Fixed point semantics and partial recursion in Coq

Bertot, Yves; Komendantsky, Vladimir

doi:10.1145/1389449.1389461

Cited by 15 publications

(15 citation statements)

References 28 publications

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“…Similar functionality was developed for Coq by Bertot and Komendantsky (2008), relying on additional classical axioms. Paulin-Mohring (2009) gives a constructive formalization of cpos in Coq, representing flat domains coinductively, similar to what we will see in the next subsection.…”

Section: Partiality In the Resultsmentioning

confidence: 99%

Partiality and recursion in interactive theorem provers – an overview

Bove¹,

Krauß²,

Sozeau³

2014

Math. Struct. Comp. Sci.

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The use of interactive theorem provers to establish the correctness of critical parts of a software development or for formalizing mathematics is becoming more common and feasible in practice. However, most mature theorem provers lack a direct treatment of partial and general recursive functions; overcoming this weakness has been the objective of intensive research during the last decades. In this article, we review several techniques that have been proposed in the literature to simplify the formalization of partial and general recursive functions in interactive theorem provers. Moreover, we classify the techniques according to their theoretical basis and their practical use. This uniform presentation of the different techniques facilitates the comparison and highlights their commonalities and differences, as well as their relative advantages and limitations. We focus on theorem provers based on constructive type theory (in particular, Agda and Coq) and higher-order logic (in particular Isabelle/HOL). Other systems and logics are covered to a certain extent, but not exhaustively. In addition to the description of the techniques, we also demonstrate tools which facilitate working with the problematic functions in particular theorem provers.

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Section: Partiality In the Resultsmentioning

confidence: 99%

Partiality and recursion in interactive theorem provers – an overview

Bove¹,

Krauß²,

Sozeau³

2014

Math. Struct. Comp. Sci.

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“…The work of Bertot and Komendantsky [9] describes a way to embed general recursive functions into Coq that does not use coinduction. They define a datatype partial A that is isomorphic to the usual Maybe A but is understood as representing a lifted CPO A ⊥ , and use classical logic axioms to provide a fixpoint combinator fixp.…”

Section: Related Workmentioning

confidence: 99%

“…For this to be valid, the property must be "admissible", and it most hold for infinite loops. An equivalent variant [9] is to allow induction on the number of recursive steps an expression takes to normalize. λ θ currently provides no such principle.…”

Section: Related Workmentioning

confidence: 99%

Combining proofs and programs in a dependently typed language

Casinghino

Sjöberg

Weirich

2014

Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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Most dependently-typed programming languages either require that all expressions terminate (e.g. Coq, Agda, and Epigram), or allow infinite loops but are inconsistent when viewed as logics (e.g. Haskell, ATS, mega). Here, we combine these two approaches into a single dependently-typed core language. The language is composed of two fragments that share a common syntax and overlapping semantics: a logic that guarantees total correctness, and a call-by-value programming language that guarantees type safety but not termination. The two fragments may interact: logical expressions may be used as programs; the logic may soundly reason about potentially nonterminating programs; programs can require logical proofs as arguments; and "mobile" program values, including proofs computed at runtime, may be used as evidence by the logic. This language allows programmers to work with total and partial functions uniformly, providing a smooth path from functional programming to dependently-typed programming. AbstractMost dependently-typed programming languages either require that all expressions terminate (e.g. Coq, Agda, and Epigram), or allow infinite loops but are inconsistent when viewed as logics (e.g. Haskell, ATS, Ωmega). Here, we combine these two approaches into a single dependently-typed core language. The language is composed of two fragments that share a common syntax and overlapping semantics: a logic that guarantees total correctness, and a call-by-value programming language that guarantees type safety but not termination. The two fragments may interact: logical expressions may be used as programs; the logic may soundly reason about potentially nonterminating programs; programs can require logical proofs as arguments; and "mobile" program values, including proofs computed at runtime, may be used as evidence by the logic. This language allows programmers to work with total and partial functions uniformly, providing a smooth path from functional programming to dependently-typed programming.

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“…The most closely related work is by Bertot and Komendantsky [5], who use fixed-points in flat pcpos to define partial recursive functions in Coq [4], augmented with some classical axioms. They also show that their extension preserves the possibility to extract programs from Coq developments and provide a Coq command that automates the definition process.…”

Section: Related Workmentioning

confidence: 99%

“…This section shows how to define partial functions in the option monad. It mainly recalls the standard fixed-point construction also used by Bertot and Komendantsky [5], and shows how it is automated in Isabelle/HOL. Later we will generalize it to the heap monad.…”

Section: Recursion In the Option Monadmentioning

confidence: 99%

Recursive Definitions of Monadic Functions

Krauß

2010

Electron. Proc. Theor. Comput. Sci.

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Using standard domain-theoretic fixed-points, we present an approach for defining recursive functions that are formulated in monadic style. The method works both in the simple option monad and the state-exception monad of Isabelle/HOL's imperative programming extension, which results in a convenient definition principle for imperative programs, which were previously hard to define.For such monadic functions, the recursion equation can always be derived without preconditions, even if the function is partial. The construction is easy to automate, and convenient induction principles can be derived automatically.

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Fixed point semantics and partial recursion in Coq

Cited by 15 publications

References 28 publications

Partiality and recursion in interactive theorem provers – an overview

Partiality and recursion in interactive theorem provers – an overview

Combining proofs and programs in a dependently typed language

Recursive Definitions of Monadic Functions

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