s acc ) mul {@(F, n), U, X, W }, which reduces to 9. lim(F ) ? {n},s acc @(F, n) which succeeds since O = T S O, F is accessible in lim(F ) and s ? {n} n by case Case 1a. Our remaining goal 10. s ? {n} {@(F, n), ) which yields goals 8 and 12 by Case 1b 8. {lim(F ) } decomposes into three goals trivially solved by Case 1e, that is 11. s ? {n} {U, X, W }, and one additional goal 12. s ? {n} @(F, n) which yields two goals by Case 1c 13. s ? {n} F , which succeeds by Case 1f, and 14. s ? {n} n which succeeds by Case 1a, thus ending the computation ,
Termination of term rewriting using dependency pairs, Theoretical Computer Science, vol.236, issue.1-2, pp.133-178, 2000. ,
DOI : 10.1016/S0304-3975(99)00207-8
Adding algebraic rewriting to the calculus of constructions : Strong normalization preserved, Proceedings of the 2nd International Workshop on Conditional and Typed Rewriting Systems, 1990. ,
DOI : 10.1007/3-540-54317-1_96
Combining first and higher order rewrite systems with type assignment systems, Proceedings of the 1st International Conference on Typed Lambda Calculi and Applications, 1993. ,
DOI : 10.1007/BFb0037098
Modularity of termination and confluence in combinations of rewrite systems with ????, Proceedings of the 20th International Colloquium on Automata , Languages and Programming, 1993. ,
DOI : 10.1007/3-540-56939-1_110
Modularity of strong normalization and confluence in the algebraic-??-cube, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science, 1994. ,
DOI : 10.1109/LICS.1994.316049
Type-based termination of recursive definitions, Mathematical Structures in Computer Science, vol.14, issue.1, pp.97-141, 2004. ,
DOI : 10.1017/S0960129503004122
The size-change principle for program termination, Proceedings of the 28th ACM Symposium on Principles of Programming Languages, 2001. ,
Definitions by rewriting in the Calculus of Constructions (extended abstract), Proceedings of the 16th IEEE Symposium on Logic in Computer Science, 2001. ,
Higher-order dependency pairs, Proceedings of the 8th International Workshop on Termination, 2006. ,
URL : https://hal.archives-ouvertes.fr/inria-00084821
Termination and Confluence of Higher-Order Rewrite Systems, Proceedings of the 11th International Conference on Rewriting Techniques and Applications, Lecture Notes in Computer Science 1833, 2000. ,
DOI : 10.1007/10721975_4
URL : https://hal.archives-ouvertes.fr/inria-00105556
A Type-Based Termination Criterion for Dependently-Typed Higher-Order Rewrite Systems, Proceedings of the 15th International Conference on Rewriting Techniques and Applications, 2004. ,
DOI : 10.1007/978-3-540-25979-4_2
URL : https://hal.archives-ouvertes.fr/inria-00100254
Definitions by rewriting in the Calculus of Constructions, Mathematical Structures in Computer Science, vol.15, issue.1, pp.37-92, 2005. ,
DOI : 10.1017/S0960129504004426
URL : https://hal.archives-ouvertes.fr/inria-00105568
Inductive Types in the Calculus of Algebraic Constructions, Fundamenta Informaticae, vol.65, issue.12, pp.61-86, 2005. ,
DOI : 10.1007/3-540-44904-3_4
URL : https://hal.archives-ouvertes.fr/inria-00105617
Computability closure: Ten years later In Rewriting, Computation and Proof ? Essays Dedicated to Jean-Pierre Jouannaud on the Occasion of His 60th Birthday, Lecture Notes in Computer Science, vol.4600, 2007. ,
Inductive-data-type systems, Theoretical Computer Science, vol.272, issue.1-2, pp.41-68, 2002. ,
DOI : 10.1016/S0304-3975(00)00347-9
URL : https://hal.archives-ouvertes.fr/inria-00105578
Higher-Order Termination: From Kruskal to Computability, Proceedings of the 13th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, 2006. ,
DOI : 10.1007/11916277_1
URL : https://hal.archives-ouvertes.fr/inria-00091308
HORPO with Computability Closure: A Reconstruction, Proceedings of the 14th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, 2007. ,
DOI : 10.1007/978-3-540-75560-9_12
URL : https://hal.archives-ouvertes.fr/inria-00168304
Termination Analysis of the untyped lambda-calculus, Proceedings of the 15th International Conference on Rewriting Techniques and Applications, pp.1-23, 2004. ,
Ordering-based methods for proving termination automatically, 2003. ,
A Monotonic Higher-Order Semantic Path Ordering, Proceedings of the 8th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, 2001. ,
DOI : 10.1007/3-540-45653-8_37
Orderings and Constraints: Theory and Practice of Proving Termination, Rewriting, Computation and Proof ? Essays Dedicated to Jean-Pierre Jouannaud on the Occasion of His 60th Birthday, 2007. ,
DOI : 10.1007/978-3-540-73147-4_2
