FPCA '89 Proceedings of the fourth international conference on Functional programming languages and computer architecture
Substitution: a formal methods case study using monads and transformations
TAPSOFT '93 Selected papers of the colloquium on Formal approaches of software engineering
Inductively Defined Types in the Calculus of Constructions
Proceedings of the 5th International Conference on Mathematical Foundations of Programming Semantics
Inductive Definitions in the system Coq - Rules and Properties
TLCA '93 Proceedings of the International Conference on Typed Lambda Calculi and Applications
Monadic Presentations of Lambda Terms Using Generalized Inductive Types
CSL '99 Proceedings of the 13th International Workshop and 8th Annual Conference of the EACSL on Computer Science Logic
MPC '98 Proceedings of the Mathematics of Program Construction
COLOG '88 Proceedings of the International Conference on Computer Logic
Type-based termination of recursive definitions
Mathematical Structures in Computer Science
de Bruijn notation as a nested datatype
Journal of Functional Programming
Iteration and coiteration schemes for higher-order and nested datatypes
Theoretical Computer Science - Foundations of software science and computation structures
Recursion on Nested Datatypes in Dependent Type Theory
CiE '08 Proceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms
An induction principle for nested datatypes in intensional type theory
Journal of Functional Programming
(Co-) iteration for higher-order nested datatypes
TYPES'02 Proceedings of the 2002 international conference on Types for proofs and programs
Initial algebra semantics is enough!
TLCA'07 Proceedings of the 8th international conference on Typed lambda calculi and applications
Recursion on Nested Datatypes in Dependent Type Theory
CiE '08 Proceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms
MPC '08 Proceedings of the 9th international conference on Mathematics of Program Construction
Map fusion for nested datatypes in intensional type theory
Science of Computer Programming
The power of parameterization in coinductive proof
POPL '13 Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
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Nested datatypes are families of datatypes that are indexed over all types and where the datatype constructors relate differentmembers of the family. This may be used to represent variable binding or to maintain certain invariants through typing.In dependent type theory, a major concern is the termination of all expressible programs, so that types that depend on object terms can still be type-checked mechanically. Therefore, we study iteration and recursion schemes that have this termination guarantee throughout. This is not based on syntactic criteria (recursive calls with "smaller" arguments) but just on types ("type-based termination"). An important concern are reasoning principles that are compatible with the ambient type theory, in our case induction principles.In previous work, the author has proposed an abstract description of nested datatypes together with a mapping operation (like map for lists) and an iterator on the term side and an induction principle on the logical side that could all be implemented within the Coq system (with impredicative Set that is just needed for the justification, not for the definition and the examples). For verification purposes, it is important to have naturality theorems for the obtained iterative functions. Although intensional type theory does not provide naturality in general, criteria for naturality could be established that are met in case studies on "bushes" and representations of lambda terms (also with explicit flattening).The new contribution is an extension of this abstract description to full primitive recursion and its illustration by way of examples that have been carried out in Coq. Unlike the iterative system, we do not yet have a justification within Coq.