منابع مشابه
Inductive-Inductive Definitions
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 inductively simultaneously with B. In other wor...
متن کاملEvolution of Inductive Definitions
In formal verification, inductive definitions of predicates are widely used when we define recursive notions and algorithms. In actual verification, we often introduce a new inductively defined predicate by modifying an existing inductive definition. In such a case, it is usual that there are some relationships between two predicates, and such relationship theorems make the properties on one pr...
متن کاملElaborating Inductive Definitions
We present an elaboration of inductive definitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive types within type theory. By elaborating an inductive definition – a syntactic artefact – to its code – its semantics – we obtain an internalised account of inductives inside the type theory itself: we claim that reasoning about inductive...
متن کاملA finite axiomatisation of inductive-inductive definitions
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. We give some examples of inductive-indu...
متن کاملFunctional interpretation and inductive definitions
Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1971
ISSN: 0002-9947
DOI: 10.2307/1995907