Induction - Recursion and initial algebras

Annals of Pure and Applied Logic

Volumn 124, Issue 1-3, 2003, Pages 1-47

  Dybjer, Peter ^a   Setzer, Anton ^b  

^a CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

^b SWANSEA UNIVERSITY   (United Kingdom)

Author keywords

Dependent type theory; Inductive definitions; Inductive recursive definitions; Initial algebras; Large cardinals; Mahlo cardinals; Mahlo universes; Martin L f type theory

Indexed keywords

EID: 0142200756     PISSN: 01680072     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-0072(02)00096-9     Document Type: Article

References (36)

1. Aczel P. Frege structures and the notions of proposition, truth, and set, in: Barwise J. Keisler H.J. Kunen K. (Eds.), The Kleene Symposium 1980 31-59 North-Holland Amsterdam
2. S. Allen, A non-type-theoretic semantics for type-theoretic language, Ph.D. Thesis, Department of Computer Science, Cornell University, 1987
3. Buchholz W. A note on the ordinal analysis of KPM, in: Oikkonen J. Väänänen J. (Eds.), Logic Colloquium '90, ASL Summer Meeting Helsinki Lecture Notes in Logic vol. 2 1993 1-9 Springer Berlin
4. Cartmell J. Generalized algebraic theories and contextual categories Ann. Pure Appl. Logic 32 1986 209-243
5. C. Coquand, The Agda homepage, February 2000. http://www.cs.chalmers.se/~catarina/agda/.
6. T. Coquand, C. Paulin, Inductively defined types, preliminary version, in: COLOG '88, Internat. Conf. on Computer Logic, Lecture Notes Computer Science, vol. 417, Springer, Berlin, 1990
7. Dybjer P. Inductive sets and families in Martin-Löf's type theory and their set-theoretic semantics, in: Huet G. Plotkin G. (Eds.), Logical Frameworks 1991 280 306 Cambridge University Press Cambridge
8. P. Dybjer, Universes and a general notion of simultaneous inductive-recursive definition in type theory, in: B. Nordström, K. Petersson, G. Plotkin (Eds.), Proceedings of the 1992 Workshop on Types for Proofs and Programs, 1992.
9. Dybjer P. Inductive families Formal Aspects Comput. 6 1994 440-465
10. P. Dybjer, Internal type theory, in: TYPES '95, Types for Proofs and Programs, Lecture Notes in Computer Science, Springer, Berlin, 1996, pp 120-134.
11. Dybjer P. A general formulation of simultaneous inductive-recursive definitions in type theory J. Symbolic Logic 65 2 2000 525-549
12. Dybjer P. Setzer A. A finite axiomatization of inductive-recursive definitions Girard J.-Y. Typed Lambda Calculi and Applications Lecture Notes in Computer Science vol. 1581 1999 129-146 Springer Berlin
13. Dybjer P. Setzer A. Indexed induction-recursion, in: Kahle R. Schroeder-Heister P. Stärk R. (Eds.), Proof Theory in Computer Science Lecture Notes in Computer Science vol. 2183 2001 93-113 Springer Berlin
14. Hofmann M. Syntax and semantics of dependent types, in: Pitts A. Dybjer P. (Eds.), Semantics and Logics of Computation 1997 79-130 Cambridge University Press Cambridge
15. Lambek J. Scott P.J. Introduction to higher order categorical logic, in: Studies in Advanced Mathematics vol. 7 1986 Cambridge University Press Cambridge
16. C. Löfwall, G. Sjödin, Strong normalizability in Martin-Löf's type theory, Technical Report R91-09, Swedish Institute of Computer Science, 1991.
17. Martin-Löf P. An intuitionistic theory of types: predicative part, in: Rose H.E. Shepherdson J.C. (Eds.), Logic Colloquium '73 1975 73-118 North-Holland Amsterdam
18. P. Martin-Löf, Constructive mathematics and computer programming, in: Logic, Methodology and Philosophy of Science, VI, 1979, North-Holland, Amsterdam, 1982, pp. 153-175.
19. P. Martin-Löf, Intuitionistic Type Theory, Bibliopolis, Napoli, 1984.
20. P. Martin-Löf, An intuitionistic theory of types, in: G. Sambin, J. Smith (Eds.), Twenty-Five Years of Constructive Type Theory, Oxford University Press, 1998, pp. 127-172 (Reprinted version of an unpublished report from 1972).
21. P.F. Mendler, Predicative type universes and primitive recursion, in: Proc. 6th Annual Symp. on Logic in Computer Science, IEEE Computer Society Press, Silver Spring, MD, 1991.
22. Nordström B. Petersson K. Smith J. Programming in Martin-Löf's Type Theory: An Introduction 1990 Oxford University Press Oxford
23. E. Palmgren, On fixed point operators, inductive definitions and universes in Martin-Löf's type theory, Ph.D. Thesis, Uppsala University, 1991.
24. Palmgren E. Type-theoretic interpretation of iterated, strictly positive inductive definitions Arch. Math. Logic 32 1992 75-99
25. Palmgren E. On universes in type theory, in: Sambin G. Smith J. (Eds.), Twenty-Five Years of Constructive Type Theory 1998 191-204 Oxford University Press Oxford
26. C. Paulin-Mohring, Inductive definitions in the system Coq-rules and properties, in: Typed lambda calculi and applications, Lecture Notes in Computer Science, vol. 664, Springer, Berlin, 1993, pp. 328-345.
27. Rathjen M. Ordinal notations based on a weakly Mahlo cardinal Arch. Math. Logic 29 1990 249-263
28. Rathjen M. Proof-theoretical analysis of KPM Arch. Math. Logic 30 1991 377-403
29. Rathjen M. Collapsing functions based on recursively large cardinals: a well-ordering proof for KPM Arch. Math. Logic 33 1994 35-55
30. Rathjen M. Griffor E.R. Palmgren E. Inaccessibility in constructive set theory and type theory Ann. Pure Appl. Logic 94 1998 181-200
31. D.S. Scott, Constructive validity, in: Symp. on Automatic Demonstration, Lecture Notes in Mathematics, vol. 125, Springer, Berlin, 1970, pp. 237-275.
32. Seely R.A.G. Locally cartesian closed categories and type theory Proc. Cambridge Philos. Soc. 95 1984 33-48
33. A. Setzer, Proof theoretical strength of Martin-Löf type theory with W-type and one universe, Ph.D. Thesis, Fakultät für Mathematik der Ludwig-Maximilians-Universität München, 1993.
34. A. Setzer, A model for a type theory with Mahlo universe, Draft, available from http://www-compsci.swan.ac.uk/~csetzer/
35. Setzer A. A type theory for Mahlo universes, Abstract for Logic Colloquium 95 Bull. Symbolic Logic 3 1997 128-129
36. Setzer A. Extending Martin-Löf type theory by one Mahlo-universe Arch. Math. Logic 39 2000 155-181