Higher-Order Dynamic Pattern Unification for Dependent Types and Records

Abel, Andreas; Pientka, Brigitte

doi:10.1007/978-3-642-21691-6_5

Cited by 20 publications

(25 citation statements)

References 11 publications

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“…This enables precise scope tracking of free variables in judgments. Furthermore it avoids the duplication of context information across judgments in worklists that occurs in other techniques [Abel and Pientka 2011;Reed 2009]. Despite the use of a different algorithm we prove the same results as DK, although with significantly different proofs and proof techniques.…”

Section: Introductionmentioning

confidence: 52%

“…For instance, Reed [2009] represents a unification problem as ∆ ⊢ P where P is a set of equations to be solved and ∆ is a (modal) context. Abel and Pientka [2011] even use multiple contexts within a unification problem. Such a problem is denoted ∆ K where the meta-context ∆ contains all the typings of meta-variables in the constraint set K. The latter consists of constraints like Ψ ⊢ M = N : C that are equipped with their individual context Ψ.…”

Section: Judgment Listsmentioning

confidence: 99%

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A mechanical formalization of higher-ranked polymorphic type inference

Zhao

Oliveira

Schrijvers

2019

Proc. ACM Program. Lang.

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Modern functional programming languages, such as Haskell or OCaml, use sophisticated forms of type inference. While an important topic in the Programming Languages research, there is little work on the mechanization of the metatheory of type inference in theorem provers. In particular we are unaware of any complete formalization of the type inference algorithms that are the backbone of modern functional languages.This paper presents the first full mechanical formalization of the metatheory for higher-ranked polymorphic type inference. The system that we formalize is the bidirectional type system by Dunfield and Krishnaswami (DK). The DK type system has two variants (a declarative and an algorithmic one) that have been manually proven sound, complete and decidable. We present a mechanical formalization in the Abella theorem prover of DK's declarative type system with a novel algorithmic system. We have a few reasons to use a new algorithm. Firstly, our new algorithm employs worklist judgments, which precisely capture the scope of variables and simplify the formalization of scoping in a theorem prover. Secondly, while DK's original formalization comes with very well-written manual proofs, there are several details missing and some incorrect proofs, which complicate the task of writing a mechanized proof. Despite the use of a different algorithm we prove the same results as DK, although with significantly different proofs and proof techniques. Since such type inference algorithms are quite subtle and have a complex metatheory, mechanical formalizations are an important advance in type-inference research.

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Section: Introductionmentioning

confidence: 52%

Section: Judgment Listsmentioning

confidence: 99%

A mechanical formalization of higher-ranked polymorphic type inference

Zhao

Oliveira

Schrijvers

2019

Proc. ACM Program. Lang.

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show abstract

“…Indeed, in general, higher-order unification of terms in the calculus of constructions is undecidable, so we cannot hope for a complete unification algorithm. Barring completeness, we might want to ensure correctness in the sense that a unification problem t ≡ u is solved only if there is a most general unifier σ (a substitution of existentials by terms) such that t[σ] ≡ βδι u[σ], like the algorithm defined by Abel et al [8]. This is however not the case of Coq's unification algorithm, because of the use of the first-order unification heuristic that can return less general unifiers.…”

Section: Unificationmentioning

confidence: 99%

Universe Polymorphism in Coq

Sozeau

Tabareau

2014

Interactive Theorem Proving

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Abstract. Universes are used in Type Theory to ensure consistency by checking that definitions are well-stratified according to a certain hierarchy. In the case of the Coq proof assistant, based on the predicative Calculus of Inductive Constructions (pCIC), this hierachy is built from an impredicative sort Prop and an infinite number of predicative Typei universes. A cumulativity relation represents the inclusion order of universes in the core theory. Originally, universes were thought to be floating levels, and definitions to implicitely constrain these levels in a consistent manner. This works well for most theories, however the globality of levels and constraints precludes generic constructions on universes that could work at different levels. Universe polymorphism extends this setup by adding local bindings of universes and constraints, supporting generic definitions over universes, reusable at different levels. This provides the same kind of code reuse facilities as ML-style parametric polymorphism. However, the structure and hierarchy of universes is more complex than bare polymorphic type variables. In this paper, we introduce a conservative extension of pCIC supporting universe polymorphism and treating its whole hierarchy. This new design supports typical ambiguity and implicit polymorphic generalization at the same time, keeping it mostly transparent to the user. Benchmarking the implementation as an extension of the Coq proof assistant on real-world examples gives encouraging results.

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“…Another example is WAM-style compilation of logic programs [2,3], which is founded on duality and currying. Standard implementation techniques such as left-to-right goal search and first-to-last clause selection have found their foundation in ordered logic [12], while unification combines logic with equality and contextual reasoning [1].…”

Section: Introductionmentioning

confidence: 99%

Proof-Theoretic Foundations of Indexing in Logic Programming

Cervesato

2014

Proceedings of the 2014 International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice

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Indexing is generally viewed as an implementation artifact, indispensable to speed up the execution of logic programs and theorem provers, but with little intrinsically logical about it. We show that indexing can be given a justification in proof theory on the basis of focusing and linearity. We demonstrate this approach on predicate indexing for Horn clauses and several formulations of hereditary Harrop formulas. We also show how to refine this approach to discriminate on function symbols as well.

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Higher-Order Dynamic Pattern Unification for Dependent Types and Records

Cited by 20 publications

References 11 publications

A mechanical formalization of higher-ranked polymorphic type inference

A mechanical formalization of higher-ranked polymorphic type inference

Universe Polymorphism in Coq

Proof-Theoretic Foundations of Indexing in Logic Programming

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