A proof of strong normalisation using domain theory

Coquand, Thierry; Spiwack, Arnaud

doi:10.2168/lmcs-3(4:12)2007

Cited by 20 publications

(44 citation statements)

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“…This includes a strong normalization theorem for Mini-TT using the denotational semantics of [4,7], non-uniform inductive families of types, universe hierarchy, proven correct compilation to abstract machine code as in [10], etc. If these routines return without producing error messages, then there are derivations that conclude corresponding judgements.…”

Section: Resultsmentioning

confidence: 99%

“…As we explained in the introduction, the work [4,7] should provide a general semantical condition ensuring termination of type-checking: it is enough that the strict denotational semantics of the program is =⊥. As in [4,7], one can ensure this by proving totality of the program.…”

Section: Metamathematical Remarksmentioning

confidence: 99%

“…As in [4,7], one can ensure this by proving totality of the program. In turn, there are sufficient purely syntactical criterion ensuring totality.…”

Section: Metamathematical Remarksmentioning

confidence: 99%

“…Our approach should also allow us to apply the results of [4,7]. This should provide a modular and semantical sufficient condition ensuring strong normalisation and hence decidability of our type-checking algorithm.…”

Section: Introductionmentioning

confidence: 99%

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A simple type-theoretic language: Mini-TT

Coquand¹,

Kinoshita²,

Nordström³

et al. 2009

From Semantics to Computer Science

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This paper presents a formal description of a small functional language with dependent types. The language contains data types, mutual recursive/inductive definitions and a universe of small types. The syntax, semantics and type system is specified in such a way that the implementation of a parser, interpreter and type checker is straightforward. The main difficulty is to design the conversion algorithm in such a way that it works for open expressions. The paper ends with a complete implementation in Haskell (around 400 lines of code). IntroductionWe are going to describe a small language with dependent types, its syntax, operational semantics and type system. This is in the spirit of the paper "A simple applicative language: Mini-ML" by Clément, Despeyroux, and Kahn [5], where they explain a small functional language. From them we have borrowed the idea of using patterns instead of variables in abstractions and let-bindings. It gives an elegant way to express mutually recursive definitions. We also share with them the view that a programming language should not only be formally specified, but it should also be possible to reason about the correctness of its implementation. There should be a small step from the formal opera-139

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Section: Resultsmentioning

confidence: 99%

Section: Metamathematical Remarksmentioning

confidence: 99%

“…As in [4,7], one can ensure this by proving totality of the program. In turn, there are sufficient purely syntactical criterion ensuring totality.…”

Section: Metamathematical Remarksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A simple type-theoretic language: Mini-TT

Coquand¹,

Kinoshita²,

Nordström³

et al. 2009

From Semantics to Computer Science

Self Cite

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“…We explain finally how we can precise further the representation of type theory as a functional programming language using some recent results in domain theory [3,5].…”

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confidence: 99%

Constructive Mathematics and Functional Programming (Abstract)

Coquand

Programming Languages and Systems

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Around thirty years ago, P. Martin-Löf [12] suggested that the intuitionistic theory of types, originally designed as a formal system for constructive mathematics, could be viewed as a programming language. The conclusion of this paper stresses the mutual benefit of relating constructive mathematics and computer programming. In one direction one gets a precise system of notations for both statements and proofs, and one obtains the computerization of abstract intuitionistic mathematics that was asked by Bishop [2]. In the other direction, computer programming "gets access to the whole conceptual apparatus of pure mathematics".In the first part of this talk we shall survey some recent works that illustrate this relation and its fruitfulness. One line of work, close to Bishop, represents real numbers and numerical functions [4,13] in type theory. Another line is concerned with algorithms on finite combinatorial structure (graphs, hypermaps, finite groups). One main example is the complete formalization of a proof of the four color theorem by G. Gonthier and B. Werner. The report on this work [9] points out as well the mutual benefits of this correspondence: "Although this work is purportedly about using computer programming to help doing mathematics, we expect that most of its fallout will be in the reverse direction -using mathematics to help programming computers". A related work, also dealing with hypermaps, aims at obtaining formal specification in geometric modeling [6], and presents algorithms that can be designed in this way [7]. There is also on-going work [8] on the formalization of finite group theory.The second part will reflect on the design of type theory as a functional programming language. The analogy between type theory and functional programming was pointed out already by (for instance the correspondence between canonical and non-canonical form of expressions and the notion of constructors and selectors, respectively, of Landin [11]). This analogy should go further and type theory should benefit in using more the powerful system of notations provided by functional programming. (In particular, where expressions correspond to local abbreviations, definitions and lemmas, functions defined by pattern-matching correspond to definitions by case and proofs by case analysis, uses of recursive definitions correspond to inductive arguments, module systems can be used to structure proofs; the system Agda [1] follows these analogies.)S.

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On the Strength of Proof-Irrelevant Type Theories

Werner

2006

Automated Reasoning

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We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the subset types of the theory of PVS. We show that in these theories, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic. Finally we describe a simple set-theoretic semantics. CCCreative Commons 2 B. WERNER was more an external feature of implemented proof systems 1 : programs certified by extraction are no longer objects of the formalism and cannot be used to assert facts like in the point above. Some related formalisms only build on some of the points above. For example PVS implements a theory whose objects are functional programs, but where proofs are not objects of the formalism.An important remark about (2) is that the more terms are identified by the conversion rule, the more powerful this rule is. In order to identify more terms it thus is tempting to combine points (2) and (3) by integrating program extraction into the formalism so that the conversion rule does not require the computationally irrelevant parts of terms to be convertible.In what follows, we present and argue in favor of a type-theory along this line. More precisely, we claim that such a feature is useful in at least two respects. For one, it gives a more comfortable type theory, especially in the way it handles equality. Furthermore it is a good starting point to build a platform for programming with dependent types, that is to use the theorem prover also as a programming environment. Finally, on a more theoretical level, we will also see that by making the theory more extensional, proof-irrelevance brings type theory closer to set-theory regarding the consequences of the axiom of choice.The central idea of this work is certainly simple enough to be adjusted to various kinds of type theories, whether they are predicative or not, with various kinds of inductive types, more refined mechanisms to distinguish the computational parts of the proofs etc. . . . In what follows we illustrate it by using a marking of the computational content which is as simple as possible. The extraction function we use is quite close to Letouzey's [21,22], except that we discard the inclusion rule Prop ⊂ Type, which would complicate the definition of the type theory and the semantics (see [29] for the last point).Related work Almost surprisingly, proof-irrelevant type theories do not seem to enjoy wide use yet. In the literature, they are often not studied for themselves, but as means for proving properties of other systems. This is the case for the work of Altenkirch [3] and Barthe [6]. One very interesting work is Pfenning's modal type theory which involves proofirrelevance and a sophisticated way to pinpoint which definitional equality is to be used for each part of a term; in comparision we here stick to much simpler extraction mechanism. The NuPRL ...

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A proof of strong normalisation using domain theory

Cited by 20 publications

References 20 publications

A simple type-theoretic language: Mini-TT

A simple type-theoretic language: Mini-TT

Constructive Mathematics and Functional Programming (Abstract)

On the Strength of Proof-Irrelevant Type Theories

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