Related papers: Formalizing relations in type theory
Dependently typed proof assistant rely crucially on definitional equality, which relates types and terms that are automatically identified in the underlying type theory. This paper extends type theory with definitional functor laws,…
We introduce basic notions in category theory to type theorists, including comprehension categories, categories with attributes, contextual categories, type categories, and categories with families along with additional discussions that are…
Process theories combine a graphical language for compositional reasoning with an underlying categorical semantics. They have been successfully applied to fields such as quantum computation, natural language processing, linear dynamical…
Refinement types sharpen systems of simple and dependent types by offering expressive means to more precisely classify well-typed terms. We present a system of refinement types for LF in the style of recent formulations where only canonical…
We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the…
Representation theorems for formal systems often take the form of an inductive translation that satisfies certain invariants, which are proved inductively. Theory morphisms and logical relations are common patterns of such inductive…
In this paper we develope a categorical theory of relations and use this formulation to define the notion of quantization for relations. Categories of relations are defined in the context of symmetric monoidal categories. They are shown to…
We use type-theoretic techniques to present an algebraic theory of $\infty$-categories with strict units. Starting with a known type-theoretic presentation of fully weak $\infty$-categories, in which terms denote valid operations, we extend…
Logical relations are one of the most powerful techniques in the theory of programming languages, and have been used extensively for proving properties of a variety of higher-order calculi. However, there are properties that cannot be…
Bidirectional typing is a discipline in which the typing judgment is decomposed explicitly into inference and checking modes, allowing to control the flow of type information in typing rules and to specify algorithmically how they should be…
The growing complexity of modern practical problems puts high demands on the mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice becomes…
We introduce judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and natural deduction as special kinds of…
Inspired by the trend on unifying theories of programming, this paper shows how the algebraic treatment of standard data dependency theory equips relational data with functional types and an associated type system which is useful for type…
We provide a treatment of isomorphism within a set-theoretic formulation of dependent type theory. Type expressions are assigned their natural set-theoretic compositional meaning. Types are divided into small and large types --- sets and…
We present gradual type theory, a logic and type theory for call-by-name gradual typing. We define the central constructions of gradual typing (the dynamic type, type casts and type error) in a novel way, by universal properties relative to…
We present a type theory combining both linearity and dependency by stratifying typing rules into a level for logics and a level for programs. The distinction between logics and programs decouples their semantics, allowing the type system…
Fiore and Hur recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a…
Logical relations and their generalizations are a fundamental tool in proving properties of lambda-calculi, e.g., yielding sound principles for observational equivalence. We propose a natural notion of logical relations able to deal with…
Refinement types are a well-studied manner of performing in-depth analysis on functional programs. The dependency pair method is a very powerful method used to prove termination of rewrite systems; however its extension to higher order…
A theory graph is a network of axiomatic theories connected with meaning-preserving mappings called theory morphisms. Theory graphs are well suited for organizing large bodies of mathematical knowledge. Traditional and formal proofs do not…