The EPI Gallinette aims at developing a new generation of proof assistants, with the belief that practical experiments must go in pair with foundational investigations:
Software quality is a requirement that is becoming more and more prevalent,
by now far exceeding the traditional scope of embedded systems. The
development of tools to construct software that respects a given specification
is a major challenge in computer science. Proof assistants
such as Coq 36 provide a formal method whose central
innovation is to produce certified programs by transforming
the very activity of programming. Programming and proving are merged
into a single development activity, informed by an elegant but rigid
mathematical theory inspired by the correspondence between programming,
logic and algebra: the Curry-Howard correspondence. For the
certification of programs, this approach has shown its effectiveness
in the development of important pieces of certified software such
as the C compiler of the CompCert project 41.
The extracted CompCert compiler is reliable and efficient, running
only 15% slower than GCC 4 at optimisation level 2 (gcc -O2),
a level of optimisation that was considered before to be unreliable from
critical applications such as embedded systems.
Proof assistants can also be used to formalise mathematical
theories: they not only provide a means of representing mathematical
theories in a form amenable to computer processing, but their internal
logic provides a language for reasoning about such theories. In the
last decade, proof assistants have been used to verify extremely large
and complicated proofs of recent mathematical results, sometimes requiring
either intensive computations 39, 40
or intricate combinations of a multitude of mathematical theories
38. But formalised mathematics is more than just proof
checking and proof assistants can help with the organisation of mathematical
knowledge or even with the discovery of new constructions and proofs.
Unfortunately, the rigidity of the theory behind proof assistants restricts their expressiveness both as programming languages and as logical systems. For instance, a program extracted from Coq only uses a purely functional subset of OCaml, leaving behind important means of expression such as side-effects and objects. Limitations also appear in the formalisation of advanced mathematics: proof assistants do not cope well with classical axioms such as excluded middle and choice which are sometimes used crucially. The fact of the matter is that the development of proof assistants cannot be dissociated from a reflection on the nature of programs and proofs coming from the Curry-Howard correspondence. In the EPC Gallinette, we propose to address several limitations of proof assistants by pushing the boundaries of this correspondence.
In the 1970's, the Curry-Howard correspondence was seen as a perfect match between functional programs, intuitionistic logic, and Cartesian closed categories. It received several generalisations over the decades, and now it is more widely understood as a fertile correspondence between computation, logic, and algebra.
Nowadays, the Curry-Howard correspondence is not perceived as a perfect match anymore, but rather as a collection of theories meant to explain similar structures at work in logic and computation, underpinned by mathematical abstractions. By relaxing the requirement of a perfect match between programs and proofs, and instead emphasising the common foundations of both, the insights of the Curry-Howard correspondence may be extended to domains for which the requirements of programming and mathematics may in fact be quite different.
Consider the following two major theories of the past decades, which were until recently thought to be irreconcilable:
We now outline a series of scientific challenges aimed at understanding of type theory, effects, and their combination.
More precisely, three key axes of improvement have been identified:
To validate the new paradigms, we propose in Section 3.5 three particular application fields in which members of the team already have a strong expertise: code refactoring, constraint programming and symbolic computation.
The experience of the team on various extensions of type theory (definitional proof irrelevant propositions, observational type theory, gradual type theory, opetopic type theory) begs naturally the question of the integration of these distinct flavours of type theory in a single type theory. At a theoretical level, we will investigate type theories with multiple universe hierarchies hosting theories with potentially incompatible principles able to express efficiently a variety of mathematical situations. At a practical level, we will develop a version of the Coq proof assistant with multiple sorts, generalizing the existing situation where the sorts of types, propositions and definitionally proof-irrelevant propositions cohabit de facto. An important challenge in that direction is to design a sound mechanism of sort polymorphism to factor away the constructions common to multiple sorts and prevent the combinatorial explosion induced by a naive implementation. A somewhat related line of research is designing an efficient decision procedure for universe level constraints, following the work of 35.
In the long quest towards a practical and extensional notion of equality, observational type theory, first introduced by 34 and further developped and studied in 46 and 45, represents an important milestone that should now be implemented in practice. We will pursue in parallel other extensionality principles to enhance the expressivity of type theories.
The investigation of extensions of CIC with side effects, in particular that of exceptions and the addition of a case analysis operator on types, yields important insights to give sound models of the cast calculi behind gradual dependent types. We plan to go beyond these relatively simple extensions and consider other widely used side effects, for instance the addition of global state to a type theory. These type theories with a primitive support for effectful operations could provide a new approach to the verification of programs exhibiting side-effects. Extending our previous work on classical sequent calculus with dependent types, we will study the integration of classical axioms such as excluded middle and choice in rich type theory. One goal is to better integrate insights of (classical) proof theory in the state of art of type theory (or in an alternative approach thereof). We also aim to look at concrete issues met in formalized mathematics stemming from the classical/intuitionist divide.
