--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Cut/CL-plugin.html Thu Mar 15 15:50:18 2012 +0000
@@ -0,0 +1,169 @@
+<HTML lang=en>
+<HEAD>
+<TITLE> Christian Urban </TITLE>
+<BASE HREF="http://www4.in.tum.de/~urbanc/Cut/">
+</HEAD>
+
+<BODY TEXT="#000000"
+ BGCOLOR="#000080"
+ LINK="#0000EF"
+ VLINK="#51188E"
+ ALINK="#FF0000">
+
+<TABLE WIDTH="100%"
+ COLS="2"
+ BGCOLOR="#000080"
+ BORDER="0"
+ FRAME="none"
+ CELLPADDING="10"
+ CELLSPACING="2"
+ RULES="COLS,ROWS">
+
+<!-- left column -->
+<TR>
+<TD BGCOLOR="#FFFFFF"
+ WIDTH="24%"
+ VALIGN="TOP"
+ ROWSPAN="4">
+
+<B>Links</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/cutapplet.html">Home</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/index.html">Applet Home</A><BR><BR><p>
+
+<B>Java Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND.html">ND</A><BR><BR>
+
+<B>Plugin Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL-plugin.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT-plugin.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND-plugin.html">ND</A><BR>
+
+<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%">
+<A NAME="Home"></A>
+<H2>Cut-Elimination in Classical Logic</H2>
+
+If you use this applet it is strongly(!) recommended to have read the the paper on
+strong normalisation of cut-elimination in classical logic by Urban and Bierman
+[<A HREF="http://www4.in.tum.de/~urbanc/Publications/fi-01.ps.gz">ps.gz</A>,
+ <A HREF="http://www4.in.tum.de/~urbanc/Publications/fi-01.pdf">pdf</A>].
+Mayor difference between the standard sequent calculus and the sequent calculus
+implemented by the applet is that the rules contraction and weakening are completely
+implicit. This means that sequents consist of two sets of (labelled) formulae,
+as opposed to lists or multisets of formulae. Although this is very simple, it
+needs some time to get used to.
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>How to use it?</B>
+If your browser is playing game with Java, then you will see a window
+with a number of examples. When pressing on one of the buttons, a new
+window will pop up. What follows is a brief explanation of all
+the features available in this window.
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>Buttons and Keys</B>
+<UL>
+<LI> The buttons
+ <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/l.jpg"> and
+ <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/r.jpg"> are for
+ commuting cuts, which can slide up either in the right or left proof branch.
+<LI> Pressing on <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/x.jpg">
+ , <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/x1.jpg"> or
+ <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/x2.jpg">
+ `fires' a logical cut. The latter two buttons correspond to the two possible
+ nestings of multiplicative logical cuts.
+<LI> If you use the <B>left mouse button</B> for pressing on those buttons,
+ then the new proof will appear inside the window. Whereas if you use
+ the <B>right mouse button</B>, a new window will pop up and
+ the new proof will be displayed in this window. Use the right mouse
+ button if you want to compare a proof and its reduct(s).
+<LI> The keys <B>Page-up</B> and <B>Page-down</B> zoom in or out of a proof
+ respectively.
+</UL>
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>Menu Functions</B>
+
+<UL>
+<LI> <B>Zoom in</B> and <B>Zoom out</B> work like Page-up and Page-down,
+ respectively.
+<LI> <B>Unicode</B> If this radiobutton is switched on, logic symbols
+ are displayed in unicode, otherwise in ascii. (This is for the
+ poor guys who have a browser and/or operating system which cannot handle
+ unicode.)
+<LI> <B>Labels</B> As mentioned earlier the sequents are composed of
+ two sets of labelled
+ formulae, not multisets of formulae as in the standard formulation
+ of sequent calculus. Enabling this radiobutton causes that labels are
+ drawn. Use this when you are unsure where implicit contractions are.
+<LI> <B>Auxiliary Substitution</B> If you have read the paper referred
+ above, you know what this does.
+<LI> <B>Gentzen's D.N.Transl.</B> and <B>Kolmogorov's D.N.Transl.</B>
+ perform double negation translations on the current proof.
+</UL>
+</TD>
+</TR>
+
+</TABLE>
+<center>
+<!--"CONVERTED_APPLET"-->
+<!-- HTML CONVERTER -->
+<OBJECT classid="clsid:CAFEEFAC-0013-0001-0000-ABCDEFFEDCBA"
+WIDTH = 1 HEIGHT = 1 codebase="http://java.sun.com/products/plugin/1.3.1/jinstall-131-win32.cab#Version=1,3,1,0">
+<PARAM NAME = CODE VALUE = "MyApplet.class" >
+<PARAM NAME = ARCHIVE VALUE = MyApplet.zip >
+
+<PARAM NAME="type" VALUE="application/x-java-applet;jpi-version=1.3.1">
+<PARAM NAME="scriptable" VALUE="false">
+<PARAM NAME = "calculi" VALUE ="A">
+<COMMENT>
+<EMBED type="application/x-java-applet;jpi-version=1.3.1" CODE = "MyApplet.class" ARCHIVE = MyApplet.zip WIDTH = 1 HEIGHT = 1 calculi = "A" scriptable=false pluginspage="http://java.sun.com/products/plugin/1.3.1/plugin-install.html"><NOEMBED>
+<p><blink><FONT COLOR="#800000">
+Applet should appear in a new window, but you don't have Java enabled in your browser.
