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Examples.html
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Examples.html
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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<title>ATS-PL-SYS</title>
<style>
.center {
margin-left: auto;
margin-right: auto;
}
.command_line {
width:96%;
margin:auto;
padding:10px;
color: #000000;
background-color: #D0D0D0;
border-radius:6px
}
#thePageLeft {
/*
parent: thePage
children:
*/
height: 100%;
} /* thePageLeft */
#thePageRHeaderTop {
/*
parent: thePageRHeader
children:
*/
} /* thePageRHeaderTop */
#thePageRHeaderSep {
/*
parent: thePageRHeader
children:
*/
background-color: rgb(143,2,34);
} /* thePageRHeaderSep */
#thePageRHeader {
/*
parent: thePageRight
children: thePageRHeaderTop, thePageRHeaderSep
*/
} /* thePageRHeader */
#thePageRBodyLHeader {
/*
parent: thePageRBodyLeft
children:
*/
} /* thePageRBodyLHeader */
#thePageRBodyLContent {
/*
parent: thePageRBodyLeft
children:
*/
} /* thePageRBodyLContent */
#thePageRBodyLeft {
/*
parent: thePageRBody
children: thePageRBodyLHeader, thePageRBodyLContent
*/
height: 100%;
} /* thePageRBodyLeft */
#thePageRBodyRight {
/*
parent: thePageRBody
children:
*/
height: 100%;
} /* thePageRBodyRight */
#thePageRBody {
/*
parent: thePageRight
children: thePageRBodyLeft, thePageRBodyRight
*/
} /* thePageRBody */
#thePageRFooterSep {
/*
parent: thePageRFooter
children:
*/
background-color: #8f0222;
} /* thePageRFooterSep */
#thePageRFooterRest {
/*
parent: thePageRFooter
children:
*/
} /* thePageRFooterRest */
#thePageRFooter {
/*
parent: thePageRight
children: thePageRFooterSep, thePageRFooterRest
*/
} /* thePageRFooter */
#thePageRight {
/*
parent: thePage
children: thePageRHeader, thePageRBody, thePageRFooter
*/
height: 100%;
} /* thePageRight */
#thePage {
/*
parent: theBodyProp
children: thePageLeft, thePageRight
*/
height: 100%;
background-color: #ffffff;
} /* thePage */
#theBodyProp {
/*
parent:
children: thePage
*/
height: 100%;
} /* theBodyProp */
body {
margin-top: 8px;
margin-bottom: 8px;
margin-left: 0px;
margin-right: 0px;
font-family: Helvetica, Arial, sans-serif;
background-color: #213449; /* dark blue */
}
#thePageRHeader {
text-align: center;
}
#thePageRFooter {
text-align: center;
}
#thePageLeft {
/*
background-color: #1e5799;
background-image: linear-gradient(to right, #1e5799, #7db9e8);
*/
background-color: rgba(30, 87, 153, 0.750);
}
#thePageRBodyRight
{
font-size: 88%;
background: #d1d360;
overflow-y: auto;
border-top-left-radius:12px;
/*
border-bottom-left-radius:12px;
*/
}
.thePageRBodyLContent
{
margin-left: 8px;
margin-right: 8px;
padding-top: 12px;
}
</style>
<link
rel="icon" type="image/gif"
href="./MYDATA/favicon_animated.gif">
</link>
<script
src="./SCRIPT/jquery-2.1.1.min.js">
</script>
<script
src="./CLIENT/MYCODE/libatscc2js_all.js">
</script>
<script
src="./CLIENT/MYCODE/libatscc2js_print_store_cats.js">
</script>
<script
src="./CLIENT/MYCODE/atslangweb_utils_dats.js">
</script>
</head>
<body>
<div id="theBodyProp">
<div id="thePage">
<style type="text/css">
.patsyntax {width:99%;margin:auto;}
.patsyntax {color:#808080;background-color:#E0E0E0;}
.patsyntax span.keyword {color:#000000;font-weight:bold;}
.patsyntax span.comment {color:#787878;font-style:italic;}
.patsyntax span.extcode {color:#A52A2A;}
.patsyntax span.neuexp {color:#800080;}
.patsyntax span.staexp {color:#0000F0;}
.patsyntax span.prfexp {color:#603030;}
.patsyntax span.dynexp {color:#F00000;}
.patsyntax span.stalab {color:#0000F0;font-style:italic}
.patsyntax span.dynlab {color:#F00000;font-style:italic}
.patsyntax span.dynstr {color:#008000;font-style:normal}
.patsyntax span.stacstdec {text-decoration:none;}
.patsyntax span.stacstuse {color:#0000CF;text-decoration:underline;}
.patsyntax span.dyncstdec {text-decoration:none;}
.patsyntax span.dyncstuse {color:#B80000;text-decoration:underline;}
.patsyntax span.dyncst_implement {color:#B80000;text-decoration:underline;}
