/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. We are asked to determine termination of the following first-order TRS. 0 : [] --> o app : [o * o] --> o cons : [o * o] --> o false : [] --> o high : [o * o] --> o ifhigh : [o * o * o] --> o iflow : [o * o * o] --> o le : [o * o] --> o low : [o * o] --> o nil : [] --> o quicksort : [o] --> o s : [o] --> o true : [] --> o le(0, X) => true le(s(X), 0) => false le(s(X), s(Y)) => le(X, Y) app(nil, X) => X app(cons(X, Y), Z) => cons(X, app(Y, Z)) low(X, nil) => nil low(X, cons(Y, Z)) => iflow(le(Y, X), X, cons(Y, Z)) iflow(true, X, cons(Y, Z)) => cons(Y, low(X, Z)) iflow(false, X, cons(Y, Z)) => low(X, Z) high(X, nil) => nil high(X, cons(Y, Z)) => ifhigh(le(Y, X), X, cons(Y, Z)) ifhigh(true, X, cons(Y, Z)) => high(X, Z) ifhigh(false, X, cons(Y, Z)) => cons(Y, high(X, Z)) quicksort(nil) => nil quicksort(cons(X, Y)) => app(quicksort(low(X, Y)), cons(X, quicksort(high(X, Y)))) 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 : [] --> nf app : [nf * nf] --> nf cons : [nf * nf] --> nf false : [] --> rd high : [nf * nf] --> nf ifhigh : [rd * nf * nf] --> nf iflow : [rd * nf * nf] --> nf le : [nf * nf] --> rd low : [nf * nf] --> nf nil : [] --> nf quicksort : [nf] --> nf s : [nf] --> nf true : [] --> rd 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] le#(s(X), s(Y)) =#> le#(X, Y) 1] app#(cons(X, Y), Z) =#> app#(Y, Z) 2] low#(X, cons(Y, Z)) =#> iflow#(le(Y, X), X, cons(Y, Z)) 3] low#(X, cons(Y, Z)) =#> le#(Y, X) 4] iflow#(true, X, cons(Y, Z)) =#> low#(X, Z) 5] iflow#(false, X, cons(Y, Z)) =#> low#(X, Z) 6] high#(X, cons(Y, Z)) =#> ifhigh#(le(Y, X), X, cons(Y, Z)) 7] high#(X, cons(Y, Z)) =#> le#(Y, X) 8] ifhigh#(true, X, cons(Y, Z)) =#> high#(X, Z) 9] ifhigh#(false, X, cons(Y, Z)) =#> high#(X, Z) 10] quicksort#(cons(X, Y)) =#> app#(quicksort(low(X, Y)), cons(X, quicksort(high(X, Y)))) 11] quicksort#(cons(X, Y)) =#> quicksort#(low(X, Y)) 12] quicksort#(cons(X, Y)) =#> low#(X, Y) 13] quicksort#(cons(X, Y)) =#> quicksort#(high(X, Y)) 14] quicksort#(cons(X, Y)) =#> high#(X, Y) Rules R_0: le(0, X) => true le(s(X), 0) => false le(s(X), s(Y)) => le(X, Y) app(nil, X) => X app(cons(X, Y), Z) => cons(X, app(Y, Z)) low(X, nil) => nil low(X, cons(Y, Z)) => iflow(le(Y, X), X, cons(Y, Z)) iflow(true, X, cons(Y, Z)) => cons(Y, low(X, Z)) iflow(false, X, cons(Y, Z)) => low(X, Z) high(X, nil) => nil high(X, cons(Y, Z)) => ifhigh(le(Y, X), X, cons(Y, Z)) ifhigh(true, X, cons(Y, Z)) => high(X, Z) ifhigh(false, X, cons(Y, Z)) => cons(Y, high(X, Z)) quicksort(nil) => nil quicksort(cons(X, Y)) => app(quicksort(low(X, Y)), cons(X, quicksort(high(X, Y)))) 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 * 3 : 0 * 4 : 2, 3 * 5 : 2, 3 * 6 : 8, 9 * 7 : 0 * 8 : 6, 7 * 9 : 6, 7 * 10 : 1 * 11 : 10, 11, 12, 13, 14 * 12 : 2, 3 * 13 : 10, 11, 12, 13, 