+</FONT></blink><p>
+</NOEMBED>
+</EMBED>
+</COMMENT>
+</OBJECT>
+
+<!--
+<APPLET CODE = "MyApplet.class" ARCHIVE = MyApplet.zip WIDTH = 1 HEIGHT = 1>
+<PARAM NAME = "calculi" VALUE ="A">
+<p><blink><FONT COLOR="#800000">
+Applet should appear in a new window, but you don't have Java enabled in your browser.
+</FONT></blink><p>
+
+</APPLET>
+-->
+
+
+<!--"END_CONVERTED_APPLET"-->
+
+
+</center>
+<P><!-- Created: Tue Jul 3 21:01:42 BST 2001 -->
+<!-- hhmts start -->
+Last modified: Sat Mar 3 05:17:41 CET 2007
+<!-- hhmts end -->
+</BODY>
+</HTML>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Cut/CL.html Thu Mar 15 15:50:18 2012 +0000
@@ -0,0 +1,142 @@
+<HTML lang=en>
+<HEAD>
+<TITLE> Christian Urban </TITLE>
+<BASE HREF="http://www4.in.tum.de/~urbanc/Cut/">
+</HEAD>
+
+<BODY TEXT="#000000"
+ BGCOLOR="#000080"
+ LINK="#0000EF"
+ VLINK="#51188E"
+ ALINK="#FF0000">
+
+<TABLE WIDTH="100%"
+ COLS="2"
+ BGCOLOR="#000080"
+ BORDER="0"
+ FRAME="none"
+ CELLPADDING="10"
+ CELLSPACING="2"
+ RULES="COLS,ROWS">
+
+<!-- left column -->
+<TR>
+<TD BGCOLOR="#FFFFFF"
+ WIDTH="24%"
+ VALIGN="TOP"
+ ROWSPAN="4">
+
+<B>Links</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/index.html">Home</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/cutapplet.html">Applet Home</A><BR><BR><p>
+
+<B>Java Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND.html">ND</A><BR><BR>
+
+<B>Plugin Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL-plugin.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT-plugin.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND-plugin.html">ND</A><BR>
+
+<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%">
+<A NAME="Home"></A>
+<H2>Cut-Elimination in Classical Logic</H2>
+
+If you use this applet it is strongly(!) recommended to have read the the paper on
+strong normalisation of cut-elimination in classical logic by Urban and Bierman
+[<A HREF="http://www4.in.tum.de/~urbanc/Publications/fi-01.ps.gz">ps.gz</A>,
+ <A HREF="http://www4.in.tum.de/~urbanc/Publications/fi-01.pdf">pdf</A>].
+Mayor difference between the standard sequent calculus and the sequent calculus
+implemented by the applet is that the rules contraction and weakening are completely
+implicit. This means that sequents consist of two sets of (labelled) formulae,
+as opposed to lists or multisets of formulae. Although this is very simple, it
+needs some time to get used to.
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>How to use it?</B>
+If your browser is playing game with Java, then you will see a window
+with a number of examples. When pressing on one of the buttons, a new
+window will pop up. What follows is a brief explanation of all
+the features available in this window.
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>Buttons and Keys</B>
+<UL>
+<LI> The buttons
+ <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/l.jpg"> and
+ <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/r.jpg"> are for
+ commuting cuts, which can slide up either in the right or left proof branch.
+<LI> Pressing on <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/x.jpg">
+ , <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/x1.jpg"> or
+ <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/x2.jpg">
+ `fires' a logical cut. The latter two buttons correspond to the two possible
+ nestings of multiplicative logical cuts.
+<LI> If you use the <B>left mouse button</B> for pressing on those buttons,
+ then the new proof will appear inside the window. Whereas if you use
+ the <B>right mouse button</B>, a new window will pop up and
+ the new proof will be displayed in this window. Use the right mouse
+ button if you want to compare a proof and its reduct(s).
+<LI> The keys <B>Page-up</B> and <B>Page-down</B> zoom in or out of a proof
+ respectively.
+</UL>
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>Menu Functions</B>
+
+<UL>
+<LI> <B>Zoom in</B> and <B>Zoom out</B> work like Page-up and Page-down,
+ respectively.
+<LI> <B>Unicode</B> If this radiobutton is switched on, logic symbols
+ are displayed in unicode, otherwise in ascii. (This is for the
+ poor guys who have a browser and/or operating system which cannot handle
+ unicode.)
