/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 edge : [o * o * o] --> o empty : [] --> o eq : [o * o] --> o false : [] --> o if!6220reach!62201 : [o * o * o * o * o] --> o if!6220reach!62202 : [o * o * o * o * o] --> o or : [o * o] --> o reach : [o * o * o * o] --> o s : [o] --> o true : [] --> o union : [o * o] --> o eq(0, 0) => true eq(0, s(X)) => false eq(s(X), 0) => false eq(s(X), s(Y)) => eq(X, Y) or(true, X) => true or(false, X) => X union(empty, X) => X union(edge(X, Y, Z), U) => edge(X, Y, union(Z, U)) reach(X, Y, empty, Z) => false reach(X, Y, edge(Z, U, V), W) => if!6220reach!62201(eq(X, Z), X, Y, edge(Z, U, V), W) if!6220reach!62201(true, X, Y, edge(Z, U, V), W) => if!6220reach!62202(eq(Y, U), X, Y, edge(Z, U, V), W) if!6220reach!62202(true, X, Y, edge(Z, U, V), W) => true if!6220reach!62202(false, X, Y, edge(Z, U, V), W) => or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) if!6220reach!62201(false, X, Y, edge(Z, U, V), W) => reach(X, Y, V, edge(Z, U, W)) 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 : [] --> cg edge : [cg * cg * bg] --> bg empty : [] --> bg eq : [cg * cg] --> cg false : [] --> cg if!6220reach!62201 : [cg * cg * cg * bg * bg] --> cg if!6220reach!62202 : [cg * cg * cg * bg * bg] --> cg or : [cg * cg] --> cg reach : [cg * cg * bg * bg] --> cg s : [cg] --> cg true : [] --> cg union : [bg * bg] --> bg 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] eq#(s(X), s(Y)) =#> eq#(X, Y) 1] union#(edge(X, Y, Z), U) =#> union#(Z, U) 2] reach#(X, Y, edge(Z, U, V), W) =#> if!6220reach!62201#(eq(X, Z), X, Y, edge(Z, U, V), W) 3] reach#(X, Y, edge(Z, U, V), W) =#> eq#(X, Z) 4] if!6220reach!62201#(true, X, Y, edge(Z, U, V), W) =#> if!6220reach!62202#(eq(Y, U), X, Y, edge(Z, U, V), W) 5] if!6220reach!62201#(true, X, Y, edge(Z, U, V), W) =#> eq#(Y, U) 6] if!6220reach!62202#(false, X, Y, edge(Z, U, V), W) =#> or#(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 7] if!6220reach!62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, W) 8] if!6220reach!62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(U, Y, union(V, W), empty) 9] if!6220reach!62202#(false, X, Y, edge(Z, U, V), W) =#> union#(V, W) 10] if!6220reach!62201#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, edge(Z, U, W)) Rules R_0: eq(0, 0) => true eq(0, s(X)) => false eq(s(X), 0) => false eq(s(X), s(Y)) => eq(X, Y) or(true, X) => true or(false, X) => X union(empty, X) => X union(edge(X, Y, Z), U) => edge(X, Y, union(Z, U)) reach(X, Y, empty, Z) => false reach(X, Y, edge(Z, U, V), W) => if!6220reach!62201(eq(X, Z), X, Y, edge(Z, U, V), W) if!6220reach!62201(true, X, Y, edge(Z, U, V), W) => if!6220reach!62202(eq(Y, U), X, Y, edge(Z, U, V), W) if!6220reach!62202(true, X, Y, edge(Z, U, V), W) => true if!6220reach!62202(false, X, Y, edge(Z, U, V), W) => or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) if!6220reach!62201(false, X, Y, edge(Z, U, V), W) => reach(X, Y, V, edge(Z, U, W)) 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 : 1 * 2 : 4, 5, 10 * 3 : 0 * 4 : 6, 7, 8, 9 * 5 : 0 * 6 : * 7 : 2, 3 * 8 : 2, 3 * 9 : 1 * 10 : 2, 3 This graph has the following strongly connected components: P_1: eq#(s(X), s(Y)) =#> eq#(X, Y) P_2: union#(edge(X, Y, Z), U) =#> union#(Z, U) P_3: reach#(X, Y, edge(Z, U, V), W) =#> if!6220reach!62201#(eq(X, Z), X, Y, edge(Z, U, V), W) if!6220reach!62201#(true, X, Y, edge(Z, U, V), W) =#> if!6220reach!62202#(eq(Y, U), X, Y, edge(Z, U, V), W) if!6220reach!62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, W) if!6220reach!62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(U, Y, union(V, W), empty) if!6220reach!62201#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, edge(Z, U, W)) 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), (P_2, R_0, m, f) and (P_3, R_0, m, f). Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_3, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_3, R_0, minimal, formative). We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44]. The usable rules of (P_3, R_0) are: eq(0, 0) => true eq(0, s(X)) => false eq(s(X), 0) => false eq(s(X), s(Y)) => eq(X, Y) union(empty, X) => X union(edge(X, Y, Z), U) => edge(X, Y, union(Z, U)) It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: reach#(X, Y, edge(Z, U, V), W) >? if!6220reach!62201#(eq(X, Z), X, Y, edge(Z, U, V), W) if!6220reach!62201#(true, X, Y, edge(Z, U, V), W) >? if!6220reach!62202#(eq(Y, U), X, Y, edge(Z, U, V), W) if!6220reach!62202#(false, X, Y, edge(Z, U, V), W) >? reach#(X, Y, V, W) if!6220reach!62202#(false, X, Y, edge(Z, U, V), W) >? reach#(U, Y, union(V, W), empty) if!6220reach!62201#(false, X, Y, edge(Z, U, V), W) >? reach#(X, Y, V, edge(Z, U, W)) eq(0, 0) >= true eq(0, s(X)) >= false eq(s(X), 0) >= false