14 * 14 : 6, 7 This graph has the following strongly connected components: P_1: le#(s(X), s(Y)) =#> le#(X, Y) P_2: app#(cons(X, Y), Z) =#> app#(Y, Z) P_3: low#(X, cons(Y, Z)) =#> iflow#(le(Y, X), X, cons(Y, Z)) iflow#(true, X, cons(Y, Z)) =#> low#(X, Z) iflow#(false, X, cons(Y, Z)) =#> low#(X, Z) P_4: high#(X, cons(Y, Z)) =#> ifhigh#(le(Y, X), X, cons(Y, Z)) ifhigh#(true, X, cons(Y, Z)) =#> high#(X, Z) ifhigh#(false, X, cons(Y, Z)) =#> high#(X, Z) P_5: quicksort#(cons(X, Y)) =#> quicksort#(low(X, Y)) quicksort#(cons(X, Y)) =#> quicksort#(high(X, 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), (P_2, R_0, m, f), (P_3, R_0, m, f), (P_4, R_0, m, f) and (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), (P_3, R_0, minimal, formative), (P_4, 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 will use the reduction pair processor with usable rules [Kop12, Thm. 7.44]. The usable rules of (P_5, R_0) are: le(0, X) => true le(s(X), 0) => false le(s(X), s(Y)) => le(X, Y) low(X, nil) => nil low(X, cons(Y, Z)) => iflow(le(Y, X), X, cons(Y, Z)) iflow(true, X, cons(Y, Z)) => cons(Y, low(X, Z)) iflow(false, X, cons(Y, Z)) => low(X, Z) high(X, nil) => nil high(X, cons(Y, Z)) => ifhigh(le(Y, X), X, cons(Y, Z)) ifhigh(true, X, cons(Y, Z)) => high(X, Z) ifhigh(false, X, cons(Y, Z)) => cons(Y, high(X, Z)) It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: quicksort#(cons(X, Y)) >? quicksort#(low(X, Y)) quicksort#(cons(X, Y)) >? quicksort#(high(X, Y)) le(0, X) >= true le(s(X), 0) >= false le(s(X), s(Y)) >= le(X, Y) low(X, nil) >= nil low(X, cons(Y, Z)) >= iflow(le(Y, X), X, cons(Y, Z)) iflow(true, X, cons(Y, Z)) >= cons(Y, low(X, Z)) iflow(false, X, cons(Y, Z)) >= low(X, Z) high(X, nil) >= nil high(X, cons(Y, Z)) >= ifhigh(le(Y, X), X, cons(Y, Z)) ifhigh(true, X, cons(Y, Z)) >= high(X, Z) ifhigh(false, X, cons(Y, Z)) >= cons(Y, high(X, Z)) We orient these requirements with a polynomial interpretation in the natural numbers. We consider usable_rules with respect to the following argument filtering: ifhigh(x_1,x_2,x_3) = ifhigh(x_2x_3) iflow(x_1,x_2,x_3) = iflow(x_2x_3) This leaves the following ordering requirements: quicksort#(cons(X, Y)) > quicksort#(low(X, Y)) quicksort#(cons(X, Y)) >= quicksort#(high(X, Y)) low(X, nil) >= nil low(X, cons(Y, Z)) >= iflow(le(Y, X), X, cons(Y, Z)) iflow(true, X, cons(Y, Z)) >= cons(Y, low(X, Z)) iflow(false, X, cons(Y, Z)) >= low(X, Z) high(X, nil) >= nil high(X, cons(Y, Z)) >= ifhigh(le(Y, X), X, cons(Y, Z)) ifhigh(true, X, cons(Y, Z)) >= high(X, Z) ifhigh(false, X, cons(Y, Z)) >= cons(Y, high(X, Z)) The following interpretation satisfies the requirements: 0 = 3 cons = \y0y1.3 + y1 false = 0 high = \y0y1.y1 ifhigh = \y0y1y2.y2 iflow = \y0y1y2.y2 le = \y0y1.0 low = \y0y1.y1 nil = 0 quicksort# = \y0.3y0 s = \y0.3 true = 0 Using this interpretation, the