--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/PhD/index.html Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,85 @@
+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"
+"http://www.w3.org/TR/REC-html40/loose.dtd">
+<HEAD>
+<TITLE>Homepage of Christian Urban</TITLE>
+<BASE HREF="http://www4.in.tum.de/~urbanc/">
+</HEAD>
+
+<BODY TEXT="#000000"
+ BGCOLOR="#4169E1"
+ LINK="#0000EF"
+ VLINK="#51188E"
+ ALINK="#FF0000">
+
+<TABLE WIDTH="100%"
+ BGCOLOR="#4169E1"
+ BORDER="0"
+ FRAME="border"
+ CELLPADDING="10"
+ CELLSPACING="2"
+ RULES="all">
+
+<!-- left column -->
+<TR>
+<TD BGCOLOR="#FFFFFF"
+ WIDTH="24%"
+ VALIGN="TOP"
+ ROWSPAN="2">
+
+<B>Links</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/">Home</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/publications.html">Publications</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/travelling.html">Recent Talks</A>
+<BR>
+<BR>
+
+<B>Handy Information</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/logic.html">People in Logic</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/programming.html">Programming Languages</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/misc.html">Miscellaneous</A>
+<BR><BR>
+
+<A HREF="http://isabelle.in.tum.de/nominal/">
+<IMG SRC="ribbon.gif" ALT="" style="width: 114px; height: 100px;" align="left">
+</A>
+</TD>
+
+<!-- right column -->
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+Postscript file of the thesis:
+<A HREF="http://www4.in.tum.de/~urbanc/Publications/Phd-Urban.ps.gz">ps.gz</A>
+</TD>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<H3>The Dinner after the Viva</H3>
+organised by Gavin in The Three Horseshoes in Maddingley. Very nice.<P>
+
+<IMG SRC="http://www4.in.tum.de/~urbanc/PhD/phd-1.jpg" ALT=""><BR>
+My examiners,
+<A HREF="http://www.cl.cam.ac.uk/~amp12">Andy Pitts</A> and
+<A HREF="http://www.cs.kun.nl/~henk/">Henk Barendregt</A>, are next to me.
+<A HREF="http://research.microsoft.com/~nick/">Nick Benton</A>, Mrs. Barendregt-Kessing (Henk's mother),
+<A HREF="http://research.microsoft.com/~gmb/">Gavin Bierman</A>,
+<A HREF="http://www.cs.bham.ac.uk/~mxj/">Mateja Jamnik</A>,
+<A HREF="http://www.dpmms.cam.ac.uk/~martin/">Martin Hyland</A> and
+<A HREF="http://www.cl.cam.ac.uk/~gw104/">Glynn Winskel</A>
+are standing behind.<P>
+
+<IMG SRC="http://www4.in.tum.de/~urbanc/PhD/phd-2.jpg" ALT=""><BR>
+There are <A HREF="http://www.statslab.cam.ac.uk/Dept/People/pitts.html">Susan Pitts</A> and Alice Benton on the left.<P>
+
+<H3>The Ceremony</H3>
+<IMG SRC="http://www4.in.tum.de/~urbanc/PhD/phd-3.jpg" ALT=""><p>
+
+</TD>
+</TR>
+
+</TABLE>
+<P><!-- Created: Tue Mar 4 00:23:25 GMT 1997 -->
+<!-- hhmts start -->
+Last modified: Thu Feb 28 20:23:06 CET 2008
+<!-- hhmts end -->
+<a href="http://validator.w3.org/check/referer">[Validate this page.]</a>
+</BODY>
+</HTML>
Binary file PhD/phd-1.jpg has changed
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Contexts.pizza Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,70 @@
+package G4ip;
+
+import java.util.Vector;
+import G4ip.Form.*;
+
+/** Context acts as a multiset.<p>
+ * Typical contexts in logical rules are Gamma, Delta, etc.
+ * @author Christian Urban
+ */
+public class Context extends Vector {
+
+ public Context() { super(); }
+
+ public Context(Context init_context) {
+ super();
+ for (int i=0;i<init_context.size();i++) {
+ addElement(init_context.elementAt(i));
+ }
+ }
+
+ /** should be toString, but this is a "final" method in Vector
+ */
+ public String makeString() {
+ String s = new String();
+ int i; // size()-1 because the last formula
+ for (i=0;i<size()-1;i++) { // should be printed without a comma
+ s=s.concat(elementAt(i).toString());
+ s=s.concat(", ");
+ }
+ if (size() > 0) { // add last formula
+ s=s.concat(elementAt(i).toString());
+ }
+ return s;
+ }
+
+ /** returns a context with one additional formula at the beginning
+ * @param new_formula a formula
+ */
+ public Context add(Form new_formula) {
+ Context new_c = new Context(this);
+ new_c.insertElementAt(new_formula,0);
+ return new_c;
+ }
+
+ /** returns a context with two additional formulae at the beginning
+ * @param new_formula1,new_formula2 two formulae
+ */
+ public Context add(Form new_formula1, Form new_formula2) {
+ Context new_c = new Context(this);
+ new_c.insertElementAt(new_formula1,0);
+ new_c.insertElementAt(new_formula2,0);
+ return new_c;
+ }
+
+ /** tests whether a context contains a specific atom
+ * @param a a formula
+ */
+ public boolean includes(Form a) {
+ if (a instanceof Atm) {
+ for (int i=0;i<size();i++) {
+ switch((Form)elementAt(i)) {
+ case Atm(String c):
+ if (c.compareTo(((Atm)a).c) == 0) return true; break;
+ }
+ }
+ }
+ return false;
+ }
+
+}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Doc/G4ip.Context.html Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,144 @@
+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2//EN">
+<!--NewPage-->
+<html>
+<head>
+<!-- generated by pizzadoc on Thu Oct 30 13:46:17 GMT+00:00 1997 -->
+<title>
+public class G4ip.Context
+</title>
+</head>
+<body bgcolor=FFFFFF>
+<a name="_top_"></a>
+<code><font size=-1>
+[<a href="index.html">all packages</a>]
+[<a href="G4ip.html">package G4ip</a>]
+[<a href="tree-pd.html">class hierarchy</a>]
+[<a href="names-pd.html">index</a>]
+</font></code>
+<hr>
+<h1>public class G4ip.Context</h1>
+(source file: Contexts.pizza)<br>
+<pre>
+<a href="http://java.sun.com:80/products/jdk/1.1/docs/api/java.lang.Object.html">java.lang.Object</a>
+ |
+ +----<a href="http://java.sun.com:80/products/jdk/1.1/docs/api/java.util.Vector.html">java.util.Vector</a>
+ |
+ +----G4ip.Context
+</pre>
+<hr>
+<pre>
+public class <a href="G4ip.Context.html">Context</a>
+ extends Vector
+</pre>
+<dl>
+ <dd> Contexts act as multiset.<p>
+ Typical contexts in logical rules are Gamma, Delta, etc.<p></dd>
+ <dd><dl><dt></dt>
+ <dt> <b>Author:</b></dt>
+ <dd> Christian Urban</dd>
+ </dl></dd>
+</dl>
+<hr>
+<p><h2><img src="p-images/constructor-index.gif" alt="Constuctor Index"></h2>
+<dl>
+ <dt><img src="p-images/yellow-ball-small.gif" alt="O "> <b><a href="G4ip.Context.html#Context(Context)">Context</a></b>(Context)
+ <dd>
+ <dt><img src="p-images/yellow-ball-small.gif" alt="O "> <b><a href="G4ip.Context.html#Context()">Context</a></b>()
+ <dd>
+</dl>
+<p><h2><img src="p-images/method-index.gif" alt="Methods"></h2>
+<dl>
+ <dt><img src="p-images/red-ball-small.gif" alt="O "> <b><a href="G4ip.Context.html#add(Form, Form)">add</a></b>(Form, Form)
+ <dd>returns a contexts with two additional formulae at the beginning
+ <dt><img src="p-images/red-ball-small.gif" alt="O "> <b><a href="G4ip.Context.html#add(Form)">add</a></b>(Form)
+ <dd>returns a context with one additional formula at the beginning
+ <dt><img src="p-images/red-ball-small.gif" alt="O "> <b><a href="G4ip.Context.html#includes(Form)">includes</a></b>(Form)
+ <dd>tests whether a context contains a specific atom
+ <dt><img src="p-images/red-ball-small.gif" alt="O "> <b><a href="G4ip.Context.html#makeString()">makeString</a></b>()
+ <dd>should be toString, but this is a "final" method in Vector
+</dl>
+<p><h2><img src="p-images/constructors.gif" alt="Constructors"></h2>
+<img src="p-images/yellow-ball.gif" alt="O "> <a name="<init>"></a><a name="Context()"></a><b>Context</b><br>
+<pre>
+public Context();
+</pre>
+<p>
+<img src="p-images/yellow-ball.gif" alt="O "> <a name="<init>"></a><a name="Context(Context)"></a><b>Context</b><br>
+<pre>
+public Context(<a href="G4ip.Context.html">Context</a> init_context);
+</pre>
+<p>
+<p><h2><img src="p-images/methods.gif" alt="Methods"></h2>
+<img src="p-images/red-ball.gif" alt="O "> <a name="makeString"></a><a name="makeString()"></a><b>makeString</b><br>
+<pre>
+public <a href="http://java.sun.com:80/products/jdk/1.1/docs/api/java.lang.String.html">String</a> makeString();
+</pre>
+<dl>
+ <dd> should be toString, but this is a "final" method in Vector <p></dd>
+ <dd><dl><dt></dt>
+ <dt> <b>Overrides:</b></dt>
+ <dd><dl>
+ <dt><b><a href="G4ip.Context.html#makeString()">makeString</a></b> in class <a href="G4ip.Context.html">Context</a></dt>
+ </dl></dd>
+ </dl></dd>
+</dl>
+<p>
+<img src="p-images/red-ball.gif" alt="O "> <a name="add"></a><a name="add(Form)"></a><b>add</b><br>
+<pre>
+public <a href="G4ip.Context.html">Context</a> add(<a href="G4ip.Form.html">Form</a> new_formula);
+</pre>
+<dl>
+ <dd> returns a context with one additional formula at the beginning <p></dd>
+ <dd><dl><dt></dt>
+ <dt> <b>Parameters:</b></dt>