The MetaCoq project currently provides bare-bones meta-programming facilities of quotation and denotation. We plan to improve this to provide a full-feature meta-programming facility, and explore the possibility to give strong specifications and verify our meta-programs. A prime example of this is the development of support for verified parametricity translations that can have many uses during formalization (elimination principles, automatic transport, etc.).
We aim at pursuing the study of representation independence principles
and the implementation of corresponding tools, so as to dramatically
reduce the practical cost of library development. The mid-term
expected outcome concerns the design of refinements libraries, which
connect proof-oriented with computation-oriented data-structures, and
better transport instruments for formalized mathematics, e.g., automating reasoning modulo structure isomorphisms.
The porting of the developement of logical relations for MLTT of Abel et al. from Agda to Coq paves the may to a much more modular library. We would like to
extend this work by developping a generic framework for dependently-typed
logical relations, and use it for a wide variety of new dependent type theories.
The main goal is to establish strong metatheoretical properties: normalization,
but also suitable forms of interoperability. Ultimately, we
believe this framework could interface with the MetaCoq project.
We will keep investigating the semantic foundations of features of
systems programming languages from a mathematical point of view. Based
on our earlier work showing a link between resources and ordered
logic, we will study resources management in the context of the
formal theory of side-effects and linearity.
Existing theorems will need to be generalized in many ways (extension
of the notions of effect and resource modalities, handling of order,
etc.). A link with linear logic will help make tighter connections
between systems programming and “linear” approaches to program
semantics (ownership, linear types, etc.). The notion of “borrowing”
will be studied from the angle of linear logic, with possible
applications to program verification. This study should also be
extended to notions of fault tolerance (exception-safety and
isolation) which might show links with modal logics.
The anticipated outcome is an understanding of advanced notions in
programming languages that better align with proven concepts from
systems programming, compared to experimental type systems originating
from pure theory, while providing clear distinctions between essential
and accidental aspects of these real-world languages. As concrete
experiments, we will keep researching ways to integrate systems
programming concepts such as resources and fault tolerance in
functional programming languages (notably OCaml and the OCaml-Rust
interface).
We will continue our work on game semantics for programming languages, with the aim of studying interoperability and compilation between languages. Indeed, these semantics are particularly well suited to studying the interaction between a program and an environment written in different languages. We believe this approach will make it possible to overcome major open problems concerning interoperability between languages equipped with abstraction properties statically enforced by parametric polymorphism, and untyped languages where such abstractions properties are enforced dynamically. We will also continue studying the automation of reasoning on these semantics, along the lines of the CAVOC project. To do this, we want to apply abstract interpretation techniques, in particular the Abstracting Abstract Machine methodology, to automatically check accessibility properties on programs, such as unverified assertions.
As part of the CANofGAS project, we also plan to apply these interactive semantics to develop compositional cost models for programs. This would provide compositional reasoning on time and space complexity for higher-order programs.
The MetaCoq project's Achille's heel is that it relies on an assumption of strong normalization for the calculus: there is ongoing work in the team on defining powerful logical relations in Coq without relying on inductive-recursion, that gives hope that a strong-normalization model for a large fragment of MetaCoq can be constructed in the future. The main scientific obstacle is to specify the syntactic guard/productivity condition at the heart of termination checking in such a way that it can be reduced to an eliminator-based definition of (co-)inductive types, which is how they are usually modelled. We anticipate difficulties with nested and indexed inductive types, which might be currently accepted by Coq but difficult to emulate with eliminators. However, this can only lead to a better understanding of the theory. As part of the ReCiProg project, we also plan to establish formal links between the validation criteria derived from circular proofs and this guard condition.
The benefits of formally verified symbolic computations is twofold: increase the trust in computer-produced mathematics and expand the automation available for users of proof assistants. The main challenge is to enable the formal verification of efficient programs, whose correctness proofs involve sophisticated mathematical ingredients rather than subtle memory or parallelism issues. This involves in particular scaling up the automatic transport of libraries, as well as the formal verification of existing imperative code from computer algebra systems (typically written in C).