eq(s(X), s(Y)) >= eq(X, Y) union(empty, X) >= X union(edge(X, Y, Z), U) >= edge(X, Y, union(Z, U)) We orient these requirements with a polynomial interpretation in the natural numbers. We consider usable_rules with respect to the following argument filtering: if!6220reach!62201#(x_1,x_2,x_3,x_4,x_5) = if!6220reach!62201#(x_2x_3,x_4,x_5) if!6220reach!62202#(x_1,x_2,x_3,x_4,x_5) = if!6220reach!62202#(x_2x_3,x_4,x_5) This leaves the following ordering requirements: reach#(X, Y, edge(Z, U, V), W) >= if!6220reach!62201#(eq(X, Z), X, Y, edge(Z, U, V), W) if!6220reach!62201#(true, X, Y, edge(Z, U, V), W) >= if!6220reach!62202#(eq(Y, U), X, Y, edge(Z, U, V), W) if!6220reach!62202#(false, X, Y, edge(Z, U, V), W) > reach#(X, Y, V, W) if!6220reach!62202#(false, X, Y, edge(Z, U, V), W) >= reach#(U, Y, union(V, W), empty) if!6220reach!62201#(false, X, Y, edge(Z, U, V), W) >= reach#(X, Y, V, edge(Z, U, W)) union(empty, X) >= X union(edge(X, Y, Z), U) >= edge(X, Y, union(Z, U)) The following interpretation satisfies the requirements: 0 = 3 edge = \y0y1y2.1 + y2 empty = 0 eq = \y0y1.0 false = 0 if!6220reach!62201# = \y0y1y2y3y4.y3 + y4 if!6220reach!62202# = \y0y1y2y3y4.y3 + y4 reach# = \y0y1y2y3.y2 + y3 s = \y0.3 true = 0 union = \y0y1.y0 + y1 Using this interpretation, the requirements translate to: [[reach#(_x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 >= 1 + x4 + x5 = [[if!6220reach!62201#(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] [[if!6220reach!62201#(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 >= 1 + x4 + x5 = [[if!6220reach!62202#(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] [[if!6220reach!62202#(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 > x4 + x5 = [[reach#(_x0, _x1, _x4, _x5)]] [[if!6220reach!62202#(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 > x4 + x5 = [[reach#(_x3, _x1, union(_x4, _x5), empty)]] [[if!6220reach!62201#(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 >= 1 + x4 + x5 = [[reach#(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] [[union(empty, _x0)]] = x0 >= x0 = [[_x0]] [[union(edge(_x0, _x1, _x2), _x3)]] = 1 + x2 + x3 >= 1 + x2 + x3 = [[edge(_x0, _x1, union(_x2, _x3))]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_3, R_0, minimal, formative) by (P_4, R_0, minimal, formative), where P_4 consists of: reach#(X, Y, edge(Z, U, V), W) =#> if!6220reach!62201#(eq(X, Z), X, Y, edge(Z, U, V), W) if!6220reach!62201#(true, X, Y, edge(Z, U, V), W) =#> if!6220reach!62202#(eq(Y, U), X, Y, edge(Z, U, V), W) if!6220reach!62201#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, edge(Z, U, W)) Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_4, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_4, 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 : 1, 2 * 1 : * 2 : 0 This graph has the following strongly connected components: P_5: reach#(X, Y, edge(Z, U, V), W) =#> if!6220reach!62201#(eq(X, Z), X, Y, edge(Z, U, V), W) if!6220reach!62201#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, edge(Z, U, W)) By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_4, R_0, m, f) by (P_5, R_0, m, f). Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_5, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_5, R_0, minimal, formative). We apply the subterm criterion with the following projection function: nu(if!6220reach!62201#) = 4 nu(reach#) = 3 Thus, we can orient the dependency pairs as follows: nu(reach#(X, Y, edge(Z, U, V), W)) = edge(Z, U, V) = edge(Z, U, V) = nu(if!6220reach!62201#(eq(X, Z), X, Y, edge(Z, U, V), W)) nu(if!6220reach!62201#(false, X, Y, edge(Z, U, V), W)) = edge(Z, U, V) |> V = nu(reach#(X, Y, V, edge(Z, U, W))) By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_5, R_0, minimal, f) by (P_6, R_0, minimal, f), where P_6 contains: reach#(X, Y, edge(Z, U, V), W) =#> if!6220reach!62201#(eq(X, Z), X, Y, edge(Z, U, V), W) Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_6, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_6, 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 : 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 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). We apply the subterm criterion with the following projection function: nu(union#) = 1 Thus, we can orient the dependency pairs as follows: nu(union#(edge(X, Y, Z), U)) = edge(X, Y, Z) |> Z = nu(union#(Z, U)) By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_2, R_0, minimal, f) by ({}, R_0, minimal, f). By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed. 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(eq#) = 1 Thus, we can orient the dependency pairs as follows: nu(eq#(s(X), s(Y))) = s(X) |> X = nu(eq#(X, Y)) 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.