/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. der : [o] --> o din : [o] --> o dout : [o] --> o plus : [o * o] --> o times : [o * o] --> o u21 : [o * o * o] --> o u22 : [o * o * o * o] --> o u31 : [o * o * o] --> o u32 : [o * o * o * o] --> o u41 : [o * o] --> o u42 : [o * o * o] --> o din(der(plus(X, Y))) => u21(din(der(X)), X, Y) u21(dout(X), Y, Z) => u22(din(der(Z)), Y, Z, X) u22(dout(X), Y, Z, U) => dout(plus(U, X)) din(der(times(X, Y))) => u31(din(der(X)), X, Y) u31(dout(X), Y, Z) => u32(din(der(Z)), Y, Z, X) u32(dout(X), Y, Z, U) => dout(plus(times(Y, X), times(Z, U))) din(der(der(X))) => u41(din(der(X)), X) u41(dout(X), Y) => u42(din(der(X)), Y, X) u42(dout(X), Y, Z) => dout(X) 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: der : [sd] --> sd din : [sd] --> yd dout : [sd] --> yd plus : [sd * sd] --> sd times : [sd * sd] --> sd u21 : [yd * sd * sd] --> yd u22 : [yd * sd * sd * sd] --> yd u31 : [yd * sd * sd] --> yd u32 : [yd * sd * sd * sd] --> yd u41 : [yd * sd] --> yd u42 : [yd * sd * sd] --> yd 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] din#(der(plus(X, Y))) =#> u21#(din(der(X)), X, Y) 1] din#(der(plus(X, Y))) =#> din#(der(X)) 2] u21#(dout(X), Y, Z) =#> u22#(din(der(Z)), Y, Z, X) 3] u21#(dout(X), Y, Z) =#> din#(der(Z)) 4] din#(der(times(X, Y))) =#> u31#(din(der(X)), X, Y) 5] din#(der(times(X, Y))) =#> din#(der(X)) 6] u31#(dout(X), Y, Z) =#> u32#(din(der(Z)), Y, Z, X) 7] u31#(dout(X), Y, Z) =#> din#(der(Z)) 8] din#(der(der(X))) =#> u41#(din(der(X)), X) 9] din#(der(der(X))) =#> din#(der(X)) 10] u41#(dout(X), Y) =#> u42#(din(der(X)), Y, X) 11] u41#(dout(X), Y) =#> din#(der(X)) Rules R_0: din(der(plus(X, Y))) => u21(din(der(X)), X, Y) u21(dout(X), Y, Z) => u22(din(der(Z)), Y, Z, X) u22(dout(X), Y, Z, U) => dout(plus(U, X)) din(der(times(X, Y))) => u31(din(der(X)), X, Y) u31(dout(X), Y, Z) => u32(din(der(Z)), Y, Z, X) u32(dout(X), Y, Z, U) => dout(plus(times(Y, X), times(Z, U))) din(der(der(X))) => u41(din(der(X)), X) u41(dout(X), Y) => u42(din(der(X)), Y, X) u42(dout(X), Y, Z) => dout(X) 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 : 2, 3 * 1 : 0, 1, 4, 5, 8, 9 * 2 : * 3 : 0, 1, 4, 5, 8, 9 * 4 : 6, 7 * 5 : 0, 1, 4, 5, 8, 9 * 6 : * 7 : 0, 1, 4, 5, 8, 9 * 8 : 10, 11 * 9 : 0, 1, 4, 5, 8, 9 * 10 : * 11 : 0, 1, 4, 5, 8, 9 This graph has the following strongly connected components: P_1: din#(der(plus(X, Y))) =#> u21#(din(der(X)), X, Y) din#(der(plus(X, Y))) =#> din#(der(X)) u21#(dout(X), Y, Z) =#> din#(der(Z)) din#(der(times(X, Y))) =#> u31#(din(der(X)), X, Y) din#(der(times(X, Y))) =#> din#(der(X)) u31#(dout(X), Y, Z) =#> din#(der(Z)) din#(der(der(X))) =#> u41#(din(der(X)), X) din#(der(der(X))) =#> din#(der(X)) u41#(dout(X), Y) =#> din#(der(X)) 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, minimal, formative) is finite. We consider the dependency pair problem (P_1, R_0, 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: din#(der(plus(X, Y))) >? u21#(din(der(X)), X, Y) din#(der(plus(X, Y))) >? din#(der(X)) u21#(dout(X), Y, Z) >? din#(der(Z)) din#(der(times(X, Y))) >? u31#(din(der(X)), X, Y) din#(der(times(X, Y))) >? din#(der(X)) u31#(dout(X), Y, Z) >? din#(der(Z)) din#(der(der(X))) >? u41#(din(der(X)), X) din#(der(der(X))) >? din#(der(X)) u41#(dout(X), Y) >? din#(der(X)) din(der(plus(X, Y))) >= u21(din(der(X)), X, Y) u21(dout(X), Y, Z) >= u22(din(der(Z)), Y, Z, X) u22(dout(X), Y, Z, U) >= dout(plus(U, X)) din(der(times(X, Y))) >= u31(din(der(X)), X, Y) u31(dout(X), Y, Z) >= u32(din(der(Z)), Y, Z, X) u32(dout(X), Y, Z, U) >= dout(plus(times(Y, X), times(Z, U))) din(der(der(X))) >= u41(din(der(X)), X) u41(dout(X), Y) >= u42(din(der(X)), Y, X) u42(dout(X), Y, Z) >= dout(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: der = \y0.0 din = \y0.0 din# = \y0.0 dout = \y0.1 plus = \y0y1.0 times = \y0y1.0 u21 = \y0y1y2.0 u21# = \y0y1y2.0 u22 = \y0y1y2y3.y0 u31 = \y0y1y2.0 u31# = \y0y1y2.0 u32 = \y0y1y2y3.2y0 u41 = \y0y1.y0 u41# = \y0y1.y0 u42 = \y0y1y2.1 Using this interpretation, the requirements translate to: [[din#(der(plus(_x0, _x1)))]] = 0 >= 0 = [[u21#(din(der(_x0)), _x0, _x1)]] [[din#(der(plus(_x0, _x1)))]] = 0 >= 0 = [[din#(der(_x0))]] [[u21#(dout(_x0), _x1, _x2)]] = 0 >= 0 = [[din#(der(_x2))]] [[din#(der(times(_x0, _x1)))]] = 0 >= 0 = [[u31#(din(der(_x0)), _x0, _x1)]] [[din#(der(times(_x0, _x1)))]] = 0 >= 0 = [[din#(der(_x0))]] [[u31#(dout(_x0), _x1, _x2)]] = 0 >= 0 = [[din#(der(_x2))]] [[din#(der(der(_x0)))]] = 0 >= 0 = [[u41#(din(der(_x0)), _x0)]] [[din#(der(der(_x0)))]] = 0 >= 0 = [[din#(der(_x0))]] [[u41#(dout(_x0), _x1)]] = 1 > 0 = [[din#(der(_x0))]] [[din(der(plus(_x0, _x1)))]] = 0 >= 0 = [[u21(din(der(_x0)), _x0, _x1)]] [[u21(dout(_x0), _x1, _x2)]] = 0 >= 0 = [[u22(din(der(_x2)), _x1, _x2, _x0)]] [[u22(dout(_x0), _x1, _x2, _x3)]] = 1 >= 1 = [[dout(plus(_x3, _x0))]] [[din(der(times(_x0, _x1)))]] = 0 >= 0 = [[u31(din(der(_x0)), _x0, _x1)]] [[u31(dout(_x0), _x1, _x2)]] = 0 >= 0 = [[u32(din(der(_x2)), _x1, _x2, _x0)]] [[u32(dout(_x0), _x1, _x2, _x3)]] = 2 >= 1 = [[dout(plus(times(_x1, _x0), times(_x2, _x3)))]] [[din(der(der(_x0)))]] = 0 >= 0 = [[u41(din(der(_x0)), _x0)]] [[u41(dout(_x0), _x1)]] = 1 >= 1 = [[u42(din(der(_x0)), _x1, _x0)]] [[u42(dout(_x0), _x1, _x2)]] = 1 >= 1 = [[dout(_x0)]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_2, R_0, minimal, formative), where P_2 consists of: din#(der(plus(X, Y))) =#> u21#(din(der(X)), X, Y) din#(der(plus(X, Y))) =#> din#(der(X)) u21#(dout(X), Y, Z) =#> din#(der(Z)) din#(der(times(X, Y))) =#> u31#(din(der(X)), X, Y) din#(der(times(X, Y))) =#> din#(der(X)) u31#(dout(X), Y, Z) =#> din#(der(Z)) din#(der(der(X))) =#> u41#(din(der(X)), X) din#(der(der(X))) =#> din#(der(X)) Thus, the original system is terminating if (P_2, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_2, 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 : 2 * 1 : 0, 1, 3, 4, 6, 7 * 2 : 0, 1, 3, 4, 6, 7 * 3 : 5 * 4 : 0, 1, 3, 4, 6, 7 * 5 : 0, 1, 3, 4, 6, 7 * 6 : * 7 : 0, 1, 3, 4, 6, 7 This graph has the following strongly connected components: P_3: din#(der(plus(X, Y))) =#> u21#(din(der(X)), X, Y) din#(der(plus(X, Y))) =#> din#(der(X)) u21#(dout(X), Y, Z) =#> din#(der(Z)) din#(der(times(X, Y))) =#> u31#(din(der(X)), X, Y) din#(der(times(X, Y))) =#> din#(der(X)) u31#(dout(X), Y, Z) =#> din#(der(Z)) din#(der(der(X))) =#> din#(der(X)) By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_2, R_0, m, f) by (P_3, R_0, m, f). Thus, the original system is terminating if (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 [Kop12, Thm. 7.16]. It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: din#(der(plus(X, Y))) >? u21#(din(der(X)), X, Y) din#(der(plus(X, Y))) >? din#(der(X)) u21#(dout(X), Y, Z) >? din#(der(Z)) din#(der(times(X, Y))) >? u31#(din(der(X)), X, Y) din#(der(times(X, Y))) >? din#(der(X)) u31#(dout(X), Y, Z) >? din#(der(Z)) din#(der(der(X))) >? din#(der(X)) din(der(plus(X, Y))) >= u21(din(der(X)), X, Y) u21(dout(X), Y, Z) >= u22(din(der(Z)), Y, Z, X) u22(dout(X), Y, Z, U) >= dout(plus(U, X)) din(der(times(X, Y))) >= u31(din(der(X)), X, Y) u31(dout(X), Y, Z) >= u32(din(der(Z)), Y, Z, X) u32(dout(X), Y, Z, U) >= dout(plus(times(Y, X), times(Z, U))) din(der(der(X))) >= u41(din(der(X)), X) u41(dout(X), Y) >= u42(din(der(X)), Y, X) u42(dout(X), Y, Z) >= dout(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: der = \y0.0 din = \y0.0 din# = \y0.0 dout = \y0.2 + 3y0 plus = \y0y1.2 times = \y0y1.0 u21 = \y0y1y2.2y0 u21# = \y0y1y2.2y0 u22 = \y0y1y2y3.2 + 3y0 u31 = \y0y1y2.y0 u31# = \y0y1y2.0 u32 = \y0y1y2y3.2 + 3y0 u41 = \y0y1.2y0 u42 = \y0y1y2.2y0 Using this interpretation, the requirements translate to: [[din#(der(plus(_x0, _x1)))]] = 0 >= 0 = [[u21#(din(der(_x0)), _x0, _x1)]] [[din#(der(plus(_x0, _x1)))]] = 0 >= 0 = [[din#(der(_x0))]] [[u21#(dout(_x0), _x1, _x2)]] = 4 + 6x0 > 0 = [[din#(der(_x2))]] [[din#(der(times(_x0, _x1)))]] = 0 >= 0 = [[u31#(din(der(_x0)), _x0, _x1)]] [[din#(der(times(_x0, _x1)))]] = 0 >= 0 = [[din#(der(_x0))]] [[u31#(dout(_x0), _x1, _x2)]] = 0 >= 0 = [[din#(der(_x2))]] [[din#(der(der(_x0)))]] = 0 >= 0 = [[din#(der(_x0))]] [[din(der(plus(_x0, _x1)))]] = 0 >= 0 = [[u21(din(der(_x0)), _x0, _x1)]] [[u21(dout(_x0), _x1, _x2)]] = 4 + 6x0 >= 2 = [[u22(din(der(_x2)), _x1, _x2, _x0)]] [[u22(dout(_x0), _x1, _x2, _x3)]] = 8 + 9x0 >= 8 = [[dout(plus(_x3, _x0))]] [[din(der(times(_x0, _x1)))]] = 0 >= 0 = [[u31(din(der(_x0)), _x0, _x1)]] [[u31(dout(_x0), _x1, _x2)]] = 2 + 3x0 >= 2 = [[u32(din(der(_x2)), _x1, _x2, _x0)]] [[u32(dout(_x0), _x1, _x2, _x3)]] = 8 + 9x0 >= 8 = [[dout(plus(times(_x1, _x0), times(_x2, _x3)))]] [[din(der(der(_x0)))]] = 0 >= 0 = [[u41(din(der(_x0)), _x0)]] [[u41(dout(_x0), _x1)]] = 4 + 6x0 >= 0 = [[u42(din(der(_x0)), _x1, _x0)]] [[u42(dout(_x0), _x1, _x2)]] = 4 + 6x0 >= 2 + 3x0 = [[dout(_x0)]] 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: din#(der(plus(X, Y))) =#> u21#(din(der(X)), X, Y) din#(der(plus(X, Y))) =#> din#(der(X)) din#(der(times(X, Y))) =#> u31#(din(der(X)), X, Y) din#(der(times(X, Y))) =#> din#(der(X)) u31#(dout(X), Y, Z) =#> din#(der(Z)) din#(der(der(X))) =#> din#(der(X)) Thus, the original system is terminating if (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 : 0, 1, 2, 3, 5 * 2 : 4 * 3 : 0, 1, 2, 3, 5 * 4 : 0, 1, 2, 3, 5 * 5 : 0, 1, 2, 3, 5 This graph has the following strongly connected components: P_5: din#(der(plus(X, Y))) =#> din#(der(X)) din#(der(times(X, Y))) =#> u31#(din(der(X)), X, Y) din#(der(times(X, Y))) =#> din#(der(X)) u31#(dout(X), Y, Z) =#> din#(der(Z)) din#(der(der(X))) =#> din#(der(X)) 