Complete Non-Orders and Fixed Points

In this paper, we develop an Isabelle/HOL library of order-theoretic concepts, such as various completeness conditions and fixed-point theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often without any property of ordering, thus complete non-orders. In particular, we generalize the Knaster - Tarski theorem so that we ensure the existence of a quasi-fixed point of monotone maps over complete non-orders, and show that the set of quasi-fixed points is complete under a mild condition - attractivity - which is implied by either antisymmetry or transitivity. This result generalizes and strengthens a result by Stauti and Maaden. Finally, we recover Kleene\u27s fixed-point theorem for omega-complete non-orders, again using attractivity to prove that Kleene\u27s fixed points are least quasi-fixed points

Fixed Points Theorems for Non-Transitive Relations

In this paper, we develop an Isabelle/HOL library of order-theoretic fixed-point theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often with only antisymmetry or attractivity, a mild condition implied by either antisymmetry or transitivity. In particular, we generalize various theorems ensuring the existence of a quasi-fixed point of monotone maps over complete relations, and show that the set of (quasi-)fixed points is itself complete. This result generalizes and strengthens theorems of Knaster-Tarski, Bourbaki-Witt, Kleene, Markowsky, Pataraia, Mashburn, Bhatta-George, and Stouti-Maaden

AC Dependency Pairs Revisited

Rewriting modulo AC, i.e., associativity and/or commutativity of certain symbols, is among the most frequently used extensions of term rewriting by equational theories. In this paper we present a generalization of the dependency pair framework for termination analysis to rewriting modulo AC. It subsumes existing variants of AC dependency pairs, admits standard dependency graph analyses, and in particular enjoys the minimality property in the standard sense. As a direct benefit, important termination techniques are easily extended; we describe usable rules and the subterm criterion for AC termination, which properly generalize the non-AC versions. We also perform these extensions within IsaFoR - the Isabelle formalization of rewriting - and thereby provide the first formalization of AC dependency pairs. Consequently, our certifier CeTA now supports checking proofs of AC termination

Recovery of Pleioblastus variegatus from Flowering in the Aso Region -Results of the 17th Year- (Advanced Studies on Sustainable Animal Production: Interrelationships among Human, Animal and Environment, 8th International Symposium of Integrated Field Science)

Poster Sessio

Extending the Period of High Feed Value in Italian Ryegrass (\u3cem\u3eLolium multiflorum\u3c/em\u3e Lam.) for Grazing in the Warm Temperate Zone of Japan

Japan has widely diverse climate conditions, from subtropical in the south to boreal in the north. In warm regions of the temperate zone in the south-west, such as low-lying areas of Kyushu Island, temperate grasses only barely survive during the summer (Area V in Fig. 1). Therefore, for year-round grazing, tropical grasses and temperate grasses have been used, respectively, for summer and winter (Fig. 2). Throughout Japan, grazing is limited for cows and their calves and fattening takes place in barns. Recently, as consumer preference for meat has diversified, the demand for lean meat with less fat from grazing cattle has been increasing. As the improved marbling and growth of Wagyu breeding demands high-value feed for fattening, grazing fattening of Wagyu also requires high-value feed. Currently, Italian ryegrass (IR; Lolium multiflorum Lam.) is used as high-value feed grass in the Southwest warm region in winter. IR presents the critical problem that feed value declines along with its ear emergence. Furthermore, the feed value of tropical grasses grown in summer is less than that of IR. Grazing cattle need more energy to cope with high temperatures of summer than they do for winter. Be-cause of these factors, intake of energy and weight gain rate decline in summer. To resolve these problems, a new pasture system is proposed to maintain high feed value throughout the year (Fig. 2): a group of cultivars of IR that grows well in winter should be sown in autumn. Then a type of winter habitat IR should be sown in early spring. In such cases, winter habitat IR sown after winter will not flower because it has not been vernalised by low temperatures. This IR exhibits superior re-growth and heat resistance. Therefore, this new system can help prevent a decline of feed value associated with ear emergence and also prolong the available period of IR. This study investigated the effectiveness of the new system for extending the period of high feed value in IR

Fixed Points Theorems for Non-Transitive Relations

In this paper, we develop an Isabelle/HOL library of order-theoretic fixed-point theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often with only antisymmetry or attractivity, a mild condition implied by either antisymmetry or transitivity. In particular, we generalize various theorems ensuring the existence of a quasi-fixed point of monotone maps over complete relations, and show that the set of (quasi-)fixed points is itself complete. This result generalizes and strengthens theorems of Knaster-Tarski, Bourbaki-Witt, Kleene, Markowsky, Pataraia, Mashburn, Bhatta-George, and Stouti-Maaden