The MetaCoq erasure pipeline, targeting C or OCaml, provides a guarantee that the evaluation
of the compiled program gives a semantically correct result. However, in general extracted programs
are linked to larger programs of the target language, where we lose guarantees of correctness
in most non-trivial cases of interoperability. We are hence interested in developing
techniques to show interoperability results between code that
is extracted through our certified compilation pipeline and external
code, e.g., in OCaml or C.
In 14, we developped a complete verified extraction pipeline
from Coq to OCaml. The goal for the future is to scale this work to allow more scenarios of
interoperability with effectful target programs, using a formal semantics for the target language.
In particular, we should be able to soundly interpret the primitive constructs that are already part
of Coq, which are fixed-width integers, IEE-754 floating point numbers and applicative arrays.
There is a point of synergy here with the previous goal of enabling the development of efficient,
formally verified symbolic computation.
The team actively participates in open science initiatives, sharing tools and results to avoid redundant efforts, further reducing the collective computational and material costs associated with formal verification research. It spends efforts on the usability of these tools.
The team promotes a socially healthy governance for the tools it develops, taking into account the voices of all stakeholders, in accordance with the principles outlined in the ACM Code of Ethics. In particular, we have been actively involved in the renaming process of the Coq proof assistant, which is now called the Rocq Prover. This renaming initiative originated from a request by the Coq user community.
The research activities of the Gallinette team at Inria have a relatively low environmental footprint compared to other computationally intensive fields, as their work primarily involves theoretical development, formal proofs, and software tool design.
Ltac2 is a member of the ML family of languages, in the sense that it is an effectful call-by-value functional language, with static typing à la Hindley-Milner. It is commonly accepted that ML constitutes a sweet spot in PL design, as it is relatively expressive while not being either too lax (unlike dynamic typing) nor too strict (unlike, say, dependent types).
The main goal of Ltac2 is to serve as a meta-language for Coq. As such, it naturally fits in the ML lineage, just as the historical ML was designed as the tactic language for the LCF prover. It can also be seen as a general-purpose language, by simply forgetting about the Coq-specific features.
Sticking to a standard ML type system can be considered somewhat weak for a meta-language designed to manipulate Coq terms. In particular, there is no way to statically guarantee that a Coq term resulting from an Ltac2 computation will be well-typed. This is actually a design choice, motivated by backward compatibility with Ltac1. Instead, well-typedness is deferred to dynamic checks, allowing many primitive functions to fail whenever they are provided with an ill-typed term.
The language is naturally effectful as it manipulates the global state of the proof engine. This allows to think of proof-modifying primitives as effects in a straightforward way. Semantically, proof manipulation lives in a monad, which allows to ensure that Ltac2 satisfies the same equations as a generic ML with unspecified effects would do, e.g. function reduction is substitution by a value.
Equations is a tool designed to help with the definition of programs in the setting of dependent type theory, as implemented in the Coq proof assistant. Equations provides a syntax for defining programs by dependent pattern-matching and well-founded recursion and compiles them down to the core type theory of Coq, using the primitive eliminators for inductive types, accessibility and equality. In addition to the definitions of programs, it also automatically derives useful reasoning principles in the form of propositional equations describing the functions, and an elimination principle for calls to this function. It realizes this using a purely definitional translation of high-level definitions to core terms, without changing the core calculus in any way, or using axioms.
The main features of Equations include:
Dependent pattern-matching in the style of Agda/Epigram, with inaccessible patterns, with and where clauses. The use of the K axiom or a proof of K is configurable, and it is able to solve unification problems without resorting to the K rule if not necessary.
Support for well-founded and mutual recursion using measure/well-foundedness annotations, even on indexed inductive types, using an automatic derivation of the subterm relation for inductive families.
Support for mutual and nested structural recursion using with and where auxilliary definitions, allowing to factor multiple uses of the same nested fixpoint definition. It proves the expected elimination principles for mutual and nested definitions.
Automatic generation of the defining equations as rewrite rules for every definition.
Automatic generation of the unfolding lemma for well-founded definitions (requiring only functional extensionality).
Automatic derivation of the graph of the function and its elimination principle. In case the automation fails to prove these principles, the user is asked to provide a proof.
A new dependent elimination tactic based on the same splitting tree compilation scheme that can advantageously replace dependent destruction and sometimes inversion as well. The as clause of dependent elimination allows to specify exactly the patterns and naming of new variables needed for an elimination.
A set of Derive commands for automatic derivation of constructions from an inductive type: its signature, no-confusion property, well-founded subterm relation and decidable equality proof, if applicable.
Equations is a function definition plugin for Coq (supporting Coq 8.18 to 8.20, with special support for the Coq-HoTT library), that allows the definition of functions by dependent pattern-matching and well-founded, mutual or nested structural recursion and compiles them into core terms. It automatically derives the clauses equations, the graph of the function and its associated elimination principle.