</style>
<table
style="width:100%;height:100%;"
cellspacing="0" cellpadding="0">
<tr height="100%">
<td style="vertical-align:top;width:14%;">
<div id="thePageLeft">
<style>
#thePageLeft
{
color: #483D8B;
/*
color: rgba(72,61,140,0.750);
*/
}
#thePageLeft
{
position: fixed;
min-width: 14%;
}
#thePageLeft td.self
{
font-weight: bold;
}
#thePageLeft td.other
{
font-weight: normal;
}
.thePageLeft_menu
{
width: 88%;
margin: auto;
padding: 12px;
border-radius: 12px;
background-color: rgba(255,255,240,0.375);
}
</style>
<table
width="100%"
>
<tr>
<td height="156px" align="center">
<img src="./MYDATA/theLogo.png" alt="ATSlogo" style="height:72%"></img>
</td>
</tr>
<tr>
<td>
<table
class="thePageLeft_menu"
>
<tr><td name="Home" class="other"><a href="Home.html">Home(<a href="http://ats-lang.sourceforge.net">old</a>)</a></td></tr>
<tr><td name="Downloads" class="other"><a href="Downloads.html">Downloads</a></td></tr>
<tr><td name="Documents" class="other"><a href="Documents.html">Documents</a></td></tr>
<tr><td name="Libraries" class="other"><a href="Libraries.html">Libraries</a></td></tr>
<tr><td name="Community" class="other"><a href="Community.html">Community</a></td></tr>
<tr><td name="Papers" class="other"><a href="Papers.html">Papers</a></td></tr>
<tr><td name="Examples" class="self">Examples</td></tr>
<tr><td name="Resources" class="other"><a href="Resources.html">Resources</a></td></tr>
<tr><td name="Implements" class="other"><a href="Implements.html">Implements</a></td></tr>
<tr><td><a href="https://sourceforge.net/projects/ats-lang/lists/ats-lang-users">Mailing-list</a></td></tr>
<tr><td><a href="https://groups.google.com/forum/#!forum/ats-lang-users">ats-lang-users</a></td></tr>
<tr><td><a href="https://groups.google.com/forum/#!forum/ats-lang-devel">ats-lang-devel</a></td></tr>
<tr><td><a href="#" onclick="Home_tryatsnow_onclick()">Try ATS on-line</td></tr>
</table>
</td>
</tr>
</table>
<!--php-->
</div><!--thePageLeft-->
</td>
<td style="vertical-align:top;width:86%;">
<div id="thePageRight">
<div id="thePageRHeader">
<div id="thePageRHeaderTop">
<style>
#theHeaderLine1 {
margin-top: 0px;
margin-bottom: 0px;
padding-top: 48px;
padding-bottom: 2px;
font-size: 250%;
font-weight: bold;
}
#theHeaderLine2 {
margin-top: 0px;
margin-bottom: 0px;
padding-top: 2px;
padding-bottom: 32px;
font-size: 160%;
font-weight: bold;
}
</style>
<div id="theHeaderLine1">The ATS Programming Language</div>
<div id="theHeaderLine2">Unleashing the Potentials of Types and Templates!</div>
</div><!--thePageRHeaderTop-->
<div id="thePageRHeaderSep">
<style>
#thePageRHeaderSep
{
color: #FFFFF0;
}
#thePageRHeaderSepL
{
text-align: left;
padding-left: 2px;
}
#thePageRHeaderSepL ul
{
margin: 0px;
padding: 0px;
list-style-type: none;
}
#thePageRHeaderSepL ul li
{
float: left;
/*
display: inline;
*/
border-width: 1px;
border-color: #ffffff;
border-right-style: double;
padding-left: 8px;
padding-right: 8px;
}
#thePageRHeaderSepL li.self
{
font-weight: bold;
}
#thePageRHeaderSepL li.other
{
font-weight: normal;
}
#thePageRHeaderSepL a:link
{
color: #FFFFFF;
}
#thePageRHeaderSepL a:visited
{
color: #80C0C0;
}
#thePageRHeaderSepL a:hover
{
color: #C0E0E0;
}
#thePageRHeaderSepL a:active
{
color: #C0C0C0;
}
.thePageRHeaderSepL_submenu
{
display: none;
position: absolute;
padding: 6px;
border-radius: 12px;
background-color: rgba(160,20,50,0.875);
}
#thePageRHeaderSepR
{
float: right;
}
#thePageRHeaderSepR_input
{
height: 100%;
padding: 6px;
font-size: 11pt;
border-radius: 8px;
background: #FFFFF0 url(//www.google.com/coop/intl/en/images/google_custom_search_watermark.gif) left no-repeat;
}
</style>
<script>
//
var
theTopmenuTable = 0;
var
theTopmenuTimeout = 0;
//
function
theTopmenuTable_hide()
{
if(theTopmenuTable)
{
theTopmenuTable.css({display:'none'});
}
}
function
theTopmenuTimeout_clear()
{
if(theTopmenuTimeout)
{
clearTimeout(theTopmenuTimeout); theTopmenuTimeout = 0;
}
}
//
function
topmenu_mouseout()
{
