/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. We are asked to determine termination of the following first-order TRS. 0 : [] --> o cons : [o * o] --> o nil : [] --> o rev : [o] --> o rev1 : [o * o] --> o rev2 : [o * o] --> o s : [o] --> o rev1(0, nil) => 0 rev1(s(X), nil) => s(X) rev1(X, cons(Y, Z)) => rev1(Y, Z) rev(nil) => nil rev(cons(X, Y)) => cons(rev1(X, Y), rev2(X, Y)) rev2(X, nil) => nil rev2(X, cons(Y, Z)) => rev(cons(X, rev(rev2(Y, Z)))) As the system is orthogonal, it is terminating if it is innermost terminating by [Gra95]. Then, by [FuhGieParSchSwi11], it suffices to prove (innermost) termination of the typed system, with sort annotations chosen to respect the rules, as follows: 0 : [] --> va cons : [va * sb] --> sb nil : [] --> sb rev : [sb] --> sb rev1 : [va * sb] --> va rev2 : [va * sb] --> sb s : [va] --> va We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs in [Kop12, Ch. 7.8]). We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative): Dependency Pairs P_0: 0] rev1#(X, cons(Y, Z)) =#> rev1#(Y, Z) 1] rev#(cons(X, Y)) =#> rev1#(X, Y) 2] rev#(cons(X, Y)) =#> rev2#(X, Y) 3] rev2#(X, cons(Y, Z)) =#> rev#(cons(X, rev(rev2(Y, Z)))) 4] rev2#(X, cons(Y, Z)) =#> rev#(rev2(Y, Z)) 5] rev2#(X, cons(Y, Z)) =#> rev2#(Y, Z) Rules R_0: rev1(0, nil) => 0 rev1(s(X), nil) => s(X) rev1(X, cons(Y, Z)) => rev1(Y, Z) rev(nil) => nil rev(cons(X, Y)) => cons(rev1(X, Y), rev2(X, Y)) rev2(X, nil) => nil rev2(X, cons(Y, Z)) => rev(cons(X, rev(rev2(Y, Z)))) Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_0, R_0, minimal, 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 : 0 * 1 : 0 * 2 : 3, 4, 5 * 3 : 1, 2 * 4 : 1, 2 * 5 : 3, 4, 5 This graph has the following strongly connected components: P_1: rev1#(X, cons(Y, Z)) =#> rev1#(Y, Z) P_2: rev#(cons(X, Y)) =#> rev2#(X, Y) rev2#(X, cons(Y, Z)) =#> rev#(cons(X, rev(rev2(Y, Z)))) rev2#(X, cons(Y, Z)) =#> rev#(rev2(Y, Z)) rev2#(X, cons(Y, Z)) =#> rev2#(Y, Z) 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) and (P_2, R_0, m, f). Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_2, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_2, R_0, minimal, formative). The formative rules of (P_2, R_0) are R_1 ::= rev(cons(X, Y)) => cons(rev1(X, Y), rev2(X, Y)) rev2(X, cons(Y, Z)) => rev(cons(X, rev(rev2(Y, Z)))) By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_2, R_0, minimal, formative) by (P_2, R_1, minimal, formative). Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_2, R_1, minimal, formative) is finite. We consider the dependency pair problem (P_2, R_1, minimal, formative). We will use the reduction pair processor [Kop12, Thm. 7.16]. It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: rev#(cons(X, Y)) >? rev2#(X, Y) rev2#(X, cons(Y, Z)) >? rev#(cons(X, rev(rev2(Y, Z)))) rev2#(X, cons(Y, Z)) >? rev#(rev2(Y, Z)) rev2#(X, cons(Y, Z)) >? rev2#(Y, Z) rev(cons(X, Y)) >= cons(rev1(X, Y), rev2(X, Y)) rev2(X, cons(Y, Z)) >= rev(cons(X, rev(rev2(Y, Z)))) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: cons = \y0y1.1 + 2y1 rev = \y0.y0 rev1 = \y0y1.0 rev2 = \y0y1.y1 rev2# = \y0y1.1 + y1 rev# = \y0.y0 Using this interpretation, the requirements translate to: [[rev#(cons(_x0, _x1))]] = 1 + 2x1 >= 1 + x1 = [[rev2#(_x0, _x1)]] [[rev2#(_x0, cons(_x1, _x2))]] = 2 + 2x2 > 1 + 2x2 = [[rev#(cons(_x0, rev(rev2(_x1, _x2))))]] [[rev2#(_x0, cons(_x1, _x2))]] = 2 + 2x2 > x2 = [[rev#(rev2(_x1, _x2))]] [[rev2#(_x0, cons(_x1, _x2))]] = 2 + 2x2 > 1 + x2 = [[rev2#(_x1, _x2)]] [[rev(cons(_x0, _x1))]] = 1 + 2x1 >= 1 + 2x1 = [[cons(rev1(_x0, _x1), rev2(_x0, _x1))]] [[rev2(_x0, cons(_x1, _x2))]] = 1 + 2x2 >= 1 + 2x2 = [[rev(cons(_x0, rev(rev2(_x1, _x2))))]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_2, R_1, minimal, formative) by (P_3, R_1, minimal, formative), where P_3 consists of: rev#(cons(X, Y)) =#> rev2#(X, Y) Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_3, R_1, minimal, formative) is finite. We consider the dependency pair problem (P_3, R_1, minimal, 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. Thus, the original system is terminating if (P_1, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_1, R_0, minimal, formative). We apply the subterm criterion with the following projection function: nu(rev1#) = 2 Thus, we can orient the dependency pairs as follows: nu(rev1#(X, cons(Y, Z))) = cons(Y, Z) |> Z = nu(rev1#(Y, Z)) By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_0, minimal, f) by ({}, R_0, minimal, f). By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed. As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination. +++ Citations +++ [FuhGieParSchSwi11] C. Fuhs, J. Giesl, M. Parting, P. Schneider-Kamp, and S. Swiderski. Proving Termination by Dependency Pairs and Inductive Theorem Proving. In volume 47(2) of Journal of Automated Reasoning. 133--160, 2011. [Gra95] B. Gramlich. Abstract Relations Between Restricted Termination and Confluence Properties of Rewrite Systems. In volume 24(1-2) of Fundamentae Informaticae. 3--23, 1995. [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.