Equations is based on a simplification engine for the dependent equalities appearing in dependent eliminations that is also usable as a separate tactic, providing an axiom-free variant of dependent destruction.
This is a minor update of Equations, now compatible with Coq 8.20. The main changes are fixes in the funelim tactic and simplification engine to avoid trying to simplify unrelated hypotheses (those under "block" markers). which sometimes led to slowing down the tactic. Also includes performance improvements by @ppedrot.
See https://github.com/mattam82/Coq-Equations/releases/tag/v1.3.1-8.20 for details
The MetaCoq project aims to provide a certified meta-programming environment in Coq. It builds on Template-Coq, a plugin for Coq originally implemented by Malecha (Extensible proof engineering in intensional type theory, Harvard University, 2014), which provided a reifier for Coq terms and global declarations, as represented in the Coq kernel, as well as a denotation command. Recently, it was used in the CertiCoq certified compiler project (Anand et al., in: CoqPL, Paris, France, 2017), as its front-end language, to derive parametricity properties (Anand and Morrisett, in: CoqPL’18, Los Angeles, CA, USA, 2018). However, the syntax lacked semantics, be it typing semantics or operational semantics, which should reflect, as formal specifications in Coq, the semantics of Coq ’s type theory itself. The tool was also rather bare bones, providing only rudimentary quoting and unquoting commands. MetaCoq generalizes it to handle the entire polymorphic calculus of cumulative inductive constructions, as implemented by Coq, including the kernel’s declaration structures for definitions and inductives, and implement a monad for general manipulation of Coq’s logical environment. The MetaCoq framework allows Coq users to define many kinds of general purpose plugins, whose correctness can be readily proved in the system itself, and that can be run efficiently after extraction. Examples of implemented plugins include a parametricity translation and a certified extraction to call-by-value lambda-calculus. The meta-theory of Coq itself is verified in MetaCoq along with verified conversion, type-checking and erasure procedures providing highly trustable alternatives to the procedures in Coq's OCaml kernel. MetaCoq is hence a foundation for the development of higher-level certified tools on top of Coq's kernel. A meta-programming and proving framework for Coq.
MetaCoq is made of 4 main components:
- The entry point of the project is the Template-Coq quoting and unquoting library for Coq which allows quotation and denotation of terms between three variants of the Coq AST: the OCaml one used by Coq's kernel, the Coq one defined in MetaCoq and the one defined by the extraction of the MetaCoq AST, allowing to extract OCaml plugins from Coq implementations.
- The PCUIC component is a full formalization of Coq's typing and reduction rules, along with proofs of important metatheoretic properties: weakening, substitution, validity, subject reduction and principality. The PCUIC calculus differs slightly from the Template-Coq one and verified translations between the two are provided.
- The checker component contains verified implementations of weak-head reduction, conversion and type inference for the PCUIC calculus, along with a verified checker for Coq theories.
- The erasure compoment contains a verified implementation of erasure/extraction from PCUIC to untyped (call-by-value) lambda calculus extended with a dummy value for erased terms.
Release 1.3.2 of the MetaCoq project for Coq 8.19 - 8.20 is available both as source and through opam. See the website for a detailed overview of the project, introductory material and related articles and presentations.
The main changes in this new version w.r.t. v1.2.1 are:
A full integration of the typed erasure phase from the ConCert project in the erasure pipeline, with a complete correctness proof, by @mattam82. Use option MetaCoq Erase -typed to switch it on. It can be configured with the "live" erasure function inside Coq (see erasure_live_test.v) Generalizations of the correctness and simulation lemmas by @yforster @mattam82 and @tabareau, showing in particular that erasure of applications of functions from firstorder types to firstorder types is compiled to applications, justifying separate compilation of functions and their arguments. Using standardization and canonicity, we also show that erased values of programs of firstorder inductive types (non-erasable inductives types for which all constructor argument types are themselves firstorder) are in direct correspondence with their Coq counterparts, allowing sound readback of these values. In other words, evaluating the erased terms under these assumptions faithfully simulates evaluation in Coq. Based on this, CertiCoq and coq-malfunction both implement an Eval variant that reads back Coq values and can be trusted. Support for primitive ints, floats and array literal values. Primitive operations are still treated as axioms to be realized in target languages and the correctness theorems do not apply in their presence yet. Optional passes have been added that replicate the Coq Extraction plugin's functionality, without proof (yet): Inlining of defined constants (e.g. Extract Inline). Reordering of constructors (e.g. part of Extract Inductive). This allows to target different representations in target languages (typically bool in OCaml). Unboxing of singleton unary constructors. For example, exist nat (fun x : nat => x = 1) 1 p : { x : nat | x = 1 } becomes exist 1 after typed erasure and removal of constructor parameters, which can be further unboxed to just 1. CoFixpoints/CoInductives to Lazy/Inductives: cofixpoints and (co)-constructors get translated to fixpoints + lazy/force constructs in lambda-box, allowing efficient evaluation of coinductive terms in target languages (supported only in coq-malfunction/ocaml extraction for now). Beta-reduction. This reduces manifest beta-redexes in the erased terms, especially useful after inlining. The preprint "Verified Extraction from Coq to OCaml" presents the development of the compilation pipeline from Coq to Malfunction/OCaml, including the correctness proofs mentioned above.