requirements translate to: [[quicksort#(cons(_x0, _x1))]] = 9 + 3x1 > 3x1 = [[quicksort#(low(_x0, _x1))]] [[quicksort#(cons(_x0, _x1))]] = 9 + 3x1 > 3x1 = [[quicksort#(high(_x0, _x1))]] [[low(_x0, nil)]] = 0 >= 0 = [[nil]] [[low(_x0, cons(_x1, _x2))]] = 3 + x2 >= 3 + x2 = [[iflow(le(_x1, _x0), _x0, cons(_x1, _x2))]] [[iflow(true, _x0, cons(_x1, _x2))]] = 3 + x2 >= 3 + x2 = [[cons(_x1, low(_x0, _x2))]] [[iflow(false, _x0, cons(_x1, _x2))]] = 3 + x2 >= x2 = [[low(_x0, _x2)]] [[high(_x0, nil)]] = 0 >= 0 = [[nil]] [[high(_x0, cons(_x1, _x2))]] = 3 + x2 >= 3 + x2 = [[ifhigh(le(_x1, _x0), _x0, cons(_x1, _x2))]] [[ifhigh(true, _x0, cons(_x1, _x2))]] = 3 + x2 >= x2 = [[high(_x0, _x2)]] [[ifhigh(false, _x0, cons(_x1, _x2))]] = 3 + x2 >= 3 + x2 = [[cons(_x1, high(_x0, _x2))]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_5, R_0) by ({}, R_0). By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed. Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative), (P_3, 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 apply the subterm criterion with the following projection function: nu(high#) = 2 nu(ifhigh#) = 3 Thus, we can orient the dependency pairs as follows: nu(high#(X, cons(Y, Z))) = cons(Y, Z) = cons(Y, Z) = nu(ifhigh#(le(Y, X), X, cons(Y, Z))) nu(ifhigh#(true, X, cons(Y, Z))) = cons(Y, Z) |> Z = nu(high#(X, Z)) nu(ifhigh#(false, X, cons(Y, Z))) = cons(Y, Z) |> Z = nu(high#(X, Z)) By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_4, R_0, minimal, f) by (P_6, R_0, minimal, f), where P_6 contains: high#(X, cons(Y, Z)) =#> ifhigh#(le(Y, X), X, cons(Y, Z)) Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative), (P_3, 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), (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 apply the subterm criterion with the following projection function: nu(iflow#) = 3 nu(low#) = 2 Thus, we can orient the dependency pairs as follows: nu(low#(X, cons(Y, Z))) = cons(Y, Z) = cons(Y, Z) = nu(iflow#(le(Y, X), X, cons(Y, Z))) nu(iflow#(true, X, cons(Y, Z))) = cons(Y, Z) |> Z = nu(low#(X, Z)) nu(iflow#(false, X, cons(Y, Z))) = cons(Y, Z) |> Z = nu(low#(X, Z)) By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_3, R_0, minimal, f) by (P_7, R_0, minimal, f), where P_7 contains: low#(X, cons(Y, Z)) =#> iflow#(le(Y, X), X, cons(Y, Z)) Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_7, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_7, 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(app#) = 1 Thus, we can orient the dependency pairs as follows: nu(app#(cons(X, Y), Z)) = cons(X, Y) |> Y = nu(app#(Y, Z)) 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(le#) = 1 Thus, we can orient the dependency pairs as follows: nu(le#(s(X), s(Y))) = s(X) |> X = nu(le#(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.