public_html: Nominal/example.html@f365ce636294 (annotated)

402 9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1	<?xml version="1.0" encoding="utf-8"?>
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9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	3	<html>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	4	<head>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	5	<title>Nominal Methods Group</title>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	6	<link rel="stylesheet" href="nominal.css">
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	7	</head>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	8
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	9	<body>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	10
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	11	<div align="right" style="position:relative; left:15%; width:80%">
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	12	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	13	<small>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	14	<SCRIPT LANGUAGE="JAVASCRIPT" type="text/javascript">
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	15	<!--
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	16	var r_text = new Array ();
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	17	r_text[0] = "<em>\"Proving theorems about substitutions (and related operations such as alpha-conversion) required far more time and HOL code than any other variety of theorem.\"<br><\/em>M. VanInwegen using a concrete representation for binders in her PhD-thesis, 1996";
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	18
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	19	r_text[1] = "<em>\"When doing the formalization, I discovered that the core part of the proof... is fairly straightforward and only requires a good understanding of the paper version. However, in completing the proof I observed that in certain places I had to invest much more work than expected, e.g. proving lemmas about substitution and weakening.\"<\/em><br>T. Altenkirch using de Bruijn indices in Proc. of TLCA, 1993";
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	20
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	21	r_text[2] = "<em>\"Technical work, however, still represents the biggest part of our implementation, mainly due to the managing of de Bruijn indexes...Of our 800 proved lemmas, about 600 are concerned with operators on free names.\"<\/em><br>D. Hirschkoff in Proc. of TPHOLs, 1997";
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	22
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	23	r_text[3] = "<em>\"It took the author many long months to complete the work on this formalization...The part concerning substitution is by far the largest part of the whole development.\"<\/em><br>A. Koprowski using de Bruijn indices in a draft paper, 2006";
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	24
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	25	r_text[4] = "<em>\"We thank T. Thacher Robinson for showing us on August 19, 1962 by a counterexample the existence of an error in our handling of bound variables.\"<\/em><br>S. Kleene in J. of Symbolic Logic 21(1):11-18, 1962";
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	26
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	27	r_text[5] = "<em>\"The main drawback in HOAS is the difficulty of dealing with metatheoretic issues concerning names in processes...As a consequence, some metatheoretic properties involving substitution and freshness of names inside proofs and processes cannot be proved inside the framework and instead have to be postulated.\"<\/em><br>F. Honsell, M. Miculan and I. Scagnetto in Theoretical Computer Science, 253(2):239-285, 2001";
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	28
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	29	r_text[6] = "<em>\"Because Twelf metatheorems are proved using totality assertions about LF type families, the class of metatheorems that can be mechanized is restricted to All/Exists-statements over LF types. On the one hand, as the successful Twelf formalizations cited in Section 5 demonstrate, these All/Exists-statements have proved to be sufficient for formalizing a wide variety of metatheorems about programming languages and logics. On the other hand, we have no way to quantify when metatheorems of this form will be sufficient, and there are some well-known examples of proofs that cannot be formalized directly using Twelf as metatheorem language. For example, proofs by logical relations often require more quantifier complexity than All/Exists-statements afford.\"<\/em><br>Robert Harper and Daniel Licata in a paper on Twelf, 2007";
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	30
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	31	r_text[7] = "<em>\"So we cannot, hand-on-heart, recommend the vanilla LN style for anything but small, kernel language developments. \"<\/em><br>in F-ing Modules by Rossberg, Russo and Dreyer, TLDI 2010";
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	32
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	33	r_text[8] = "<em>\"Higher-order abstract syntax is a convenient way to approach languages with binding, but it is possible to imagine a problem where manipulating a fully concrete object without binding is simpler. In these cases, it is possible to establish a bijection between your HOAS terms and de Bruijn versions of the same terms. \"<\/em><br>Interesting responses from the <A HREF=\"http://twelf.plparty.org/wiki/Ask_Twelf_Elf\">Twelf wiki.</A> (To be honest, the same comment applies to Nominal. --cu)";
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	34
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	35
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	36	var i = Math.floor(r_text.length*Math.random());
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	37	document.write(r_text[i]);
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	38	//-->
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	39	</SCRIPT>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	40	</small>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	41	</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	42	</div>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	43
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	44
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	45	<H1>Barendregt's Substitution Lemma</H1>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	46
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	47	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	48	Let us explain one of our results with a simple proof about the lambda calculus.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	49	An informal "pencil-and-paper" proof there looks typically as follows (this one is taken from <A
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	50	HREF="http://www.cs.ru.nl/~henk/" target="_top">Barendregt's</A> classic book
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	51	on the lambda calculus):
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	52	</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	53
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	54	<!-- Barendregt's proof -->
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	55	<CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	56	<TABLE style="text-align: left; width: 90%;" BORDER=0 CELLSPACING=0 CELLPADDING=5>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	57	<TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	58	<TD style="background-color: rgb(180, 180, 180);">
