Related papers: Impredicative Encodings of (Higher) Inductive Type…

This paper presents a case study of formalizing a normalization proof for Leivant's Predicative System F using the Equations package. Leivant's Predicative System F is a stratified version of System F, where type quantification is annotated…

The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of…

To ensure decidability and consistency of its type theory, a proof assistant should only accept terminating recursive functions and productive corecursive functions. Most proof assistants enforce this through syntactic conditions, which can…

We propose a new positional encoding method for a neural network architecture called the Transformer. Unlike the standard sinusoidal positional encoding, our approach is based on solid mathematical grounds and has a guarantee of not losing…

In this paper, we define indexed type theories which are related to indexed ($\infty$-)categories in the same way as (homotopy) type theories are related to ($\infty$-)categories. We define several standard constructions for such theories…

We study the coherence and conservativity of extensions of dependent type theories by additional strict equalities. By considering notions of congruences and quotients of models of type theory, we reconstruct Hofmann's proof of the…

We present a rich type system with subtyping for an extension of System F. Our type constructors include sum and product types, universal and existential quantifiers, inductive and coinductive types. The latter two size annotations allowing…

Let $G$ be a finite group, $H$ be a normal subgroup of prime index $p$. Let $F$ be a field of either characteristic $0$ or prime to $|G|$. Let $\eta$ be an irreducible $F$-representation of $H$. If $F$ is an algebraically closed field of…

The paper is organized as a self-contained literate Prolog program that implements elements of an executable finite set theory with focus on combinatorial generation and arithmetic encodings. The complete Prolog code is available at…

Homotopy type theory is an interpretation of Martin-L\"of's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for…

The program of internal type theory seeks to develop the categorical model theory of dependent type theory using the language of dependent type theory itself. In the present work we study internal homotopical type theory by relaxing the…

Expansion is an operation on typings (i.e., pairs of typing environments and result types) defined originally in type systems for the lambda-calculus with intersection types in order to obtain principal (i.e., most informative, strongest)…

We present an extension of System F with call-by-name exceptions. The type system is enriched with two syntactic constructs: a union type for programs whose execution may raise an exception at top level, and a corruption type for programs…

We present a coinductive framework for defining and reasoning about the infinitary analogues of equational logic and term rewriting in a uniform, coinductive way. The setup captures rewrite sequences of arbitrary ordinal length, but it has…

We study a type checking algorithm that is able to type check a nontrivial subclass of functional programs that use features such as higher-rank, impredicative and second-order types. The only place the algorithm requires type annotation is…

This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories with possibly…

In type theories, universe hierarchies are commonly used to increase the expressive power of the theory while avoiding inconsistencies arising from size issues. There are numerous ways to specify universe hierarchies, and theories may…

We describe a way to represent computable functions between coinductive types as particular transducers in type theory. This generalizes earlier work on functions between streams by P. Hancock to a much richer class of coinductive types.…

To be usable in practice, interactive theorem provers need to provide convenient and efficient means of writing expressions, definitions, and proofs. This involves inferring information that is often left implicit in an ordinary…

We propose an extension of pure type systems with an algebraic presentation of inductive and co-inductive type families with proper indices. This type theory supports coercions toward from smaller sorts to bigger sorts via explicit type…