1.44/1.32 YES 1.44/1.33 We consider the system theBenchmark. 1.44/1.33 1.44/1.33 Alphabet: 1.44/1.33 1.44/1.33 app : [] --> arrab -> a -> b 1.44/1.33 lam : [] --> (a -> b) -> arrab 1.44/1.33 pair : [] --> a -> b -> prodab 1.44/1.33 pia : [] --> prodab -> a 1.44/1.33 pib : [] --> prodab -> b 1.44/1.33 1.44/1.33 Rules: 1.44/1.33 1.44/1.33 app (lam (/\x.f x)) y => f y 1.44/1.33 lam (/\x.app y x) => y 1.44/1.33 pia (pair x y) => x 1.44/1.33 pib (pair x y) => y 1.44/1.33 pair (pia x) (pib x) => x 1.44/1.33 1.44/1.33 Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term. 1.44/1.33 1.44/1.33 We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0). This leads to the following AFSM: 1.44/1.33 1.44/1.33 Alphabet: 1.44/1.33 1.44/1.33 app : [arrab * a] --> b 1.44/1.33 lam : [a -> b] --> arrab 1.44/1.33 pair : [a * b] --> prodab 1.44/1.33 pia : [prodab] --> a 1.44/1.33 pib : [prodab] --> b 1.44/1.33 ~AP1 : [a -> b * a] --> b 1.44/1.33 1.44/1.33 Rules: 1.44/1.33 1.44/1.33 app(lam(/\x.~AP1(F, x)), X) => ~AP1(F, X) 1.44/1.33 lam(/\x.app(X, x)) => X 1.44/1.33 pia(pair(X, Y)) => X 1.44/1.33 pib(pair(X, Y)) => Y 1.44/1.33 pair(pia(X), pib(X)) => X 1.44/1.33 app(lam(/\x.app(X, x)), Y) => app(X, Y) 1.44/1.33 ~AP1(F, X) => F X 1.44/1.33 1.44/1.33 Symbol ~AP1 is an encoding for application that is only used in innocuous ways. We can simplify the program (without losing non-termination) by removing it. Additionally, we can remove some (now-)redundant rules. This gives: 1.44/1.33 1.44/1.33 Alphabet: 1.44/1.33 1.44/1.33 app : [arrab * a] --> b 1.44/1.33 lam : [a -> b] --> arrab 1.44/1.33 pair : [a * b] --> prodab 1.44/1.33 pia : [prodab] --> a 1.44/1.33 pib : [prodab] --> b 1.44/1.33 1.44/1.33 Rules: 1.44/1.33 1.44/1.33 app(lam(/\x.X(x)), Y) => X(Y) 1.44/1.33 lam(/\x.app(X, x)) => X 1.44/1.33 pia(pair(X, Y)) => X 1.44/1.33 pib(pair(X, Y)) => Y 1.44/1.33 pair(pia(X), pib(X)) => X 1.44/1.33 1.44/1.33 We observe that the rules contain a first-order subset: 1.44/1.33 1.44/1.33 pia(pair(X, Y)) => X 1.44/1.33 pib(pair(X, Y)) => Y 1.44/1.33 pair(pia(X), pib(X)) => X 1.44/1.33 1.44/1.33 Moreover, the system is finitely branching. Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is Ce-terminating when seen as a many-sorted first-order TRS. 1.44/1.33 1.44/1.33 According to the external first-order termination prover, this system is indeed Ce-terminating: 1.44/1.33 1.44/1.33 || proof of resources/system.trs 1.44/1.33 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616 1.44/1.33 || 1.44/1.33 || 1.44/1.33 || Termination w.r.t. Q of the given QTRS could be proven: 1.44/1.33 || 1.44/1.33 || (0) QTRS 1.44/1.33 || (1) QTRSRRRProof [EQUIVALENT] 1.44/1.33 || (2) QTRS 1.44/1.33 || (3) RisEmptyProof [EQUIVALENT] 1.44/1.33 || (4) YES 1.44/1.33 || 1.44/1.33 || 1.44/1.33 || ---------------------------------------- 1.44/1.33 || 1.44/1.33 || (0) 1.44/1.33 || Obligation: 1.44/1.33 || Q restricted rewrite system: 1.44/1.33 || The TRS R consists of the following rules: 1.44/1.33 || 1.44/1.33 || pia(pair(%X, %Y)) -> %X 1.44/1.33 || pib(pair(%X, %Y)) -> %Y 1.44/1.33 || pair(pia(%X), pib(%X)) -> %X 1.44/1.33 || ~PAIR(%X, %Y) -> %X 1.44/1.33 || ~PAIR(%X, %Y) -> %Y 1.44/1.33 || 1.44/1.33 || Q is empty. 