+<LI> <B>Labels</B> As mentioned earlier the sequents are composed of
+ two sets of labelled
+ formulae, not multisets of formulae as in the standard formulation
+ of sequent calculus. Enabling this radiobutton causes that labels are
+ drawn. Use this when you are unsure where implicit contractions are.
+<LI> <B>Auxiliary Substitution</B> If you have read the paper referred
+ above, you know what this does.
+<LI> <B>Gentzen's D.N.Transl.</B> and <B>Kolmogorov's D.N.Transl.</B>
+ perform double negation translations on the current proof.
+</UL>
+</TD>
+</TR>
+
+</TABLE>
+<center>
+<applet archive=MyApplet.zip code="MyApplet.class" width=1 height=1>
+<param name="calculi" value="A">
+<p><blink><FONT COLOR="#800000">
+Applet should appear in a new window, but you don't have Java enabled in your browser.
+</FONT></blink><p>
+</applet>
+
+</center>
+<P><!-- Created: Tue Jul 3 21:01:42 BST 2001 -->
+<!-- hhmts start -->
+Last modified: Sat Mar 3 05:17:31 CET 2007
+<!-- hhmts end -->
+</BODY>
+</HTML>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Cut/LJT-plugin.html Thu Mar 15 15:50:18 2012 +0000
@@ -0,0 +1,161 @@
+<HTML lang=en>
+<HEAD>
+<TITLE> Christian Urban </TITLE>
+<BASE HREF="http://www4.in.tum.de/~urbanc/Cut/">
+</HEAD>
+
+<BODY TEXT="#000000"
+ BGCOLOR="#000080"
+ LINK="#0000EF"
+ VLINK="#51188E"
+ ALINK="#FF0000">
+
+<TABLE WIDTH="100%"
+ COLS="2"
+ BGCOLOR="#000080"
+ BORDER="0"
+ FRAME="none"
+ CELLPADDING="10"
+ CELLSPACING="2"
+ RULES="COLS,ROWS">
+
+<!-- left column -->
+<TR>
+<TD BGCOLOR="#FFFFFF"
+ WIDTH="24%"
+ VALIGN="TOP"
+ ROWSPAN="4">
+
+<B>Links</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/index.html">Home</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/cutapplet.html">Applet Home</A><BR><BR><p>
+
+<B>Java Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND.html">ND</A><BR><BR>
+
+<B>Plugin Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL-plugin.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT-plugin.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND-plugin.html">ND</A><BR>
+
+<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%">
+<A NAME="Home"></A>
+<H2>Cut-Elimination in Herbelin's calculus for intuitionistic logic</H2>
+
+Recommended reading for this applet is a paper by Dyckhoff and Urban,
+[<A HREF="http://www-theory.dcs.st-and.ac.uk/~rd/publications/MJES/DyckhoffUrbanWestApp01.rev.ps">ps</A>,
+ <A HREF="http://www-theory.dcs.st-and.ac.uk/~rd/publications/MJES/DyckhoffUrbanWestApp01.rev.ps">pdf</A>].
+Mayor difference between the standard sequent calculus and Herbelin's sequent calculus
+is that in the cut-free fragment "there are no semantically
+trivial permutations of the inference rules, where by 'semantically trivial' we mean
+'interpreted in NJ as true equations between to (normal) deductions'"
+(from [<A HREF="http://www-theory.dcs.st-and.ac.uk/~rd/publications/SLPaper.corr.ps">ps</A>]).
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>How to use it?</B>
+If your browser is working correctly, then you will see a window
+with two examples (more added if you suggest some). When pressing on
+one of the buttons, a new window will pop up. What follows
+is a brief explanation of all the features available in this window.
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>Buttons and Keys</B>
+<UL>
+<LI> The buttons
+ <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/x.jpg">
+ `fire' cuts (not all cuts necessarily have such buttons).
+<LI> If you use the <B>left mouse button</B> for pressing on those buttons,
+ then the new proof will appear inside the window. Whereas if you use
+ the <B>right mouse button</B>, a new window will pop up and
+ the new proof will be displayed in this window. Use the right mouse
+ button if you want to compare a proof and its reduct.
+<LI> The keys <B>Page-up</B> and <B>Page-down</B> zoom in or out of a proof
+ respectively.
+</UL>
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>Menu Functions</B>
+
+<UL>
+<LI> <B>Zoom in</B> and <B>Zoom out</B> work like Page-up and Page-down,
+ respectively.
+<LI> <B>Unicode</B> If this radiobutton is switched on, logic symbols
+ are displayed in unicode, otherwise in ascii. (This is for the
+ poor guys who have a browser and/or operating system which cannot handle
+ unicode.)
+<LI> <B>Labels</B> As mentioned earlier the sequents are composed of
+ two sets of labelled
+ formulae, not multisets of formulae as in the standard formulation
+ of sequent calculus. Enabling this radiobutton causes that labels are
+ drawn. Use this when you are unsure where implicit contractions are.
