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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 -- GitLab