topmenu_mouseout2(500); return;
}
function
topmenu_mouseout2(msec)
{
//
if(theTopmenuTimeout) clearTimeout(theTopmenuTimeout);
theTopmenuTimeout =
setTimeout (function(){theTopmenuTimeout=0;theTopmenuTable_hide();}, msec);
//
}
//
function
topmenu_mouseover(x0)
{
var jqi = jQuery(x0);
theTopmenuTable_hide();
theTopmenuTimeout_clear();
theTopmenuTable = jqi.next('table');
theTopmenuTable.css({display:'table'});
theTopmenuTable.css (
{top:jqi.position().top+jqi.outerHeight(true)+10}
) ; // end of [theTopmenuTable.css]
theTopmenuTable.css({left:jqi.position().left});
return;
}
//
function
submenu_table_mouseout() { topmenu_mouseout(); }
function
submenu_table_mouseover() { theTopmenuTimeout_clear(); }
//
</script>
<table width="100%" border="0" cellspacing="0" cellpadding="0">
<tr>
<td width="75%">
<div
id="thePageRHeaderSepL"
>
<ul>
<li name="Home" class="other" onmouseout="topmenu_mouseout()" onmouseover="topmenu_mouseover(this)"><a href="Home.html">Home</a></li>
<table class="thePageRHeaderSepL_submenu" onmouseout="submenu_table_mouseout()" onmouseover="submenu_table_mouseover()">
<tr><td><a href="Home.html#What_is_ATS">What is ATS?<a></td></tr>
<tr><td><a href="Home.html#What_is_ATS_good_for">What is ATS good for?<a></td></tr>
<tr><td><a href="Home.html#Suggestion_on_learning_ATS">Suggestion on learning ATS<a></td></tr>
<tr><td><a href="Home.html#Acknowledgments">Acknowledgments</a></td></tr>
</table>
<li name="Downloads" class="other" onmouseout="topmenu_mouseout()" onmouseover="topmenu_mouseover(this)"><a href="Downloads.html">Downloads</a></li>
<table class="thePageRHeaderSepL_submenu" onmouseout="submenu_table_mouseout()" onmouseover="submenu_table_mouseover()">
<tr><td><a href="Downloads.html#ATS_packages">ATS packages</a></td></tr>
<tr><td><a href="Downloads.html#Uninstall_for_ATS">Uninstallation for ATS</a></td></tr>
<tr><td><a href="Downloads.html#Requirements_install">Requirements for installation</a></td></tr>
<tr><td><a href="Downloads.html#Precompiledpack_install">Precompiled packages for installation</a></td></tr>
<tr><td><a href="Downloads.html#Install_source_compile">Installation through source compilation</a></td></tr>
<tr><td><a href="Downloads.html#Install_of_ATS2_contrib">Installation of ATS2-contrib</a></td></tr>
<tr><td><a href="Downloads.html#Install_of_ATS2_include">Installation of ATS2-include</a></td></tr>
<tr><td><a href="Downloads.html#Scripts_for_installing_ATS_Postiats">Scripts for installing ATS-Postiats</a></td></tr>
</table>
<li name="Documents" class="other" onmouseout="topmenu_mouseout()" onmouseover="topmenu_mouseover(this)"><a href="Documents.html">Documents</a></li>
<table class="thePageRHeaderSepL_submenu" onmouseout="submenu_table_mouseout()" onmouseover="submenu_table_mouseover()">
<tr><td><a href="Documents.html#INT2PROGINATS">Introduction to Programming in ATS</a></td></tr>
<tr><td><a href="Documents.html#ATS2TUTORIAL0">A Tutorial on Programming Features in ATS</a></td></tr>
<tr><td><a href="Documents.html#ATS2FUNCRASH0">A Crash into Functional Programming via ATS</a></td></tr>
<tr><td><a href="Documents.html#EFF2ATSPROGEX">Effective Programming in ATS through Examples</a></td></tr>
</table>
<li name="Libraries" class="other" onmouseout="topmenu_mouseout()" onmouseover="topmenu_mouseover(this)"><a href="Libraries.html">Libraries</a></li>
<li name="Community" class="other" onmouseout="topmenu_mouseout()" onmouseover="topmenu_mouseover(this)"><a href="Community.html">Community</a></li>
<table class="thePageRHeaderSepL_submenu" onmouseout="submenu_table_mouseout()" onmouseover="submenu_table_mouseover()">
<tr><td><a href="Community.html#ATS_wikipage">Wiki for ATS users</a></td></tr>
<tr><td><a href="Community.html#ATS_subreddit">ATS news links at reddit</a></td></tr>
<tr><td><a href="Community.html#ATS_IRC_channel">IRC channel for ATS users</a></td></tr>
<tr><td><a href="Community.html#ATS_stackoverflow_tag">StackOverflow tag for ATS</a></td></tr>
