Generic datatypes à la carte

Keuchel, Steven; Schrijvers, Tom

doi:10.1145/2502488.2502491

Cited by 10 publications

(7 citation statements)

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“…From the point of view of possible applications to semantics and verification in frameworks such as OTT [15], the relational style that can be supported seems to fit in well with SOS and in particular with component-based approaches, such as the one proposed by Churchill, Mosses, Sculthorpe and Torrini [4]. Our plans for future work include integrating our technique in MTC/3MT, and comparing this approach with the container-based one proposed by Keuchel and Schrijvers [11].…”

Section: Resultsmentioning

confidence: 69%

“…Motivated by the importance of modularity in program development, semantics and verification, we have discussed the use of MDTs, their semantic foundations and their impredicative encoding along the lines of existing work [7,11,16]. We have shown how impredicative MDT encodings based on Mendler algebras can be used to reason about inductively defined relations, in a way that is comparatively close to a more conventional style of reasoning based on closed datatypes, by providing a simpler notion of proof algebra, if less general, than the one proposed by Delaware et al [7].…”

Section: Resultsmentioning

confidence: 99%

“…Notice that the definition of fold C F C f does not guarantee the uniqueness of the mediating map -it rather corresponds to a condition called quasi-initiality by Wadler [19]. In order to obtain uniqueness, hence to ensure that in C is an isomorphism, the following implication needs to be proved for F [7,11,10].…”

Section: Datatypes A-la-cartementioning

confidence: 99%

“…One of the main approaches to represent MDT in Coq, due to Delaware, Oliveira and Schrijvers [7] and implemented in the MTC/3MT framework [6], takes advantage of impredicativity, and relies on the Church encoding of fixed points (as recalled in Section 3). Another promising approach, due to Keuchel and Schrijvers [11], relies on containers -it is predicative, but it involves a more indirect representation of types. Church encodings are purely based on CC and do not involve any extra-logical machinery -however, they rather complicate inductive reasoning.…”

Section: Introductionmentioning

confidence: 99%

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Reasoning about modular datatypes with Mendler induction

Torrini

Schrijvers

2015

Electron. Proc. Theor. Comput. Sci.

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In functional programming, datatypes a la carte provide a convenient modular representation of recursive datatypes, based on their initial algebra semantics. Unfortunately it is highly challenging to implement this technique in proof assistants that are based on type theory, like Coq. The reason is that it involves type definitions, such as those of type-level fixpoint operators, that are not strictly positive. The known work-around of impredicative encodings is problematic, insofar as it impedes conventional inductive reasoning. Weak induction principles can be used instead, but they considerably complicate proofs. This paper proposes a novel and simpler technique to reason inductively about impredicative encodings, based on Mendler-style induction. This technique involves dispensing with dependent induction, ensuring that datatypes can be lifted to predicates and relying on relational formulations. A case study on proving subject reduction for structural operational semantics illustrates that the approach enables modular proofs, and that these proofs are essentially similar to conventional ones.Comment: In Proceedings FICS 2015, arXiv:1509.0282

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Section: Resultsmentioning

confidence: 69%

Section: Resultsmentioning

confidence: 99%

Section: Datatypes A-la-cartementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Reasoning about modular datatypes with Mendler induction

Torrini

Schrijvers

2015

Electron. Proc. Theor. Comput. Sci.

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“…She got in contact with a group of sages in the area of verification stating her problem concerning Coq's restriction. These experts pointed her to a manuscript by Keuchel et al [16]. Keuchel and Schrijvers wanted to define the following data type in Coq that can be used to define the fix point of a functor.…”