+<LI> <B>Compact printing</B> In the paper referred above weakening
+ is done only in axioms. If this radiobutton is switched on (default),
+ then, in order to save space, formulae are weakened as soon as possible.
+</UL>
+</TD>
+</TR>
+
+</TABLE>
+<center>
+<!--"CONVERTED_APPLET"-->
+<!-- HTML CONVERTER -->
+<OBJECT classid="clsid:CAFEEFAC-0013-0001-0000-ABCDEFFEDCBA"
+WIDTH = 1 HEIGHT = 1 codebase="http://java.sun.com/products/plugin/1.3.1/jinstall-131-win32.cab#Version=1,3,1,0">
+<PARAM NAME = CODE VALUE = "MyApplet.class" >
+<PARAM NAME = ARCHIVE VALUE = MyApplet.zip >
+
+<PARAM NAME="type" VALUE="application/x-java-applet;jpi-version=1.3.1">
+<PARAM NAME="scriptable" VALUE="false">
+<PARAM NAME = "calculi" VALUE ="B">
+<COMMENT>
+<EMBED type="application/x-java-applet;jpi-version=1.3.1" CODE = "Cut/MyApplet.class" ARCHIVE = MyApplet.zip WIDTH = 1 HEIGHT = 1 calculi = "B" scriptable=false pluginspage="http://java.sun.com/products/plugin/1.3.1/plugin-install.html"><NOEMBED>
+<p><blink><FONT COLOR="#800000">
+Applet should appear in a new window, but you don't have Java enabled in your browser.
+</FONT></blink><p>
+</NOEMBED>
+</EMBED>
+</COMMENT>
+</OBJECT>
+
+<!--
+<APPLET CODE = "Cut/MyApplet.class" ARCHIVE = MyApplet.zip WIDTH = 1 HEIGHT = 1>
+<PARAM NAME = "calculi" VALUE ="B">
+<p><blink><FONT COLOR="#800000">
+Applet should appear in a new window, but you don't have Java enabled in your browser.
+</FONT></blink><p>
+
+</APPLET>
+-->
+
+
+<!--"END_CONVERTED_APPLET"-->
+
+
+</center>
+<P><!-- Created: Tue Jul 3 21:01:42 BST 2001 -->
+<!-- hhmts start -->
+Last modified: Sat Mar 3 05:17:20 CET 2007
+<!-- hhmts end -->
+</BODY>
+</HTML>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Cut/LJT.html Thu Mar 15 15:50:18 2012 +0000
@@ -0,0 +1,134 @@
+<HTML lang=en>
+<HEAD>
+<TITLE> Christian Urban </TITLE>
+<BASE HREF="http://www4.in.tum.de/~urbanc/Cut/">
+</HEAD>
+
+<BODY TEXT="#000000"
+ BGCOLOR="#000080"
+ LINK="#0000EF"
+ VLINK="#51188E"
+ ALINK="#FF0000">
+
+<TABLE WIDTH="100%"
+ COLS="2"
+ BGCOLOR="#000080"
+ BORDER="0"
+ FRAME="none"
+ CELLPADDING="10"
+ CELLSPACING="2"
+ RULES="COLS,ROWS">
+
+<!-- left column -->
+<TR>
+<TD BGCOLOR="#FFFFFF"
+ WIDTH="24%"
+ VALIGN="TOP"
+ ROWSPAN="4">
+
+<B>Links</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/index.html">Home</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/cutapplet.html">Applet Home</A><BR><BR><p>
+
+<B>Java Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND.html">ND</A><BR><BR>
+
+<B>Plugin Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL-plugin.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT-plugin.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND-plugin.html">ND</A><BR>
+
+<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%">
+<A NAME="Home"></A>
+<H2>Cut-Elimination in Herbelin's calculus for intuitionistic logic</H2>
+
+Recommended reading for this applet is a paper by Dyckhoff and Urban,
+[<A HREF="http://www-theory.dcs.st-and.ac.uk/~rd/publications/MJES/DyckhoffUrbanWestApp01.rev.ps">ps</A>,
+ <A HREF="http://www-theory.dcs.st-and.ac.uk/~rd/publications/MJES/DyckhoffUrbanWestApp01.rev.pdf">pdf</A>].
+Mayor difference between the standard sequent calculus and Herbelin's sequent calculus
+is that in the cut-free fragment "there are no semantically
+trivial permutations of the inference rules, where by 'semantically trivial' we mean
+'interpreted in NJ as true equations between to (normal) deductions'"
+(from [<A HREF="http://www-theory.dcs.st-and.ac.uk/~rd/publications/SLPaper.corr.ps">ps</A>]).
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>How to use it?</B>
+If your browser is working correctly, then you will see a window
+with two examples (more added if you suggest some). When pressing on
+one of the buttons, a new window will pop up. What follows
+is a brief explanation of all the features available in this window.