+ <dd><b>new_formula</b> - a formula</dd>
+ <dt> <b>Overrides:</b></dt>
+ <dd><dl>
+ <dt><b><a href="G4ip.Context.html#add(Form)">add</a></b> in class <a href="G4ip.Context.html">Context</a></dt>
+ </dl></dd>
+ </dl></dd>
+</dl>
+<p>
+<img src="p-images/red-ball.gif" alt="O "> <a name="add"></a><a name="add(Form, Form)"></a><b>add</b><br>
+<pre>
+public <a href="G4ip.Context.html">Context</a> add(<a href="G4ip.Form.html">Form</a> new_formula1,
+ <a href="G4ip.Form.html">Form</a> new_formula2);
+</pre>
+<dl>
+ <dd> returns a contexts with two additional formulae at the beginning <p></dd>
+ <dd><dl><dt></dt>
+ <dt> <b>Parameters:</b></dt>
+ <dd><b>new_formula1,new_formula2</b> - two formulae</dd>
+ <dt> <b>Overrides:</b></dt>
+ <dd><dl>
+ <dt><b><a href="G4ip.Context.html#add(Form, Form)">add</a></b> in class <a href="G4ip.Context.html">Context</a></dt>
+ </dl></dd>
+ </dl></dd>
+</dl>
+<p>
+<img src="p-images/red-ball.gif" alt="O "> <a name="includes"></a><a name="includes(Form)"></a><b>includes</b><br>
+<pre>
+public boolean includes(<a href="G4ip.Form.html">Form</a> a);
+</pre>
+<dl>
+ <dd> tests whether a context contains a specific atom <p></dd>
+ <dd><dl><dt></dt>
+ <dt> <b>Parameters:</b></dt>
+ <dd><b>a</b> - a formula</dd>
+ <dt> <b>Overrides:</b></dt>
+ <dd><dl>
+ <dt><b><a href="G4ip.Context.html#includes(Form)">includes</a></b> in class <a href="G4ip.Context.html">Context</a></dt>
+ </dl></dd>
+ </dl></dd>
+</dl>
+<p>
+<hr>
+<code><font size=-1>
+[<a href="index.html">all packages</a>]
+[<a href="G4ip.html">package G4ip</a>]
+[<a href="tree-pd.html">class hierarchy</a>]
+[<a href="names-pd.html">index</a>]
+</font></code>
+<h6>G4ip.Context.html</h6>
+</body>
+</html>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Doc/G4ip.Form.html Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,108 @@
+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2//EN">
+<!--NewPage-->
+<html>
+<head>
+<!-- generated by pizzadoc on Thu Oct 30 13:46:18 GMT+00:00 1997 -->
+<title>
+public class G4ip.Form
+</title>
+</head>
+<body bgcolor=FFFFFF>
+<a name="_top_"></a>
+<code><font size=-1>
+[<a href="index.html">all packages</a>]
+[<a href="G4ip.html">package G4ip</a>]
+[<a href="tree-pd.html">class hierarchy</a>]
+[<a href="names-pd.html">index</a>]
+</font></code>
+<hr>
+<h1>public class G4ip.Form</h1>
+(source file: Formulae.pizza)<br>
+<pre>
+<a href="http://java.sun.com:80/products/jdk/1.1/docs/api/java.lang.Object.html">java.lang.Object</a>
+ |
+ +----G4ip.Form
+</pre>
+<hr>
+<pre>
+public class <a href="G4ip.Form.html">Form</a>
+</pre>
+<dl>
+ <dd> Class for propositional formulae.<p><p></dd>
+ <dd><dl><dt></dt>
+ <dt> <b>Author:</b></dt>
+ <dd> Christian Urban</dd>
+ </dl></dd>
+</dl>
+<hr>
+<p><h2><img src="p-images/constructor-index.gif" alt="Constuctor Index"></h2>
+<dl>
+ <dt><img src="p-images/yellow-ball-small.gif" alt="O "> <b><a href="G4ip.Form.html#And(Form, Form)">And</a></b>(Form, Form)
+ <dd>
+ <dt><img src="p-images/yellow-ball-small.gif" alt="O "> <b><a href="G4ip.Form.html#Atm(String)">Atm</a></b>(String)
+ <dd>
+ <dt><img src="p-images/yellow-ball-small.gif" alt="O "> <b><a href="G4ip.Form.html#Equ(Form, Form)">Equ</a></b>(Form, Form)
+ <dd>
+ <dt><img src="p-images/yellow-ball-small.gif" alt="O "> <b><a href="G4ip.Form.html#False()">False</a></b>()
+ <dd>
+ <dt><img src="p-images/yellow-ball-small.gif" alt="O "> <b><a href="G4ip.Form.html#Imp(Form, Form)">Imp</a></b>(Form, Form)
+ <dd>
+ <dt><img src="p-images/yellow-ball-small.gif" alt="O "> <b><a href="G4ip.Form.html#Or(Form, Form)">Or</a></b>(Form, Form)
+ <dd>
+</dl>
+<p><h2><img src="p-images/method-index.gif" alt="Methods"></h2>
+<dl>
+ <dt><img src="p-images/red-ball-small.gif" alt="O "> <b><a href="G4ip.Form.html#toString()">toString</a></b>()
+ <dd>
+</dl>
+<p><h2><img src="p-images/constructors.gif" alt="Constructors"></h2>
+<img src="p-images/yellow-ball.gif" alt="O "> <a name="False"></a><a name="False()"></a><b>False</b><br>
+<pre>
+public case False();
+</pre>
+<p>
+<img src="p-images/yellow-ball.gif" alt="O "> <a name="And"></a><a name="And(Form, Form)"></a><b>And</b><br>
+<pre>
+public case And(<a href="G4ip.Form.html">Form</a> c1,
+ <a href="G4ip.Form.html">Form</a> c2);
+</pre>
+<p>
+<img src="p-images/yellow-ball.gif" alt="O "> <a name="Or"></a><a name="Or(Form, Form)"></a><b>Or</b><br>
+<pre>
+public case Or(<a href="G4ip.Form.html">Form</a> c1,
+ <a href="G4ip.Form.html">Form</a> c2);
+</pre>
+<p>
+<img src="p-images/yellow-ball.gif" alt="O "> <a name="Imp"></a><a name="Imp(Form, Form)"></a><b>Imp</b><br>
+<pre>
+public case Imp(<a href="G4ip.Form.html">Form</a> c1,
+ <a href="G4ip.Form.html">Form</a> c2);
+</pre>
+<p>
+<img src="p-images/yellow-ball.gif" alt="O "> <a name="Equ"></a><a name="Equ(Form, Form)"></a><b>Equ</b><br>
+<pre>
+public case Equ(<a href="G4ip.Form.html">Form</a> c1,
+ <a href="G4ip.Form.html">Form</a> c2);
+</pre>
+<p>
+<img src="p-images/yellow-ball.gif" alt="O "> <a name="Atm"></a><a name="Atm(String)"></a><b>Atm</b><br>
+<pre>
+public case Atm(<a href="http://java.sun.com:80/products/jdk/1.1/docs/api/java.lang.String.html">String</a> c);
+</pre>
+<p>
+<p><h2><img src="p-images/methods.gif" alt="Methods"></h2>
+<img src="p-images/red-ball.gif" alt="O "> <a name="toString"></a><a name="toString()"></a><b>toString</b><br>
+<pre>
+public <a href="http://java.sun.com:80/products/jdk/1.1/docs/api/java.lang.String.html">String</a> toString();
+</pre>
+<p>
+<hr>
+<code><font size=-1>
+[<a href="index.html">all packages</a>]
+[<a href="G4ip.html">package G4ip</a>]
+[<a href="tree-pd.html">class hierarchy</a>]
+[<a href="names-pd.html">index</a>]
+</font></code>
+<h6>G4ip.Form.html</h6>
+</body>
+</html>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Doc/G4ip.Parser.html Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,133 @@
+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2//EN">
+<!--NewPage-->
+<html>
+<head>
+<!-- generated by pizzadoc on Thu Oct 30 13:46:18 GMT+00:00 1997 -->
+<title>
+public class G4ip.Parser
+</title>
+</head>
+<body bgcolor=FFFFFF>
+<a name="_top_"></a>
+<code><font size=-1>
+[<a href="index.html">all packages</a>]
+[<a href="G4ip.html">package G4ip</a>]
+[<a href="tree-pd.html">class hierarchy</a>]
+[<a href="names-pd.html">index</a>]
+</font></code>
+<hr>
+<h1>public class G4ip.Parser</h1>
+(source file: Parser.pizza)<br>
+<pre>
+<a href="http://java.sun.com:80/products/jdk/1.1/docs/api/java.lang.Object.html">java.lang.Object</a>
+ |
+ +----G4ip.Parser
+</pre>
+<hr>
+<pre>
+public class <a href="G4ip.Parser.html">Parser</a>
+</pre>
+<dl>
+ <dd> A left-to-right, rightmost-derivation parser.<p>
+ The following grammar is implemented:<p>
+ <dl>
+ <dd><u>Formula</u> ::= <u>Id</u> </dd>
+ <dd><code>| false</code> </dd>
+ <dd><code>|</code> ( <u>Formula</u> ) </dd>
+ <dd><code>|</code> <u>Formula</u> <u>Binop</u> <u>Formula</u></dd>
+ </dl><p>
+ <dl>
+ <dd><u>Id</u> is a string of lower case letters</dd>
+ </dl><p>
+ <dl>
+ <dd><u>Binop</u> is either <code>&, v, -></code> or <code><-></code></dd>
+ </dl><p>
+ <dl>
+ <dd><u>FormulaList</u> ::= <em>empty</em></dd>
+ <dd><code>|</code> [ <u>FormulaList</u> ,]* <u>Formula</u></dd>
+ <dl><p>
+ The parser uses a stack where two actions are performed:
+ <dl>
+ <dd><b>shift</b> moves the next token to the top of the stack (getNextToken)</dd>
+ <dd><b>reduce</b> chooses a grammar rule, X -> A B C; pops A, B, C from the
+ top of the stack and pushes X onto the stack
+ </dl><p></dd>
+ <dd><dl><dt></dt>
+ <dt> <b>Author:</b></dt>
+ <dd> Christian Urban</dd>
+ </dl></dd>
+</dl>
+<hr>
+<p><h2><img src="p-images/constructor-index.gif" alt="Constuctor Index"></h2>
+<dl>
+ <dt><img src="p-images/yellow-ball-small.gif" alt="O "> <b><a href="G4ip.Parser.html#Parser(String)">Parser</a></b>(String)
+ <dd>
+</dl>
+<p><h2><img src="p-images/method-index.gif" alt="Methods"></h2>
+<dl>
+ <dt><img src="p-images/red-ball-small.gif" alt="O "> <b><a href="G4ip.Parser.html#getNextToken()">getNextToken</a></b>()
+ <dd>tokens are: identifiers<code>( ) , & v -> <-> false</code>
+ and <code>EOI</code>
+ <dt><img src="p-images/red-ball-small.gif" alt="O "> <b><a href="G4ip.Parser.html#parseFormula()">parseFormula</a></b>()
+ <dd>parses a single formula
+ <dt><img src="p-images/red-ball-small.gif" alt="O "> <b><a href="G4ip.Parser.html#parseFormulae()">parseFormulae</a></b>()
+ <dd>parses a list of formulae separated by commas
+ <dt><img src="p-images/red-ball-small.gif" alt="O "> <b><a href="G4ip.Parser.html#reduce()">reduce</a></b>()
+ <dd>Implements the grammar rules.
