摘要：Inductive-inductive types (IITs) are a generalisation of inductive types in type theory. They allow the mutual definition of types with multiple sorts where later sorts can be indexed by previous ones. An example is the Chapman-style syntax of type theory with conversion relations for each sort where e.g. the sort of types is indexed by contexts. In this paper we show that if a model of extensional type theory (ETT) supports indexed W-types, then it supports finitely branching IITs. We use a small internal type theory called the theory of signatures to specify IITs. We show that if a model of ETT supports the syntax for the theory of signatures, then it supports all IITs. We construct this syntax from indexed W-types using preterms and typing relations and prove its initiality following Streicher. The construction of the syntax and its initiality proof were formalised in Agda.