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	59	<B>2.1.16. Substitution Lemma:</B> If <I>x≠y</I> and <I>x</I> not free in <I>L</I>,
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	60	then
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	61	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	62	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	63	<TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	64	<TD style="background-color: rgb(180, 180, 180);">
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	65	<CENTER><I>M[x:=N][y:=L] = M[y:=L][x:=N[y:=L]]</I>.</CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	66	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	67	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	68	<TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	69	<TD style="background-color: rgb(210, 210, 210);">
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	70	<B>Proof:</B> By induction on the structure of <I>M</I>.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	71	<DL>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	72	<DT>Case 1. <I>M</I> is a variable.<DD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	73	<DL>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	74	<DT>Case 1.1. <I>M=x</I>. Then both sides equal <I>N[y:=L]</I> since <I>x≠y</I>.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	75	<DT>Case 1.2. <I>M=y</I>. Then both sides equal <I>L</I>, for <I>x</I> not free in <I>L</I>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	76	implies <I>L[x:=...]=L</I>.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	77	<DT>Case 1.3. <I>M=z≠x,y</I>. Then both sides equal <I>z</I>.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	78	</DL></DD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	79	<DT>Case 2. <I>M=λz.M<SUB>1</SUB></I>. <DD>By the variable convention we may assume that
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	80	<I>z≠x,y</I> and <I>z</I> is not free in <I>N</I>, <I>L</I>. Then by the induction hypothesis<BR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	81	<CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	82	<TABLE>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	83	<TR><TD ALIGN=RIGHT><I>(λz.M<SUB>1</SUB>)[x:=N][y:=L]</I></TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	84	<TD ALIGN=CENTER><I>=</I></TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	85	<TD ALIGN=Left><I>λz.M<SUB>1</SUB>[x:=N][y:=L]</I></TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	86	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	87	<TR><TD ALIGN=RIGHT> </TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	88	<TD ALIGN=CENTER><I>=</I></TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	89	<TD ALIGN=Left><I>λz.M<SUB>1</SUB>[y:=L][x:=N[y:=L]]</I></TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	90	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	91	<TR><TD ALIGN=RIGHT> </TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	92	<TD ALIGN=CENTER><I>=</I></TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	93	<TD ALIGN=Left><I>(λz.M<SUB>1</SUB>)[y:=L][x:=N[y:=L]]</I>.</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	94	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	95	</TABLE>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	96	</CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	97	<DT>Case 3. <I>M=M<SUB>1</SUB> M<SUB>2</SUB></I>.<DD>Then the statement follows
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	98	again from the induction hypothesis.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	99	</DL>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	100	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	101	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	102	</TABLE>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	103	</CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	104
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	105	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	106	We want to make it as easy as possible to formalise such informal proofs (and
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	107	more complicated ones). Inspired by the <A
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	108	HREF="http://fling-l.seas.upenn.edu/~plclub/cgi-bin/poplmark/"
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	109	target="_top">PoplMark Challenge</A>, we want that masses use theorem
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	110	assistants to do their formal proofs.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	111	</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	112
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	113	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	114	Since the kind of informal reasoning illustrated by Barendregt's proof is very
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	115	common in the literature on programming languages, it might be surprising that
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	116	implementing his proof
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	117	in a theorem assistant is not a trivial task. This is because he relies
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	118	implicitly on some assumptions and conventions. For example he states in his
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	119	book:
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	120	</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	121
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	122	<CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	123	<TABLE style="text-align: left; width: 90%;" BORDER=0 CELLSPACING=0 CELLPADDING=5>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	124	<TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	125	<TD style="background-color: rgb(180, 180, 180);">
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	126	<B>2.1.12. Convention.</B> Terms that are α-congruent are identified. So now we
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	127	write <I>λx.x=λy.y</I>, etcetera.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	128	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	129	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	130	</TABLE>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	131	</CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	132
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	133	<CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	134	<TABLE style="text-align: left; width: 90%;" BORDER=0 CELLSPACING=0 CELLPADDING=5>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	135	<TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	136	<TD style="background-color: rgb(180, 180, 180);">
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	137	<B>2.1.13. Variable Convention.</B> If <I>M<SUB>1</SUB>,...,M<SUB>n</SUB></I> occur
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	138	in a certain mathematical context (e.g. definition, proof), then in these terms all
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	139	bound variables are chosen to be different from the free variables.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	140	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	141	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	142	</TABLE>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	143	</CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	144
