Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
The ALF proof editor and its proof engine
TYPES '93 Proceedings of the international workshop on Types for proofs and programs
Inductive Definitions in the system Coq - Rules and Properties
TLCA '93 Proceedings of the International Conference on Typed Lambda Calculi and Applications
A Finite Axiomatization of Inductive-Recursive Definitions
TLCA '99 Proceedings of the 4th International Conference on Typed Lambda Calculi and Applications
Nested General Recursion and Partiality in Type Theory
TPHOLs '01 Proceedings of the 14th International Conference on Theorem Proving in Higher Order Logics
PTCS '01 Proceedings of the International Seminar on Proof Theory in Computer Science
PTCS '01 Proceedings of the International Seminar on Proof Theory in Computer Science
Universes for generic programs and proofs in dependent type theory
Nordic Journal of Computing
Constructing strictly positive families
CATS '07 Proceedings of the thirteenth Australasian symposium on Theory of computing - Volume 65
Proceedings of the 3rd workshop on Programming languages meets program verification
Electronic Notes in Theoretical Computer Science (ENTCS)
Testing and proving distributed algorithms in constructive type theory
TAP'07 Proceedings of the 1st international conference on Tests and proofs
Outrageous but meaningful coincidences: dependent type-safe syntax and evaluation
Proceedings of the 6th ACM SIGPLAN workshop on Generic programming
Proceedings of the 15th ACM SIGPLAN international conference on Functional programming
Partial recursive functions in martin-löf type theory
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
Exploring the regular tree types
TYPES'04 Proceedings of the 2004 international conference on Types for Proofs and Programs
GMETA: a generic formal metatheory framework for first-order representations
ESOP'12 Proceedings of the 21st European conference on Programming Languages and Systems
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We give two finite axiomatizations of indexed inductive-recursive definitions in intuitionistic type theory. They extend our previous finite axiomatizations of inductive-recursive definitions of sets to indexed families of sets and encompass virtually all definitions of sets which have been used in intuitionistic type theory. The more restricted of the two axiomatization arises naturally by considering indexed inductive-recursive definitions as initial algebras in slice categories, whereas the other admits a more general and convenient form of an introduction rule.The class of indexed inductive-recursive definitions properly contains the class of indexed inductive definitions (so called "inductive families"). Such definitions are ubiquitous when using intuitionistic type theory for formalizing mathematics and program correctness. A side effect of the present paper is to get compact finite axiomatizations of indexed inductive definitions in intuitionistic type theory as special cases.Proper indexed inductive-recursive definitions (those which do not correspond to indexed inductive definitions) are useful in intuitionistic metamathematics, and as an example we formalize Tait-style computability predicates for dependent types. We also show that Palmgren's prooftheoretically strong construction of higher-order universes is an example of a proper indexed inductive-recursive definition. A third interesting example is provided by Bove and Capretta's definition of termination predicates for functions defined by nested recursion.Our axiomatizations form a powerful foundation for generic programming with dependent types by introducing a type of codes for indexed inductive-recursive definitions and making it possible to define generic functions by recursion on this type.