Combining algebra and higher-order types, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science, 1988. ,
DOI : 10.1109/LICS.1988.5103
Polymorphic rewriting conserves algebraic strong normalization, Proceedings of the 16th International Colloquium on Automata, Languages and Programming, 1989. ,
DOI : 10.1016/0304-3975(91)90037-3
URL : http://doi.org/10.1016/0304-3975(91)90037-3
Calculating sized types, Higher-Order and Symbolic Computation, vol.14, issue.2/3, pp.261-300, 2001. ,
DOI : 10.1023/A:1012996816178
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.21.4005
Orderings for term-rewriting systems, Theoretical Computer Science, vol.17, issue.3, pp.279-301, 1982. ,
DOI : 10.1016/0304-3975(82)90026-3
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.404.4058
Adding algebraic rewriting to the untyped lambda calculus, Information and Computation, vol.101, issue.2, pp.251-267, 1992. ,
DOI : 10.1016/0890-5401(92)90064-M
Automated Termination Analysis for Haskell: From Term Rewriting to Programming Languages, Proceedings of the 17th International Conference on Rewriting Techniques and Applications, 2006. ,
DOI : 10.1007/11805618_23
The Dependency Pair Framework: Combining Techniques for Automated Termination Proofs, Proceedings of the 11th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, 2004. ,
DOI : 10.1007/978-3-540-32275-7_21
Well-Founded Recursive Relations, Proceedings of the 15th International Conference on Computer Science Logic, 2001. ,
DOI : 10.1007/3-540-44802-0_34
Proving the correctness of reactive systems using sized types, Proceedings of the 23rd ACM SIGPLAN-SIGACT symposium on Principles of programming languages , POPL '96, 1996. ,
DOI : 10.1145/237721.240882
The pattern calculus, ACM Transactions on Programming Languages and Systems, vol.26, issue.6, pp.911-937, 2004. ,
DOI : 10.1145/1034774.1034775
A computation model for executable higher-order algebraic specification languages, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science, 1991. ,
DOI : 10.1109/LICS.1991.151659
Abstract data type systems, Theoretical Computer Science, vol.173, issue.2, pp.349-391, 1997. ,
DOI : 10.1016/S0304-3975(96)00161-2
URL : http://doi.org/10.1016/s0304-3975(96)00161-2
Higher-Order Orderings for Normal Rewriting, Proceedings of the 17th International Conference on Rewriting Techniques and Applications, 2006. ,
DOI : 10.1007/11805618_29
The higher-order recursive path ordering, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158), 1999. ,
DOI : 10.1109/LICS.1999.782635
Rewrite orderings for higher-order terms in ??-long ??-normal form and the recursive path ordering, Theoretical Computer Science, vol.208, issue.1-2, pp.33-58, 1998. ,
DOI : 10.1016/S0304-3975(98)00078-4
Higher-order recursive path orderings " ` a la carte, 2001. ,
Polymorphic higher-order recursive path orderings, Journal of the ACM, vol.54, issue.1, pp.1-48, 2007. ,
DOI : 10.1145/1206035.1206037
Two generalizations of the Recursive Path Ordering, 1980. ,
On recursive path ordering, Theoretical Computer Science, vol.40, issue.2-3, pp.323-328, 1985. ,
DOI : 10.1016/0304-3975(85)90175-6
Termination of combined (rewrite and ??-calculus) systems, Proceedings of the 3rd International Workshop on Conditional and Typed Rewriting Systems, 1992. ,
DOI : 10.1007/3-540-56393-8_10
Strong normalizability for the combined system of the typed lambda calculus and an arbitrary convergent term rewrite system, Proceedings of the 1989 International Symposium on Symbolic and Algebraic Computation ,
On dependency pair method for proving termination of higherorder rewrite systems, IEICE Transactions on Information and Systems, issue.3, pp.88-583, 2005. ,
An extension of dependency pair method for proving termination of higher-order rewrite systems, IEICE Transactions on Information and Systems, issue.8, pp.84-1025, 2001. ,
Termination proofs for higher-order rewrite systems, Proceedings of the 1st International Workshop on Higher-Order Algebra, Logic and Term Rewriting, 1993. ,
Termination of higher-order rewrite systems, 1996. ,
Strict functionals for termination proofs, Proceedings of the 2nd International Conference on Typed Lambda Calculi and Applications, 1995. ,
DOI : 10.1007/BFb0014064
Termination of rewriting in the Calculus of Constructions, Journal of Functional Programming, vol.13, issue.02, pp.339-414, 2003. ,
DOI : 10.1017/S0956796802004641
Dependent types for program termination verification, Higher-Order and Symbolic Computation, vol.15, issue.1, pp.91-131, 2002. ,
DOI : 10.1023/A:1019916231463