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Higman's Lemma in Type Theory
 PROCEEDINGS OF THE 1996 WORKSHOP ON TYPES FOR PROOFS AND PROGRAMS
, 1997
"... This thesis is about exploring the possibilities of a limited version of MartinLöf's type theory. This exploration consists both of metatheoretical considerations and of the actual use of that version of type theory to prove Higman's lemma. The thesis is organized in two papers, one in which type t ..."
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This thesis is about exploring the possibilities of a limited version of MartinLöf's type theory. This exploration consists both of metatheoretical considerations and of the actual use of that version of type theory to prove Higman's lemma. The thesis is organized in two papers, one in which type theory itself is studied and one in which it is used to prove Higman's lemma. In the first paper, A Lambda Calculus Model of MartinLöf's Theory of Types with Explicit Substitution, we present the formal calculus in complete detail. It consists of MartinLof's logical framework with explicit substitution extended with some inductively defined sets, also given in complete detail. These inductively defined sets are precisely those we need in the second paper of this thesis for the formal proof of Higman's lemma. The limitations of the formalism come from the fact that we do not introduce universes. It is known that for other versions of type theory, the absence of universes implies the impossib...
An Inductive Version of NashWilliams’ MinimalBadSequence Argument for Higman’s Lemma
 In P. Callaghan, e.al., Types for Proofs and Programs, Lecture Notes in Computer Science 2277
, 2001
"... Abstract. Higman’s lemma has a very elegant, nonconstructive proof due to NashWilliams [NW63] using the socalled minimalbadsequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was done ..."
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Abstract. Higman’s lemma has a very elegant, nonconstructive proof due to NashWilliams [NW63] using the socalled minimalbadsequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was done by Coquand and Fridlender [CF94]. Here we present a proof in a theory of inductive definitions that works for arbitrary decidable well quasiorders. 1
A constructive proof of Higman’s lemma
 SME Conference Proceedings Bethlelem
, 1984
"... Abstract. Higman’s lemma, a specific instance of Kruskal’s theorem, is an interesting result from the area of combinatorics, which has often been used as a test case for theorem provers. We present a constructive proof of Higman’s lemma in the theorem prover Isabelle, based on a paper proof by Coqua ..."
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Abstract. Higman’s lemma, a specific instance of Kruskal’s theorem, is an interesting result from the area of combinatorics, which has often been used as a test case for theorem provers. We present a constructive proof of Higman’s lemma in the theorem prover Isabelle, based on a paper proof by Coquand and Fridlender. Making use of Isabelle’s newlyintroduced infrastructure for program extraction, we show how a program can automatically be extracted from this proof, and analyze its computational behaviour. 1
Applications of inductive definitions and choice principles to program synthesis
"... Abstract. We describe two methods of extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and nonconstructive choice principles. The first method removes any reference to infinite sequences and transforms the theorem into a system of inductive defi ..."
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Abstract. We describe two methods of extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and nonconstructive choice principles. The first method removes any reference to infinite sequences and transforms the theorem into a system of inductive definitions, the other applies a combination of Gödel’s negativeand Friedman’s Atranslation. Both approaches are explained by means of a case study on Higman’s Lemma and its wellknown classical proof due to NashWilliams. We also discuss some prooftheoretic optimizations that were crucial for the formalization and implementation of this work in the interactive proof system Minlog. 1
A Constructive Proof of the Topological Kruskal Theorem
"... Abstract. We give a constructive proof of Kruskal’s Tree Theorem— precisely, of a topological extension of it. The proof is in the style of a constructive proof of Higman’s Lemma due to Murthy and Russell (1990), and illuminates the role of regular expressions there. In the process, we discover an e ..."
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Abstract. We give a constructive proof of Kruskal’s Tree Theorem— precisely, of a topological extension of it. The proof is in the style of a constructive proof of Higman’s Lemma due to Murthy and Russell (1990), and illuminates the role of regular expressions there. In the process, we discover an extension of Dershowitz ’ recursive path ordering to a form of cyclic terms which we call µterms. This all came from recent research on Noetherian spaces, and serves as a teaser for their theory. 1
Proofs, Lambda Terms and Control Operators
, 1995
"... ed M : V and typed by M A : V :A ffi ) and context unwrapping (denoted V E and typed by requiring V to be of type :B ffi and the evaluation context E[] to be of type B with the `hole' of type A). Here we essentially give an exposition of Griffin's result, with some simplifications and exte ..."
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ed M : V and typed by M A : V :A ffi ) and context unwrapping (denoted V E and typed by requiring V to be of type :B ffi and the evaluation context E[] to be of type B with the `hole' of type A). Here we essentially give an exposition of Griffin's result, with some simplifications and extensions based on work of Sabry and Felleisen [18]. In particular we stress its connection with questions of termination of different normalization strategies for minimal, intuitionistic and classical logic, or more precisely their fragments in implicational propositional logic. We also give some examples (due to Hirokawa) of derivations in minimal and classical logic which reproduce themselves under certain reasonable conversion rules. This work clearly owes a lot to other people. Robert Const