S[0]|) \Rightarrow
+  \mbox{invalid\_argument})$
+\end{itemize}
+
+\noindent set($p, v$):
+\begin{itemize}
+\item transition: \wss{?}$s[p.y][p.x] := v$
+\item exception: \wss{Generate an exception if the point lies outside of the
+    map.}$exc := (\neg \mbox{validPoint}(p) \Rightarrow \mbox{outside\_bounds})$
+\end{itemize}
+
+\noindent get($p$):
+\begin{itemize}
+\item output: \wss{?}$out := s[p.y][p.x]$
+\item exception: \wss{Generate an exception if the point lies outside of the
+    map.}$exc := (\neg \mbox{validPoint}(p) \Rightarrow \mbox{outside\_bounds})$
+\end{itemize}
+
+\noindent getNumRow():
+\begin{itemize}
+\item output: $out := \mbox{nRow}$
+\item exception: None
+\end{itemize}
+
+\noindent getNumCol():
+\begin{itemize}
+\item output: $out := \mbox{nCol}$
+\item exception: None
+\end{itemize}
+
+\noindent getScale():
+\begin{itemize}
+\item output: $out := \mbox{scale}$
+\item exception: None
+\end{itemize}
+
+\noindent count($t$: T):
+\begin{itemize}
+\item output: \wss{Count the number of times the value $t$ occurs in the 2D
+    sequence.}$out := +(i, j: \mathbb{N}| \mbox{validRow}(i) \land
+  \mbox{validCol}(j) \wedge s[i][j] = t : 1)$
+\item exception: None
+\end{itemize}
+
+\noindent count($l$: LineT, $t$: T):
+\begin{itemize}
+\item output: \wss{Count the number of times the value $t$ occurs in the line
+    $l$.}$out := +(p: \mbox{PointT} | p \in \mbox{pointsInLine}(l) \wedge
+  s[p.y][p.x] = t : 1)$
+\item exception: \wss{Exception if any point on the line lies off of the 2D
+    sequence (map)}$exc := (\neg\mbox{validLine}(l) \Rightarrow \mbox{invalid\_argument})$
+\end{itemize}
+
+\noindent count($\mathit{pth}$: PathT, $t$: T):
+\begin{itemize}
+\item output: \wss{Count the number of times the value $t$ occurs in the path
+    $pth$.}$out := +(p: \mbox{PointT} | p \in \mbox{pointsInPath}(\mathit{pth})
+  \wedge s[p.y][p.x] = t : 1)$
+\item exception: \wss{Exception if any point on the path lies off of the 2D
+    sequence (map)}$exc := (\neg\mbox{validPath}(\mathit{pth}) \Rightarrow \mbox{invalid\_argument})$
+\end{itemize}
+
+\noindent length($\mathit{pth}$: PathT):
+\begin{itemize}
+\item output: \wss{Use the scale to find the length of the
+    path.}$out := \mathit{pth}.\mbox{len} \cdot \mbox{scale}$
+\item exception: \wss{Exception if any point on the path lies off of the 2D
+    sequence (map)}$exc := (\neg\mbox{validPath}(\mathit{pth}) \Rightarrow \mbox{invalid\_argument})$
+\end{itemize}
+
+\noindent connected($p_1$: PointT, $p_2$: PointT):
+\begin{itemize}
+\item output: \wss{Return true if a path exists between $p_1$ and $p_2$ with all
+    of the points on the path being of the same value.  $p_1$ and $p_2$ are
+    considered to be part of the path.}
+  $out := \exists (\mathit{pth}: \mbox{PathT} | \mbox{validPath}(\mathit{pth})
+  \wedge \mathit{pth}.\mbox{strt} = p_1 \wedge \mathit{pth}.\mbox{end} = p_2 :
+  \mbox{count}(\mathit{pth}, s[p_1.y][p_1.x]) = \mathit{pth}.\mbox{len})$
+\item exception: \wss{Return an exception if either of the input points is not
+    valid.}$exc := (\neg\mbox{validPoint}(p_1) \vee \neg\mbox{validPoint}(p_1)
+  \Rightarrow \mbox{invalid\_argument})$
+
+\end{itemize}
+
+\subsection*{Local Functions}
+
+\noindent validRow: $\mathbb{N} \rightarrow \mathbb{B}$\\
+\noindent \wss{returns true if the given natural number is a valid row
+  number.}$\mbox{validRow}(i) \equiv 0 \leq i \leq (\mbox{nRow} - 1)$\\
+
+\noindent validCol: $\mathbb{N} \rightarrow \mathbb{B}$\\
+\noindent \wss{returns true if the given natural number is a valid column
+  number.}$\mbox{validCol}(j) \equiv 0 \leq j \leq (\mbox{nCol} - 1)$\\
+
+\noindent validPoint: $\mbox{PointT} \rightarrow \mathbb{B}$\\
+\noindent \wss{Returns true if the given point lies within the boundaries of the
+  map.}$\mbox{validPoint}(p) \equiv \mbox{validRow}(p.y) \wedge \mbox{validCol}(p.x)$\\
+
+\noindent validLine: $\mbox{LineT} \rightarrow \mathbb{B}$\\
+\noindent \wss{Returns true if all of the points for the given line lie within the boundaries of the
+  map.}$\mbox{validLine}(l) \equiv \forall(p: \mbox{PointT} | p \in \mbox{pointsInLine}(l):
+  \mbox{validPoint}(p))$\\
+
+\noindent validPath: $\mbox{PathT} \rightarrow \mathbb{B}$\\
+\noindent \wss{Returns true if all of the points for the given path lie within the boundaries of the
+  map.}$\mbox{validPath}(\mathit{pth}) \equiv \forall(p: \mbox{PointT} | p \in \mbox{pointsInPath}(\mathit{pth}):
+  \mbox{validPoint}(p))$\\
+
+\noindent pointsInLine: $\mbox{LineT} \rightarrow \mbox{(set of PointT)}$\\
+\noindent pointsInLine ($l$) 
+\wss{The same local function as given in the Path module.}
+\begin{multline*}
+\equiv \{ i: \mathbb{N} | i \in [0
+  .. (l.\mbox{len} - 1)] : l.\mbox{strt}.\mbox{translate}(\\
+(l.\mbox{orient}=\mbox{W} \Rightarrow -i |
+  l.\mbox{orient}=\mbox{E} \Rightarrow i | \mbox{True} \Rightarrow 0), (l.\mbox{orient}=\mbox{N} \Rightarrow i |
+  l.\mbox{orient}=\mbox{S} \Rightarrow -i | \mbox{True} \Rightarrow 0) ) \}
+\end{multline*}
+
+\noindent pointsInPath: $\mbox{PathT} \rightarrow \mbox{(set of PointT)}$\\
+\wss{Return the set of points that make up the input path.}
+\noindent pointsInPath($p$) 
+$\equiv \cup (i: \mathbb{N} | i \in [0..p.\mbox{size}]:
+\mbox{pointsInLine}(p.\mbox{line}(i)))$
+
+\newpage
+
+\section* {LanduseMap Module}
+
+\subsection* {Template Module}
+
+LanduseMapT is Seq2D(LanduseT)
+
+\section* {DEM Module}
+
+\subsection* {Template Module}
+
+DEMT is Seq2D($\mathbb{Z}$)
+
+\newpage
+
+\section{Critique of Design}
+
+Write a critique of the interface for the modules in this project.  Is there
+anything missing?  Is there anything you would consider changing?  Why?
+
+\end {document}
\ No newline at end of file
-- 
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