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>Buttons and Keys</B>
+<UL>
+<LI> The buttons
+ <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/x.jpg">
+ `fire' cuts (not all cuts necessarily have such buttons).
+<LI> If you use the <B>left mouse button</B> for pressing on those buttons,
+ then the new proof will appear inside the window. Whereas if you use
+ the <B>right mouse button</B>, a new window will pop up and
+ the new proof will be displayed in this window. Use the right mouse
+ button if you want to compare a proof and its reduct.
+<LI> The keys <B>Page-up</B> and <B>Page-down</B> zoom in or out of a proof
+ respectively.
+</UL>
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>Menu Functions</B>
+
+<UL>
+<LI> <B>Zoom in</B> and <B>Zoom out</B> work like Page-up and Page-down,
+ respectively.
+<LI> <B>Unicode</B> If this radiobutton is switched on, logic symbols
+ are displayed in unicode, otherwise in ascii. (This is for the
+ poor guys who have a browser and/or operating system which cannot handle
+ unicode.)
+<LI> <B>Labels</B> As mentioned earlier the sequents are composed of
+ two sets of labelled
+ formulae, not multisets of formulae as in the standard formulation
+ of sequent calculus. Enabling this radiobutton causes that labels are
+ drawn. Use this when you are unsure where implicit contractions are.
+<LI> <B>Compact printing</B> In the paper referred above weakening
+ is done only in axioms. If this radiobutton is switched on (default),
+ then, in order to save space, formulae are weakened as soon as possible.
+</UL>
+</TD>
+</TR>
+
+</TABLE>
+<center>
+<applet archive=MyApplet.zip code="MyApplet.class" width=1 height=1>
+<param name="calculi" value="B">
+<p><blink><FONT COLOR="#800000">
+Applet should appear in a new window, but you don't have Java enabled in your browser.
+</FONT></blink><p>
+</applet>
+
+</center>
+<P><!-- Created: Tue Jul 3 21:01:42 BST 2001 -->
+<!-- hhmts start -->
+Last modified: Sat Mar 3 05:17:09 CET 2007
+<!-- hhmts end -->
+</BODY>
+</HTML>
Binary file Cut/Main.zip has changed
Binary file Cut/MyApplet.zip has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Cut/ND-plugin.html Thu Mar 15 15:50:18 2012 +0000
@@ -0,0 +1,157 @@
+<HTML lang=en>
+<HEAD>
+<TITLE> Christian Urban </TITLE>
+<BASE HREF="http://www4.in.tum.de/~urbanc/Cut/">
+</HEAD>
+
+<BODY TEXT="#000000"
+ BGCOLOR="#000080"
+ LINK="#0000EF"
+ VLINK="#51188E"
+ ALINK="#FF0000">
+
+<TABLE WIDTH="100%"
+ COLS="2"
+ BGCOLOR="#000080"
+ BORDER="0"
+ FRAME="none"
+ CELLPADDING="10"
+ CELLSPACING="2"
+ RULES="COLS,ROWS">
+
+<!-- left column -->
+<TR>
+<TD BGCOLOR="#FFFFFF"
+ WIDTH="24%"
+ VALIGN="TOP"
+ ROWSPAN="4">
+
+<B>Links</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/index.html">Home</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/cutapplet.html">Applet Home</A><BR><BR><p>
+
+<B>Java Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND.html">ND</A><BR><BR>
+
+<B>Plugin Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL-plugin.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT-plugin.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND-plugin.html">ND</A><BR>
+
+<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%">
+<A NAME="Home"></A>
+<H2>Normalisation in intuitionistic natural deduction</H2>
+
+This applet implements the bog-standard version of intuitionistic
+natural deduction. If you are familiar with any textbook on proof
+theory (e.g. this <A HREF="http://www.amazon.co.uk/exec/obidos/ASIN/0521572231/ref=ed_ra_of_dp/026-0727529-0066055">one</A>), then most things will look very familiar to you.
+Note that natural deduction is here implemented in `sequent style'.
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>How to use it?</B>
+If your browser is working correctly, then you will see a window
+with two examples (more added if I have time or you suggest one).
+When you press on one of the buttons, a new window will pop up. What
+follows is a brief explanation of all the features available in this window.
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>Buttons and Keys</B>
+<UL>
+<LI> The buttons
+ <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/x.jpg">
+ `fire' beta-redexes (nothing surprising here).
+<LI> If you use the <B>left mouse button</B> for pressing on those buttons,
+ then the new proof will appear inside the window. Whereas if you use
+ the <B>right mouse button</B>, a new window will pop up and
+ the new proof will be displayed in this window. Use the right mouse
+ button if you want to compare a proof and its reduct.
+<LI> The keys <B>Page-up</B> and <B>Page-down</B> zoom in or out of a proof
+ respectively.
+</UL>
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>Menu Functions</B>
+
+<UL>
+<LI> <B>Zoom in</B> and <B>Zoom out</B> work like Page-up and Page-down,
+ respectively.