+</dl>
+<p><h2><img src="p-images/constructors.gif" alt="Constructors"></h2>
+<img src="p-images/yellow-ball.gif" alt="O "> <a name="<init>"></a><a name="Parser(String)"></a><b>Parser</b><br>
+<pre>
+public Parser(<a href="http://java.sun.com:80/products/jdk/1.1/docs/api/java.lang.String.html">String</a> init_in);
+</pre>
+<p>
+<p><h2><img src="p-images/methods.gif" alt="Methods"></h2>
+<img src="p-images/red-ball.gif" alt="O "> <a name="getNextToken"></a><a name="getNextToken()"></a><b>getNextToken</b><br>
+<pre>
+public <a href="G4ip.Token.html">Token</a> getNextToken()
+ throws <a href="http://java.sun.com:80/products/jdk/1.1/docs/api/java.lang.Exception.html">Exception</a>;
+</pre>
+<dl>
+ <dd> tokens are: identifiers<code>( ) , & v -> <-> false</code>
+ and <code>EOI</code> <em>(end of input)</em> <p></dd>
+ <dd><dl><dt></dt>
+ <dt> <b>Overrides:</b></dt>
+ <dd><dl>
+ <dt><b><a href="G4ip.Parser.html#getNextToken()">getNextToken</a></b> in class <a href="G4ip.Parser.html">Parser</a></dt>
+ </dl></dd>
+ </dl></dd>
+</dl>
+<p>
+<img src="p-images/red-ball.gif" alt="O "> <a name="reduce"></a><a name="reduce()"></a><b>reduce</b><br>
+<pre>
+public void reduce();
+</pre>
+<dl>
+ <dd> Implements the grammar rules.<p></dd>
+ <dd><dl><dt></dt>
+ <dt> <b>Overrides:</b></dt>
+ <dd><dl>
+ <dt><b><a href="G4ip.Parser.html#reduce()">reduce</a></b> in class <a href="G4ip.Parser.html">Parser</a></dt>
+ </dl></dd>
+ </dl></dd>
+</dl>
+<p>
+<img src="p-images/red-ball.gif" alt="O "> <a name="parseFormula"></a><a name="parseFormula()"></a><b>parseFormula</b><br>
+<pre>
+public <a href="G4ip.Form.html">Form</a> parseFormula()
+ throws <a href="http://java.sun.com:80/products/jdk/1.1/docs/api/java.lang.Exception.html">Exception</a>;
+</pre>
+<dl>
+ <dd> parses a single formula<p></dd>
+ <dd><dl><dt></dt>
+ <dt> <b>Overrides:</b></dt>
+ <dd><dl>
+ <dt><b><a href="G4ip.Parser.html#parseFormula()">parseFormula</a></b> in class <a href="G4ip.Parser.html">Parser</a></dt>
+ </dl></dd>
+ </dl></dd>
+</dl>
+<p>
+<img src="p-images/red-ball.gif" alt="O "> <a name="parseFormulae"></a><a name="parseFormulae()"></a><b>parseFormulae</b><br>
+<pre>
+public <a href="G4ip.Context.html">Context</a> parseFormulae()
+ throws <a href="http://java.sun.com:80/products/
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Doc/G4ip.html Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,47 @@
+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2//EN">
+<!--NewPage-->
+<html>
+<head>
+<!-- generated by pizzadoc on Thu Oct 30 13:46:19 GMT+00:00 1997 -->
+<title>
+package G4ip
+</title>
+</head>
+<body bgcolor=FFFFFF>
+<a name="_top_"></a>
+<code><font size=-1>
+[<a href="index.html">all packages</a>]
+[<a href="tree-pd.html">class hierarchy</a>]
+[<a href="names-pd.html">index</a>]
+</font></code>
+<hr>
+<h1>package G4ip</h1>
+<!--pd-start-I-->
+
+<!--pd-start-C-->
+<h2><img src="p-images/class-index.gif" alt="Class Index"></h2>
+<ul>
+<!--pd-C-G4ip.Context--><li> <b>public class <a href="G4ip.Context.html">Context</a></b><br><font size=-1>Contexts act as multiset.</font>
+<!--pd-C-G4ip.Form--><li> <b>public class <a href="G4ip.Form.html">Form</a></b><br><font size=-1>Class for propositional formulae.</font>
+<!--pd-C-G4ip.Parser--><li> <b>public class <a href="G4ip.Parser.html">Parser</a></b><br><font size=-1>A left-to-right, rightmost-derivation parser.</font>
+<!--pd-C-G4ip.ProofDisplay--><li> <b>public class <a href="G4ip.ProofDisplay.html">ProofDisplay</a></b><br><font size=-1>Draws a frame on the screen containing a proof.</font>
+<!--pd-C-G4ip.Prover--><li> <b>public class <a href="G4ip.Prover.html">Prover</a></b><br><font size=-1>The Gi4p prover.</font>
+<!--pd-C-G4ip.ProverApplet--><li> <b>public class <a href="G4ip.ProverApplet.html">ProverApplet</a></b><br><font size=-1>The applet for the prover.</font>
+<!--pd-C-G4ip.Sequent--><li> <b>public class <a href="G4ip.Sequent.html">Sequent</a></b><br><font size=-1>An intuitionistic sequent of the form
+ "Gamma => G".</font>
+<!--pd-C-G4ip.Token--><li> <b>public class <a href="G4ip.Token.html">Token</a></b><br><font size=-1></font>
+<!--pd-end-C-->
+</ul>
+<!--pd-start-EX-->
+
+<!--pd-start-ER-->
+
+<hr>
+<code><font size=-1>
+[<a href="index.html">all packages</a>]
+[<a href="tree-pd.html">class hierarchy</a>]
+[<a href="names-pd.html">index</a>]
+</font></code>
+<h6>G4ip.html</h6>
+</body>
+</html>
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Formulae.pizza Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,48 @@
+
+package G4ip;
+
+/** Class for propositional formulae.<p>
+ * @author Christian Urban
+ */
+public class Form {
+
+ public case False();
+ public case And(Form c1,Form c2);
+ public case Or(Form c1,Form c2);
+ public case Imp(Form c1,Form c2);
+ //public case Equ(Form c1,Form c2);
+ public case Atm(String c);
+
+ public String toString() {
+ switch(this) {
+ case False():
+ return "false";
+ case And(Form c1,Form c2):
+ return "(" + c1.toString() + " & " + c2.toString() + ")";
+ case Or(Form c1,Form c2):
+ return "(" + c1.toString() + " v " + c2.toString() + ")";
+ case Imp(Form c1,Form c2):
+ return "(" + c1.toString() + " -> " + c2.toString() + ")";
+ case Atm(String c):
+ return c;
+ }
+ }
+
+ // for testing
+ public String internalString() {
+ switch(this) {
+ case False():
+ return "false";
+ case And(Form c1,Form c2):
+ return "And(" + c1.internalString() + "," + c2.internalString() + ")";
+ case Or(Form c1,Form c2):
+ return "Or(" + c1.internalString() + "," + c2.internalString() + ")";
+ case Imp(Form c1,Form c2):
+ return "Imp(" + c1.internalString() + "," + c2.internalString() + ")";
+ case Atm(String c):
+ return "Atm(\"p"+ c + "\")";
+ }
+ }
+
+}
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip.bbl Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,9 @@
+\begin{thebibliography}{}
+
+\bibitem[Dyckhoff, 1992]{Dyckhoff92}
+Dyckhoff, R.
+\newblock {C}ontraction-{F}ree {S}equent {C}alculi for {I}ntuitionistic
+ {L}ogic.
+\newblock {\em Journal of Symbolic Logic}, 57(3):795--807.
+
+\end{thebibliography}
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip.html Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,27 @@
+<HTML>
+<HEAD>
+ <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
+ <TITLE>G4ip Infernce Rules</TITLE>
+</HEAD>
+<BODY TEXT="#000000" BGCOLOR="#C7C3C7" LINK="#0000EF" VLINK="#51188E" ALINK="#FF0000">
+
+<H2>The Inference Rules of G4ip (LJT) </H2>
+
+A <A HREF="G4ip.dvi">dvi</A> and <A HREF="G4ip.ps">ps</A> version; the rules can be found in
+<A HREF="index.html#Dyckhoff92">[Dyckhoff, 1992]</a>.
+
+<HR>
+
+<img src="G4ip.gif" width=467 height=559>
+
+<HR>
+<ADDRESS>
+<A HREF="mailto:Christian.Urban@cl.cam.ac.uk">Christian Urban</A></ADDRESS>
+
+
+<P><!-- Created: Tue Mar 4 00:23:25 GMT 1997 -->
+<!-- hhmts start -->
+Last modified: Wed Jan 28 22:14:28 GMT 1998
+<!-- hhmts end -->
+</BODY>
+</HTML>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip.mod Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,118 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% An Implementation of G4ip for Terzo
+% author: Christian.Urban@cl.cam.ac.uk
+%
+% some solvable sample queries:
+%
+% prove (nil |- p imp p).
+% prove (nil |- (p imp p) imp (p imp p)).
+% prove (nil |- ((((p imp q) imp p) imp p) imp q) imp q).
+% prove (nil |- (a imp (b imp c)) imp ((a imp b) imp (a imp c))).
+% prove (nil |- (a or (a imp b)) imp (((a imp b) imp a) imp a)).
+%
+% two non-solvable queries
+%
+% prove (nil |- a or (a imp false)).
+% prove (nil |- ((a imp b) imp a) imp a).
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+module G4ip.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% atomic formulae
+kind form type.
+
+type p form.
+type q form.
+type a form.
+type b form.
+type c form.
+
+type isatomic form -> o.
+
+isatomic p.
+isatomic q.
+isatomic a.
+isatomic b.
+isatomic c.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% logical operators
+type false form.
+type and form -> form -> form.
+type or form -> form -> form.
+type imp form -> form -> form.
+
+infixr and 9.
+infixr or 9.
+infixr imp 9.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% sequent constructor; sequents are of the form: (list |- formula)
+kind seq type.
+
+type |- list form -> form -> seq.
+infixl |- 4.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% prove predicate; prints "solvable" if seq is provable, otherwise "no"
+type prove seq -> o.
+
+prove (Gamma |- G) :- ( membNrest P Gamma Gamma',
+ left (P::Gamma' |- G) );
+ right (Gamma |- G).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% rightrules
+type right seq -> o.
+
+right (Gamma |- B and C) :- prove (Gamma |- B), %% and-R
+ prove (Gamma |- C).
+
+right (Gamma |- B imp C) :- prove (B::Gamma |- C). %% imp-R
+
+right (Gamma |- B or C) :- prove (Gamma |- B); %% or-R
+ prove (Gamma |- C).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% leftrules
+type left seq -> o.
+
+left (false :: Gamma |- G). %% false-L
+
+left (A :: Gamma |- A) :- isatomic A. %% axiom
+
+left (B and C::Gamma |- G) :- prove (B::C::Gamma |- G). %% and-L
+
+left (B or C::Gamma |- G) :- prove (B::Gamma |- G), %% or-L
+ prove (C::Gamma |- G).
+
+
+left (A imp B::Gamma |- G) :- %% imp-L1
+ isatomic A, ismember A Gamma, prove (B::Gamma |- G).
+
+left ((B and C) imp D::Gamma |- G) :- %% imp-L2
+ prove (B imp (C imp D)::Gamma |- G).
+
+left ((B or C) imp D::Gamma |- G) :- %% imp-L3
+ prove (B imp D::C imp D::Gamma |- G).
+
+left ((B imp C) imp D::Gamma |- G) :- %% imp-L4
+ prove (C imp D::Gamma |- B imp C),
+ prove (D::Gamma |- G).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% returns a member and the remainder of a list
+type membNrest A -> list A -> list A -> o.
+
+membNrest X (X::Rest) Rest.
+membNrest X (Y::Tail) (Y::Rest) :- membNrest X Tail Rest.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% succeeds only once if A is element in the list
+type ismember A -> list A -> o.
+
+ismember X (X::Rest) :- !.
+ismember X (Y::Tail) :- ismember X Tail.