<tr><td><a href="Community.html#ATS_QandA_forum">Q&A forum for ATS users</a></td></tr>
<tr><td><a href="Community.html#ATS_devel_forum">Discussion forum for ATS developers</a></td></tr>
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<h1>Programming Examples</h1>
<ul name="Examples">
<li><a href="#Hello_world">Hello, world!</a></li>
<li><a href="#Copying_files">Copying files</a></li>
<li><a href="#Fibonacci_numbers">Computing Fibonacci numbers</a></li>
<li><a href="#EffectiveATS">Effective Programming in ATS</a></li>
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<!--php-->
<hr></hr>
<h2><a id="Hello_world">Hello, world!</a></h2>
<p>
Following is a tiny program in ATS for printing out
the string "Hello, world!" plus a newline: </p>
<pre class="patsyntax">
<span class="dynexp"><span class="keyword">implement</span>
main0 <span class="keyword">(</span><span class="keyword">)</span> <span class="keyword">=</span> print<span class="keyword">(</span><span class="dynstr">"Hello, world!\n"</span><span class="keyword">)</span></span></pre>
<!--php-->
<p>
Assume that the program is stored in a file named <u>hello.dats</u>. We
can generate an executable named <u>hello</u> by executing the following
command-line: </p>
<div
class="command_line"
>patscc -o hello hello.dats</div>
<p>
If executed, <u>hello</u> does what is mentioned above. The command
<u>patscc</u> first compiles the ATS program contained in <u>hello.dats</u>
into some C code and then invokes a C compiler (gcc in this case) to
translate the generated C code into some binary object code. </p>
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<hr></hr>
<h2><a id="Copying_files">Copying files</a></h2>
<p>
Given an input channel and an output channel, the following function fcopy
writes to the latter the characters read from the former:
</p>
<pre class="patsyntax">
<span class="dynexp"><span class="keyword">fun</span>
fcopy <span class="keyword">(</span>
inp<span class="keyword">:</span> <span class="staexp">FILEref</span>
<span class="keyword">,</span> out<span class="keyword">:</span> <span class="staexp">FILEref</span>
<span class="keyword">)</span> <span class="keyword">:</span> <span class="staexp">void</span> <span class="keyword">=</span> <span class="keyword">let</span>
<span class="dynexp"><span class="keyword">val</span> c <span class="keyword">=</span> fileref_getc <span class="keyword">(</span>inp<span class="keyword">)</span></span>
<span class="keyword">in</span>
<span class="keyword">if</span> c >= <span class="dynexp">0</span> <span class="keyword">then</span> <span class="keyword">let</span>
<span class="dynexp"><span class="keyword">val</span> <span class="keyword">(</span><span class="keyword">)</span> <span class="keyword">=</span> fileref_putc <span class="keyword">(</span>out<span class="keyword">,</span> c<span class="keyword">)</span></span> <span class="keyword">in</span> fcopy <span class="keyword">(</span>inp<span class="keyword">,</span> out<span class="keyword">)</span>
<span class="keyword">end</span> <span class="comment">// end of [if]</span>
<span class="keyword">end</span></span> <span class="comment">(* end of [fcopy] *)</span></pre>
<!--php-->
<p>
Note that loops in imperative languages such as C and Java are often
implemented as tail-recursive functions in ATS, which are then compiled
back into loops in C. For instance, fcopy as is implemented above is such
a tail-recursive function.
</p>
<p>
Further reading about copying files in ATS can be found
<a href="http://ats-lang.sourceforge.net/EXAMPLE/EFFECTIVATS/file-copying/.">on-line</a>.
</p>
<hr></hr>
<h2><a id="Fibonacci_numbers">Computing Fibonacci numbers</a></h2>
<p>
ATS advocates a programming paradigm in which programs and proofs can be
constructed in a syntactically intertwined manner. This paradigm is often
referred to as <em>programming with theorem-proving</em> (PwTP), and it
plays a central indispensible role in the development of ATS.