Section: Free Monadsmentioning

confidence: 99%

One Monad to Prove Them All

Dylus¹,

Christiansen

Teegen

2019

Programming

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One Monad to Prove Them All is a modern fairy tale about curiosity and perseverance, two important properties of a successful PhD student. We follow the PhD student Mona on her adventure of proving properties about Haskell programs in the proof assistant Coq.On the one hand, as a PhD student in computer science Mona observes an increasing demand for correct software products. In particular, because of the large amount of existing software, verifying existing software products becomes more important. Verifying programs in the functional programming language Haskell is no exception. On the other hand, Mona is delighted to see that communities in the area of theorem proving are becoming popular. Thus, Mona sets out to learn more about the interactive theorem prover Coq and verifying Haskell programs in Coq.To prove properties about a Haskell function in Coq, Mona has to translate the function into Coq code. As Coq programs have to be total and Haskell programs are often not, Mona has to model partiality explicitly in Coq. In her quest for a solution Mona finds an ancient manuscript that explains how properties about Haskell functions can be proven in the proof assistant Agda by translating Haskell programs into monadic Agda programs. By instantiating the monadic program with a concrete monad instance the proof can be performed in either a total or a partial setting. Mona discovers that the proposed transformation does not work in Coq due to a restriction in the termination checker. In fact the transformation does not work in Agda anymore as well, as the termination checker in Agda has been improved.We follow Mona on an educational journey through the land of functional programming where she learns about concepts like free monads and containers as well as basics and restrictions of proof assistants like Coq. These concepts are well-known individually, but their interplay gives rise to a solution for Mona's problem based on the originally proposed monadic tranformation that has not been presented before. When Mona starts to test her approach by proving a statement about simple Haskell functions, she realizes that her approach has an additional advantage over the original idea in Agda. Mona's final solution not only works for a specific monad instance but even allows her to prove monad-generic properties. Instead of proving properties over and over again for specific monad instances she is able to prove properties that hold for all monads representable by a container-based instance of the free monad. In order to strengthen her confidence in the practicability of her approach, Mona evaluates her approach in a case study that compares two implementations for queues. In order to share the results with other functional programmers the fairy tale is available as a literate Coq file.If you are a citizen of the land of functional programming or are at least familiar with its customs, had a journey that involved reasoning about functional programs of your own, or are just a curious soul looking for the next story about monads ...

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Completeness Theorems for First-Order Logic Analysed in Constructive Type Theory

Forster

Kirst

Wehr

2019

Lecture Notes in Computer Science

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We study various formulations of the completeness of firstorder logic phrased in constructive type theory and mechanised in the Coq proof assistant. Specifically, we examine the completeness of variants of classical and intuitionistic natural deduction and sequent calculi with respect to model-theoretic, algebraic, and game-theoretic semantics. As completeness with respect to the standard model-theoretic semantics à la Tarski and Kripke is not readily constructive, we analyse connections of completeness theorems to Markov's Principle and Weak Kőnig's Lemma and discuss non-standard semantics admitting assumption-free completeness. We contribute a reusable Coq library for first-order logic containing all results covered in this paper. 1 Accepted in Russian constructivism while in conflict with Brouwer's intuitionism arXiv:2006.04399v1 [cs.LO] 8 Jun 2020 Fact 6. T and F are data types and Γ ϕ and Γ c ϕ are enumerable. Proof. By the techniques discussed in [16], e.g. Fact 3.19. The standard model-theoretic completeness proofs analysed in Section 3 require the assumption of Markov's Principle. A proposition P : P is called stable if ¬¬P → P and, analogously, a predicate p : X → P is called stable if p x is stable for all x. A synthetic version of Markov's Principle states that satisfiability of Boolean sequences is stable (cf. [51]): MP := ∀f : N → B. ¬¬(∃n. f n = tt) → ∃n. f n = tt Note that MP is trivially implied by Excluded Middle EM := ∀P : P. P ∨ ¬P . Moreover, MP regulates the behaviour of computationally tractable predicates: Fact 7. MP holds iff all enumerable predicates on discrete types are stable. Proof. The direction from left to right is Fact 2.18 in [16]. For the reverse direction assume that enumerable predicates on discrete types are stable. Let f : N → B and let p : 1 → P be defined by p x := ∃n. f n = tt. The predicate p is enumerable by f n := if f n then else ∅. Stability of p is now equivalent to ¬¬(∃n. f n = tt) → (∃n. f n = tt).As a consequence of Fact 6 and Fact 7, MP implies that the deduction systems Γ ϕ and Γ c ϕ are stable. In fact, only these stabilities are required for the standard model-theoretic completeness proofs discussed in the next section and they are equivalent to MP L , a version of Markov's Principle stated for the call-by-value λ-calculus L [57,22] and its halting problem E: ¬¬Es → EsWe will prove the following in Section 3.5: Lemma 8. MP L , stability of Γ ϕand stability of Γ c ϕ are all equivalent. Completeness Theorems for FOL Analysed in Constructive Type Theory 73 Model-Theoretic SemanticsThe first variant of semantics we consider is based on the idea of interpreting terms as objects in a model and embedding the logical connectives into the metalogic. A formula is considered valid if it is satisfied by all models. The simplest case is Tarski semantics, coinciding with classical deduction via Henkin's completeness proof factoring through a (constructive) model-existence theorem [30]. Kripke semantics, coinciding with intuitionistic deduction, add more structure by connec...

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Generic datatypes à la carte

Cited by 10 publications

References 28 publications

Reasoning about modular datatypes with Mendler induction

Reasoning about modular datatypes with Mendler induction

One Monad to Prove Them All

Completeness Theorems for First-Order Logic Analysed in Constructive Type Theory

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