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A proof of Higman's lemma by structural induction
, 1993
"... This paper gives an example of such an inductive proof for a combinatorial problem. While there exist other constructive proofs of Higman's lemma (see for instance [10, 14]), the present argument has been recorded for its extreme formal simplicity. This simplicity allows us to give a complete d ..."
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This paper gives an example of such an inductive proof for a combinatorial problem. While there exist other constructive proofs of Higman's lemma (see for instance [10, 14]), the present argument has been recorded for its extreme formal simplicity. This simplicity allows us to give a complete description of the computational content of the proof, first in term of a functional program, which follows closely the structure of the proof, and then in term of a program with state. The second program has an intuitive algorithmic meaning. In order to show that these two programs are equivalent, we introduce an intermediary program, which is a firstorder operational interpretation of the functional program. The relation between this program and the program with state is simple to establish. We can thus claim that we understand completely the computational behaviour of the proof. It is possible to give still another description of this algorithm, in term of process computing in parallel. In this form, the connection with NashWilliams non constructive argument is quite clear (though this algorithm was found first only as an alternative description of the computational content of the inductive proof). This inductive proof was actually found from the usual non constructive argument by using the technique described in [3]. These two facts give strong indication that this algorithm can be considered as the computational content of the NashWilliams argument.
Ramsey's Theorem in Type Theory
, 1993
"... We present formalizations of constructive proofs of the Intuitionistic Ramsey Theorem and Higman's Lemma in MartinLof's Type Theory. We analyze the computational content of these proofs and we compare it with programs extracted out from some classical proofs. Contents 1 Introduction 2 2 ..."
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We present formalizations of constructive proofs of the Intuitionistic Ramsey Theorem and Higman's Lemma in MartinLof's Type Theory. We analyze the computational content of these proofs and we compare it with programs extracted out from some classical proofs. Contents 1 Introduction 2 2 The proofs 4 2.1 An inductive formulation of almostfullness (AF ID ) : : : : : : : : : : 5 2.1.1 Intuitionistic Ramsey Theorem (IRT ID ) : : : : : : : : : : : : 7 2.1.2 Higman's Lemma (HL ID ) : : : : : : : : : : : : : : : : : : : : 12 2.2 A negationless inductive formulation of almostfullness (AF I ) : : : : : 17 2.2.1 Intuitionistic Ramsey Theorem (IRT I ) : : : : : : : : : : : : : 17 2.3 Equivalence between the various formulations of almostfullness : : : 20 3 The programs 22 3.1 A higher order program : : : : : : : : : : : : : : : : : : : : : : : : : 24 3.2 A first order program : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 4 Computational content of classical proofs 28 4.1 A cl...
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
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
Stop when you are AlmostFull Adventures in constructive termination
"... Disjunctive wellfoundedness (used in Terminator), sizechange termination, and wellquasiorders (used in supercompilation and termrewrite systems) are examples of techniques that have been successfully applied to automatic proofs of program termination and online termination testing, respectively ..."
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Disjunctive wellfoundedness (used in Terminator), sizechange termination, and wellquasiorders (used in supercompilation and termrewrite systems) are examples of techniques that have been successfully applied to automatic proofs of program termination and online termination testing, respectively. Although these works originate in different communities, there is an intimate connection between them – they rely on closely related principles and both employ similar arguments from Ramsey theory. At the same time there is a notable absence of these techniques in programming systems based on constructive type theory. In this paper we’d like to highlight the aforementioned connection and make the core ideas widely accessible to theoreticians and Coq programmers, by offering a Coq development which culminates in some novel tools for performing induction. The benefit is nice composability properties of termination arguments at the cost of intuitive and lightweight user obligations. Inevitably, we have to present some Ramseylike arguments: Though similar proofs are typically classical, we offer an entirely constructive development standing on the shoulders of Veldman and Bezem, and Richman and Stolzenberg. 1.
A proof of Higman's Lemma by open induction
, 1996
"... We use Raoult's principle of open induction to give a new proof of Higman's Lemma. In contrast to previous proofs, it directly uses the property that every infinite sequence has an infinite ordered subsequence. For a slightly more complex order we exchange a fairly straight inductive argum ..."
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We use Raoult's principle of open induction to give a new proof of Higman's Lemma. In contrast to previous proofs, it directly uses the property that every infinite sequence has an infinite ordered subsequence. For a slightly more complex order we exchange a fairly straight inductive argument.