Abel, A. (2004) Termination checking with types, Theor. Inform. Appl., 38 (4): 277–319.

Abel, A. (2006a) Implementing a normalizer using sized heterogeneous types. In Proceedings of the Workshop on Mathematically Structured Functional Programming, MSFP 2006 (Kuressaare, July 2006), McBride, C. & Uustalu, T. (eds), Electronic Workshops in Computing. BCS.

Abel, A. (2006b) *A Polymorphic Lambda-Calculus with Sized Higher-Order Types*, PhD thesis. Ludwig-Maximilians-Universität München.

Abel, A. (2006c) Semi-continuous sized types and termination. In Proceedings of the 20th International Workshop on Computer Science Logic, CSL 2006 (Szeged, September 2006), Ésik, Z. (ed), Lecture Notes in Computer Science, vol. 4207. Springer, pp. 72–88.

Abel, A. (2008) Normalization for the simply-typed lambda-calculus in Twelf. In Proceedings of the 4th International Workshop on Logical Frameworks and Meta-Languages, LFM 2004 (Cork, July 2004), Schürmann, C. (ed), Electronic Notes in Theoretical Computer Science, vol. 199. Elsevier, pp. 3–16.

Adams, R. (2006) Formalized metatheory with terms represented by an indexed family of types. In Revised Selected Papers from 1st International Workshop on Types for Proofs and Programs, TYPES 2004 (Jouy-en-Josas, December 2004), Filliâtre, J.-C., Paulin-Mohring, C. & Werner, B. (eds), Lecture Notes in Computer Science, vol. 3839. Springer, pp. 1–16.

Altenkirch, T. (1993) A formalization of the strong normalization proof for System F in LEGO. In Proceedings of the 1st International Conference on Typed Lambda Calculi and Applications, TLCA '93 (Utrecht, March 1993), Bezem, M. & Groote, J. F. (eds), Lecture Notes in Computer Science, vol. 664. Springer, pp. 13–28.

Altenkirch, T. & Reus, B. (1999) Monadic presentations of lambda terms using generalized inductive types. In Proceedings of the 13th International Workshop on Computer Science Logic, CSL '99 (Madrid, September 1999), Flum, J. & Rodríguez-Artalejo, M. (eds), Lecture Notes in Computer Science, vol. 1683. Springer, pp. 453–468.

Amadio, R. & Coupet-Grimal, S. (1998) Analysis of a guard condition in type theory (extended abstract). In Proceedings of the 1st International Conference on Foundations of Software Science and Computation Structures, FoSSaCS '98 (Lisbon, March/April 1998), Nivat, M. (ed), Lecture Notes in Computer Science, vol. 1378. Springer, pp. 48–62.

Barthe, G., Frade, M. J., Giménez, E., Pinto, L. & Uustalu, T. (2004) Type-based termination of recursive definitions, Math. Struct Comp. Sci., 14 (1): 97–141.

Barthe, G., Grégoire, B. & Pastawski, F. (2005) Practical inference for type-based termination in a polymorphic setting. In Proceedings of the 7th International Conference on Typed Lambda Calculi and Applications, TLCA 2005 (Nara, April 2005), Urzyczyn, P. (ed), Lecture Notes in Computer Science, vol. 3461. Springer, pp. 71–85.

Barthe, G., Grégoire, B. & Pastawski, F. (2006) : Type-based termination of recursive definitions in the calculus of inductive constructions. In Proceedings of the 13th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR 2006 (Phnom Penh, November 2006), Hermann, M. & Voronkov, A. (eds), Lecture Notes in Artificial Intelligence, vol. 4246. Springer, pp. 257–271. Bellegarde, F. & Hook, J. (1994) Substitution: A formal methods case study using monads and transformations, Sci. Comput. Program., 23 (2–3): 287–311.

Berghofer, S. (2006) Extracting a normalization algorithm in Isabelle/HOL. In Revised Selected Papers from 1st International Workshop on Types for Proofs and Programs, TYPES 2004 (Jouy-en-Josas, December 2004), Filliâtre, J.-C., Paulin-Mohring, C. & Werner, B. (eds), Lecture Notes in Computer Science, vol. 3839. Springer, pp. 50–65.