Let us now see a simple and concrete example that clearly illustrates PwTP
as is supported in ATS.
</p>
<p>
A function fib can be specified as follows for computing Fibonacci numbers:
</p>
<pre
class="patsyntax"
style="color:#000000;"
>
fib(0) = 0
fib(1) = 1
fib(n+2) = fib(n) + fib(n+1) for n >= 0
</pre>
<p>
Following is a direct implementation of this specified function in ATS:
</p>
<pre class="patsyntax">
<span class="dynexp"><span class="keyword">fun</span>
fib <span class="keyword">(</span>
n<span class="keyword">:</span> <span class="staexp">int</span>
<span class="keyword">)</span> <span class="keyword">:</span> <span class="staexp">int</span> <span class="keyword">=</span>
<span class="keyword">if</span> n >= <span class="dynexp">2</span> <span class="keyword">then</span> fib<span class="keyword">(</span>n-<span class="dynexp">2</span><span class="keyword">)</span> + fib<span class="keyword">(</span>n-<span class="dynexp">1</span><span class="keyword">)</span> <span class="keyword">else</span> n</span>
<span class="comment">// end of [fib]</span></pre>
<!--php-->
<p>
Clearly, this is a terribly inefficient implementation of exponential
time-complexity. An implementation of fib in C is given as follows that is
of linear time-complexity:
</p>
<pre
class="patsyntax"
style="color:#000000;"
>
int
fibc (int n) {
int tmp, f0 = 0, f1 = 1 ;
while (n-- > 0) { tmp = f1 ; f1 = f0 + f1 ; f0 = tmp ; } ; return f0 ;
} // end of [fibc]
</pre>
<p>
If translated into ATS, the function fibc can essentially be implemented as
follows:
</p>
<pre class="patsyntax">
<span class="dynexp"><span class="keyword">fun</span>
fibc <span class="keyword">(</span>
n<span class="keyword">:</span> <span class="staexp">int</span>
<span class="keyword">)</span> <span class="keyword">:</span> <span class="staexp">int</span> <span class="keyword">=</span> <span class="keyword">let</span>
<span class="comment">//</span>
<span class="dynexp"><span class="keyword">fun</span>
loop<span class="keyword">(</span>n<span class="keyword">:</span> <span class="staexp">int</span><span class="keyword">,</span> f0<span class="keyword">:</span> <span class="staexp">int</span><span class="keyword">,</span> f1<span class="keyword">:</span> <span class="staexp">int</span><span class="keyword">)</span><span class="keyword">:</span> <span class="staexp">int</span> <span class="keyword">=</span>
<span class="keyword">if</span> n <span class="keyword">></span> <span class="dynexp">0</span> <span class="keyword">then</span> loop<span class="keyword">(</span>n-<span class="dynexp">1</span><span class="keyword">,</span> f1<span class="keyword">,</span> f0+f1<span class="keyword">)</span> <span class="keyword">else</span> f0</span>
<span class="comment">// end of [loop]</span>
<span class="comment">//</span>
<span class="keyword">in</span>
loop<span class="keyword">(</span>n<span class="keyword">,</span> <span class="dynexp">0</span><span class="keyword">,</span> <span class="dynexp">1</span><span class="keyword">)</span>
<span class="keyword">end</span></span> <span class="comment">// end of [fibc]</span></pre>
<!--php-->
<p>
There is obviously a logic gap between the defintion of fib and its
implementation as is embodied in fibc. In ATS, an implementation of fib can
be given that completely bridges this gap. First, the specification of fib
needs to be encoded into ATS, which is fulfilled by the declaration of the
following dataprop:
</p>
<pre class="patsyntax">
<span class="keyword">dataprop</span> <span class="staexp">FIB <span class="keyword">(</span>int<span class="keyword">,</span> int<span class="keyword">)</span> <span class="keyword">=</span>
<span class="keyword">|</span> <span class="prfexp">FIB0 <span class="keyword">(</span><span class="staexp">0</span><span class="keyword">,</span> <span class="staexp">0</span><span class="keyword">)</span></span> <span class="keyword">|</span> <span class="prfexp">FIB1 <span class="keyword">(</span><span class="staexp">1</span><span class="keyword">,</span> <span class="staexp">1</span><span class="keyword">)</span></span>