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	145	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	146	The first convention is crucial for the proof above as it allows one to deal
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	147	with the variable case by using equational reasoning - one can just calculate
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	148	what the results of the substitutions are. If one uses un-equated, or raw, lambda-terms,
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	149	the same kind of reasoning cannot be performed (the reasoning then has to be
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	150	modulo α-equivalence, which causes a lot of headaches in
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	151	the lambda-case.) But if the data-structure over which the proof is
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	152	formulated is α-equivalence classes of lambda-terms, then what is the
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	153	principle "by induction over the structure of <I>M</I>"? There is an
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	154	induction principle "over the structure" for (un-equated) lambda-terms. But
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	155	quotening lambda-terms by α-equivalence does not automatically lead to
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	156	such a principle for α-equivalence classes. This seems to be a point
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	157	that is nearly always ignored in the literature. In fact it takes, as we have
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	158	shown in [1] and [2], some serious work to provide such an induction principle
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	159	for α-equivalence classes.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	160	</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	161
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	162	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	163	The second problem for an implementation of Barendregt's proof is his use of
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	164	the variable convention: there is just no proof-principle "by convention" in a
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	165	theorem assistant. Taking a closer look at Barendregt's reasoning, it turns
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	166	out that for a proof obligation of the form "for all α-equated
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	167	lambda-terms <I>λz.M<SUB>1</SUB></I>...", he does not establish this
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	168	proof obligation for all <I>λz.M<SUB>1</SUB></I>, but only for some
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	169	carefully chosen α-equated lambda-terms, namely the ones for which
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	170	<I>z</I> is not free in <I>x,y,N</I> and <I>L</I>. This style of reasoning
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	171	clearly needs some justification and in fact depends on some assumptions of
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	172	the "context" of the induction. By "context" of the induction we mean the
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	173	variables <I>x,y,N</I> and <I>L</I>. When employing the variable convention in
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	174	a formal proof, one always implicitly assumes that one can choose a fresh name
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	175	for this context. This might, however, not always be possible, for example
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	176	when the context already mentions all names. Also we found out recently that the
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	177	use of the variable convention in proofs by rule-induction can lead to
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	178	faulty reasoning [5]. So our work introduces safeguards that ensure that the
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	179	use of the variable convention is always safe.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	180	</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	181
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	182	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	183	One might conclude from our comments about Barendregt's proof that it is no
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	184	proof at all. This is, however, not the case! With Nominal Isabelle
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	185	and its infrastructure one can easily formalise his reasoning. One first
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	186	has to declare the structure of <U>α-equated</U>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	187	lambda-terms as a nominal datatype:
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	188	</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	189
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	190
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	191	<div class="codedisplay"> atom_decl name
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	192	nominal_datatype term = Var "name"
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	193	\| App "term" "term"
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	194	\| Lam "«name»term"
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	195	</div>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	196
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	197	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	198	Note though, that nominal datatypes are not datatypes in the traditional
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	199	sense, but stand for α-equivalence classes. Indeed we have for terms of
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	200	type <code>term</code> the equation(!)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	201	</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	202
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	203	<div class="codedisplay"> lemma alpha: "Lam [a].(Var a) = Lam [b].(Var b)"
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	204	</div>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	205
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	206
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	207	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	208	which does not hold for traditional datatypes (note that we write
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	209	lambda-abstractions as <code>Lam [a].t</code>). The proof of the substitution
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	210	lemma can then be formalised as follows:
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	211	</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	212
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	213	<div class="codedisplay"> lemma substitution_lemma:
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	214	assumes asm: "x≠y" "x#L"
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	215	shows "M[x:=N][y:=L] = M[y:=L][x:=N[y:=L]]"
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	216	using asm
