Normalization by Completeness with Heyting Algebras

International audienceUsual normalization by evaluation techniques have a strong relationship with completeness with respect to Kripke structures. But Kripke structures is not the only semantics that ts intuitionistic logic: Heyting algebras are a more algebraic alternative.In this paper, we focus on this less investigated area: how completeness with respect to Heyting algebras generate a normalization algorithm for a natural deduction calculus, in the propositional fragment. Our main contributions is that we prove in a direct way completeness of natural deduction with respect to Heyting algebras, that the underlying algorithm natively deals with disjunction, that we formalized those proofs in Coq, and give an extracted algorithm

Formalising Real Numbers in Homotopy Type Theory

International audienceCauchy reals can be defined as a quotient of Cauchy sequences of rationals. In this case, the limit of a Cauchy sequence of Cauchy reals is defined through lifting it to a sequence of Cauchy sequences of rationals. This lifting requires the axiom of countable choice or excluded middle, neither of which is available in homotopy type theory. To address this, the Univalent Foundations Program uses a higher inductive-inductive type to define the Cauchy reals as the free Cauchy complete metric space generated by the rationals. We generalize this construction to define the free Cauchy complete metric space generated by an arbitrary metric space. This forms a monad in the category of metric spaces with Lipschitz functions. When applied to the rationals it defines the Cauchy reals. Finally, we can use Altenkirch and Danielson (2016)'s partiality monad to define a semi-decision procedure comparing a real number and a rational number

Molecular apocrine differentiation is a common feature of breast cancer in patients with germline PTEN mutations

International audienceINTRODUCTION: Breast carcinoma is the main malignant tumor occurring in patients with Cowden disease, a cancer-prone syndrome caused by germline mutation of the tumor suppressor gene PTEN characterized by the occurrence throughout life of hyperplastic, hamartomatous and malignant growths affecting various organs. The absence of known histological features for breast cancer arising in a PTEN-mutant background prompted us to explore them for potential new markers. METHODS: We first performed a microarray study of three tumors from patients with Cowden disease in the context of a transcriptomic study of 74 familial breast cancers. A subsequent histological and immunohistochemical study including 12 additional cases of Cowden disease breast carcinomas was performed to confirm the microarray data. RESULTS: Unsupervised clustering of the 74 familial tumors followed the intrinsic gene classification of breast cancer except for a group of five tumors that included the three Cowden tumors. The gene expression profile of the Cowden tumors shows considerable overlap with that of a breast cancer subgroup known as molecular apocrine breast carcinoma, which is suspected to have increased androgenic signaling and shows frequent ERBB2 amplification in sporadic tumors. The histological and immunohistochemical study showed that several cases had apocrine histological features and expressed GGT1, which is a potential new marker for apocrine breast carcinoma. CONCLUSIONS: These data suggest that activation of the ERBB2-PI3K-AKT pathway by loss of PTEN at early stages of tumorigenesis promotes the formation of breast tumors with apocrine features

The Rewster: The Coq Proof Assistant with Rewrite Rules

International audienceDependently 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 [PT18]. Coq modulo theory [Str10] 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 [CA16] a few years ago, and later extended to provide more guarantees with a modular confluence checker [CTW19, CTW21]. We present a work-in-progress extension 1 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. We will take advantage of this talk to expose our ideas on how to solve the different issues that arise when adding user-defined rewrite rules to a proof assistant by integrating 2 rewrite rules in MetaCoq [SAB + 20, SBF + 20], building on previous work [CTW19, CTW21]

Definitional Proof-Irrelevance without K

International audienceDefinitional equality—or conversion—for a type theory with a decidable type checking is the simplest tool to prove that two objects are the same, letting the system decide just using computation. Therefore, the more things are equal by conversion, the simpler it is to use a language based on type theory. Proof-irrelevance, stating that any two proofs of the same proposition are equal, is a possible way to extend conversion to make a type theory more powerful. However, this new power comes at a price if we integrate it naively, either by making type checking undecidable or by realizing new axioms—such as uniqueness of identity proofs (UIP)—that are incompatible with other extensions, such as univalence. In this paper, taking inspiration from homotopy type theory, we propose a general way to extend a type theory with definitional proof irrelevance, in a way that keeps type checking decidable and is compatible with univalence. We provide a new criterion to decide whether a proposition can be eliminated over a type (correcting and improving the so-called singleton elimination of Coq) by using techniques coming from recent development on dependent pattern matching without UIP. We show the generality of our approach by providing implementations for both Coq and Agda, both of which are planned to be integrated in future versions of those proof assistants

The Advantages of Maintaining a Multitask, Project-Specific Bot: An Experience Report

International audienceBots are becoming a popular method for automating basic everyday tasks in many software projects. This is true in particular because of the availability of many off-the-shelf task-specific bots that teams can quickly adopt (which are sometimes completed with additional task-specific custom bots). Based on our experience in the Coq project, where we have developed and maintained a multi-task project-specific bot, we argue that this alternative approach to project automation should receive more attention because it strikes a good balance between productivity and adaptibility. In this article, we describe the kind of automation that our bot implements, what advantages we have gained by maintaining a project-specific bot, and the technology and architecture choices that have made it possible. We draw conclusions that should generalize to other medium-sized software teams willing to invest in project automation without disrupting their workflows

Setoid type theory - a syntactic translation

International audienceWe introduce setoid type theory, an intensional type theory with a proof-irrelevant universe of propositions and an equality type satisfying function extensionality, propositional extensionality and a definitional computation rule for transport. We justify the rules of setoid type theory by a syntactic translation into a pure type theory with a universe of propositions. We conjecture that our syntax is complete with regards to this translation