1.40/0.63 YES 1.40/0.64 We consider the system theBenchmark. 1.40/0.64 1.40/0.64 Alphabet: 1.40/0.64 1.40/0.64 abs : [] --> (a -> b) -> Arrab 1.40/0.64 app : [] --> Arrab -> a -> b 1.40/0.64 box : [] --> a -> Boxa 1.40/0.64 unbox : [] --> Boxa -> a 1.40/0.64 1.40/0.64 Rules: 1.40/0.64 1.40/0.64 app (abs (/\x.f x)) y => f y 1.40/0.64 abs (/\x.app y x) => y 1.40/0.64 unbox (box x) => x 1.40/0.64 box (unbox x) => x 1.40/0.64 1.40/0.64 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. 1.40/0.64 1.40/0.64 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: 1.40/0.64 1.40/0.64 Alphabet: 1.40/0.64 1.40/0.64 abs : [a -> b] --> Arrab 1.40/0.64 app : [Arrab * a] --> b 1.40/0.64 box : [a] --> Boxa 1.40/0.64 unbox : [Boxa] --> a 1.40/0.64 ~AP1 : [a -> b * a] --> b 1.40/0.64 1.40/0.64 Rules: 1.40/0.64 1.40/0.64 app(abs(/\x.~AP1(F, x)), X) => ~AP1(F, X) 1.40/0.64 abs(/\x.app(X, x)) => X 1.40/0.64 unbox(box(X)) => X 1.40/0.64 box(unbox(X)) => X 1.40/0.64 app(abs(/\x.app(X, x)), Y) => app(X, Y) 1.40/0.64 ~AP1(F, X) => F X 1.40/0.64 1.40/0.64 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: 1.40/0.64 1.40/0.64 Alphabet: 1.40/0.64 1.40/0.64 abs : [a -> b] --> Arrab 1.40/0.64 app : [Arrab * a] --> b 1.40/0.64 box : [a] --> Boxa 1.40/0.64 unbox : [Boxa] --> a 1.40/0.64 1.40/0.64 Rules: 1.40/0.64 1.40/0.64 app(abs(/\x.X(x)), Y) => X(Y) 1.40/0.64 abs(/\x.app(X, x)) => X 1.40/0.64 unbox(box(X)) => X 1.40/0.64 box(unbox(X)) => X 1.40/0.64 1.40/0.64 We observe that the rules contain a first-order subset: 1.40/0.64 1.40/0.64 unbox(box(X)) => X 1.40/0.64 box(unbox(X)) => X 1.40/0.64 1.40/0.64 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. 1.40/0.64 1.40/0.64 According to the external first-order termination prover, this system is indeed Ce-terminating: 1.40/0.64 1.40/0.64 || proof of resources/system.trs 1.40/0.64 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616 1.40/0.64 || 1.40/0.64 || 1.40/0.64 || Termination w.r.t. Q of the given QTRS could be proven: 1.40/0.64 || 1.40/0.64 || (0) QTRS 1.40/0.64 || (1) QTRSRRRProof [EQUIVALENT] 1.40/0.64 || (2) QTRS 1.40/0.64 || (3) RisEmptyProof [EQUIVALENT] 1.40/0.64 || (4) YES 1.40/0.64 || 1.40/0.64 || 1.40/0.64 || ---------------------------------------- 1.40/0.64 || 1.40/0.64 || (0) 1.40/0.64 || Obligation: 1.40/0.64 || Q restricted rewrite system: 1.40/0.64 || The TRS R consists of the following rules: 1.40/0.64 || 1.40/0.64 || unbox(box(%X)) -> %X 1.40/0.64 || box(unbox(%X)) -> %X 1.40/0.64 || ~PAIR(%X, %Y) -> %X 1.40/0.64 || ~PAIR(%X, %Y) -> %Y 1.40/0.64 || 1.40/0.64 || Q is empty. 1.40/0.64 || 1.40/0.64 || ---------------------------------------- 1.40/0.64 || 1.40/0.64 || (1) QTRSRRRProof (EQUIVALENT) 1.40/0.64 || Used ordering: 1.40/0.64 || Polynomial interpretation [POLO]: 1.40/0.64 || 1.40/0.64 || POL(box(x_1)) = 2 + x_1 1.40/0.64 || POL(unbox(x_1)) = 1 + 2*x_1 1.40/0.64 || POL(~PAIR(x_1, x_2)) = 2 + x_1 + x_2 1.40/0.64 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 1.40/0.64 || 1.40/0.64 || unbox(box(%X)) -> %X 1.40/0.64 || box(unbox(%X)) -> %X 1.40/0.64 || ~PAIR(%X, %Y) -> %X 1.40/0.64 || ~PAIR(%X, %Y) -> %Y 1.40/0.64 || 1.40/0.64 || 1.40/0.64 || 1.40/0.64 || 1.40/0.64 || ---------------------------------------- 1.40/0.64 || 1.40/0.64 || (2) 1.40/0.64 || Obligation: 1.40/0.64 || Q restricted rewrite system: 1.40/0.64 || R is empty. 1.40/0.64 || Q is empty. 1.40/0.64 || 1.40/0.64 || ---------------------------------------- 1.40/0.64 || 1.40/0.64 || (3) RisEmptyProof (EQUIVALENT) 1.40/0.64 || The TRS R is empty. Hence, termination is trivially proven. 1.40/0.64 || ---------------------------------------- 1.40/0.64 || 1.40/0.64 || (4) 1.40/0.64 || YES 1.40/0.64 || 1.40/0.64 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]). 1.40/0.64 1.40/0.64 We thus obtain the following dependency pair problem (P_0, R_0, static, all): 1.40/0.64 1.40/0.64 Dependency Pairs P_0: 1.40/0.64 1.40/0.64 1.40/0.64 Rules R_0: 1.40/0.64 1.40/0.64 app(abs(/\x.X(x)), Y) => X(Y) 1.40/0.64 abs(/\x.app(X, x)) => X 1.40/0.64 unbox(box(X)) => X 1.40/0.64 box(unbox(X)) => X 1.40/0.64 1.40/0.64 Thus, the original system is terminating if (P_0, R_0, static, all) is finite. 1.40/0.64 1.40/0.64 We consider the dependency pair problem (P_0, R_0, static, all). 1.40/0.64 1.40/0.64 We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: 1.40/0.64 1.40/0.64 1.40/0.64 This graph has no strongly connected components. By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem. 1.40/0.64 1.40/0.64 As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination. 1.40/0.64 1.40/0.64 1.40/0.64 +++ Citations +++ 1.40/0.64 1.40/0.64 [Kop11] C. Kop. Simplifying Algebraic Functional Systems. In Proceedings of CAI 2011, volume 6742 of LNCS. 201--215, Springer, 2011. 1.40/0.64 [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. 1.40/0.64 [Kop13] C. Kop. Static Dependency Pairs with Accessibility. Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013. 1.40/0.64 [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. 1.40/0.64 EOF