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Induction-induction is a principle for defining data types in Martin-Löf Type Theory. An inductive-inductive definition consists of a set A, together with an A-indexed family B : A Ñ Set, where both A and B are inductively defined in such a way that the constructors for A can refer to B and vice versa. In addition, the constructors for B can refer to the… (More)

We present a principle for introducing new types in type theory which generalises strictly positive indexed inductive data types. In this new principle a set A is defined inductively simultaneously with an A-indexed set B, which is also defined inductively. Compared to indexed inductive definitions, the novelty is that the index set A is generated… (More)

In the 1980s, John Reynolds postulated that a parametrically polymorphic function is an ad-hoc polymorphic function satisfying a uniformity principle. This allowed him to prove that his set-theoretic semantics has a relational lifting which satisfies the Identity Extension Lemma and the Abstraction Theorem. However, his definition (and subsequent variants)… (More)

This paper combines reflexive-graph-category structure for relational parametricity with fibrational models of impredicative poly-morphism. To achieve this, we modify the definition of fibrational model of impredicative polymorphism by adding one further ingredient to the structure: comprehension in the sense of Lawvere. Our main result is that such… (More)

Minlog is a proof assistant which automatically extracts computational content in an extension of Gödel's T from formalized proofs. We report on extending Minlog to deal with predicates defined using a particular combination of induction and coinduction, via so-called nested definitions. In order to increase the efficiency of the extracted programs, we have… (More)

- Robert Atkey, Neil Ghani, Fredrik Nordvall Forsberg, Timothy Revell, Sam Staton, Nordvall Forsberg
- 2015

We provide a categorical framework for models of a type theory that has special types for physical quantities. The types are indexed by the physical dimensions that they involve. Fibrations are used to organize this index structure in the models of the type theory. We develop some informative models of this type theory: firstly, a model based on group… (More)

This article is concerned with the application of the program extraction technique to a new class of problems: the synthesis of decision procedures for the classical satis-fiability problem that are correct by construction. To this end, we formalize a completeness proof for the DPLL proof system and extract a SAT solver from it. When applied to a… (More)