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip.ps Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,427 @@
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+\documentclass[11pt]{article}
+\usepackage{dina4}
+\usepackage{my}
+\usepackage{latexsym,amssymb}
+%\usepackage[final]{graphics}
+
+
+\newcommand{\FN}[1]{FN(#1)} % free name
+\newcommand{\FC}[1]{FC(#1)} % free co-name
+\renewcommand{\refname}{\-}
+
+\begin{document}
+\section*{Inference Rules of G4ip (LJT)}
+
+\begin{figure}[h]
+ \begin{center}
+ \begin{FramedTitled}{\-}
+ \centering
+ \def\arraystretch{3.0}
+ \begin{tabular}{cc}
+
+ \infer[Axiom]{\sequ{A,\Gamm}{A}}{} &
+
+ \infer[${\em false}-L$]{\sequ{\mbox{\em false},\Gamm}{G}}{}\\[-8mm]
+
+ ($A$ being atomic) \\
+
+ \infer[\&$-L$]{\sequ{B \& C,\Gamm}{G}}
+ {\sequ{B,C,\Gamm}{G}} &
+
+ \infer[\&$-R$]{\sequ{\Gamm}{B \& C}}
+ { \sequ{\Gamm}{B}
+ & \sequ{\Gamm}{C}}\\
+
+ \infer[\OR$-L$]
+ {\sequ{B\OR C,\Gamm}{G}}
+ { \sequ{B,\Gamm}{G}
+ &\sequ{C,\Gamm}{G}} &
+
+ \infer[\OR$-R$_1]
+ {\sequ{\Gamm}{B\OR C}}
+ {\sequ{\Gamm}{B}}\hspace{3mm}
+ \infer[\OR$-R$_2]
+ {\sequ{\Gamm}{B\OR C}}
+ {\sequ{\Gamm}{C}}\\
+
+ \infer[\mbox{\tt ->}$-L$_1]
+ {\sequ{A\mbox{\tt ->} B,A,\Gamm}{G}}
+ {\sequ{B,A,\Gamm}{G}} &
+
+ \infer[\mbox{\tt ->}$-R$]
+ {\sequ{\Gamm}{B\mbox{\tt ->} C}}
+ { \sequ{B,\Gamm}{C}}\\[-8mm]
+
+ ($A$ being atomic)\\
+
+ \multicolumn{2}{c}{
+ \infer[\mbox{\tt ->}$-L$_2]
+ {\sequ{(C \& D)\mbox{\tt ->} B,\Gamm}{G}}
+ {\sequ{C \mbox{\tt ->} (D \mbox{\tt ->} B)\Gamm}{G}}} \\
+
+ \multicolumn{2}{c}{
+ \infer[\mbox{\tt ->}$-L$_3]
+ {\sequ{(C\OR D)\mbox{\tt ->} B\Gamm}{G}}
+ {\sequ{C\mbox{\tt ->}B,D\mbox{\tt ->}B,\Gamm}{G}}}\\
+
+ \multicolumn{2}{c}{
+ \infer[\mbox{\tt ->}$-L$_4]
+ {\sequ{(C\mbox{\tt ->} D)\mbox{\tt ->} B\Gamm}{G}}
+ {\sequ{D\mbox{\tt ->}B,\Gamm}{C\mbox{\tt ->} D} &
+ \sequ{B,\Gamm}{G}} }\\
+
+ %\infer[\mbox{\tt <->}$-L$]
+ % {\sequ{B\mbox{\tt <->} C,\Gamm}{G}}
+ % {\sequ{B\mbox{\tt ->} C, C\mbox{\tt ->} B,\Gamm}{G}
+ % } &
+
+ %\infer[\mbox{\tt <->}$-R$]
+ % {\sequ{\Gamm}{B\mbox{\tt <->} C}}
+ % { \sequ{\Gamm}{B\mbox{\tt ->} C} &
+ % \sequ{\Gamm}{C\mbox{\tt ->} B} }\\
+
+ \multicolumn{2}{c}
+ { $B\mbox{\tt <->}C$ is defined as $B\mbox{\tt ->}C \& C\mbox{\tt ->}B$
+ }
+ \end{tabular}
+ \end{FramedTitled}
+ \caption{Inference rules for G4ip; see for example \cite{Dyckhoff92}.}
+ \label{fig:CL}
+ \end{center}
+\end{figure}
+
+\bibliography{/home/cu200/tex/bib/all}
+\bibliographystyle{fullnames}
+
+
+\end{document}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% Local Variables:
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+%%% TeX-master: "G4ip.tex"
+%%% End:
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/GettingStarted.html Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,1 @@
+Not yet available!
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Parser.pizza Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,183 @@
+package G4ip;
+
+import java.util.*;
+import pizza.lang.Pair;
+import G4ip.Form.*;
+import G4ip.Token.*;
+
+public class Token {
+ public static final int AND = 0;
+ public static final int OR = 1;
+ public static final int IMP = 2;
+ public static final int EQU = 3;
+
+ /** End-of-Input */
+ public case EOI(); // end-of-input
+ public case LPAREN();
+ public case RPAREN();
+ public case ID(String id);
+ public case FALSE();
+ public case BINOP(int op); // binary operation
+ public case COMMA();
+}
+
+/** A left-to-right, rightmost-derivation parser.<p>
+ * The following grammar is implemented:<p>
+ * <dl>
+ * <dd><u>Formula</u> ::= <u>Id</u> </dd>
+ * <dd><code>| false</code> </dd>
+ * <dd><code>|</code> ( <u>Formula</u> ) </dd>
+ * <dd><code>|</code> <u>Formula</u> <u>Binop</u> <u>Formula</u></dd>
+ * </dl><p>
+ * <dl>
+ * <dd><u>Id</u> is a string of lower case letters</dd>
+ * </dl><p>
+ * <dl>
+ * <dd><u>Binop</u> is either <code>&, v, -></code> or <code><-></code></dd>
+ * </dl><p>
+ * <dl>
+ * <dd><u>FormulaList</u> ::= <em>empty</em></dd>
+ * <dd><code>|</code> [ <u>FormulaList</u> ,]* <u>Formula</u></dd>
+ * <dl><p>
+ * The parser uses a stack where two actions are performed:
+ * <dl>
+ * <dd><b>shift</b> moves the next token to the top of the stack (getNextToken)</dd>
+ * <dd><b>reduce</b> chooses a grammar rule, X -> A B C; pops A, B, C from the
+ * top of the stack and pushes X onto the stack
+ * </dl>
+ * @author Christian Urban
+ */
+public class Parser {
+ final Pair<String,Token> keywords[] =
+ { new Pair("(", LPAREN()),
+ new Pair(")", RPAREN()),
+ new Pair(",", COMMA()),
+ new Pair("&", BINOP(Token.AND)),
+ new Pair("v ", BINOP(Token.OR)),
+ new Pair("->", BINOP(Token.IMP)),
+ new Pair("<->", BINOP(Token.EQU)),
+ new Pair("false",FALSE()) };
+ String in; // string to be parsed
+ int index; // position of the current input
+ Stack stack; // stack for the left-to-right parser 'LR(0)'
+
+ public Parser(String init_in){
+ index = 0;
+ in = init_in;
+ stack = new Stack();
+ }
+
+ /** tokens are: identifiers<code>( ) , & v -> <-> false</code>
+ * and <code>EOI</code> <em>(end of input)</em>
+ */
+ public Token getNextToken() throws Exception {
+ while ( index < in.length() && Character.isSpace(in.charAt(index)) )
+ { index++; } //delete white-spaces
+ if (index == in.length()) { return EOI(); } //end-of-string
+ for (int i=0;i<keywords.length;i++) //keywords
+ { if (in.startsWith(keywords[i].fst,index)) {
+ index=index+keywords[i].fst.length();
+ return keywords[i].snd;
+ }
+ }
+ if (Character.isLowerCase(in.charAt(index)) || Character.isDigit(in.charAt(index)))
+ { //reads identifiers
+ String s = "";
+ while ( index < in.length() &&
+ ( Character.isLowerCase(in.charAt(index))
+ || Character.isDigit(in.charAt(index))
+ )
+ )
+ { s=s.concat(in.substring(index,index+1));
+ index++;
+ }
+ return ID(s);
+ }
+ throw new Exception("Syntax error at: '" + in.charAt(index) + "'");
+ // no match at all: probably an unknown character
+ return null;
+ }
+
+ /** Implements the grammar rules. */
+ public void reduce()
+ { boolean again = false;
+ /* ID -> Atm(string) */
+ if (stack.size() > 0 &&
+ (stack.elementAt(stack.size()-1) instanceof ID))
+ { ID id = (ID)stack.pop();
+ stack.push(Atm(id.id));
+ again = true;
+ }
+ /* FALSE -> False() */
+ if (stack.size() > 0 &&
+ (stack.elementAt(stack.size()-1) instanceof FALSE))
+ { stack.pop();
+ stack.push(False());
+ again = true;
+ }
+ /* ( Formula ) -> Formula */
+ if (stack.size() > 2 &&
+ (stack.elementAt(stack.size()-3) instanceof LPAREN) &&
+ (stack.elementAt(stack.size()-2) instanceof Form) &&
+ (stack.elementAt(stack.size()-1) instanceof RPAREN))
+ { stack.pop();
+ Form form = (Form)stack.pop();
+ stack.pop();
+ stack.push(form);
+ again = true;
+ }
+ /* Formula BINOP Formula -> Formula */
+ if (stack.size() > 2 &&
+ (stack.elementAt(stack.size()-3) instanceof Form) &&
+ (stack.elementAt(stack.size()-2) instanceof BINOP) &&
+ (stack.elementAt(stack.size()-1) instanceof Form))
+ { Form c2 = (Form)stack.pop();
+ BINOP op = (BINOP)stack.pop();
+ Form c1 = (Form)stack.pop();
+ switch(op.op) {
+ case Token.AND: stack.push(new And(c1,c2)); again = true;break;
+ case Token.OR: stack.push(new Or(c1,c2)); again = true;break;
+ case Token.IMP: stack.push(new Imp(c1,c2)); again = true;break;
+ case Token.EQU: stack.push(new And(Imp(c1,c2),Imp(c2,c1))); again = true;break;
+ }
+ }
+ if (again == true) { reduce(); } // do as many "reduces" as possible
+ }
+
+ /** parses a single formula
+ */
+ public Form parseFormula() throws Exception {
+ Token tok;
+ while (!((tok = getNextToken()) instanceof EOI)) {
+ stack.push(tok);
+ reduce();
+ }
+ if (stack.size() == 1 &&
+ (stack.elementAt(stack.size()-1) instanceof Form))
+ { return (Form)stack.pop(); }
+ else throw new Exception("Grammar error");
+ return null;
+ }
+
+ /** parses a list of formulae separated by commas
+ */
+ public Context parseFormulae() throws Exception {
+ Token tok;
+ Context erg = new Context();
+ while (!((tok = getNextToken()) instanceof EOI)) {
+ stack.push(tok);
+ reduce();
+ }
+ if (stack.empty()) return erg; // LHS can be empty !!
+ stack.push(new COMMA());
+ for(int i=0;i<stack.size()-1;i=i+2){
+ if (stack.elementAt(i) instanceof Form)
+ { erg.addElement(stack.elementAt(i));
+ } else throw new Exception("Grammar error");
+ if (stack.elementAt(i+1) instanceof COMMA)
+ {} else throw new Exception("Grammar error");
+ }
+ return erg;
+ }
+
+}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/ProofDisplay.pizza Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,106 @@
+package G4ip;
+
+import pizza.util.Hashtable;
+import java.awt.*;
+import G4ip.Form.*;
+
+
+/** A label with a surrounding box.
+ */
+class MyLabel extends Label {
+ public MyLabel() { super(); }
+ public MyLabel(String s) { super(s); }
+
+ public void paint(Graphics g) {
+ Dimension d = this.size();
+ g.setColor(Color.gray);
+ g.drawRect(0,0,d.width-1,d.height-1);
+ }
+}
+
+
+/** Draws a frame on the screen containing a proof.
+ */
+public class ProofDisplay extends Frame {
+ Button ok;
+ Hashtable<int,Sequent> proof; // contains the indexed sequents
+
+ /** The proof is represented in the hashtable.
+ * @param proof is a hashtable; the structure of the proof (or tree)
+ * is mapped onto a sequence of integers; each integer is a key for
+ * a sequent stored in the hashtable.
+ */
+ public ProofDisplay(Hashtable<int,Sequent> iproof) {
+ super("Proof");
+ ok = new Button("Close");
+ proof = iproof;
+
+ setLayout(new BorderLayout());
+ // SOUTH
+ Panel p = new Panel();
+ p.add(ok);
+ add("South",p);
+
+ //CENTER
+ Panel display = new Panel();
+ printproof(display,1);
+ add("Center", display);
+
+ pack();
+ show();
+ }
+
+ /** Recovers the structure of the proofs from the hashtable representation.