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	217	by (nominal_induct M avoiding: x y N L rule: term.induct)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	218	(auto simp add: forget fresh_fact)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	219	</div>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	220
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	221
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	222	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	223	where the assumption "<I>x</I> is fresh for <I>L</I>", written <code>x#L</code>,
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	224	encodes the usual relation of "<I>x</I> not free in <I>L</I>". The method
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	225	<code>nominal_induct</code> takes as arguments the variable over which the
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	226	induction is
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	227	performed (here <I>M</I>), and the context of the induction, which consists of
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	228	the variables mentioned in the variable convention (that is the part in
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	229	Barendregt's proof where he writes "...we may assume that <I>z≠x,y</I> and
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	230	<I>z</I> is not free in <I>N,L</I>"). The last argument of <code>nominal_induct</code>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	231	specifies which induction rule should be applied - in this case induction over
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	232	α-equated lambda-terms, an induction-principle Nominal Isabelle provides
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	233	automatically when the nominal datatype <code>term</code> is defined. The
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	234	implemented proof of the substitution lemma proceeds then completely
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	235	automatically, except for the need of having to mention the facts <code>forget</code> and
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	236	<code>fresh_fact</code>, which are proved separately (also by induction over
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	237	α-equated lambda-terms).</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	238
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	239
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	240	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	241	The lemma <code>forget</code> shows that if <I>x</I> is not
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	242	free in <I>L</I>, then <I>L[x:=...]=L</I> (Barendregt's Case 1.2). Its formalised proof
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	243	is as follows:
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	244	</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	245
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	246	<div class="codedisplay"> lemma forget:
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	247	assumes asm: "x#L"
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	248	shows "L[x:=P] = L"
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	249	using asm
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	250	by (nominal_induct L avoiding: x P rule: term.induct)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	251	(auto simp add: abs_fresh fresh_atm)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	252	</div>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	253
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	254
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	255	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	256	In this proof <code>abs_fresh</code> is an automatically generated lemma that
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	257	establishes when <I>x</I> is fresh for a lambda-abstraction, namely <I>x#Lam
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	258	[z].P'</I> if and only if <I>x=z</I> or (<I>x≠z</I> and <I>x#P'</I>);
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	259	<code>fresh_atm</code> states that <I>x#y</I> if and only if <I>x≠y</I>. The lemma
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	260	<code>fresh_fact</code> proves the property that if <I>z</I> does not occur
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	261	freely in <I>N</I> and <I>L</I> then it also does not occur freely in
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	262	<I>N[y:=L]</I>. This fact can be formalised as follows:</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	263
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	264	<div class="codedisplay"> lemma fresh_fact:
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	265	assumes asm: "z#N" "z#L"
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	266	shows "z#N[y:=L]"
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	267	using asm
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	268	by (nominal_induct N avoiding: z y L rule: term.induct)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	269	(auto simp add: abs_fresh fresh_atm)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	270	</div>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	271
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	272	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	273	Although the latter lemma does not appear explicitly in Barendregt's reasoning, it is required
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	274	in the last step of the lambda-case (Case 2) where he pulls the substitution from under
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	275	the binder <I>z</I> (the interesting step is marked with a •):</P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	276	<CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	277	<TABLE>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	278	<TR><TD> </TD><TD><I>λz.(M<SUB>1</SUB>[y:=L][x:=N[y:=L]])</I></TD><TD> </TD></TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	279	<TR><TD>=</TD><TD><I>(λz.M<SUB>1</SUB>[y:=L])[x:=N[y:=L]]</I></TD><TD>  •</TD></TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	280	<TR><TD>=</TD><TD><I>(λz.M<SUB>1</SUB>)[y:=L][x:=N[y:=L]]</I></TD><TD> </TD></TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	281	</TABLE>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	282	</CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	283
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	284	<P>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	285	After these 22 lines one has a completely formalised proof of the substitution
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	286	lemma. This proof does not rely on any axioms, apart from the ones on which
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	287	HOL is built.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	288	</P><BR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	289
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	290	<B>References</B><BR><BR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	291	<CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	292	<TABLE>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	293	<TR><TD WIDTH="7%" VALIGN=Top>[1]</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	294	<TD ALIGN=Left>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	295	<B>Nominal Reasoning Techniques in Isabelle/HOL.</B> In
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	296	Journal of Automatic Reasoning, 2008, Vol. 40(4), 327-356.