<span class="keyword">|</span> <span class="prfexp"><span class="staexp"><span class="keyword">{</span>n<span class="keyword">:</span>nat<span class="keyword">}</span></span> <span class="staexp"><span class="keyword">{</span>r0<span class="keyword">,</span>r1<span class="keyword">:</span>int<span class="keyword">}</span></span> FIB2 <span class="keyword">(</span><span class="staexp">n</span><span class="staexp">+</span><span class="staexp">2</span><span class="keyword">,</span> <span class="staexp">r0</span><span class="staexp">+</span><span class="staexp">r1</span><span class="keyword">)</span> <span class="keyword">of</span> <span class="keyword">(</span><span class="staexp">FIB</span> <span class="keyword">(</span><span class="staexp">n</span><span class="keyword">,</span> <span class="staexp">r0</span><span class="keyword">)</span><span class="keyword">,</span> <span class="staexp">FIB</span> <span class="keyword">(</span><span class="staexp">n</span><span class="staexp">+</span><span class="staexp">1</span><span class="keyword">,</span> <span class="staexp">r1</span><span class="keyword">)</span><span class="keyword">)</span></span></span>
<span class="comment">// end of [FIB] // end of [dataprop]</span></pre>
<!--php-->
<p>
This declaration introduces a type FIB for proofs, and such a type is
referred to as a prop in ATS. Intuitively, if a proof can be assgined the
type FIB(n,r) for some integers n and r, then fib(n) equals r. In other
words, FIB(n,r) encodes the relation fib(n)=r. There are three constructors
FIB0, FIB1 and FIB2 associated with FIB, which are given the following
types corresponding to the three equations in the definition of fib:
</p>
<pre class="patsyntax">
<span class="dynexp">FIB0 <span
class="keyword">:</span> <span class="keyword">(</span><span
class="keyword">)</span> <span class="staexp"><span
class="keyword">-></span></span> <span class="staexp">FIB</span> <span
class="keyword">(</span><span class="staexp">0</span><span
class="keyword">,</span> <span class="staexp">0</span><span
class="keyword">)</span></span>
<span class="dynexp">FIB1 <span
class="keyword">:</span> <span class="keyword">(</span><span
class="keyword">)</span> <span class="staexp"><span
class="keyword">-></span></span> <span class="staexp">FIB</span> <span
class="keyword">(</span><span class="staexp">1</span><span
class="keyword">,</span> <span class="staexp">1</span><span
class="keyword">)</span></span>
<span class="dynexp">FIB2 <span
class="keyword">:</span> <span class="staexp"><span
class="keyword">{</span>n<span class="keyword">:</span>nat<span
class="keyword">}</span></span><span class="staexp"><span
class="keyword">{</span>r0<span class="keyword">,</span>r1<span
class="keyword">:</span>int<span class="keyword">}</span></span> <span
class="keyword">(</span><span class="staexp">FIB</span> <span
class="keyword">(</span><span class="staexp">n</span><span
class="keyword">,</span> <span class="staexp">r0</span><span
class="keyword">)</span><span class="keyword">,</span> <span
class="staexp">FIB</span> <span class="keyword">(</span><span
class="staexp">n+1</span><span class="keyword">,</span> <span
class="staexp">r1</span><span class="keyword">)</span><span
class="keyword">)</span> <span class="staexp"><span
class="keyword">-></span></span> <span class="staexp">FIB</span> <span
class="keyword">(</span><span class="staexp">n</span><span
class="staexp">+</span><span class="staexp">2</span><span
class="keyword">,</span> <span class="staexp">r0</span><span
class="staexp">+</span><span class="staexp">r1</span><span
class="keyword">)</span></span></pre>
<!--php-->
<p>
Note that {...} is the concrete syntax in ATS for universal
quantification. For instance, FIB2(FIB0(), FIB1()) is a term of the type
FIB(2,1), attesting to fib(2)=1.