+ */
+ public void printproof(Panel p,int index) {
+ MyLabel l = new MyLabel(proof.get(index).toString());
+ l.setAlignment(MyLabel.CENTER);
+ p.setLayout(new BorderLayout());
+ p.add("South",l);
+ // axiom (do nothing)
+ if ((proof.get(2*index) == null) && (proof.get(2*index+1) == null)) {
+ return;
+ }
+ // rule with a single premise
+ if ((proof.get(2*index) != null) && (proof.get(2*index+1) == null)) {
+ Panel np = new Panel();
+ printproof(np,2*index);
+ p.add("Center",np);
+ return;
+ }
+ // rule with two premises
+ if ((proof.get(2*index) != null) && (proof.get(2*index+1) != null)) {
+ Panel np1 = new Panel();
+ Panel np2 = new Panel();
+ printproof(np1,2*index);
+ printproof(np2,2*index+1);
+ p.add("West",np1);
+ p.add("East",np2);
+ return;
+ }
+
+ }
+
+ /** If either the "Close" or the "Window-destroy" button
+ * is pressed, then close the window.
+ */
+ public boolean handleEvent(Event e) {
+ if (e.id == Event.ACTION_EVENT && e.target == ok) {
+ this.finalize();
+ return true;
+ }
+ if (e.id == Event.WINDOW_DESTROY) {
+ this.finalize();
+ return true;
+ }
+ return false;
+ }
+
+ /** Closes the window.
+ */
+ public void finalize()
+ { this.dispose(); }
+
+}
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Prover.pizza Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,223 @@
+package G4ip;
+
+
+import pizza.util.Hashtable;
+import java.util.Vector;
+import G4ip.Form.*;
+import G4ip.ProofDisplay.*;
+import G4ip.Sequent.*;
+
+
+/** The Gi4p prover.<p>
+ * The prover is a thread in order to suspend the proof search
+ * after one proof is found and to let the user make a choice
+ * of whether to stop or to continue with the proof search.
+ * @author Christian Urban
+ */
+public class Prover extends Thread
+{ ProverApplet parent; // the parent of the prover;
+ boolean once; // if true then at least one proof was found
+ Sequent sequ; // the root sequent to be proved
+ Vector frames; // to record all frames which have been opened
+ int index ;
+ Hashtable<int,Sequent> proof; // keeps a record of the proof
+
+
+ /** The constructor creates a prover which proves a specific sequent.<p>
+ * The parameter parent is for access to the applet buttons.
+ */
+ public Prover(Sequent init_sequ, ProverApplet init_parent) {
+ super("Prover");
+ parent = init_parent;
+ once = false;
+ sequ = new Sequent(init_sequ);
+ frames = new Vector();
+ index = 1;
+ proof = new Hashtable();
+ }
+
+ /** Starts the thread; calls prove and stops the thread when
+ * the proof search is finished.
+ */
+ public void run()
+ { prove(1,sequ,initial_sc);
+ if (once == true) {
+ parent.messages.setText("No more proofs.");
+ parent.messages.repaint();
+ parent.repaint();
+ }
+ else {
+ parent.messages.setText("Not provable.");
+ parent.messages.repaint();
+ parent.repaint();
+ }
+ parent.switch_into_input_mode();
+ this.stop(); // everything is done, stop the thread
+ }
+
+ /** Closes all frames which have been opened.
+ */
+ public void finalize() {
+ for (int i=0; i<frames.size();i++) {
+ if (frames.elementAt != null)
+ { ((ProofDisplay)frames.elementAt(i)).dispose(); }
+ }
+ }
+
+ /** The tactic of the proof search.<p>
+ * Method prove first enumerates all formulae on the LHS
+ * as being the principal formula and attempts to apply left-rules,
+ * subsequently it analyses the goal formula.
+ */
+ public void prove(int index, Sequent is,(() -> void) sc) {
+ Form principal;
+ proof.put(index,new Sequent(is)); // add sequence to proof
+ rightrules(index,is,sc);
+ for (int i=0;i<is.Gamma.size();i++) {
+ principal = (Form)is.Gamma.elementAt(i);
+ is.Gamma.removeElement(principal); // Gamma minus principal formula
+ leftrules(index,principal,is,sc);
+ is.Gamma.insertElementAt(principal,i); // put principal formula
+ } // back into Gamma
+ //rightrules(index,is,sc);
+ proof.remove(index); // remove sequence from proof
+ }
+
+ /** Analyses the goal formula.
+ */
+ public void rightrules(int index, Sequent sequ,(() -> void) sc) {
+ Context Gamma = sequ.Gamma;
+ switch(sequ.G) {
+ /** And-R
+ * Gamma => A Gamma => B
+ * --------------------------
+ * Gamma => A and B
+ */
+ case And(Form A,Form B):
+ prove(2*index,new Sequent(Gamma,A),
+ fun() -> void { prove(2*index+1,new Sequent(Gamma,B),sc); } );
+ break;
+
+ /** Imp-R
+ * Gamma,A => B
+ * -----------------
+ * Gamma => A imp B
+ */
+ case Imp(Form A,Form B):
+ prove(2*index,new Sequent(Gamma.add(A),B),sc); break;
+
+ /** Or-R
+ * Gamma => A Gamma => B
+ * ----------------- or -----------------
+ * Gamma => A or B Gamma => A or B
+ */
+ case Or(Form A,Form B):
+ prove(2*index,new Sequent(Gamma,A),sc);
+ prove(2*index,new Sequent(Gamma,B),sc); break;
+
+ /** Equ-R
+ * Gamma => A -> B Gamma => B -> A
+ * --------------------------------
+ * Gamma => A <-> B
+ */
+ }
+ }
+
+ /** Analyses the principal formula on the LHS.
+ */
+ public void leftrules(int index,Form principal,Sequent sequ,(() -> void) sc){
+ Context Gamma = sequ.Gamma;
+ Form G = sequ.G;
+ switch(principal) {
+ /** false-L
+ * --------------------
+ * false, Gamma => G
+ */
+ case False(): sc(); break;
+
+ /** Axiom
+ * ---------------
+ * Gamma, G => G G being atomic
+ */
+ case Atm(String c):
+ if (G instanceof Atm) {
+ if (((Atm)G).c.compareTo(c) == 0) { sc(); }
+ }
+ break;
+
+ /** And-L
+ * Gamma, A, B => G
+ * --------------------
+ * Gamma, A and B => G
+ */
+
+ case And(Form A, Form B):
+ prove(2*index,new Sequent(Gamma.add(A,B),G),sc); break;
+
+ /** Or-R
+ * Gamma, A => G Gamma, B => G
+ * -------------------------------
+ * Gamma, A or B => G
+ */
+ case Or(Form A, Form B):
+ prove(2*index,new Sequent(Gamma.add(A),G),
+ fun() -> void {prove(2*index+1,new Sequent(Gamma.add(B),G),sc);});
+ break;
+
+ /** Imp-L 2
+ * Gamma, A imp (B imp C) => G
+ * ------------------------------
+ * Gamma, (A and B) imp C => G
+ */
+ case Imp(And(Form A, Form B), Form C):
+ prove(2*index,new Sequent(Gamma.add(Imp(A,Imp(B,C))),G),sc); break;
+
+
+ /** Imp-L 3
+ * Gamma, (A imp C), (B imp C) => G
+ * ----------------------------------
+ * Gamma, (A or B) imp C => G
+ */
+ case Imp(Or(Form A, Form B), Form C):
+ prove(2*index,new Sequent(Gamma.add(Imp(A,C),Imp(B,C)),G),sc); break;
+
+ /** Imp-L 4
+ * Gamma, (B imp C) => (A imp B) Gamma, C => G
+ * ------------------------------------------------
+ * Gamma, (A imp B) imp C => G
+ */
+ case Imp(Imp(Form A, Form B), Form C):
+ prove(2*index,new Sequent(Gamma.add(Imp(B,C)),Imp(A,B)),
+ fun() -> void { prove(2*index+1,new Sequent(Gamma.add(C),G),sc);});
+ break;
+ /** Imp-L 1
+ * Gamma(A), B => G A being atomic
+ * ------------------------ Gamma(A) means:
+ * Gamma(A), A imp B => G Gamma contains A
+ */
+ case Imp(Form A, Form B):
+ if (A instanceof Atm) {
+ if (Gamma.includes(A)) {
+ { prove(2*index,new Sequent(Gamma.add(B),G),sc); }
+ }
+ }
+ break;
+ }
+
+ }
+
+
+ /** The initial success continuation.
+ * Suspends the thread when a proof is found.
+ */
+ public void initial_sc() {
+ once = true;
+ ProofDisplay p = new ProofDisplay(proof);
+ frames.addElement(p); // keep a record for later disposal
+ try { // suspend the proof search,
+ suspend(); } // it might be resumed later on by the user
+ catch(SecurityException sec_exc)
+ { /* this catch is neccessary for Netscape 3.0 (Linux) */ }
+ }
+
+}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/ProverApplet.html Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,108 @@
+<HTML>
+<HEAD>
+ <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
+ <TITLE>G4ip Prover</TITLE>
+</HEAD>
+<BODY TEXT="#000000" BGCOLOR="#C7C3C7" LINK="#0000EF" VLINK="#51188E" ALINK="#FF0000">
+
+<H2>Prover for G4ip (LJT) </H2>
+
+<A HREF="README">Readme</A>
+
+
+<HR>
+<B>Formulae:</B>
+ F ::= <CODE>false</CODE>
+ | A
+ | F <CODE>&</CODE> F
+ | F <CODE>v</CODE> F
+ | F <CODE>-></CODE> F
+ | F <CODE><-></CODE> F
+ | (F)<BR>
+<B>Sequents:</B> <CODE>=></CODE> F or
+ [F,]* F <CODE>=></CODE> F
+
+<HR>
+<B>Inference Rules:</B>
+<A HREF="G4ip.html">html</A>,
+<A HREF="G4ip.dvi">dvi</A> or
+<A HREF="G4ip.ps">ps</A>.
+
+<HR>
+<B>Start:</B> starts the proof search,
+<B>Clear:</B> clears the text fields,<BR>
+<B>Next:</B> searches for the next proof,
+<B>Stop:</B> stops the proof search.