465 4dac76eb27d9 updated Christian Urban <urbanc@in.tum.de> parents: 411 diff changeset	297	[<A HREF="http://nms.kcl.ac.uk/christian.urban/Publications/nom-tech.ps" target="_top">ps</A>].
402 9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	298	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	299	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	300	<TR><TD VALIGN=Top>[2]</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	301	<TD ALIGN=Left>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	302	<B>A Formal Treatment of the Barendregt Variable Convention in Rule Inductions</B>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	303	(Christian Urban and Michael Norrish)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	304	Proceedings of the ACM Workshop on Mechanized Reasoning about Languages with Variable
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	305	Binding and Names (MERLIN 2005). Tallinn, Estonia. September 2005. Pages 25-32. &copy ACM, Inc.
465 4dac76eb27d9 updated Christian Urban <urbanc@in.tum.de> parents: 411 diff changeset	306	[<A HREF="http://nms.kcl.ac.uk/christian.urban/Publications/merlin-05.ps" target="_top">ps</A>]
4dac76eb27d9 updated Christian Urban <urbanc@in.tum.de> parents: 411 diff changeset	307	[<A HREF="http://nms.kcl.ac.uk/christian.urban/Publications/merlin-05.pdf" target="_top">pdf</A>]
402 9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	308	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	309	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	310	<TR><TD VALIGN=Top>[3]</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	311	<TD ALIGN=Left>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	312	<B>A Recursion Combinator for Nominal Datatypes Implemented in Isabelle/HOL</B>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	313	(Christian Urban and Stefan Berghofer)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	314	Proceedings of the 3rd
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	315	International Joint Conference on Automated Deduction (IJCAR 2006). In volume 4130 of
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	316	Lecture Notes in Artificial Intelligence. Seattle, USA. August 2006. Pages 498-512.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	317	&copy <A HREF="http://link.springer.de/link/service/series/0558/" target="_top">Springer Verlag</A>
465 4dac76eb27d9 updated Christian Urban <urbanc@in.tum.de> parents: 411 diff changeset	318	[<A HREF="http://nms.kcl.ac.uk/christian.urban/Publications/ijcar-06.ps" target="_top">ps</A>]
402 9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	319	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	320	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	321
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	322	<TR><TD VALIGN=Top>[4]</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	323	<TD ALIGN=Left>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	324	<B>A Head-to-Head Comparison of de Bruijn Indices and Names.</B>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	325	(Stefan Berghofer and Christian Urban)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	326	Proceedings of the International Workshop on Logical Frameworks and
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	327	Meta-Languages: Theory and Practice (LFMTP 2006). Seattle, USA. ENTCS. Pages 53-67.