The preprint "Correct and Complete Type Checking and Certified Erasure for Coq, in Coq" presents the development of the sound and complete type checker based on bidirectional typing, the meta-theoretical results (subject reduction, standardization, canonicity and consistency) and the verified erasure procedure of this version of MetaCoq.
MetaCoq integrates Template-Coq, a reification and denotation plugin for Coq terms and global declarations, a Template monad for metaprogramming (including the ability to extract these metaprograms to OCaml for efficiency), a formalisation of Coq's calculus PCUIC in Coq, a relatively efficient, sound and complete type checker for PCUIC, a verified type and proof erasure procedure from PCUIC to untyped lambda calculus and a quotation library. MetaCoq provides a low-level interface to develop certified plugins like translations, compilers or tactics in Coq itself.
You can install MetaCoq directly from sources or using opam install coq-metacoq. This release will be included in an upcoming Coq Platform.
The current release includes several subpackages, which can be compiled and installed separately if desired:
the utils library contains extensions to the standard library (notably for reasoning with All/All-n type-valued predicates) (in directory utils, and as coq-metacoq-utils). the common libraries of basic definitions for the abstract syntax trees shared by multiple languages (common, coq-metacoq-common) the Template-Coq quoting library and plugin (template-coq / coq-metacoq-template) a formalisation of meta-theoretical properties of PCUIC, the calculus underlying Coq (pcuic / coq-metacoq-pcuic) a verified equivalence between Template-Coq and PCUIC typing (in directory template-pcuic and as coq-metacoq-template-pcuic) a total verified type-checker for Coq (safechecker / coq-metacoq-safechecker), usable inside Coq. a plugin interfacing with the extracted type-checker in OCaml, providing the MetaCoq SafeCheck <term> command (safechecker-plugin, coq-metacoq-safechecker-plugin) a verified type and proof erasure function for Coq (erasure / coq-metacoq-erasure), usable inside Coq. a plugin interfacing with the extracted erasure pipeline in OCaml, providing the MetaCoq Erase <term> command (erasure-plugin, coq-metacoq-erasure-plugin) a quoting library, allowing the quotation of terms and type derivations along with associated data structures as ASTs/terms (quotation / coq-metacoq-quotation). a set of example translations from Type Theory to Type Theory (translation/ coq-metacoq-translations). A good place to start are the files demo.v, safechecker_test.v, erasure_test.v in the test-suite directory.
This version of MetaCoq was developed by Yannick Forster, Jason Gross, Yann Leray, Matthieu Sozeau and Nicolas Tabareau with contributions from Yishuai Li. You are welcome to contribute by opening issues and PRs. A MetaCoq Zulip stream is also available.
The MetaCoq Team
Memprof-limits is an implementation of per-thread global memory limits, and per-thread allocation limits à la Haskell, and CPU-bound thread cancellation, for OCaml, compatible with multiple threads.
Memprof-limits interrupts the execution by raising an asynchronous exception: an exception that can arise at almost any location in the program. It is provided with a guide on how to recover from asynchronous exceptions and other unexpected exceptions, summarising for the first time practical knowledge acquired in OCaml by the Coq proof assistant as well as in other programming languages.
Memprof-limits is probabilistic, as it is based on the statistical memory accountant memprof. It is provided with a statistical analysis that the user can rely on to have guarantees about the enforcement of limits.
Memprof-limits is an implementation of (per-thread) global memory limits, (per-thread) allocation limits, and cancellation of CPU-bound threads, for OCaml. Memprof-limits interrupts a computation by raising an exception asynchronously and offers features to recover from them such as interrupt-safe resources.
It is provided with an extensive documentation with examples which explains what must be done to ensure one recovers from an interrupt. This documentation summarises for the first time the experience acquired in OCaml in the Coq proof assistant, as well as in other situations in other programming languages.
