Structuring Metatheory on Inductive Definitions

Structuring Metatheory on Inductive Definitions

We examine a problem in formal metatheory: if theories are structured hierarchically, there are metatheorems which hold in only some extensions. We illustrate this using modal logics and the deduction theorem. We show how statements of metatheorems in such hierarchies can take account of possible theory extensions; i.e. a metatheorem formalizes not only the theory in which it holds, but also un...

Structuring Metatheory on Inductive Definitions

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...

Evolution 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...

Elaborating 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...