465 4dac76eb27d9 updated Christian Urban <urbanc@in.tum.de> parents: 411 diff changeset	328	[<A HREF="http://nms.kcl.ac.uk/christian.urban/Publications/lfmtp-06.ps" target="_top">ps</A>]
402 9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	329	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	330	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	331
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	332	<TR><TD VALIGN=Top>[5]</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	333	<TD ALIGN=Left>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	334	<B>Barendregt's Variable Convention in Rule Inductions.</B> (Christian
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	335	Urban, Stefan Berghofer and Michael Norrish) Proceedings of the 21th
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	336	Conference on Automated Deduction (CADE 2007). In volume 4603 of Lecture
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	337	Notes in Artificial Intelligence. Bremen, Germany. July 2007. Pages 35-50.
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	338	&copy <A HREF="http://link.springer.de/link/service/series/0558/tocs/t4603.htm"
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	339	target="_top">Springer Verlag</A>
465 4dac76eb27d9 updated Christian Urban <urbanc@in.tum.de> parents: 411 diff changeset	340	[<A HREF="http://nms.kcl.ac.uk/christian.urban/Publications/cade07.ps" target="_top">ps</A>]
402 9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	341	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	342	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	343
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	344	<TR><TD VALIGN=Top>[6]</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	345	<TD ALIGN=Left>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	346	<B>Mechanising the Metatheory of LF.</B>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	347	(Christian Urban, James Cheney and Stefan Berghofer)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	348	In Proc. of the 23rd IEEE Symposium on Logic in Computer Science (LICS 2008), IEEE Computer Society,
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	349	June 2008. Pages 45-56.
465 4dac76eb27d9 updated Christian Urban <urbanc@in.tum.de> parents: 411 diff changeset	350	[<A HREF="http://nms.kcl.ac.uk/christian.urban/Publications/lics-08.pdf">pdf</A>] More
4dac76eb27d9 updated Christian Urban <urbanc@in.tum.de> parents: 411 diff changeset	351	information <A HREF="http://nms.kcl.ac.uk/christian.urban/Nominal/LF/index.html">elsewhere</A>.
402 9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	352	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	353	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	354	<TR><TD VALIGN=Top>[7]</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	355	<TD ALIGN=Left>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	356	<B>Proof Pearl: A New Foundation for Nominal Isabelle.</B>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	357	(Brian Huffman and Christian Urban)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	358	In Proc. of the 1st Conference on Interactive Theorem Proving (ITP 2010). In volume 6172 in
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	359	Lecture Notes in Computer Science, Pages 35-50, 2010.
465 4dac76eb27d9 updated Christian Urban <urbanc@in.tum.de> parents: 411 diff changeset	360	[<A HREF="http://nms.kcl.ac.uk/christian.urban/Publications/nominal-atoms.pdf">pdf</A>]
402 9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	361	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	362	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	363	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	364	<TR><TD VALIGN=Top>[8]</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	365	<TD ALIGN=Left>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	366	<B>General Bindings and Alpha-Equivalence in Nominal Isabelle.</B>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	367	(Christian Urban and Cezary Kaliszyk)
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	368	In Proc. of the 20th European Symposium on Programming (ESOP 2011).
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	369	In Volume 6602 of Lecture Notes in Computer Science, Pages 480-500, 2011.
465 4dac76eb27d9 updated Christian Urban <urbanc@in.tum.de> parents: 411 diff changeset	370	[<A HREF="http://nms.kcl.ac.uk/christian.urban/Publications/esop-11.pdf">pdf</A>]
402 9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	371	</TD>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	372	</TR>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	373
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	374	</TABLE>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	375	</CENTER>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	376
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	377	<P><!-- Created: Tue Mar 4 00:23:25 GMT 1997 -->
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	378	<!-- hhmts start -->
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	379	Last modified: Mon May 9 05:35:17 BST 2011
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	380	<!-- hhmts end -->
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	381	<a href="http://validator.w3.org/check/referer" target="_top">[Validate this page.]</a>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	382
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	383	</body>
9e089afe5086 added Nominal Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	384	</html>

author	Christian Urban <christian.urban@kcl.ac.uk>
	Sat, 08 Jul 2023 21:24:18 +0100
changeset 634	f365ce636294
parent 465	4dac76eb27d9
permissions	-rw-r--r--