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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)

Data types are undergoing a major leap forward in their sophistication driven by a conjunction of i) theoretical advances in the foundations of data types, and ii) requirements of programmers for ever more control of the data structures they work with. In this paper we develop a theory of indexed data types where, crucially, the indices are generated… (More)

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)

Induction-induction is a principle for mutually defining data types A ∶ Set and B ∶ A → Set. Both A and B are defined inductively, and the constructors for A can refer to B and vice versa. In addition, the constructor for B can refer to the constructor for A. Induction-induction occurs in a natural way when formalising dependent type theory in type theory.… (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)

Homotopy Type Theory extends intensional Martin-Löf Type Theory with two new features: Voevodsky's Univalence Axiom and higher inductive types (HITs). Both are motivated by interpretations of Type Theory into abstract homotopy theory, where intuitively types are interpreted as topological spaces, terms are interpreted as points, and the identity type is… (More)

- Hubert Comon-Lundh, Joseph Y. Halpern, Prakash Panangaden, Rajeev Alur, Loris D’Antoni, Jyotirmoy Deshmukh +37 others
- 2013

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)