+<LI> <B>Unicode</B> If this radiobutton is switched on, logic symbols
+ are displayed in unicode, otherwise in ascii. (This is for the
+ poor guys who have a browser and/or operating system which cannot handle
+ unicode.)
+<LI> <B>Labels</B> As mentioned earlier the sequents are composed of
+ two sets of labelled
+ formulae, not multisets of formulae as in the standard formulation
+ of sequent calculus. Enabling this radiobutton causes that labels are
+ drawn. Use this when you are unsure where implicit contractions are.
+<LI> <B>Compact printing</B> Usually formulae are weakened only in axiom.
+ If this radiobutton is switched on (default), then, in order to save space,
+ formulae are weakened as soon as possible.
+</UL>
+</TD>
+</TR>
+
+</TABLE>
+<center>
+<!--"CONVERTED_APPLET"-->
+<!-- HTML CONVERTER -->
+<OBJECT classid="clsid:CAFEEFAC-0013-0001-0000-ABCDEFFEDCBA"
+WIDTH = 1 HEIGHT = 1 codebase="http://java.sun.com/products/plugin/1.3.1/jinstall-131-win32.cab#Version=1,3,1,0">
+<PARAM NAME = CODE VALUE = "MyApplet.class" >
+<PARAM NAME = ARCHIVE VALUE = MyApplet.zip >
+
+<PARAM NAME="type" VALUE="application/x-java-applet;jpi-version=1.3.1">
+<PARAM NAME="scriptable" VALUE="false">
+<PARAM NAME = "calculi" VALUE ="C">
+<COMMENT>
+<EMBED type="application/x-java-applet;jpi-version=1.3.1" CODE = "MyApplet.class" ARCHIVE = MyApplet.zip WIDTH = 1 HEIGHT = 1 calculi = "C" scriptable=false pluginspage="http://java.sun.com/products/plugin/1.3.1/plugin-install.html"><NOEMBED>
+<p><blink><FONT COLOR="#800000">
+Applet should appear in a new window, but you don't have Java enabled in your browser.
+</FONT></blink><p>
+</NOEMBED>
+</EMBED>
+</COMMENT>
+</OBJECT>
+
+<!--
+<APPLET CODE = "MyApplet.class" ARCHIVE = MyApplet.zip WIDTH = 1 HEIGHT = 1>
+<PARAM NAME = "calculi" VALUE ="C">
+<p><blink><FONT COLOR="#800000">
+Applet should appear in a new window, but you don't have Java enabled in your browser.
+</FONT></blink><p>
+
+</APPLET>
+-->
+
+
+<!--"END_CONVERTED_APPLET"-->
+
+
+</center>
+<P><!-- Created: Tue Jul 3 21:01:42 BST 2001 -->
+<!-- hhmts start -->
+Last modified: Sat Mar 3 05:16:58 CET 2007
+<!-- hhmts end -->
+</BODY>
+</HTML>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Cut/ND.html Thu Mar 15 15:50:18 2012 +0000
@@ -0,0 +1,130 @@
+<HTML lang=en>
+<HEAD>
+<TITLE> Christian Urban </TITLE>
+<BASE HREF="http://www4.in.tum.de/~urbanc/Cut/">
+</HEAD>
+
+<BODY TEXT="#000000"
+ BGCOLOR="#000080"
+ LINK="#0000EF"
+ VLINK="#51188E"
+ ALINK="#FF0000">
+
+<TABLE WIDTH="100%"
+ COLS="2"
+ BGCOLOR="#000080"
+ BORDER="0"
+ FRAME="none"
+ CELLPADDING="10"
+ CELLSPACING="2"
+ RULES="COLS,ROWS">
+
+<!-- left column -->
+<TR>
+<TD BGCOLOR="#FFFFFF"
+ WIDTH="24%"
+ VALIGN="TOP"
+ ROWSPAN="4">
+
+<B>Links</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/index.html">Home</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/cutapplet.html">Applet Home</A><BR><BR><p>
+
+<B>Java Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND.html">ND</A><BR><BR>
+
+<B>Plugin Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL-plugin.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT-plugin.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND-plugin.html">ND</A><BR>
+
+<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%">
+<A NAME="Home"></A>
+<H2>Normalisation in intuitionistic natural deduction</H2>
+
+This applet implements the bog-standard version of intuitionistic
+natural deduction. If you are familiar with any textbook on proof
+theory (e.g. this <A HREF="http://www.amazon.co.uk/exec/obidos/ASIN/0521572231/ref=ed_ra_of_dp/026-0727529-0066055">one</A>), then most things will look very familiar to you.
+Note that natural deduction is here implemented in `sequent style'.
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>How to use it?</B>
+If your browser is working correctly, then you will see a window
+with two examples (more added if I have time or you suggest one).
+When you press on one of the buttons, a new window will pop up. What
+follows is a brief explanation of all the features available in this window.
