Type Isomorphisms and Proof Reuse in Dependent Type Theory

Barthe, Gilles; Pons, Olivier

doi:10.1007/3-540-45315-6_4

Cited by 26 publications

(26 citation statements)

References 24 publications

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“…Composition and identities are as in (3) and (4). Further, we define F + [T ] analogously by omitting the empty type together with its associated constructor and equations throughout.…”

Section: Product and Sum Typesmentioning

confidence: 99%

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Objective Representation

2017

Film and Video Intermediality

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Tarski asked whether the arithmetic identities taught in high school are complete for showing all arithmetic equations valid for the natural numbers. The answer to this question for the language of arithmetic expressions using a constant for the number one and the operations of product and exponentiation is affirmative, and the complete equational theory also characterises isomorphism in the typed lambda calculus, where the constant for one and the operations of product and exponentiation respectively correspond to the unit type and the product and arrow type constructors. This paper studies isomorphisms in typed lambda calculi with empty and sum types from this viewpoint. Our main contribution is to show that a family of so-called Wilkie-Gurevič identities, that plays a pivotal role in the study of Tarski's high school algebra problem, arises from type-theoretic isomorphisms. We thus close an open problem by establishing that the theory of type isomorphisms in the presence of product, arrow, and sum types (with or without the unit type) is not finitely axiomatisable. Further, we observe that for type theories with arrow, empty and sum types the correspondence between isomorphism and arithmetic equality generally breaks down, but that it still holds in some particular cases including that of type isomorphism with the empty type and equality with zero.

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“…Composition and identities are as in (3) and (4). Further, we define F + [T ] analogously by omitting the empty type together with its associated constructor and equations throughout.…”

Section: Product and Sum Typesmentioning

confidence: 99%

“…In proof assistants they are used to find proofs in libraries up to irrelevant syntactical details [10,4].…”

Section: Introductionmentioning

confidence: 99%

Objective Representation

2017

Film and Video Intermediality

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“…They are used in functional programming to provide a means to search functions by types [20,21,22,39,40,41,42] and to match modules by specifications [7,19].…”

Section: Type Isomorphismsmentioning

confidence: 99%

Memoization in Type-Directed Partial Evaluation

Balat¹,

Danvy²

2002

BRICS

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We use a code generator-type-directed partial evaluation-to verify conversions between isomorphic types, or more precisely to verify that a composite function is the identity function at some complicated type. A typed functional language such as ML provides a natural support to express the functions and type-directed partial evaluation provides a convenient setting to obtain the normal form of their composition. However, off-the-shelf type-directed partial evaluation turns out to yield gigantic normal forms.We identify that this gigantism is due to redundancies, and that these redundancies originate in the handling of sums, which uses delimited continuations. We successfully eliminate these redundancies by extending type-directed partial evaluation with memoization capabilities. The result only works for pure functional programs, but it provides an unexpected use of code generation and it yields orders-of-magnitude improvements both in time and in space for type isomorphisms. *

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“…Sharing knowledge among different proof assistants at the logical level is usually less difficult since the relationships between various logical frameworks are known and some techniques that facilitate knowledge reuse have been provided [5].…”

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confidence: 99%

Procedural Representation of CIC Proof Terms

Guidi

2009

J Autom Reasoning

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We propose an effective procedure, the first one to our knowledge, for translating a proof term of the Calculus of Inductive Constructions (CIC), into a tactical expression of the high-level specification language of a CIC-based proof assistant like coq (Coq development team 2008) or matita (Asperti et al., J Autom Reason 39: 2007). This procedure, which should not be considered definitive at its present stage, is intended for translating the logical representation of a proof coming from any source, i.e. from a digital library or from another proof development system, into an equivalent proof presented in the proof assistant's editable high-level format. To testify to effectiveness of our procedure, we report on its implementation in matita and on the translation of a significant set of proofs (Guidi, ACM Trans Comput Log 2009) from their logical representation as coq 7.3.1 (Coq development team 2002) CIC proof terms to their high-level representation as tactical expressions of matita's user interface language.

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Type Isomorphisms and Proof Reuse in Dependent Type Theory

Abstract: Abstract. We propose a theoretical foundation for proof reuse, based on the novel idea of a computational interpretation of type isomorphisms.

Cited by 26 publications

References 24 publications

Objective Representation

Objective Representation

Memoization in Type-Directed Partial Evaluation

Procedural Representation of CIC Proof Terms

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