</p>
<p>
A fully verified implementaion of the fib function in ATS can now be given
as follows:
</p>
<pre class="patsyntax">
<span class="dynexp"><span class="keyword">fun</span>
fibats<span class="staexp"><span class="keyword">{</span>n<span class="keyword">:</span>nat<span class="keyword">}</span></span>
<span class="keyword">(</span>n<span class="keyword">:</span> <span class="staexp">int</span> <span class="keyword">(</span><span class="staexp">n</span><span class="keyword">)</span><span class="keyword">)</span>
<span class="keyword">:</span> <span class="staexp"><span class="keyword">[</span>r<span class="keyword">:</span>int<span class="keyword">]</span></span> <span class="keyword">(</span><span class="staexp">FIB</span> <span class="keyword">(</span><span class="staexp">n</span><span class="keyword">,</span> <span class="staexp">r</span><span class="keyword">)</span> <span class="keyword">|</span> <span class="staexp">int</span> <span class="staexp">r</span><span class="keyword">)</span> <span class="keyword">=</span> <span class="keyword">let</span>
<span class="comment">//</span>
<span class="dynexp"><span class="keyword">fun</span>
loop
<span class="staexp"><span class="keyword">{</span>i<span class="keyword">:</span>nat <span class="keyword">|</span> i <= n<span class="keyword">}</span></span><span class="staexp"><span class="keyword">{</span>r0<span class="keyword">,</span>r1<span class="keyword">:</span>int<span class="keyword">}</span></span>
<span class="keyword">(</span>
<span class="prfexp">pf0<span class="keyword">:</span> <span class="staexp">FIB</span><span class="keyword">(</span><span class="staexp">i</span><span class="keyword">,</span> <span class="staexp">r0</span><span class="keyword">)</span></span><span class="keyword">,</span> <span class="prfexp">pf1<span class="keyword">:</span> <span class="staexp">FIB</span><span class="keyword">(</span><span class="staexp">i</span><span class="staexp">+</span><span class="staexp">1</span><span class="keyword">,</span> <span class="staexp">r1</span><span class="keyword">)</span></span>
<span class="keyword">|</span> n_i<span class="keyword">:</span> <span class="staexp">int</span><span class="keyword">(</span><span class="staexp">n</span><span class="staexp">-</span><span class="staexp">i</span><span class="keyword">)</span><span class="keyword">,</span> r0<span class="keyword">:</span> <span class="staexp">int</span> <span class="staexp">r0</span><span class="keyword">,</span> r1<span class="keyword">:</span> <span class="staexp">int</span> <span class="staexp">r1</span>
<span class="keyword">)</span> <span class="keyword">:</span> <span class="staexp"><span class="keyword">[</span>r<span class="keyword">:</span>int<span class="keyword">]</span></span> <span class="keyword">(</span><span class="staexp">FIB</span><span class="keyword">(</span><span class="staexp">n</span><span class="keyword">,</span> <span class="staexp">r</span><span class="keyword">)</span> <span class="keyword">|</span> <span class="staexp">int</span><span class="keyword">(</span><span class="staexp">r</span><span class="keyword">)</span><span class="keyword">)</span> <span class="keyword">=</span>
<span class="keyword">(</span>
<span class="keyword">if</span> <span class="keyword">(</span>n_i <span class="keyword">></span> <span class="dynexp">0</span><span class="keyword">)</span>
<span class="keyword">then</span>
loop<span class="staexp"><span class="keyword">{</span>i+1<span class="keyword">}</span></span>
<span class="keyword">(</span>
<span class="prfexp">pf1</span><span class="keyword">,</span> <span class="prfexp">FIB2<span class="keyword">(</span>pf0<span class="keyword">,</span> pf1<span class="keyword">)</span></span> <span class="keyword">|</span> n_i-<span class="dynexp">1</span><span class="keyword">,</span> r1<span class="keyword">,</span> r0+r1
<span class="keyword">)</span> <span class="comment">(* then *)</span>
<span class="keyword">else</span> <span class="keyword">(</span><span class="prfexp">pf0</span> <span class="keyword">|</span> r0<span class="keyword">)</span>
<span class="comment">// end of [if]</span>
<span class="keyword">)</span></span> <span class="comment">(* end of [loop] *)</span>
<span class="keyword">in</span>
loop<span class="staexp"><span class="keyword">{</span>0<span class="keyword">}</span></span><span class="keyword">(</span><span class="prfexp">FIB0</span><span class="comment">(*void*)</span><span class="keyword">,</span> <span class="prfexp">FIB1</span><span class="comment">(*void*)</span> <span class="keyword">|</span> n<span class="keyword">,</span> <span class="dynexp">0</span><span class="keyword">,</span> <span class="dynexp">1</span><span class="keyword">)</span>
<span class="keyword">end</span></span> <span class="comment">// end of [fibats]</span></pre>
<!--php-->
<p>
Note that fibats is given the following declaration:
</p>
<pre class="patsyntax">
<span class="dynexp"><span class="keyword">fun</span> fibats <span class="keyword">:</span> <span class="staexp"><span class="keyword">{</span>n<span class="keyword">:</span>nat<span class="keyword">}</span></span> <span class="staexp">int</span><span class="keyword">(</span><span class="staexp">n</span><span class="keyword">)</span> <span class="staexp"><span class="keyword">-></span></span> <span class="staexp"><span class="keyword">[</span>r<span class="keyword">:</span>int<span class="keyword">]</span></span> <span class="keyword">(</span><span class="staexp">FIB</span><span class="keyword">(</span><span class="staexp">n</span><span class="keyword">,</span><span class="staexp">r</span><span class="keyword">)</span> <span class="keyword">|</span> <span class="staexp">int</span><span class="keyword">(</span><span class="staexp">r</span><span class="keyword">)</span><span class="keyword">)</span></span></pre>
<!--php-->
<p>
where the concrete syntax [...] is for existential quantification
and the bar symbol (|) is just a separator (like a comma) for separating
proofs from values. For each integer I, int(I) is a singleton type for the
only integer whose value equals I. When fibats is applied to an integer of
value n, it returns a pair consisting of a proof and an integer value r such
that the proof, which is of the type FIB(n,r), asserts fib(n)=r.
