Copatterns

Abstract

Inductive datatypes provide mechanisms to define finite data such as finite lists and trees via constructors and allow programmers to analyze and manipulate finite data via pattern matching. In this paper, we develop a dual approach for working with infinite data structures such as streams. Infinite data inhabits coinductive datatypes which denote greatest fixpoints. Unlike finite data which is defined by constructors we define infinite data by observations. Dual to pattern matching, a tool for analyzing finite data, we develop the concept of copattern matching, which allows us to synthesize infinite data. This leads to a symmetric language design where pattern matching on finite and infinite data can be mixed.We present a core language for programming with infinite structures by observations together with its operational semantics based on (co)pattern matching and describe coverage of copatterns. Our language naturally supports both call-by-name and call-by-value interpretations and can be seamlessly integrated into existing languages like Haskell and ML. We prove type soundness for our language and sketch how copatterns open new directions for solving problems in the interaction of coinductive and dependent types.

Coinduction in Uniform: Foundations for Corecursive Proof Search with Horn Clauses

“…To save space, when we build a proof in iFOL using (∀-I), (∀-E) or (Conv), etc., we may omit the condition branch, which is x : τ Γ, Γ M : τ or ψ ≡ ψ respectively, if and only if we know that the condition holds. Now let P denote the singleton set of clause (2). In Fig.…”

“…Related Work. It may be surprising that automated proof search for coinductive predicates in rstorder logic does not have a coherent and comprehensive theory, even after three decades [3,60], despite all the attention that it received as programming [2,28,41,43] and proof [32,33,37,38,44,59,65,66,67,68] method. The work that comes close to algorithmic proof search is the system CIRC [64], but it cannot handle general coinductive predicates and corecursive programming.…”

Up-to techniques using sized types

“…We can define values in the delay monad corecursively, using copatterns [Abel et al 2013]. A copattern corresponding to a record field specifies the result of projecting the given field from the value.…”

Normalization by Evaluation in the Delay Monad: A Case Study for Coinduction via Copatterns and Sized Types

“…Formally, we have defined a member of the record type ∞Delay i A by giving the contents of all of its fields, here only force. The use of a projection like force on the left hand side of a defining equation is called a copattern [5]. Corecursive definitions by copatterns are the latest addition to Agda, and can be activated since version 2.3.2 via the flag --copatterns.…”