Trocq is a prototype of a modular parametricity plugin for Coq, aiming to perform proof transfer by translating the goal into an associated goal featuring the target data structures as well as a rich parametricity witness from which a function justifying the goal substitution can be extracted.
The plugin features a hierarchy of parametricity witness types, ranging from structure-less relations to a new formulation of type equivalence, gathering several pre-existing parametricity translations, including univalent parametricity and CoqEAL, in the same framework.
This modular translation performs a fine-grained analysis and generates witnesses that are rich enough to preprocess the goal yet are not always a full-blown type equivalence, allowing to perform proof transfer with the power of univalent parametricity, but trying not to pull in the univalence axiom in cases where it is not required.
The translation is implemented in Coq-Elpi and features transparent and readable code with respect to a sequent-style theoretical presentation.
Proof assistants based on dependent type theory, such as Coq, Lean and Agda, use different universes to classify types, typically combining a predicative hierarchy of universes for computationally-relevant types, and an impredicative universe of proof-irrelevant propositions. In general, a universe is characterized by its sort, such as Type or Prop, and its level, in the case of a predicative sort. Recent research has also highlighted the potential of introducing more sorts in the type theory of the proof assistant as a structuring means to address the coexistence of different logical or computational principles, such as univalence, exceptions, or definitional proof irrelevance. This diversity raises concrete and subtle issues from both theoretical and practical perspectives. In particular, in order to avoid duplicating definitions to inhabit all (combinations of) universes, some sort of polymorphism is needed. Universe level polymorphism is well-known and effective to deal with hierarchies, but the handling of polymorphism between sorts is currently ad hoc and limited in all major proof assistants, hampering reuse and extensibility. In 16, we develop sort polymorphism and its metatheory, studying in particular monomorphization, large elimination, and parametricity. Sort polymorphism is a natural solution that effectively addresses the limitations of current approaches and prepares the ground for future multi-sorted type theories.
Equality is at the heart of dependent type theory, as it plays a fundamental role in specifications and mathematical reasoning. The standard way to handle it in mainstream proof assistants such as Agda , Lean or Coq is based on Martin-Löf’s identity type, which comes straight out of the ’70s—its elegance and simplicity have earned it a long-standing use, despite a major discrepancy with traditional mathematical formulations: it does not satisfy any extensionality principles. Recently, the work on observational equality has regained interest as a new way to encode equality in proof assistants that support a universe of definitionally proof-irrelevant propositions; however it has yet to be integrated in any major proof assistant, because it is not fully compatible with another important feature of type theory: indexed inductive types. In 28, we propose a systematic integration of indexed inductive types with an observational equality, and show that this integration can only be completely satisfactory if the observational equality satisfies the computational rule of Martin-Löf’s identity type. The second contribution of this paper is a formal proof that this additional computation rule, although not present in previous works on observational equality, can be integrated to the system without compromising the decidability of conversion.
We report in 19 on a mechanization in the Coq proof assistant of the decidability of conversion and type-checking for Martin-Löf Type Theory (MLTT), extending a previous Agda formalization. Our development proves the decidability not only of conversion, but also of type-checking, using bidirectional derivations that are canonical for typing. Moreover, we wish to narrow the gap between the object theory we formalize (currently MLTT with Π, Σ, N and one universe) and the metatheory used to prove the normalization result, e.g., MLTT, to a mere difference of universe levels. We thus avoid induction-recursion or impredicativity, which are central in previous work. Working in Coq, we also investigate how its features, including universe polymorphism and the metaprogramming facilities provided by tactics, impact the development of the formalization compared to the development style in Agda. The development is freely accessible on GitHub.
The internal Church thesis (CT) is a logical principle stating that one can associate to any function
Indexed inductive types are essential in dependently-typed programming languages, enabling precise and expressive specifications of data structures and properties. Recognizing that programming and proving with dependent types could benefit from the smooth integration of static and dynamic checking that gradual typing offers, recent efforts have studied gradual dependent types. Gradualizing indexed inductive types however remains mostly unexplored: the standard encodings of indexed inductive types in intensional type theory, e.g., using type-level fixpoints or subset types, break in the presence of gradual features; and previous work on gradual dependent types focus on very specific instances of indexed inductive types. In 15, we contribute a general framework, named PUNK, specifically designed for exploring the design space of gradual indexed inductive types. PUNK is a versatile framework, enabling the exploration of the space between eager and lazy cast reduction semantics that arise from the interaction between casts and the inductive eliminator, allowing them to coexist and interoperate in a single system. Our work provides significant insights into the intersection of dependent types and gradual typing, by proposing a criteria for well-behaved gradual indexed inductive types, systematically addressing the outlined challenges of integrating these types. The contributions of this paper are a step forward in the quest for making gradual theorem proving and gradual dependently-typed programming a reality.