1.44/1.33 || 1.44/1.33 || ---------------------------------------- 1.44/1.33 || 1.44/1.33 || (1) QTRSRRRProof (EQUIVALENT) 1.44/1.33 || Used ordering: 1.44/1.33 || Knuth-Bendix order [KBO] with precedence:~PAIR_2 > pib_1 > pair_2 > pia_1 1.44/1.33 || 1.44/1.33 || and weight map: 1.44/1.33 || 1.44/1.33 || pia_1=1 1.44/1.33 || pib_1=1 1.44/1.33 || pair_2=0 1.44/1.33 || ~PAIR_2=0 1.44/1.33 || 1.44/1.33 || The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 1.44/1.33 || 1.44/1.33 || pia(pair(%X, %Y)) -> %X 1.44/1.33 || pib(pair(%X, %Y)) -> %Y 1.44/1.33 || pair(pia(%X), pib(%X)) -> %X 1.44/1.33 || ~PAIR(%X, %Y) -> %X 1.44/1.33 || ~PAIR(%X, %Y) -> %Y 1.44/1.33 || 1.44/1.33 || 1.44/1.33 || 1.44/1.33 || 1.44/1.33 || ---------------------------------------- 1.44/1.33 || 1.44/1.33 || (2) 1.44/1.33 || Obligation: 1.44/1.33 || Q restricted rewrite system: 1.44/1.33 || R is empty. 1.44/1.33 || Q is empty. 1.44/1.33 || 1.44/1.33 || ---------------------------------------- 1.44/1.33 || 1.44/1.33 || (3) RisEmptyProof (EQUIVALENT) 1.44/1.33 || The TRS R is empty. Hence, termination is trivially proven. 1.44/1.33 || ---------------------------------------- 1.44/1.33 || 1.44/1.33 || (4) 1.44/1.33 || YES 1.44/1.33 || 1.44/1.33 We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [Kop13]). 1.44/1.33 1.44/1.33 We thus obtain the following dependency pair problem (P_0, R_0, static, all): 1.44/1.33 1.44/1.33 Dependency Pairs P_0: 1.44/1.33 1.44/1.33 1.44/1.33 Rules R_0: 1.44/1.33 1.44/1.33 app(lam(/\x.X(x)), Y) => X(Y) 1.44/1.33 lam(/\x.app(X, x)) => X 1.44/1.33 pia(pair(X, Y)) => X 1.44/1.33 pib(pair(X, Y)) => Y 1.44/1.33 pair(pia(X), pib(X)) => X 1.44/1.33 1.44/1.33 Thus, the original system is terminating if (P_0, R_0, static, all) is finite. 1.44/1.33 1.44/1.33 We consider the dependency pair problem (P_0, R_0, static, all). 1.44/1.33 1.44/1.33 We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: 1.44/1.33 1.44/1.33 1.44/1.33 This graph has no strongly connected components. By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem. 1.44/1.33 1.44/1.33 As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination. 1.44/1.33 1.44/1.33 1.44/1.33 +++ Citations +++ 1.44/1.33 1.44/1.33 [Kop11] C. Kop. Simplifying Algebraic Functional Systems. In Proceedings of CAI 2011, volume 6742 of LNCS. 201--215, Springer, 2011. 1.44/1.33 [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. 1.44/1.33 [Kop13] C. Kop. Static Dependency Pairs with Accessibility. Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013. 1.44/1.33 [KusIsoSakBla09] K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui. Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems. In volume 92(10) of IEICE Transactions on Information and Systems. 2007--2015, 2009. 1.44/1.33 EOF