We consider the system AotoYamada_05__016. Alphabet: 0 : [] --> a cons : [a * c] --> c false : [] --> b filter : [a -> b] --> c -> c filtersub : [b * a -> b * c] --> c neq : [a] --> a -> b nil : [] --> c nonzero : [] --> c -> c s : [a] --> a true : [] --> b Rules: neq(0) 0 => false neq(0) s(x) => true neq(s(x)) 0 => true neq(s(x)) s(y) => neq(x) y filter(f) nil => nil filter(f) cons(x, y) => filtersub(f x, f, cons(x, y)) filtersub(true, f, cons(x, y)) => cons(x, filter(f) y) filtersub(false, f, cons(x, y)) => filter(f) y nonzero => filter(neq(0)) This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). We observe that the rules contain a first-order subset: neq(0, 0) => false neq(0, s(X)) => true neq(s(X), 0) => true neq(s(X), s(Y)) => neq(X, Y) Moreover, the system is orthogonal. 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 terminating when seen as a many-sorted first-order TRS. According to the external first-order termination prover, this system is indeed terminating: || proof of resources/system.trs || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616 || || || Termination w.r.t. Q of the given QTRS could be proven: || || (0) QTRS || (1) QTRSRRRProof [EQUIVALENT] || (2) QTRS || (3) RisEmptyProof [EQUIVALENT] || (4) YES || || || ---------------------------------------- || || (0) || Obligation: || Q restricted rewrite system: || The TRS R consists of the following rules: || || neq(0, 0) -> false || neq(0, s(%X)) -> true || neq(s(%X), 0) -> true || neq(s(%X), s(%Y)) -> neq(%X, %Y) || || Q is empty. || || ---------------------------------------- || || (1) QTRSRRRProof (EQUIVALENT) || Used ordering: || Polynomial interpretation [POLO]: || || POL(0) = 0 || POL(false) = 1 || POL(neq(x_1, x_2)) = 2 + 2*x_1 + x_2 || POL(s(x_1)) = 1 + 2*x_1 || POL(true) = 0 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: || || neq(0, 0) -> false || neq(0, s(%X)) -> true || neq(s(%X), 0) -> true || neq(s(%X), s(%Y)) -> neq(%X, %Y) || || || || || ---------------------------------------- || || (2) || Obligation: || Q restricted rewrite system: || R is empty. || Q is empty. || || ---------------------------------------- || || (3) RisEmptyProof (EQUIVALENT) || The TRS R is empty. Hence, termination is trivially proven. || ---------------------------------------- || || (4) || YES || 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 [FuhKop19]). In order to do so, we start by eta-expanding the system, which gives: neq(0, 0) => false neq(0, s(X)) => true neq(s(X), 0) => true neq(s(X), s(Y)) => neq(X, Y) filter(F, nil) => nil filter(F, cons(X, Y)) => filtersub(F X, F, cons(X, Y)) filtersub(true, F, cons(X, Y)) => cons(X, filter(F, Y)) filtersub(false, F, cons(X, Y)) => filter(F, Y) nonzero(X) => filter(/\x.neq(0, x), X) We thus obtain the following dependency pair problem (P_0, R_0, computable, formative): Dependency Pairs P_0: 0] filter#(F, cons(X, Y)) =#> filtersub#(F X, F, cons(X, Y)) 1] filtersub#(true, F, cons(X, Y)) =#> filter#(F, Y) 2] filtersub#(false, F, cons(X, Y)) =#> filter#(F, Y) 3] nonzero#(X) =#> filter#(/\x.neq(0, x), X) 4] nonzero#(X) =#> neq#(0, Y) Rules R_0: neq(0, 0) => false neq(0, s(X)) => true neq(s(X), 0) => true neq(s(X), s(Y)) => neq(X, Y) filter(F, nil) => nil filter(F, cons(X, Y)) => filtersub(F X, F, cons(X, Y)) filtersub(true, F, cons(X, Y)) => cons(X, filter(F, Y)) filtersub(false, F, cons(X, Y)) => filter(F, Y) nonzero(X) => filter(/\x.neq(0, x), X) Thus, the original system is terminating if (P_0, R_0, computable, formative) is finite. We consider the dependency pair problem (P_0, R_0, computable, formative). We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: * 0 : 1, 2 * 1 : 0 * 2 : 0 * 3 : 0 * 4 : This graph has the following strongly connected components: P_1: filter#(F, cons(X, Y)) =#> filtersub#(F X, F, cons(X, Y)) filtersub#(true, F, cons(X, Y)) =#> filter#(F, Y) filtersub#(false, F, cons(X, Y)) =#> filter#(F, Y) By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f). Thus, the original system is terminating if (P_1, R_0, computable, formative) is finite. We consider the dependency pair problem (P_1, R_0, computable, formative). We apply the subterm criterion with the following projection function: nu(filter#) = 2 nu(filtersub#) = 3 Thus, we can orient the dependency pairs as follows: nu(filter#(F, cons(X, Y))) = cons(X, Y) = cons(X, Y) = nu(filtersub#(F X, F, cons(X, Y))) nu(filtersub#(true, F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter#(F, Y)) nu(filtersub#(false, F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter#(F, Y)) By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_1, R_0, computable, f) by (P_2, R_0, computable, f), where P_2 contains: filter#(F, cons(X, Y)) =#> filtersub#(F X, F, cons(X, Y)) Thus, the original system is terminating if (P_2, R_0, computable, formative) is finite. We consider the dependency pair problem (P_2, R_0, computable, formative). We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: * 0 : This graph has no strongly connected components. By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem. As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination. +++ Citations +++ [FuhKop19] C. Fuhs, and C. Kop. A static higher-order dependency pair framework. In Proceedings of ESOP 2019, 2019. [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. [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.