Continuity principles stating that all functions are continuous play a central role in some schools of constructive mathematics. However, there are different ways to formalise the property of being continuous in constructive foundations. We analyse these continuity properties from the perspective of constructive reverse mathematics. We work in constructive type theory, which can be seen as a minimal foundation for constructive reverse mathematics. We treat continuity of functions
Dependently typed languages such as Coq or Agda are very convenient tools to program with strong invariants and develop mathematical proofs. However, a user might be inconvenienced by things such as the fact that n and n+0 are not considered definitionally equal, or the inability to postulate one's own constructs with computation rules such as exceptions. Coq modulo theory solves the first of the two problems by extending Coq's conversion with decision procedures, e.g., for linear integer arithmetic. Rewrite rules can be used to deal with directed equalities for natural numbers, but also to implement exceptions that compute. They were introduced in Agda a few years ago, and later extended to provide more guarantees with a modular confluence checker. We present in 25 a work-in-progress extension of Coq which supports user-defined rewrite rules. While we mostly follow in the footsteps of the Agda implementation, we also have to face new issues due to the differences in the implementation and meta-theory of Coq and Agda. The most prominent one being the different treatment of universes as Coq supports cumulativity but no first-class universe levels.
In interactive theorem proving, a range of different representations may be available for a single mathematical concept, and some proofs may rely on several representations. Without automated support such as proof transfer, theorems available with different representations cannot be combined, without light to major manual input from the user. Tools with such a purpose exist, but in proof assistants based on dependent type theory, it still requires human effort to prove transfer, whereas it is obvious and often left implicit on paper. In 21, 20, we present Trocq, a new proof transfer framework, based on a generalization of the univalent parametricity translation, thanks to a new formulation of type equivalence. This translation takes care to avoid dependency on the axiom of univalence for transfers in a delimited class of statements, and may be used with relations that are not necessarily isomorphisms. We motivate and apply our framework on a set of examples designed to show that it unifies several existing proof transfer tools. The article also discusses an implementation of this translation for the Coq proof assistant, in the Coq-Elpi metalanguage.
Coq is built around a well-delimited kernel that performs type checking for definitions in a variant of the Calculus of Inductive Constructions (CIC). Although the metatheory of CIC is very stable and reliable, the correctness of its implementation in Coq is less clear. Indeed, implementing an efficient type checker for CIC is a rather complex task, and many parts of the code rely on implicit invariants which can easily be broken by further evolution of the code. Therefore, on average, one critical bug has been found every year in Coq. In 17, 18, we present the first implementation of a type checker for the kernel of Coq (without the module system, template polymorphism and
We continued previous work establishing a formal link between resource-management features from systems programming (C++/Rust), and ordered (or non-commutative) logic.
This is an ongoing work to propose a sound design for asynchronous exceptions (or interrupts, in ML parlance) in the OCaml language, inspired by fault-tolerance concepts and Rust's design for exceptions, in collaboration with researchers from Jane Street. In 32, we first describe the problems of recovering from interrupts in Standard ML and OCaml, and how these language deficiencies have been worked around, notably in the implementation of proof assistants and other interactive tools. We then aim to explain more conceptually how safe interrupt handling is possible by evolving ML exceptions. We emphasize key notions: exception-safety concepts originating from C++ and Rust's extension of Erlang's "let it fail" approach to shared mutable state. Lastly we will discuss considerations specific to the OCaml language. This work suggests, in the spirit of structured programming, that while the ML notion of exception lacks structure, it refines into better-structured abstractions for error handling, suggested by its relevant usage by programmers.
A result by Fürhmann and by Hasegawa is important in the type-theoretic semantics of side-effects and linearity, as it characterises what it means to be without side-effects in the context of classical and linear logics. It was admitted to be true, but lacked a written proof, for lack of a conceptual approach to the result. We proposed such a conceptual proof with Éléonore Mangel.
In 26, we discuss the formalization of proofs "by diagram chasing", a standard technique for proving properties in abelian categories. We discuss how the essence of diagram chases can be captured by a simple many-sorted first-order theory, and we study the models and decidability of this theory. The longer-term motivation of this work is the design of a computer-aided instrument for writing reliable proofs in homological algebra, based on interactive theorem provers.