+
+<HR>
+<APPLET code="G4ip/ProverApplet.class" width=600 height=100>
+<PARAM name="LHS_size" value="22">
+<PARAM name="RHS_size" value="37">
+</APPLET>
+
+
+<HR>
+<B>Provable Examples:</B>
+<TT>
+<DL>
+<DD> => (p v (q v r)) <-> ((p v q) v r)</DD>
+<DD> => (p & (q & r)) <-> ((p & q) & r)</DD>
+<DD> => (p & (q v r)) <-> ((p & q) v (p & r))</DD>
+<DD> => (p v (q & r)) <-> ((p v q) & (p v r))</DD>
+<DD> => (p -> p) -> (p -> p)</DD>
+<DD> => (a -> (b -> c)) -> ((a -> b) -> (a -> c))</DD>
+<DD> => a -> ((a -> b) -> a)</DD>
+<DD> => b -> ((a -> b) -> b)</DD>
+<DD> => (a & b) -> (b & a)</DD>
+<DD> => (a -> (a -> b)) -> (a -> b)</DD>
+<DD> => ((((p -> q) -> p) -> p) -> q) -> q</DD>
+<DD> => (a v (a -> b)) -> (((a -> b) -> a)-> a)</DD>
+<DD> => (a -> (b -> false)) -> (b -> (a -> false))</DD>
+<DD> => ((a v (a -> false))-> false)-> false</DD>
+<DD>a & b, c & d => b & c</DD>
+<DD>(a v (a-> false)) -> false => false</DD>
+</DL>
+</TT>
+
+<B>Non-Provable Examples:</B>
+<TT>
+<DL>
+<DD> => ((a -> b) -> a) -> a</DD>
+<DD> => a v (a -> false)</DD>
+<DD> => (a & b v (((( a-> flase) -> false) -> q) v (b -> q)) -> q) -> q</DD>
+</DL>
+</TT>
+
+
+<A HREF="fun.html">Some advanced examples</A>
+
+<HR>
+<B>Source Code:</B>
+<A HREF="Formulae.pizza">Formulae</A>,
+<A HREF="Contexts.pizza">Contexts</A>,
+<A HREF="Sequents.pizza">Sequents</A>,
+<A HREF="ProofDisplay.pizza">ProofDisplay</A>,
+<A HREF="Prover.pizza">Prover</A>,
+<A HREF="Parser.pizza">Parser</A>,
+<A HREF="ProverApplet.pizza">Applet</A><BR>
+
+<HR>
+<B>Lambda Prolog Version:</B> The programm
+<A HREF="G4ip.mod">G4ip.mod</A> can be executed using
+<A HREF="http://www.cis.upenn.edu/~dale/lProlog/terzo/index.html">Terzo</A>
+Another implementation of G4ip by Joshua Hodas and Dale Miller written in
+<A HREF="http://www.cs.hmc.edu/~hodas/research/lolli/">Lolli</a> can be found
+<A HREF="ftp://ftp.cse.psu.edu/pub/dale/ic94-code/index.html">here</A>.
+
+
+<HR>
+<ADDRESS>
+<A HREF="mailto:Christian.Urban@cl.cam.ac.uk">Christian Urban</A></ADDRESS>
+
+
+<P><!-- Created: Tue Mar 4 00:23:25 GMT 1997 -->
+<!-- hhmts start -->
+Last modified: Sun May 10 18:18:13 GMT 1998
+<!-- hhmts end -->
+</BODY>
+</HTML>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/ProverApplet.pizza Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,150 @@
+package G4ip;
+
+import java.applet.*;
+import java.awt.*;
+import G4ip.Sequent.*;
+import G4ip.Prover.*;
+import G4ip.Parser.*;
+
+/** The applet for the prover.
+ * @author Christian Urban
+ */
+public class ProverApplet extends Applet {
+ Prover prover;
+ Parser parser;
+
+ Button start;
+ Button clear;
+ Button next;
+ Button stop;
+ TextField lhs_text; // place for the program formulae
+ TextField rhs_text; // place for the goal formula
+ /** A field for various messages (e.g. error messages).
+ */
+ public Label messages; // to be verbose
+
+ public void init() {
+ setLayout(new BorderLayout());
+ // NORTH
+ Panel textfields = new Panel();
+ add("North", textfields);
+ // the parameters LHS_size and RHS_size are given in the html-file
+ int LHS_size = Integer.parseInt(this.getParameter("LHS_size"));
+ int RHS_size = Integer.parseInt(this.getParameter("RHS_size"));
+ lhs_text = new TextField(LHS_size); // program textfield
+ rhs_text = new TextField(RHS_size); // goal textfield
+ Label separation = new Label("=>"); // for separation
+
+ if (LHS_size > 0) { // if someone prefears only single side sequents
+ textfields.add(lhs_text); }
+ textfields.add(separation);
+ textfields.add(rhs_text);
+
+ // CENTER
+ Panel buttonfield = new Panel();
+ add("Center", buttonfield);
+ start = new Button("Start");
+ clear = new Button("Clear");
+ next = new Button("Next");
+ stop = new Button("Stop");
+
+ switch_into_input_mode(); // start in input mode
+ buttonfield.add(start);
+ buttonfield.add(clear);
+ buttonfield.add(next);
+ buttonfield.add(stop);
+
+ // SOUTH
+ messages = new Label(""); // for any kind of message
+ messages.setAlignment(Label.CENTER);
+ add("South",messages);
+ }
+
+ /** When start is pressed;
+ * start and clear button are disabled;
+ * next and stop button are enabled.
+ */
+ public void switch_into_prove_mode() {
+ start.disable(); next.enable();
+ clear.disable(); stop.enable();
+ repaint();
+ }
+
+ /** When start is pressed
+ * and (hopefully) when prover finishes.
+ */
+ public void switch_into_input_mode() {
+ start.enable(); next.disable();
+ clear.enable(); stop.disable();
+ repaint();
+ }
+
+ /** Initiates the parse and prove process.
+ */
+ void initiate_proving() {
+ Sequent sequ;
+ messages.setText("");
+ messages.repaint();
+ repaint();
+ Parser rhs_parser = new Parser(rhs_text.getText());
+ Parser lhs_parser = new Parser(lhs_text.getText());
+ try { // try to parse the input
+ sequ = new Sequent(lhs_parser.parseFormulae(),
+ rhs_parser.parseFormula());
+ }
+ catch (Exception exc) { // there was a parser exception
+ messages.setText(exc.getMessage());
+ messages.repaint();
+ repaint();
+ return; // in case there was an exception
+ } // don't run the following code
+ System.out.println(sequ.G.internalString());
+ switch_into_prove_mode();
+ prover = new Prover(sequ,this);
+ prover.start();
+ }
+
+ /** Forces the prover to dispose all open frames.
+ */
+ public void destroy() {
+ if (prover != null && prover.isAlive()) {
+ prover.stop();
+ prover.finalize();
+ }
+ }
+
+
+ /** The actions for the buttons:
+ * this bit of code would be different in Java 1.1
+ */
+ public boolean action(Event e, Object arg) {
+ // clear button has been pressed
+ if (e.target == clear) {
+ messages.setText("");
+ lhs_text.setText("");
+ rhs_text.setText("");
+ messages.repaint();
+ repaint();
+ }
+ // start button has been pressed
+ if (e.target == start) {
+ initiate_proving();
+ return true;
+ }
+ // stop button has been pressed
+ if (e.target == stop) {
+ switch_into_input_mode();
+ if (prover != null && prover.isAlive()) { prover.stop(); }
+ return true;
+ }
+ // next button has been pressed
+ if (e.target == next) {
+ if (prover != null && prover.isAlive()) { prover.resume();}
+ return true;
+ }
+ return false;
+ }
+
+}
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/ProverAppletJar.html Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,110 @@
+<HTML>
+<HEAD>
+ <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
+ <TITLE>G4ip Prover</TITLE>
+</HEAD>
+<BODY TEXT="#000000" BGCOLOR="#C7C3C7" LINK="#0000EF" VLINK="#51188E" ALINK="#FF0000">
+
+<H2>Prover for G4ip (LJT) </H2>
+
+<A HREF="README">Readme</A>
+
+
+<HR>
+<B>Formulae:</B>
+ F ::= <CODE>false</CODE>
+ | A
+ | F <CODE>&</CODE> F
+ | F <CODE>v</CODE> F
+ | F <CODE>-></CODE> F
+ | F <CODE><-></CODE> F
+ | (F)<BR>
+<B>Sequents:</B> <CODE>=></CODE> F or
+ [F,]* F <CODE>=></CODE> F
+
+<HR>
+<B>Inference Rules:</B>
+<A HREF="G4ip.html">html</A>,
+<A HREF="G4ip.dvi">dvi</A> or
+<A HREF="G4ip.ps">ps</A>.
+
+<HR>
+<B>Start:</B> starts the proof search,
+<B>Clear:</B> clears the text fields,<BR>
+<B>Next:</B> searches for the next proof,
+<B>Stop:</B> stops the proof search.
+
+<HR>
+<!-- APPLET code="G4ip/ProverApplet.class" width=600 height=100 -->
+<APPLET code="G4ip/ProverApplet.class" archive="G4ip/ProverApplet.jar" width=600 height=100>
+<PARAM name="LHS_size" value="22">
+<PARAM name="RHS_size" value="37">
+</APPLET>
+
+
+<HR>
+<B>Provable Examples:</B>
+<TT>
+<DL>
+<DD> => (p v (q v r)) <-> ((p v q) v r)</DD>
+<DD> => (p & (q & r)) <-> ((p & q) & r)</DD>
+<DD> => (p & (q v r)) <-> ((p & q) v (p & r))</DD>
+<DD> => (p v (q & r)) <-> ((p v q) & (p v r))</DD>
+<DD> => (p -> p) -> (p -> p)</DD>
+<DD> => (a -> (b -> c)) -> ((a -> b) -> (a -> c))</DD>
+<DD> => a -> ((a -> b) -> a)</DD>
+<DD> => b -> ((a -> b) -> b)</DD>
+<DD> => (a & b) -> (b & a)</DD>
+<DD> => (a -> (a -> b)) -> (a -> b)</DD>
+<DD> => ((((p -> q) -> p) -> p) -> q) -> q</DD>
+<DD> => (a v (a -> b)) -> (((a -> b) -> a)-> a)</DD>
+<DD> => (a -> (b -> false)) -> (b -> (a -> false))</DD>
+<DD> => ((a v (a -> false))-> false)-> false</DD>
+<DD>a & b, c & d => b & c</DD>
+<DD>(a v (a-> false)) -> false => false</DD>
+</DL>
+</TT>
+
+<B>Non-Provable Examples:</B>
+<TT>
+<DL>
+<DD> => ((a -> b) -> a) -> a</DD>
+<DD> => a v (a -> false)</DD>
+<DD> => (a & b v (((( a-> flase) -> false) -> q) v (b -> q)) -> q) -> q</DD>
+</DL>
+</TT>
+
+
+<A HREF="fun.html">Some advanced examples</A>
+
+<HR>
+<B>Source Code:</B>
+<A HREF="Formulae.pizza">Formulae</A>,
+<A HREF="Contexts.pizza">Contexts</A>,
+<A HREF="Sequents.pizza">Sequents</A>,
+<A HREF="ProofDisplay.pizza">ProofDisplay</A>,
+<A HREF="Prover.pizza">Prover</A>,
+<A HREF="Parser.pizza">Parser</A>,
+<A HREF="ProverApplet.pizza">Applet</A><BR>
+
+
+<HR>
+<B>Lambda Prolog Version:</B> The programm
+<A HREF="G4ip.mod">G4ip.mod</A> can be executed using
+<A HREF="http://www.cis.upenn.edu/~dale/lProlog/terzo/index.html">Terzo</A>
+Another implementation of G4ip by Joshua Hodas and Dale Miller written in
+<A HREF="http://www.cs.hmc.edu/~hodas/research/lolli/">Lolli</a> can be found
+<A HREF="ftp://ftp.cse.psu.edu/pub/dale/ic94-code/index.html">here</A>.
+
+
+<HR>
+<ADDRESS>
+<A HREF="mailto:Christian.Urban@cl.cam.ac.uk">Christian Urban</A></ADDRESS>
+
+
+<P><!-- Created: Tue Mar 4 00:23:25 GMT 1997 -->
+<!-- hhmts start -->
+Last modified: Sun May 10 18:18:28 GMT 1998
+<!-- hhmts end -->
+</BODY>
+</HTML>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/README Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,76 @@
+
+Write once, run everywhere
+==========================
+ Good idea, but still wishful thinking.
+
+ From my experience I would like to add:
+ "Write once, run everywhere, show the same behaviour!"
+
+
+Known bugs and failures:
+========================
+
+- Cut-and-paste
+ I have not included any code for cut-and-paste. However,
+ cut-and-paste should work fine under Unix. Cut-and-Paste
+ also works in MS IE 3.0 under Mickeysoft Windows. For most of
+ the other platforms (MacOS etc.) and browsers (Netscape etc.)