Bird, R. S. & Paterson, R. (1999) De Bruijn notation as a nested datatype, J. Funct. Program., 9 (1): 77–91.

Blanqui, F. (2004) A type-based termination criterion for dependently-typed higher-order rewrite systems. In Proceedings of the 15th International Conference on Rewriting Techniques and Applications, RTA 2004 (Aachen, June 2004), van Oostrom, V. (ed), Lecture Notes in Computer Science, vol. 3091. Springer, pp. 24–39.

Blanqui, F. (2005) Decidability of type-checking in the Calculus of Algebraic Constructions with size annotations. In Proceedings of the 19th International Workshop on Computer Science Logic, CSL 2005 (Oxford, August 2005), Ong, C.-H. L. (ed), Lecture Notes in Computer Science, vol. 3634. Springer, pp. 135–150.

David, R. & Nour, K. (2005) Arithmetical proofs of strong normalization results for the symmetric lambda-mu-calculus. In Proceedings of the 7th International Conference on Typed Lambda Calculi and Applications, TLCA 2005 (Nara, April 2005), Urzyczyn, P. (ed), Lecture Notes in Computer Science, vol. 3461. Springer, pp. 162–178.

Gentzen, G. (1935) Untersuchungen über das logische Schließen, *Math. Z.*, **39**, 176–210: 405–431. English translation (1969) in The Collected Papers of Gerhard Gentzen, Szabo, M. E. (ed), North-Holland, pp. 68–131.

Giménez, E. (1998) Structural recursive definitions in type theory. In Proceedings of the 25th International Colloquium on Automata, Languages and Programming, ICALP '98 (Aalborg, July 1998), Larsen, K. G., Skyum, S. & Winskel, G. (eds), Lecture Notes in Computer Science, vol. 1443. Springer, pp. 397–408.

Harper, R. & Licata, D. (2007) Mechanizing metatheory in a logical framework, J. Funct. Program., 17 (4–5): 613–673.

Hughes, J., Pareto, L. & Sabry, A. (1996) Proving the correctness of reactive systems using sized types. In Conference Record of 23rd ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL '96 (St. Petersburg Beach, FL, January 1996). ACM Press, pp. 410–423.

Joachimski, F. & Matthes, R. (2003) Short proofs of normalization, Arch. Math. Logic, 42 (1): 59–87.

Mac Lane, S. (1971) Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5. Springer.

Manes, E. G. (1976) Algebraic Theories, Graduate Texts in Mathematics, vol. 26. Springer.

McBride, C. (2006) *Type-Preserving Renaming and Substitution*. Manuscript.

Mendler, N. P. (1987) Recursive types and type constraints in second-order lambda calculus. In Proceedings of the 2nd Annual IEEE Symposium on Logic in Computer Science, LICS '87 (Ithaca, NY, June 1987). IEEE CS Press, pp. 30–36.

Mendler, N. P. (1991) Inductive types and type constraints in the second-order lambda calculus, Ann. Pure Appl. Logic, 51 (1–2): 159–172.

Pareto, L. (2000) *Types for Crash Prevention*, PhD thesis. Chalmers University of Technology.

Prawitz, D. (1965) Natural Deduction: A Proof-Theoretic Study, Stockholm Studies in Philosophy, vol. 3. Almqvist & Wiksell.

Tait, W. W. (1967) Intensional interpretations of functionals of finite type I, J. Symb. Logic, 32 (2): 198–212.

Watkins, K., Cervesato, I., Pfenning, F. & Walker, D. (2004) A concurrent logical framework: the propositional fragment. In Revised Selected Papers from 3rd International Workshop on Types for Proofs and Programs, TYPES 2003 (Torino, April/May 2003), Berardi, S., Coppo, M. & Damiani, F. (eds), Lecture Notes in Computer Science, vol. 3085. Springer, pp. 355–377.