Relative Termination via Dependency Pairs

[EN] A term rewrite system is terminating when no infinite reduction sequences are possible. Relative termination generalizes termination by permitting infinite reductions as long as some distinguished rules are not applied infinitely many times. Relative termination is thus a fundamental notion that has been used in a number of different contexts, like analyzing the confluence of rewrite systems or the termination of narrowing. In this work, we introduce a novel technique to prove relative termination by reducing it to dependency pair problems. To the best of our knowledge, this is the first significant contribution to Problem #106 of the RTA List of Open Problems. We first present a general approach that is then instantiated to provide a concrete technique for proving relative termination. The practical significance of our method is illustrated by means of an experimental evaluation.Open access funding provided by Austrian Science Fund (FWF). We would like to thank Nao Hirokawa, Keiichirou Kusakari, and the anonymous reviewers for their helpful comments and suggestions in early stages of this work.Iborra, J.; Nishida, N.; Vidal Oriola, G.; Yamada, A. (2017). Relative Termination via Dependency Pairs. Journal of Automated Reasoning. 58(3):391-411. https://doi.org/10.1007/s10817-016-9373-5391411583Alarcón, B., Lucas, S., Meseguer, J.: A dependency pair framework for A $$\vee$$ ∨ C-termination. In: WRLA 2010, LNCS, vol. 6381, pp. 36–52. Springer (2010)Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236(1–2), 133–178 (2000)Arts, T., Giesl, J.: A collection of examples for termination of term rewriting using dependency pairs. Technical report AIB-2001-09, RWTH Aachen (2001)Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Bachmair, L., Dershowitz, N.: Critical pair criteria for completion. J. Symb. Comput. 6, 1–18 (1988)Bonacina, M., Hsiang, J.: On fairness of completion-based theorem proving strategies. In: RTA 1991, LNCS, vol. 488, pp. 348–360. Springer (1991)Dershowitz, N.: Termination of rewriting. J. Symb. Comput. 3(1&2), 69–115 (1987)Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. J. Autom. Reason. 40(2–3), 195–220 (2008)Geser, A.: Relative Termination. Dissertation, Fakultät für Mathematik und Informatik. Universität Passau, Germany (1990)Giesl, J., Kapur, D.: Dependency pairs for equational rewriting. In: RTA 2001, LNCS, vol. 2051, pp. 93–107. Springer (2001)Giesl, J., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: automatic termination proofs in the dependency pair framework. In: IJCAR 2006, LNCS, vol. 4130, pp. 281–286. Springer (2006)Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency Pairs. J. Autom. Reason. 37(3), 155–203 (2006)Hirokawa, N., Middeldorp, A.: Dependency pairs revisited. In: RTA 2004, LNCS, vol. 3091, pp. 249–268. Springer (2004)Hirokawa, N., Middeldorp, A.: Polynomial interpretations with negative coefficients. In: AISC 2004, LNAI, vol. 3249, pp. 185–198. Springer (2004)Hirokawa, N., Middeldorp, A.: Tyrolean termination tool: techniques and features. Inf. Comput. 205(4), 474–511 (2007)Hirokawa, N., Middeldorp, A.: Decreasing diagrams and relative termination. J. Autom. Reason. 47(4), 481–501 (2011)Hullot, J.M.: Canonical forms and unification. In: CADE 1980, LNCS, vol. 87, pp. 318–334. Springer (1980)Iborra, J., Nishida, N., Vidal, G.: Goal-directed and relative dependency pairs for proving the termination of narrowing. In: LOPSTR 2009, LNCS, vol. 6037, pp. 52–66. Springer (2010)Iborra, J., Nishida, N., Vidal, G., Yamada, A.: Reducing relative termination to dependency pair problems. In: CADE-25, LNAI, vol. 9195, pp. 163–178. Springer (2015)Kamin, S., Lévy, J.J.: Two generalizations of the recursive path ordering (1980). Unpublished noteKlop, J.W.: Term rewriting systems: a tutorial. Bull. Eur. Assoc. Theor. Comput. Sci. 32, 143–183 (1987)Koprowski, A.: TPA: termination proved automatically. In: RTA 2006, LNCS, vol. 4098, pp. 257–266. Springer (2006)Koprowski, A., Zantema, H.: Proving liveness with fairness using rewriting. In: FroCoS 2005, LNCS, vol. 3717, pp. 232–247. Springer (2005)Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean termination tool 2. In: RTA 2009, LNCS, vol. 5595, pp. 295–304. Springer (2009)Kusakari, K., Toyama, Y.: On proving AC-termination by AC-dependency pairs. IEICE Trans. Inf. Syst. E84–D(5), 439–447 (2001)Lankford, D.: Canonical algebraic simplification in computational logic. Technical report ATP-25, University of Texas (1975)Marché, C., Urbain, X.: Modular and incremental proofs of AC-termination. J. Symb. Comput. 38(1), 873–897 (2004)Nishida, N., Sakai, M., Sakabe, T.: Narrowing-based simulation of term rewriting systems with extra variables. ENTCS 86(3), 52–69 (2003)Nishida, N., Vidal, G.: Termination of narrowing via termination of rewriting. Appl. Algebra Eng. Commun. Comput. 21(3), 177–225 (2010)Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer, London (2002)Slagle, J.: Automated theorem-proving for theories with simplifiers commutativity and associativity. J. ACM 21(4), 622–642 (1974)Thiemann, R., Allais, G., Nagele, J.: On the formalization of termination techniques based on multiset orderings. In: RTA 2012, LIPIcs, vol. 15, pp. 339–354. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)Vidal, G.: Termination of narrowing in left-linear constructor systems. In: FLOPS 2008, LNCS, vol. 4989, pp. 113–129. Springer (2008)Yamada, A., Kusakari, K., Sakabe, T.: Nagoya termination tool. In: RTA-TLCA 2014, LNCS, pp. 466–475. Springer (2014)Yamada, A., Kusakari, K., Sakabe, T.: A unified ordering for termination proving. Sci. Comput. Program. 111, 110–134 (2015)Zantema, H.: Termination of term rewriting by semantic labelling. Fundam. Inf. 24(1/2), 89–105 (1995)Zantema, H.: Termination. In: Bezem, M., Klop, J. W., de Vrijer, R. (eds.) Term Rewriting Systems, Cambridge Tracts in Theoretical Computer Science, chap. 6, vol. 55, pp. 181–259. Cambridge University Press, Cambridge (2003