+ cut-and-paste does not work.
+
+- Security exception
+ Various versions of Netscape throw a security exception when
+ I try to suspend the thread of the proof search. I have caught
+ this exception, but my literature does not tell me how to resume
+ safely from this exception. In effect the proof search enumerates
+ all possible proofs for a sequent and subsequently stops. The user
+ has no control over how many proofs are shown (the corresponding
+ windows might appear over each other). This unintended behaviour
+ has been encountered in:
+
+ - Netscape Navigator 3.0 Gold under MS Windows'45
+ - Netscape Navigator 3.0 under MacOS
+ - Netscape Communicator 4.0 under MacOS
+ - Netscape Navigator 3.01 Gold under Linux
+ (But here a "catch" of this exception and a "do nothing"
+ helps. God knows why.)
+ - possible others
+
+- Boxes around the sequents in proofs
+ I have modified the "Label" class such that the "MyLabel" class
+ has a surrounding box. This improves the readability of the proofs.
+ However, this feature works ONLY under Unix. The boxes do not
+ appear in MS Windows and MacOS (reason unknown).
+
+- Flickering messages
+ The applet prints some messages ("Grammar error", "No more proofs",
+ "Not provable", etc.) in a field below the control buttons.
+ In some browsers the messages only flicker for a short time.
+ Other browsers show the messages correctly (reason unknown).
+
+
+The applet has been tested under (but see restrictions and bugs):
+=================================================================
+
+Linux:
+ Netscape Navigator 3.01 Gold
+ Netscape Communicator 4.03, 4.04, 4.07
+ HotJava 1.1 beta2 (Linux-port)
+ SUN's appletviewer for JDK 1.1.3
+
+Win'95:
+ Netscape Navigator 3.0 Gold
+ Netscape Communicator 4.01a
+ MS Internet Explorer 3.0.2
+ MS Internet Explorer 4.0
+
+MacOS:
+ Netscape Navigator 3.0
+ Netscape Communicator 4.0
+ MS Internet Explorer 3.0.1b1 (PowerPC)
+
+
+Any comments on success or failure for other
+environments and platforms are appreciated.
+
+Christian.Urban@cl.cam.ac.uk
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Sequents.pizza Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,29 @@
+package G4ip;
+
+import G4ip.Form.*;
+
+/** An intuitionistic sequent of the form
+ * "Gamma => G".
+ */
+public class Sequent {
+ /** Goal formula on the right-hand side. */
+ public Form G;
+ /** Context (multiset of formulae) on the left-hand side. */
+ public Context Gamma;
+
+ public Sequent(Context init_Gamma, Form init_G) {
+ G = init_G;
+ Gamma = new Context(init_Gamma);
+ }
+
+ public Sequent(Sequent sequ) {
+ G = sequ.G;
+ Gamma = new Context(sequ.Gamma);
+ }
+
+ /** Prints a sequent. */
+ public String toString() {
+ String s = Gamma.makeString() + " => " + G.toString();
+ return s;
+ }
+}
Binary file Prover/Tableaux98.dvi.gz has changed
Binary file Prover/Tableaux98.ps.gz has changed
Binary file Prover/execution.gif has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/fun.html Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,32 @@
+<HTML>
+<HEAD>
+ <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
+ <TITLE>Advanced Examples</TITLE>
+</HEAD>
+<BODY TEXT="#000000" BGCOLOR="#C7C3C7" LINK="#0000EF" VLINK="#51188E" ALINK="#FF0000">
+
+<H2>Some Advanced Examples </H2>
+
+Be cautious: some of the following formulae have big proofs!
+
+<HR>
+<TT>
+<DL>
+<DD> => (((a & b) v (a->f) v (b->f))->f)->f</DD>
+<DD> => (a & b & c v ((a->f) v (b->f) v (c->f))->f)->f</DD>
+<DD> => d&(b->b->a)&(c->c->b)&(d->d->c)->a</DD>
+</DL>
+</TT>
+
+
+<HR>
+<ADDRESS>
+<A HREF="mailto:Christian.Urban@cl.cam.ac.uk">Christian Urban</A></ADDRESS>
+
+
+<P><!-- Created: Tue Mar 4 00:23:25 GMT 1997 -->
+<!-- hhmts start -->
+Last modified: Sun Nov 23 20:20:31 GMT 1997
+<!-- hhmts end -->
+</BODY>
+</HTML>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/index.html Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,384 @@
+<HTML>
+<HEAD>
+ <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
+ <TITLE>G4ip</TITLE>
+</HEAD>
+<BODY TEXT="#000000" BGCOLOR="#C7C3C7" LINK="#0000EF" VLINK="#51188E" ALINK="#FF0000">
+
+<H2>An Implementation of G4ip in Pizza</H2>
+
+<FONT COLOR="#800000"><B>Warning:</B></FONT>
+This page is now rather old! While you might still be interested
+in the algorithms, Robert Macdonald reported that Pizza and the current Java
+implementation (version 1.3.0) do not work together. This means you need to
+install an older Java version if you want to recompile the files given below.
+I am happy to answer all question concerning the prover, but be aware that
+currently for any kind of Java stuff I am using MLJ, which as of writing
+this note has not yet been made available for the general audience (maybe
+in the future also its OCaml equivalent). So I am not very fluent in Pizza
+anymore. <B>Update</B> Pizza development is continued and starting from version
+<A HREF="http://pizzacompiler.sourceforge.net/">0.40</A> it should work
+with recent Java implementations.<P>
+
+
+Jump to the <A HREF="#Implementation">implementation.</a>
+
+
+<H4>Introduction</H4>
+
+
+A convenient representation of intuitionistic logic is Getzen's
+sequent calculus LJ (also G1i). A sequent of LJ can be proved
+by applying inference rules until one reaches axioms, or can make no further
+progress in which case one must backtrack or even abandon the search.
+Unfortunately an interpreter for LJ using this depth-first strategy cannot
+guarantee termination of the proof search. Several modifications can be
+made to LJ's inference rules without loss of soundness and completeness.
+As result an efficient depth-first proof search can be designed for the
+propositional fragment of intuitionistic logic. The name G4ip has been
+assigned to the corresponding calculus in
+<A HREF="#TroelstraSchwichtenberg96">[Troelstra and Schwichtenberg, 1996]</a>.
+This calculus is also known as LJT which has been studied thoroughly
+in <A HREF="#Dyckhoff92">[Dyckhoff, 1992]</a>. The inference rules of
+G4ip are given <A HREF="G4ip.html">here</A>.<P>
+
+It is not very complicated to implement an interpreter for G4ip using a logic
+programming language (backtracking is directly supported within the language).
+Our first implementation is written in the logic programming language
+<A HREF="http://www.cis.upenn.edu/~dale/lProlog/terzo/index.html">Lambda Prolog</A>
+and can be found <A HREF="G4ip.mod">here</A>. Another implementation by
+Hodas and Miller written in <A HREF="http://www.cs.hmc.edu/~hodas/research/lolli/">Lolli</a>
+can be found <A HREF="ftp://ftp.cse.psu.edu/pub/dale/ic94-code/index.html">here</A>
+(see <A HREF="#HodasMiller94">[Hodas and Miller, 1994]</a>). These are simple and
+straightforward implementations of G4ip's rules. On the other hand it seems
+that imperative languages need a rather high overhead of code when implementing
+a logic calculus. For example choice points are usually implemented with stacks.
+We shall demonstrate the implementation technique of success
+continuations which provides an equally simple method for implementing logic calculi
+in imperative languages. This technique is not new: it has been introduced in
+<A HREF="#Carlsson84">[Carlsson, 1984]</a>. This paper presents a rather technical
+implementation of Prolog in LISP. Later an excellent paper,
+<A HREF="#ElliotPfenning91">[Elliot and Pfenning, 1991]</a>, appeared which
+describes a full-fledged implementation of Lambda Prolog in SML.
+We demonstrate the technique of success continuations for G4ip in
+<A HREF="http://www.cis.unisa.edu.au/~pizza/">Pizza</A>.<P>
+
+Pizza is an object-oriented programming language and an attractive extension of
+<A HREF="http://www.javasoft.com/">Java</A>. Although Pizza is a superset of
+Java, Pizza programs can be translated into Java or compiled into ordinary
+Java Byte Code (see <A HREF="#OderskyWadler97">[Odersky and Wadler, 1997]</a>
+for a technical introduction to Pizza). We make use of the following two new
+features of Pizza:
+<UL>
+<LI> higher-order functions, i.e. functions may be passed as parameters or returned
+from methods,
+<LI> class cases and pattern matching: this allows much simpler and more readable code.
+</UL>
+
+These features are not directly present in Java, but Pizza makes them accessible by
+translating them into Java. Pizza provides the programmer with the same
+extensive libraries for graphic and network applications as Java. The higher-order
+functions are essential for the technique of success continuations. The success
+continuations are functions passed as parameters or returned as values.<BR>
+
+
+<H4>The Representation of Formulae and Sequents</H4>
+
+Amongst the new language features of Pizza are class cases and pattern
+matching, which provide a very pleasant syntax for algebraic data types. The
+formulae of G4ip are specified by the following grammar:<P>
+
+<CODE>F ::= false | A | F & F | F v F | F -> F</CODE><P>
+
+
+The class cases allow a straightforward implementation of this specification;
+it is analogous to the SML implementation of
+<A HREF="http://www.cis.upenn.edu/~dale/lProlog/terzo/index.html">Lambda Prolog's</A>
+formulae in <A HREF="#ElliotPfenning91">[Elliot and Pfenning, 1991]</A>. The class
+of formulae for G4ip is given below:<P>
+
+<TT>
+<DL>
+<DD>public class Form { </DD>
+<DD> case False(); </DD>
+<DD> case Atm(String c); </DD>
+<DD> case And(Form c1,Form c2); </DD>
+<DD> case Or(Form c1,Form c2); </DD>
+<DD> case Imp(Form c1,Form c2); </DD>
+<DD>} </DD>
+</DL>
+</TT>
+
+Two examples that illustrate the use of the representation are as follows:<P>
+
+
+<CODE> p -> p </CODE>
+is represented as <CODE> Imp(Atm("p"),Atm("p"))</CODE><BR>
+<CODE>a v (a -> false) </CODE> is represented as <CODE> Or(Atm("a"),Imp(Atm("a"),False()))</CODE><P>
+
+The class cases of Pizza also support an implementation of formulae specified
+by a mutually recursive grammar. This is required, for example, when
+implementing hereditary Harrop formulae.<P>
+
+The sequents of G4ip, which have the form <CODE>Gamma=>G</CODE>, are represented
+by means of the class below. The left-hand side of each sequent is specified by a multiset
+of formulae. Therefore, we do not need to worry about the order in which the
+formulae occur.<P>
+
+<TT>
+<DL>
+<DD>public class Sequent { </DD>
+<DD> Form G; </DD>
+<DD> Context Gamma; </DD>
+<DD> public Sequent(Context _Gamma, Form _G) {...};</DD>
+<DD>} </DD>
+</DL>
+</TT>
+
+We have a constructor for generating new sequents during proof search.
+<CODE>Context</CODE> is a class which represents multisets; it is a simple
+extension of the class <CODe>Vector</CODE> available in the Java libraries.
+This class provides methods for adding elements to a multiset (<CODE>add</CODE>),
+taking out elements from a multiset (<CODE>removeElement</CODE>) and testing
+the membership of an element in a multiset (<CODE>includes</CODE>).