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>Buttons and Keys</B>
+<UL>
+<LI> The buttons
+ <IMG SRC="http://www4.in.tum.de/~urbanc/Cut/x.jpg">
+ `fire' beta-redexes (nothing surprising here).
+<LI> If you use the <B>left mouse button</B> for pressing on those buttons,
+ then the new proof will appear inside the window. Whereas if you use
+ the <B>right mouse button</B>, a new window will pop up and
+ the new proof will be displayed in this window. Use the right mouse
+ button if you want to compare a proof and its reduct.
+<LI> The keys <B>Page-up</B> and <B>Page-down</B> zoom in or out of a proof
+ respectively.
+</UL>
+</TD>
+</TR>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+<B>Menu Functions</B>
+
+<UL>
+<LI> <B>Zoom in</B> and <B>Zoom out</B> work like Page-up and Page-down,
+ respectively.
+<LI> <B>Unicode</B> If this radiobutton is switched on, logic symbols
+ are displayed in unicode, otherwise in ascii. (This is for the
+ poor guys who have a browser and/or operating system which cannot handle
+ unicode.)
+<LI> <B>Labels</B> As mentioned earlier the sequents are composed of
+ two sets of labelled
+ formulae, not multisets of formulae as in the standard formulation
+ of sequent calculus. Enabling this radiobutton causes that labels are
+ drawn. Use this when you are unsure where implicit contractions are.
+<LI> <B>Compact printing</B> Usually formulae are weakened only in axiom.
+ If this radiobutton is switched on (default), then, in order to save space,
+ formulae are weakened as soon as possible.
+</UL>
+</TD>
+</TR>
+
+</TABLE>
+<center>
+<applet archive=MyApplet.zip code="MyApplet.class" width=1 height=1>
+<param name="calculi" value="C">
+<p><blink><FONT COLOR="#800000">
+Applet should appear in a new window, but you don't have Java enabled in your browser.
+</FONT></blink><p>
+</applet>
+
+</center>
+<P><!-- Created: Tue Jul 3 21:01:42 BST 2001 -->
+<!-- hhmts start -->
+Last modified: Sat Mar 3 05:16:48 CET 2007
+<!-- hhmts end -->
+</BODY>
+</HTML>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Cut/START.txt Thu Mar 15 15:50:18 2012 +0000
@@ -0,0 +1,1 @@
+java -classpath Main.zip Main
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Cut/cutapplet.html Thu Mar 15 15:50:18 2012 +0000
@@ -0,0 +1,137 @@
+<HTML lang=en>
+<HEAD>
+<TITLE> Christian Urban </TITLE>
+<BASE HREF="http://www4.in.tum.de/~urbanc/">
+</HEAD>
+
+<BODY TEXT="#000000"
+ BGCOLOR="#000080"
+ LINK="#0000EF"
+ VLINK="#51188E"
+ ALINK="#FF0000">
+
+<TABLE WIDTH="100%"
+ COLS="2"
+ BGCOLOR="#000080"
+ BORDER="0"
+ FRAME="none"
+ CELLPADDING="10"
+ CELLSPACING="2"
+ RULES="COLS,ROWS">
+
+<!-- left column -->
+<TR>
+<TD BGCOLOR="#FFFFFF"
+ WIDTH="24%"
+ VALIGN="TOP"
+ ROWSPAN="2">
+
+<B>Links</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/index.html">Home</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/cutapplet.html">Applet Home</A><BR><BR>
+
+<B>Java Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND.html">ND</A><BR><BR>
+
+<B>Plugin Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL-plugin.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT-plugin.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND-plugin.html">ND</A><BR>
+
+<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%">
+<A NAME="Home"></A>
+<H2>Applet for Cut-Elimination and Normalisation </H2>
+
+<FONT COLOR="#800000"><B>Warning:</B></FONT>
+These pages are still under construction and designed for the conoscenti
+only. If you encounter any problems, email me. Some general information
+about the applet is given below.<p>
+
+The applet is outcome of my interest in cut-elimination. It helps to
+explore reduction trees in a slight variant of the sequent calculus LK,
+and also reduction trees in Herbelin's calculus LJT (as presented
+<A HREF="http://www-theory.dcs.st-and.ac.uk/~rd/publications/MJES/">here</A>)
+and in the standard formulation of natural deduction for intuitionistic
+logic. Given for example the following classical sequent proof<p>
+
+<center>
+<IMG SRC="Cut/ex1.jpg" ALT="" ALIGN=CENTER>
+</center><p>
+
+you can produce by simply pressing the buttons L and R the following
+two cut-free proofs.<p>
+
+<center>
+<IMG SRC="Cut/ex2.jpg" ALT="" ALIGN=CENTER>
+<IMG SRC="Cut/ex3.jpg" ALT="" ALIGN=CENTER>
+</center><p>
+
+If you ever have done cut-elimination by hand, then you
+know how useful this applet is.<p>
+
+<img ALIGN=BOTTOM ALT="" SRC="new.gif"> The applet is particularly useful for doing
+some calculations finding out what the (proof-theoretical) semantics for classical logic
+should look like. Some informal notes are
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/notes.html">here</A>.
