Reduction of Inductive-Inductive Types in Agda
These agda files attempt to reduce inductive-inductive types to inductive Families. They are supposed to be checked using Agda version 2.5.4.1. Later versions' REWRITE pragmas might not be compatible with the formalization anymore, sorry.
II.agda
Contains the syntax for closed inductive-inductive types
IF.agda
Contains the syntax for indexed inductive types with contexts for sort (Scon
) and for point constructors (Con
).
IFA.agda
Contains the Set interpretation of the syntax described in IF.agda, describing algebras of indexed inductive types.
IFM.agda
Contains the model for the indexed inductive types which describes morphisms.
IFD.agda
Contains displayed algebras of indexed inductive types. These algebras depend on an algebra as described in IFA.agda.
IFS.agda
Contains the indexed inductive type interpretation for the section of the aforementioned displayed algebras.
IFEx.agda
Shows the existence of inductive families given the internalization of the syntax as given in IF.agda.
EWRSg.agda
Contains the definitions on how to obtain the the erasure E
, the wellformedness w
, the eliminator relation R
and the initial object Sg
for inductive-inductive types.
II2IF.agda
Encapsulates the previous files by assuming the existence of indexed inductive types and deriving the existence of inductive-inductive types.
TestNat.agda
The types of natural numbers as a test case.
TestVec.agda
The type of vectors as a test case.
TestConTy.agda
The example of contexts and types of a type theoretical syntax as a test case.