In 23, we describe a formal proof library, developed using the Coq proof assistant, designed to assist users in writing correct diagrammatic proofs, for 1-categories. This library proposes a deep-embedded, domain-specific formal language, which features dedicated proof commands to automate the synthesis, and the verification, of the technical parts often eluded in the literature.
In 31, we have constructed a dynamical theory that is as complete as possible to describe the algebraic properties of the real number field in constructive mathematics without a dependent choice axiom.
One of the central claims of fame of the Coq proof assistant is extraction, i.e., the ability to obtain efficient programs in industrial programming languages such as OCaml, Haskell, or Scheme from programs written in Coq's expressive dependent type theory. Extraction is of great practical usefulness, used crucially e.g., in the CompCert project. However, for such executables obtained by extraction, the extraction process is part of the trusted code base (TCB), as are Coq's kernel and the compiler used to compile the extracted code. The extraction process contains intricate semantic transformation of programs that rely on subtle operational features of both the source and target language. Its code has also evolved since the last theoretical exposition in the seminal PhD thesis of Pierre Letouzey. Furthermore, while the exact correctness statements for the execution of extracted code are described clearly in academic literature, the interoperability with unverified code has never been investigated formally, and yet is used in virtually every project relying on extraction. In 14, we describe the development of a novel extraction pipeline from Coq to OCaml, implemented and verified in Coq itself, with a clear correctness theorem and guarantees for safe interoperability. We build our work on the MetaCoq project, which aims at decreasing the TCB of Coq's kernel by reimplementing it in Coq itself and proving it correct w.r.t. a formal specification of Coq's type theory in Coq. Since OCaml does not have a formal specification, we make use of the Malfunction project specifying the semantics of the intermediate language of the OCaml compiler. Our work fills some gaps in the literature and highlights important differences between the operational semantics of Coq programs and their extracted variants. In particular, we focus on the guarantees that can be provided for interoperability with unverified code, identify guarantees that are infeasible to provide, and raise interesting open question regarding semantic guarantees that could be provided. As central result, we prove that extracted programs of first-order data type are correct and can safely interoperate, whereas for higher-order programs already simple interoperations can lead to incorrect behaviour and even outright segfaults.
Given a table with a minimal set of input columns that functionally determines an output column, we introduce in 22 a method that tries to gradually decompose the corresponding minimal functional dependency (mfd) to acquire a formula expressing the output column in terms of the input columns. A first key element of the method is to create sub-problems that are easier to solve than the original formula acquisition problem, either because it learns formulae with fewer inputs parameters, or as it focuses on formulae of a particular class, such as Boolean formulae; as a result, the acquired formulae can mix different learning biases such as polynomials, conditionals or Boolean expressions. A second key feature of the method is that it can be applied recursively to find formulae that combine polynomial, conditional or Boolean subterms in a nested manner. The method was tested on data for eight families of combinatorial objects; new conjectures were found that were previously unattainable. The method often creates conjectures that combine several formulae into one with a limited number of automatically found Boolean terms.
The bilateral contracts listed below ended in 2023, but the associated overhead continued for an additional 12 months.
FRESCO project on cordis.europa.eu
The use of computers for formulating conjectures, but also for substantiating proof steps, pervades mathematics, even in its most abstract fields. Most computer proofs are produced by symbolic computations, using computer algebra systems. Sadly, these systems suffer from severe, intrinsic flaws, key to their amazing efficiency, but preventing any flavor of post-hoc verification.
But can computer algebra become reliable while remaining fast? Bringing a positive answer to this question represents an outstanding scientific challenge per se, which this project aims at solving.
Our starting point is that interactive theorem provers are the best tools for representing mathematics in silico. But we intend to disrupt their architecture, shaped by decades of applications in computer science, so as to dramatically enrich their programming features, while remaining compatible with their logical foundations.
We will then design a novel generation of mathematical software, based on the firm grounds of modern programming language theory. This environment will feature a new, high-level, performance-oriented programming language, devised for writing efficient and correct code easily, and for serving the frontline of research in computational mathematics. Users will have access to fast implementations, and to powerful proving technologies for verifying any component à la carte, with high productivity. Logic- and computer-based formal proofs will prevent run-time errors, and incorrect mathematical semantics.
We will maintain a close, continuous collaboration with interested high-profile mathematicians, on the verification of cutting-edge research results, today beyond the reach of formal proofs. We ambition to empower mathematical journals to install high-quality artifact evaluation, when peer-reviewing falls short of assessing computer proofs. This project will eventually impact the use of formal methods in engineering, in areas like cryptography or signal-processing.