+
+
+<H4>The Technique of Success Continuations</H4>
+
+We have to distinguish between the concepts of proof obligations (which must
+be proved) and choice points (which can be tried out to construct a proof).
+The first argument of the method <CODE>prove</CODE> is the sequent being
+proved; the second argument is an anonymous function. The function <CODE>prove</CODE> is now
+of the form <CODE>prove(sequent,sc)</CODE>. Somewhat simplified the
+first argument is the leftmost premise and the second argument <CODE>sc</CODE>,
+the success continuation, represents the other proof obligations. In case we
+succeed in proving the first premise we then can attempt to prove the other
+premises. The technique of success continuations will be explained using the following
+proof (each sequent is marked with a number):<P>
+
+<UL><img src="proof.gif" width=337 height=112></UL>
+<BR><P>
+
+The inference rules fall into three groups:
+
+<UL>
+<LI> inference rules with a single premise (e.g. ->_R, &_L),
+<LI> inference rules with two premises (e.g. v_L) and
+<LI> inference rules without premises (e.g. Axiom).
+</UL>
+
+The following picture shows the order in which the sequents are being proved.
+
+<UL><img src="execution.gif" width=358 height=191></UL>
+<BR><P>
+
+Suppose we have called <CODE>prove</CODE> with a sequent <B>s</B> and a
+success continuation <B>is</B>. The inference rules of the first
+group manipulate <B>s</B> obtaining <B>s'</B> and call <CODE>prove</CODE>
+again with the new sequent <B>s'</B> and the current success continuation
+(Steps 1-2, 3-4 and 5-6). The inference rules
+of the second group have two premises, <B>s1</B> and <B>s2</B>.
+These rules call <CODE>prove</CODE> with <B>s1</B> and a new success
+continuation <CODE>prove(s2,is)</CODE> (Step 2-3).
+The third group of inference rules only invoke the success continuation
+if the rule was applicable (Steps 4-5 and 6-7).<P>
+
+
+We are going to give a detailed description of the code for the rules: &_L,
+->_R, v_Ri, v_L and Axiom. The function <CODE>prove</CODE> receives as arguments
+a sequent <CODE>Sequent(Gamma,G)</CODE> and a success continuation
+<CODE>sc</CODE>. It enumerates all formulae as being principal and
+two switch statements select a corresponding case depending on the form
+and the occurrence of the principal formula.<P>
+
+The &_L rule is in the first group; it modifies the sequent being proved and calls
+<CODE>prove</CODE> again with the current success continuation sc. The code is as
+follows (<CODE>Gamma</CODE> stands for the set of formulae on the left-hand
+side of a sequent excluding the principal formula; <CODE>G</CODE> stands
+for the goal formula of a sequent; <CODE>B</CODE> and <CODE>C</CODE> stand
+for the two components of the principal formula).<P>
+
+<TT>
+<DL>
+<DD>case And(Form B, Form C):</DD>
+<DD> prove(new Sequent(Gamma.add(B,C),G),sc); break;</DD>
+</DL>
+</TT>
+
+The code for the ->_R rule is similar:<P>
+
+<TT>
+<DL>
+<DD>case Imp(Form B, Form C):</DD>
+<DD> prove(new Sequent(Gamma.add(A),B),sc); break;</DD>
+</DL>
+</TT>
+
+The v_Ri rule is an exception in the first group. It breaks up a goal
+formula of the form <CODE>B1 v B2</CODE> and proceeds with one of its component.
+Since we do not know in advance which component leads to a successful proof we have
+to try both. Therefore this rule acts as a choice point, which is encoded by a
+recursive call of <CODE>prove</CODE> for each case.
+
+<TT>
+<DL>
+<DD>case Or(Form B1,Form B2):</DD>
+<DD> prove(new Sequent(Gamma,B1),sc);</DD>
+<DD> prove(new Sequent(Gamma,B2),sc); break;</DD>
+</DL>
+</TT>
+
+The v_L rule falls into the second group where the current success
+continuation, sc, is modified. It calls <CODE>prove</CODE> with the first premise,
+<CODE>B,Gamma=>G</CODE>, and wraps up the success continuation with the
+new proof obligation, <CODE>C,Gamma=>G</CODE>. The construction
+<CODE>fun()->void {...}</CODE> defines an anonymous function: the new
+success continuation. In case the sequent <CODE>B,Gamma=>G</CODE> can be
+proved, this function is invoked.
+
+<TT>
+<DL>
+<DD>case Or(Form B,Form C):</DD>
+<DD> prove(new Sequent(Gamma.add(B),G),</DD>
+<DD>
+ fun()->void {prove(new Sequent(Gamma.add(C),G),sc);}</DD>
+<DD> ); break</DD>
+</DL>
+</TT>
+
+The Axiom rule falls into the third group. It first checks if the
+principal formula (which is an atom) matches with the goal formula and
+then invokes the success continuation sc in order to prove all remaining
+proof obligations.
+
+<TT>
+<DL>
+<DD>case Atm(String c):</DD>
+<DD> if (G instanceof Atm) { </DD>
+<DD> if (G.c.compareTo(c) == 0) { sc(); }</DD>
+<DD> } break;</DD>
+</DL>
+</TT>
+
+The proof search is started with an initial success continuation <B>is</B>.
+This initial success continuation is invoked when a proof has been found.
+In this case we want to give some response to the user, an
+example for the initial success continuation could be as follows:
+
+<TT>
+<DL>
+<DD> public void initial_sc() { System.out.println("Provable!"); } </DD>
+</DL>
+</TT>
+
+
+Suppose we attempt to start the proof search with <CODE>prove(p,p => p,is)</CODE>.
+We would find that the prover responds twice with <CODE>"Provable!"</CODE>, because
+it finds two proofs. In our implementation this problem is avoided by encoding
+the proof search as a thread. Whenever a proof is found, the initial success
+continuation displays the proof and suspends the thread. The user can
+decide to resume with the proof search or abandon the search.
+
+
+<H4>Conclusion</H4>
+
+The implementation cannot be considered as optimal in terms of speed.
+A much more efficient algorithm for G4ip (but less clear) has been
+implemented by Dyckhoff in Prolog. Similar ideas can be encoded in our
+Pizza implementation; but our point was not the efficiency but the clarity
+of the implementation using success continuations.
+The technique is applicable elsewhere whenever backtracking is required. We
+compared the code of our implementation with an implementation in
+<A HREF="http://www.cis.upenn.edu/~dale/lProlog/terzo/index.html">Lambda Prolog</A>:
+the ratio of code is approximately 2 to 1.
+(see <A HREF="G4ip.mod">LambdaProlog code</A> and
+<A HREF="minimal/MinProver.pizza">Pizza code</A>).
+This result is partly due to the fact that we had to implement a class for
+multisets. In a future version of Java, we could have accessed a package
+in the library. The technique of success continuation can also be applied
+to a first-order calculus as shown in <A HREF="#ElliotPfenning91">[Elliot and Pfenning, 1991]</a>,
+but the required mechanism of substitution needs to be implemented separately.
+However, we think the technique of success continuations provides a remarkable
+simple implementation for logic calculi.<P>
+
+We had to make some compromises in order to support as many platforms
+as possible. This should change with the release of new browsers and a stable
+Java-specification (resp. Pizza-specification).<P>
+
+A paper about the implementation appeared in the LNAI series No 1397,
+Automated Reasoning with Analytic Tableaux and Related Methods,
+ed. Harry de Swart, International Conference Tableaux'98 in Oisterwijk,
+The Netherlands. The title is: Implementation of Proof Search in
+the Imperative Programming Language Pizza (pp. 313-319). The paper can be
+found here: <A HREF="Tableaux98.dvi.gz">DVI</A>, <A HREF="Tableaux98.ps.gz">Postscript</A>
+(© Springer-Verlag <A HREF="http://www.springer.de/comp/lncs/index.html">LNCS</A>).<P>
+
+
+<B>Acknowledgements:</B> I am very grateful for Dr Roy Dyckhoff's constant
+encouragement and many comments on my work. I thank Dr Gavin Bierman who
+helped me to test the prover applet.
+
+<HR>
+<A NAME="Implementation"></A><H4>Implementation</H4>
+
+<A HREF="README">Readme</A><p>
+
+<A HREF="ProverApplet.html"><B>Prover Applet</B></A><BR>
+<A HREF="ProverAppletJar.html"><B>Jar Version</B></A>
+(slightly faster, but requires Netscape 4 or MS Explorer 4).<P>
+
+
+<HR>
+<B>References</B>
+<UL>
+<LI> <A NAME="Carlsson84"></A>
+ [Carlsson, 1984]<BR>
+ M. Carlsson, On Implementing Prolog in Functional Programming,
+ New Generation Computing, pp 347-359.
+<LI> <A NAME="Dyckhoff92"></A>
+ [Dyckhoff, 1992]<BR>
+ <A HREF="http://www-theory.dcs.st-and.ac.uk/~rd/">R. Dyckhoff</A>,
+ Contraction-Free Sequent Calculi for Intuitionistic Logic,
+ Journal of Symbolic Logic 57(3), pp 795-807.
+<LI> <A NAME="ElliotPfenning91"></A>
+ [Elliot and Pfenning, 1991]<BR>
+ C. Elliot,
+ <A HREF="http://foxnet.cs.cmu.edu/people/fp/homepage.html">F. Pfenning</A>,
+ A Semi-Functional Implementation of a Higher-Order Programming Language,
+ In Peter Lee, editor, Topics in Advanced Language Implementation, MIT Press,
+ pp 289-352.
+ <A HREF="http://www.cs.cmu.edu/~fp/papers/elpsml-paper.tar.gz">Available electronically</a>.
+<LI> <A NAME="HodasMiller94"></A>
+ [Hodas and Miller, 1994]<BR>
+ <A HREF="http://www.cs.hmc.edu/~hodas/">J. Hodas</A>,
+ <A HREF="http://www.cse.psu.edu/~dale/">D. Miller</A>,
+ Logic Programming in a Fragment of Intuitionistic Linear Logic,
+ Information and Computation 110(2), pp 327-365.
+ <A HREF="ftp://ftp.cse.psu.edu/pub/dale/ic94.ps.Z">Available electronically</a>.
+<LI> <A NAME="OderskyWadler97"></A>
+ [Odersky and Wadler, 1997]<BR>
+ <A HREF="http://www.cis.unisa.edu.au/~cismxo">M. Odersky</A>,
+ <A HREF="http://cm.bell-labs.com/cm/cs/who/wadler/">P. Wadler</A>,
+ Pizza into Java: Translating Theory into Practice,
+ In Proceedings of the 24th ACM Symposium on Principles of Programming Languages.
+ <A HREF="http://www.cis.unisa.edu.au/~cismxo/papers/popl97.dvi.gz">Available electronically</a>.
+<LI> <A NAME="TroelstraSchwichtenberg96"></A>
+ [Troelstra and Schwichtenberg, 1996]<BR>
+ <A HREF="http://turing.wins.uva.nl/~anne/">A. Troelstra</A>,
+ <A HREF="http://www.mathematik.uni-muenchen.de/~gadmin6/professoren/schwichtenberg">H. Schwichtenberg</A>,
+ Basic Proof Theory, Cambridge Tracts in Theoretical Computer Science,
+ Cambridge University Press.
+</UL>
+
+
+<HR>
+<ADDRESS>
+<A HREF="mailto:Christian.Urban@cl.cam.ac.uk">Christian Urban</A></ADDRESS>
+
+
+<P><!-- Created: Tue Mar 4 00:23:25 GMT 1997 -->
+<!-- hhmts start -->
+Last modified: Sun Sep 23 12:04:47 BST 2001
+<!-- hhmts end -->
+</BODY>
+</HTML>
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