Elaborating dependent (co)pattern matching

Cockx, Jesper; Abel, Andreas

doi:10.1145/3236770

Cited by 12 publications

(21 citation statements)

References 25 publications

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“…Our main technical contribution is an elaboration algorithm for (co)pattern matching definitions in the presence of HITs and path applications. Following Cockx and Abel [2018] we formulate our algorithm as a translation from (co)pattern matching clauses to case trees. The main challenges are generating the computational behavior on hcomp elements of HITs and making sure clauses for path constructors agree with what the function does at the endpoints of the path.…”

Section: Higher Inductive Types and Pattern Matchingmentioning

confidence: 99%

“…We recall some definitions from Cockx and Abel [2018], extended to allow for the new cubical primitives. Expressions (Fig.…”

Section: Syntax Of the Core Type Theorymentioning

confidence: 99%

“…2. Declaration forms are due to Cockx and Abel [2018] except for datatype constructors c ∆ [ī | b]. These take a telescope of arguments ∆, i.e.…”

Section: Syntax Of the Core Type Theorymentioning

confidence: 99%

“…The core definition of Cockx and Abel [2018] is the elaboration judgment Σ; Γ ⊢ P | fq := Q : C Σ ′ which performs type and coverage checking for the user supplied clauses of f represented by the problem set P. It further computes the corresponding case tree Q and checked clauses cls ⊕ in Σ ′ . Case trees Q are previously specified by a typing judgment Σ; Γ ⊢ fq := Q : C Σ ′ which follows the same structure but takes the case tree as input.…”

Section: Syntax Of the Core Type Theorymentioning

confidence: 99%

“…Moreover we need to consider the case for hcomp r u u 0 : we check Q hc by (1) replacing x by the pattern hcomp r u u 0 in the left-hand side and (2) extending Θ with (r = i1) since that is the face where hcomp r u u 0 computes to u i1 1=1. The remaining rules do not directly involve any of the new features of Cubical Agda, so we ask the reader to refer back to the corresponding judgment in Cockx and Abel [2018].…”

Section: Case Treesmentioning

confidence: 99%

See 4 more Smart Citations

Cubical agda: a dependently typed programming language with univalence and higher inductive types

Vezzosi

Mörtberg

Abel

2019

Proc. ACM Program. Lang.

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Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types. This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of higher inductive types. These new primitives make function and propositional extensionality as well as quotient types directly definable with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. This extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.or invariance up to equivalence, in the sense that equivalent types will share the same structures and properties. The fact that equality is proof relevant in dependent type theory is the key to enabling this; the data of an equality proof can store the equivalence and transporting along this equality should then apply the function underlying the equivalence. In particular, this allows programs and properties to be transported between equivalent types, hereby increasing modularity and decreasing code duplication. A concrete example are the equivalent representations of natural numbers in unary and binary format. In a univalent system it is possible develop theory about natural numbers using the unary representation, but compute using the binary representation, and as the two representations are equivalent they share the same properties.The principle of univalence is the major new addition in Homotopy Type Theory and Univalent Foundations (HoTT/UF) [Univalent Foundations Program 2013]. However, these new type theoretic foundations add univalence as an axiom which disrupts the good constructive properties of type theory. In particular, if we transport addition on binary numbers to the unary representation we will not be able to compute with it as the system would not know how to reduce the univalence axiom. Cubical type theory [Cohen et al. 2018] addresses this problem by introducing a novel representation of equality proofs and thereby providing computational content to univalence. This makes it possible to constructively transport programs and properties between equivalent types. This representation of equality proofs has many other useful consequences, in particular function and propositional extensionality (pointwise equal functions and logically equivalent propositions are equal), and the equivalence between bisi...

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Section: Higher Inductive Types and Pattern Matchingmentioning

confidence: 99%

“…We recall some definitions from Cockx and Abel [2018], extended to allow for the new cubical primitives. Expressions (Fig.…”

Section: Syntax Of the Core Type Theorymentioning

confidence: 99%

“…2. Declaration forms are due to Cockx and Abel [2018] except for datatype constructors c ∆ [ī | b]. These take a telescope of arguments ∆, i.e.…”

Section: Syntax Of the Core Type Theorymentioning

confidence: 99%

Section: Syntax Of the Core Type Theorymentioning

confidence: 99%

Section: Case Treesmentioning

confidence: 99%

See 3 more Smart Citations

Cubical agda: a dependently typed programming language with univalence and higher inductive types

Vezzosi

Mörtberg

Abel

2019

Proc. ACM Program. Lang.

Self Cite

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Codata in Action

Downen

Sullivan

Ariola

et al. 2019

Lecture Notes in Computer Science

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Computer scientists are well-versed in dealing with data structures. The same cannot be said about their dual: codata. Even though codata is pervasive in category theory, universal algebra, and logic, the use of codata for programming has been mainly relegated to representing infinite objects and processes. Our goal is to demonstrate the benefits of codata as a general-purpose programming abstraction independent of any specific language: eager or lazy, statically or dynamically typed, and functional or object-oriented. While codata is not featured in many programming languages today, we show how codata can be easily adopted and implemented by offering simple intercompilation techniques between data and codata. We believe codata is a common ground between the functional and object-oriented paradigms; ultimately, we hope to utilize the Curry-Howard isomorphism to further bridge the gap.

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Constructing Infinitary Quotient-Inductive Types

Fiore

Pitts

Steenkamp

2020

Lecture Notes in Computer Science

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This paper introduces an expressive class of quotient-inductive types, called QW-types. We show that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types can be encoded using the combination of inductive-inductive definitions, Hofmann-style quotient types, and Abel's size types. The latter, which provide a convenient constructive abstraction of what classically would be accomplished with transfinite ordinals, are used to prove termination of the recursive definitions of the elimination and computation properties of our encoding of QW-types. The development is formalized using the Agda theorem-prover.

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Elaborating dependent (co)pattern matching

Cited by 12 publications

References 25 publications

Cubical agda: a dependently typed programming language with univalence and higher inductive types

Cubical agda: a dependently typed programming language with univalence and higher inductive types

Codata in Action

Constructing Infinitary Quotient-Inductive Types

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