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 (P_5, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_5, R_0, 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: din#(der(plus(X, Y))) >? din#(der(X)) din#(der(times(X, Y))) >? u31#(din(der(X)), X, Y) din#(der(times(X, Y))) >? din#(der(X)) u31#(dout(X), Y, Z) >? din#(der(Z)) din#(der(der(X))) >? din#(der(X)) din(der(plus(X, Y))) >= u21(din(der(X)), X, Y) u21(dout(X), Y, Z) >= u22(din(der(Z)), Y, Z, X) u22(dout(X), Y, Z, U) >= dout(plus(U, X)) din(der(times(X, Y))) >= u31(din(der(X)), X, Y) u31(dout(X), Y, Z) >= u32(din(der(Z)), Y, Z, X) u32(dout(X), Y, Z, U) >= dout(plus(times(Y, X), times(Z, U))) din(der(der(X))) >= u41(din(der(X)), X) u41(dout(X), Y) >= u42(din(der(X)), Y, X) u42(dout(X), Y, Z) >= dout(X) We orient these requirements with a polynomial interpretation in the natural numbers. We consider usable_rules with respect to the following argument filtering: u31#(x_1,x_2,x_3) = u31#(x_2x_3) This leaves the following ordering requirements: din#(der(plus(X, Y))) >= din#(der(X)) din#(der(times(X, Y))) > u31#(din(der(X)), X, Y) din#(der(times(X, Y))) >= din#(der(X)) u31#(dout(X), Y, Z) >= din#(der(Z)) din#(der(der(X))) >= din#(der(X)) The following interpretation satisfies the requirements: der = \y0.3y0 din = \y0.0 din# = \y0.2 + y0 dout = \y0.0 plus = \y0y1.3y0 times = \y0y1.1 + 2y1 + 3y0 u21 = \y0y1y2.0 u22 = \y0y1y2y3.0 u31 = \y0y1y2.0 u31# = \y0y1y2.3 + 3y2 u32 = \y0y1y2y3.0 u41 = \y0y1.0 u42 = \y0y1y2.0 Using this interpretation, the requirements translate to: [[din#(der(plus(_x0, _x1)))]] = 2 + 9x0 >= 2 + 3x0 = [[din#(der(_x0))]] [[din#(der(times(_x0, _x1)))]] = 5 + 6x1 + 9x0 > 3 + 3x1 = [[u31#(din(der(_x0)), _x0, _x1)]] [[din#(der(times(_x0, _x1)))]] = 5 + 6x1 + 9x0 > 2 + 3x0 = [[din#(der(_x0))]] [[u31#(dout(_x0), _x1, _x2)]] = 3 + 3x2 > 2 + 3x2 = [[din#(der(_x2))]] [[din#(der(der(_x0)))]] = 2 + 9x0 >= 2 + 3x0 = [[din#(der(_x0))]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_5, R_0, minimal, formative) by (P_6, R_0, minimal, formative), where P_6 consists of: din#(der(plus(X, Y))) =#> din#(der(X)) din#(der(der(X))) =#> din#(der(X)) Thus, the original system is terminating if (P_6, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_6, R_0, minimal, formative). This combination (P_6, R_0) has no formative rules! We will name the empty set of rules:R_1. By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_6, R_0, minimal, formative) by (P_6, R_1, minimal, formative). Thus, the original system is terminating if (P_6, R_1, minimal, formative) is finite. We consider the dependency pair problem (P_6, 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: din#(der(plus(X, Y))) >? din#(der(X)) din#(der(der(X))) >? din#(der(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: der = \y0.3 + 3y0 din# = \y0.y0 plus = \y0y1.3 + 3y0 Using this interpretation, the requirements translate to: [[din#(der(plus(_x0, _x1)))]] = 12 + 9x0 > 3 + 3x0 = [[din#(der(_x0))]] [[din#(der(der(_x0)))]] = 12 + 9x0 > 3 + 3x0 = [[din#(der(_x0))]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_6, R_1) by ({}, R_1). 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.