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POPL '89 Proceedings of the 16th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
COLOG-88 Proceedings of the international conference on Computer logic
Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
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Science of Computer Programming
Inductively defined types in the calculus of constructions
Proceedings of the fifth international conference on Mathematical foundations of programming semantics
Recursive programming with proofs
Theoretical Computer Science - Special issue on discrete mathematics and applications to computer science
A framework for defining logics
Journal of the ACM (JACM)
The extended calculus of constructions (ECC) with inductive types
Information and Computation
FPCA '93 Proceedings of the conference on Functional programming languages and computer architecture
Fixed points and extensionality in typed functional programming languages
Fixed points and extensionality in typed functional programming languages
Type fixpoints: iteration vs. recursion
Proceedings of the fourth ACM SIGPLAN international conference on Functional programming
Programming with Proofs: A Second Order Type Theory
ESOP '88 Proceedings of the 2nd European Symposium on Programming
Positive Recursive Type Assignment
MFCS '95 Proceedings of the 20th International Symposium on Mathematical Foundations of Computer Science
Inductive Definitions in the system Coq - Rules and Properties
TLCA '93 Proceedings of the International Conference on Typed Lambda Calculi and Applications
Structural Recursive Definitions in Type Theory
ICALP '98 Proceedings of the 25th International Colloquium on Automata, Languages and Programming
Codifying Guarded Definitions with Recursive Schemes
TYPES '94 Selected papers from the International Workshop on Types for Proofs and Programs
Mendler-style inductive types, categorically
Nordic Journal of Computing
Programming with Inductive and Co-Inductive Types
Programming with Inductive and Co-Inductive Types
A categorical programming language
A categorical programming language
CPS translating inductive and coinductive types
PEPM '02 Proceedings of the 2002 ACM SIGPLAN workshop on Partial evaluation and semantics-based program manipulation
Monotone Inductive and Coinductive Constructors of Rank 2
CSL '01 Proceedings of the 15th International Workshop on Computer Science Logic
Type-based termination of recursive definitions
Mathematical Structures in Computer Science
An induction principle for nested datatypes in intensional type theory
Journal of Functional Programming
Some Remarks on Type Systems for Course-of-value Recursion
Electronic Notes in Theoretical Computer Science (ENTCS)
Verification of programs on truly nested datatypes in intensional type theory
MSFP'06 Proceedings of the 2006 international conference on Mathematically Structured Functional Programming
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This paper is a comparative study of a number of (intensional-semantically distinct) least and greatest fixed point operators that natural-deduction proof systems for intuitionistic logics can be extended with in a proof-theoretically defendable way. Eight pairs of such operators are analysed. The exposition is centred around a cube-shaped classification where each node stands for an axiomatization of one pair of operators as logical constants by intended proof and reduction rules and each arc for a proof- and reduction-preserving encoding of one pair in terms of another. The three dimensions of the cube reflect three orthogonal binary options: conventional-style vs. Mendler-style, basic ("[co]iterative") vs. enhanced ("primitive-[co]recursive"), simple vs. course-of-value [co]induction. Some of the axiomatizations and encodings are well known; others, however, are novel; the classification into a cube is also new. The differences between the least fixed point operators considered are illustrated on the example of the corresponding natural number types.