+<p>
+
+
+The applet needs at least <B>Java 1.2</B> including the
+<B>Swing</B> libraries. Some browsers support these requirements
+directly, and if you are one of the lucky to have such a browser
+you can access the three calculi by pressing on one of the
+following links.
+
+<UL>
+<LI><A HREF="http://www4.in.tum.de/~urbanc/Cut/CL.html">CL - cut-elimination
+in classical logic</A>
+<LI><A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT.html">LJT - cut-elimination in
+intuitionistic logic</A>
+<LI><A HREF="http://www4.in.tum.de/~urbanc/Cut/ND.html">ND - normalisation in
+intuitionistic natural deduction</A><p>
+</UL><p>
+
+If not, then your browser can most probably run the applet with
+the help of a Java plugin. In this case follow the links below.
+
+<UL>
+<LI><A HREF="http://www4.in.tum.de/~urbanc/Cut/CL-plugin.html">CL - cut-elimination
+in classical logic</A>
+<LI><A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT-plugin.html">LJT - cut-elimination in
+intuitionistic logic</A>
+<LI><A HREF="http://www4.in.tum.de/~urbanc/Cut/ND-plugin.html">ND - normalisation in
+intuitionistic natural deduction</A><p>
+</UL><p>
+
+<TR>
+<TD BGCOLOR="#FFFFFF" WIDTH="75%">
+For the curious: the applet is written in MLJ, which is being
+developed by
+<A HREF="http://research.microsoft.com/~nick/">Benton</A>,
+<A HREF="http://research.microsoft.com/~akenn/">Kennedy</A> and
+<A HREF="http://research.microsoft.com/~crusso/">Russo</A>.
+MLJ is a dialect of SML that provides access to Java libraries; in my
+opinion it is a really nifty language! I am using an improved version of
+the MLJ-0.2 compiler.
+</TD>
+</TR>
+
+</TABLE>
+<P><!-- Created: Tue Jul 3 21:01:42 BST 2001 -->
+<!-- hhmts start -->
+Last modified: Sat Mar 3 05:16:36 CET 2007
+<!-- hhmts end -->
+</BODY>
+</HTML>
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Cut/notes.html Thu Mar 15 15:50:18 2012 +0000
@@ -0,0 +1,91 @@
+<HTML lang=en>
+<HEAD>
+<TITLE> Christian Urban </TITLE>
+<BASE HREF="http://www4.in.tum.de/~urbanc/">
+</HEAD>
+
+<BODY TEXT="#000000"
+ BGCOLOR="#000080"
+ LINK="#0000EF"
+ VLINK="#51188E"
+ ALINK="#FF0000">
+
+<TABLE WIDTH="100%"
+ COLS="2"
+ BGCOLOR="#000080"
+ BORDER="0"
+ FRAME="none"
+ CELLPADDING="10"
+ CELLSPACING="2"
+ RULES="COLS,ROWS">
+
+<!-- left column -->
+<TR>
+<TD BGCOLOR="#FFFFFF"
+ WIDTH="24%"
+ VALIGN="TOP"
+ ROWSPAN="2">
+
+<B>Links</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/index.html">Home</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/cutapplet.html">Applet Home</A><BR><BR>
+
+<B>Java Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND.html">ND</A><BR><BR>
+
+<B>Plugin Versions</B><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/CL-plugin.html">CL</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/LJT-plugin.html">LJT</A><BR>
+<A HREF="http://www4.in.tum.de/~urbanc/Cut/ND-plugin.html">ND</A><BR>
+
+<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%">
+
+<H2>What is the Semantics for Classical Logic?</H2>
+
+In the intuitionistic case, the answer to this question is well-understood -
+essentially a semantics is given to all proofs in normal form and all other
+proofs inherit this semantics from the normal forms they reduce to. This
+implies that:
+
+<blockquote>
+(1) reductions preserving the semantics of a proof.
+</blockquote>
+
+
+It is widely known that this assumption causes any reasonable semantics
+for classical logic to collaps. In the literature Lafont's example is
+often cited for this observation.<p>
+
+<center>
+<IMG SRC="Cut/lafont1.jpg" ALT="" ALIGN=CENTER>
+</center><p>
+
+
+<center>
+<IMG SRC="Cut/lafont2.jpg" ALT="" ALIGN=CENTER>
+<IMG SRC="Cut/lafont3.jpg" ALT="" ALIGN=CENTER>
+</center><p>
+
+
+</TD>
+</TR>
+
+</TABLE>
+<P><!-- Created: Tue Jul 3 21:01:42 BST 2001 -->
+<!-- hhmts start -->
+Last modified: Sat Mar 3 05:17:53 CET 2007
+<!-- hhmts end -->
+</BODY>
+</HTML>
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