Therefore, fibats is a verified implementation of fib as is encoded by FIB.
Note that the inner function loop directly corresponds to the while-loop in
the body of the function fibc (written in C).
</p>
<p>
Lastly, it should be emphasized that proofs are completely erased after they
pass typechecking. In particular, there is no proof construction at run-time.
</p>
<p>
Please click
<a href="./SERVER/MYCODE/Patsoptaas_serve.php?mycode_fil=fibats">here</a> if you
are interested in compiling and running this example on-line.
</p>
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Here is a button for trying this example on-line:
<button
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>Try-it-yourself</button>
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<hr></hr>
<h2><a id="EffectiveATS">Effective Programming in ATS</a></h2>
<p>
Please find <a
href="http://ats-lang.sourceforge.net/EXAMPLE/EFFECTIVATS/">on-line</a>
various (short) programs in ATS plus detailed explanation that are written
for the purpose of advocating effective programming in ATS.
</p>
<hr></hr>
</div>
</div><!--thePageRBodyLContent-->
</div><!--thePageRBodyLeft-->
</td>
<td style="vertical-align:top;width:32%;height:0px;">
<div id="thePageRBodyRight">
<div
style="margin-top:8px;padding:8px;"
><!--div-->
<h2 style="display:inline;">Try examples on-line</h2>
<hr></hr>
In ATS, a single-file program refers to one fully contained in a
single file that does not make use of any other files except those
provided by ATS in its standard compilation environment.
<hr></hr>
<a href="http://www.ats-lang.org/Patsoptaas.html">Patsoptaas</a>
(patsopt-as-a-service) is a free on-line service provided for compiling
single-file programs in ATS. In the case where Patsoptaas can successfully
compile a single-file program into JS code, then the JS code may be
evaluated inside the browser. Some interesting single-file programs in ATS
are listed as follows that can be serviced by Patsoptaas:
<table
width="100%"
style="margin-left:8px"
><!--table-->
<tr><td>
</tr></td>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats/master/doc/EXAMPLE/ARITH/Bezout_identity.dats">Proving Bézout's identity</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats/master/doc/EXAMPLE/ARITH/tally-of-powers.dats">Proving Identities for tally-of-powers</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats/master/doc/EXAMPLE/ATSLF/SMI.dats">Proving the principle of SMI</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats/master/doc/EXAMPLE/ATSLF/PHP.dats">Proving the pigenhole principle</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats/master/doc/EXAMPLE/ATSLF/infprime.dats">Proving that primes are unbounded</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats/master/doc/BOOK/INT2PROGINATS/CODE/CHAP_FUNCTION/JS/queens.dats">Solving the 8-queen puzzle</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats-contrib/master/projects/SMALL/JSmydraw/Queen_puzzle/Queen_puzzle_php.dats">Solving the 8-queen puzzle (anim)</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats/master/doc/EXAMPLE/MISC/JS/ferryman.dats">Solving the ferryman puzzle</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats-contrib/master/projects/SMALL/JSmydraw/Sierpinski_3angle/Sierpinski_3angle_php.dats">Drawing a Sierpinski triangle</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats-contrib/master/projects/SMALL/JSclock/myclock1-2.dats">Drawing an analog clock (1)</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats-contrib/master/projects/SMALL/JSmydraw/Sorting_merge/Sorting_merge_php.dats">Animating list-merge-sort (1)</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats-contrib/master/projects/SMALL/JSmydraw/Sorting_quick/Sorting_quick_php.dats">Animating array-quick-sort (1)</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats-contrib/master/projects/SMALL/JSmydraw/Sorting_quick2/Sorting_quick2_php.dats">Animating array-quick-sort (2)</a>
</td></tr>
<tr><td>
<a
href="http://www.ats-lang.org/SERVER/MYCODE/Patsoptaas_serve.php?mycode_url=https://raw.githubusercontent.com/githwxi/ATS-Postiats-contrib/master/projects/SMALL/JSmydraw/Sorting_insert/Sorting_insert_php.dats">Animating array-insertion-sort</a>
</td></tr>
<tr><td>
</tr></td>
</table>
<hr></hr>
</div>