We consider the system Applicative_first_order_05__#3.6. Alphabet: 0 : [] --> b cons : [c * d] --> d false : [] --> a filter : [c -> a * d] --> d filter2 : [a * c -> a * c * d] --> d gcd : [b * b] --> b if!fac6220gcd : [a * b * b] --> b le : [b * b] --> a map : [c -> c * d] --> d minus : [b * b] --> b nil : [] --> d pred : [b] --> b s : [b] --> b true : [] --> a Rules: le(0, x) => true le(s(x), 0) => false le(s(x), s(y)) => le(x, y) pred(s(x)) => x minus(x, 0) => x minus(x, s(y)) => pred(minus(x, y)) gcd(0, x) => x gcd(s(x), 0) => s(x) gcd(s(x), s(y)) => if!fac6220gcd(le(y, x), s(x), s(y)) if!fac6220gcd(true, s(x), s(y)) => gcd(minus(x, y), s(y)) if!fac6220gcd(false, s(x), s(y)) => gcd(minus(y, x), s(x)) map(f, nil) => nil map(f, cons(x, y)) => cons(f x, map(f, y)) filter(f, nil) => nil filter(f, cons(x, y)) => filter2(f x, f, x, y) filter2(true, f, x, y) => cons(x, filter(f, y)) filter2(false, f, x, y) => filter(f, y) This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). 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]). We thus obtain the following dependency pair problem (P_0, R_0, computable, formative): Dependency Pairs P_0: 0] le#(s(X), s(Y)) =#> le#(X, Y) 1] minus#(X, s(Y)) =#> pred#(minus(X, Y)) 2] minus#(X, s(Y)) =#> minus#(X, Y) 3] gcd#(s(X), s(Y)) =#> if!fac6220gcd#(le(Y, X), s(X), s(Y)) 4] gcd#(s(X), s(Y)) =#> le#(Y, X) 5] if!fac6220gcd#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) 6] if!fac6220gcd#(true, s(X), s(Y)) =#> minus#(X, Y) 7] if!fac6220gcd#(false, s(X), s(Y)) =#> gcd#(minus(Y, X), s(X)) 8] if!fac6220gcd#(false, s(X), s(Y)) =#> minus#(Y, X) 9] map#(F, cons(X, Y)) =#> map#(F, Y) 10] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y) 11] filter2#(true, F, X, Y) =#> filter#(F, Y) 12] filter2#(false, F, X, Y) =#> filter#(F, Y) Rules R_0: le(0, X) => true le(s(X), 0) => false le(s(X), s(Y)) => le(X, Y) pred(s(X)) => X minus(X, 0) => X minus(X, s(Y)) => pred(minus(X, Y)) gcd(0, X) => X gcd(s(X), 0) => s(X) gcd(s(X), s(Y)) => if!fac6220gcd(le(Y, X), s(X), s(Y)) if!fac6220gcd(true, s(X), s(Y)) => gcd(minus(X, Y), s(Y)) if!fac6220gcd(false, s(X), s(Y)) => gcd(minus(Y, X), s(X)) map(F, nil) => nil map(F, cons(X, Y)) => cons(F X, map(F, Y)) filter(F, nil) => nil filter(F, cons(X, Y)) => filter2(F X, F, X, Y) filter2(true, F, X, Y) => cons(X, filter(F, Y)) filter2(false, F, X, Y) => filter(F, Y) 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 : 0 * 1 : * 2 : 1, 2 * 3 : 5, 6, 7, 8 * 4 : 0 * 5 : 3, 4 * 6 : 1, 2 * 7 : 3, 4 * 8 : 1, 2 * 9 : 9 * 10 : 11, 12 * 11 : 10 * 12 : 10 This graph has the following strongly connected components: P_1: le#(s(X), s(Y)) =#> le#(X, Y) P_2: minus#(X, s(Y)) =#> minus#(X, Y) P_3: gcd#(s(X), s(Y)) =#> if!fac6220gcd#(le(Y, X), s(X), s(Y)) if!fac6220gcd#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) if!fac6220gcd#(false, s(X), s(Y)) =#> gcd#(minus(Y, X), s(X)) P_4: map#(F, cons(X, Y)) =#> map#(F, Y) P_5: filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y) filter2#(true, F, X, Y) =#> filter#(F, Y) filter2#(false, F, 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), (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, computable, formative), (P_2, R_0, computable, formative), (P_3, R_0, computable, formative), (P_4, R_0, computable, formative) and (P_5, R_0, computable, formative) is finite. We consider the dependency pair problem (P_5, R_0, computable, formative). We apply the subterm criterion with the following projection function: nu(filter2#) = 4 nu(filter#) = 2 Thus, we can orient the dependency pairs as follows: nu(filter#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter2#(F X, F, X, Y)) nu(filter2#(true, F, X, Y)) = Y = Y = nu(filter#(F, Y)) nu(filter2#(false, F, X, Y)) = Y = Y = nu(filter#(F, Y)) By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_5, R_0, computable, f) by (P_6, R_0, computable, f), where P_6 contains: filter2#(true, F, X, Y) =#> filter#(F, Y) filter2#(false, F, X, Y) =#> filter#(F, Y) Thus, the original system is terminating if each of (P_1, R_0, computable, formative), (P_2, R_0, computable, formative), (P_3, R_0, computable, formative), (P_4, R_0, computable, formative) and (P_6, R_0, computable, formative) is finite. We consider the dependency pair problem (P_6, 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 : 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, computable, formative), (P_2, R_0, computable, formative), (P_3, R_0, computable, formative) and (P_4, R_0, computable, formative) is finite. We consider the dependency pair problem (P_4, R_0, computable, formative). We apply the subterm criterion with the following projection function: nu(map#) = 2 Thus, we can orient the dependency pairs as follows: nu(map#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(map#(F, Y)) By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_4, R_0, computable, f) by ({}, R_0, computable, f). 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, computable, formative), (P_2, R_0, computable, formative) and (P_3, R_0, computable, formative) is finite. We consider the dependency pair problem (P_3, R_0, computable, formative). The formative rules of (P_3, R_0) are R_1 ::= le(0, X) => true le(s(X), 0) => false le(s(X), s(Y)) => le(X, Y) pred(s(X)) => X minus(X, 0) => X minus(X, s(Y)) => pred(minus(X, Y)) gcd(0, X) => X gcd(s(X), 0) => s(X) gcd(s(X), s(Y)) => if!fac6220gcd(le(Y, X), s(X), s(Y)) if!fac6220gcd(true, s(X), s(Y)) => gcd(minus(X, Y), s(Y)) if!fac6220gcd(false, s(X), s(Y)) => gcd(minus(Y, X), s(X)) By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_3, R_0, computable, formative) by (P_3, R_1, computable, formative). Thus, the original system is terminating if each of (P_1, R_0, computable, formative), (P_2, R_0, computable, formative) and (P_3, R_1, computable, formative) is finite. We consider the dependency pair problem (P_3, R_1, computable, formative). We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44]. The usable rules of (P_3, R_1) are: le(0, X) => true le(s(X), 0) => false le(s(X), s(Y)) => le(X, Y) pred(s(X)) => X minus(X, 0) => X minus(X, s(Y)) => pred(minus(X, Y)) It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: gcd#(s(X), s(Y)) >? if!fac6220gcd#(le(Y, X), s(X), s(Y)) if!fac6220gcd#(true, s(X), s(Y)) >? gcd#(minus(X, Y), s(Y)) if!fac6220gcd#(false, s(X), s(Y)) >? gcd#(minus(Y, X), s(X)) le(0, X) >= true le(s(X), 0) >= false le(s(X), s(Y)) >= le(X, Y) pred(s(X)) >= X minus(X, 0) >= X minus(X, s(Y)) >= pred(minus(X, Y)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 3 false = 1 gcd# = \y0y1.y1 + 2y0 if!fac6220gcd# = \y0y1y2.y1 + y2 + 2y0 le = \y0y1.y1 minus = \y0y1.y0 pred = \y0.y0 s = \y0.2y0 true = 0 Using this interpretation, the requirements translate to: [[gcd#(s(_x0), s(_x1))]] = 2x1 + 4x0 >= 2x1 + 4x0 = [[if!fac6220gcd#(le(_x1, _x0), s(_x0), s(_x1))]] [[if!fac6220gcd#(true, s(_x0), s(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[gcd#(minus(_x0, _x1), s(_x1))]] [[if!fac6220gcd#(false, s(_x0), s(_x1))]] = 2 + 2x0 + 2x1 > 2x0 + 2x1 = [[gcd#(minus(_x1, _x0), s(_x0))]] [[le(0, _x0)]] = x0 >= 0 = [[true]] [[le(s(_x0), 0)]] = 3 >= 1 = [[false]] [[le(s(_x0), s(_x1))]] = 2x1 >= x1 = [[le(_x0, _x1)]] [[pred(s(_x0))]] = 2x0 >= x0 = [[_x0]] [[minus(_x0, 0)]] = x0 >= x0 = [[_x0]] [[minus(_x0, s(_x1))]] = x0 >= x0 = [[pred(minus(_x0, _x1))]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_3, R_1, computable, formative) by (P_7, R_1, computable, formative), where P_7 consists of: gcd#(s(X), s(Y)) =#> if!fac6220gcd#(le(Y, X), s(X), s(Y)) if!fac6220gcd#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) Thus, the original system is terminating if each of (P_1, R_0, computable, formative), (P_2, R_0, computable, formative) and (P_7, R_1, computable, formative) is finite. We consider the dependency pair problem (P_7, R_1, computable, formative). We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44]. The usable rules of (P_7, R_1) are: le(0, X) => true le(s(X), 0) => false le(s(X), s(Y)) => le(X, Y) pred(s(X)) => X minus(X, 0) => X minus(X, s(Y)) => pred(minus(X, Y)) It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: gcd#(s(X), s(Y)) >? if!fac6220gcd#(le(Y, X), s(X), s(Y)) if!fac6220gcd#(true, s(X), s(Y)) >? gcd#(minus(X, Y), s(Y)) le(0, X) >= true le(s(X), 0) >= false le(s(X), s(Y)) >= le(X, Y) pred(s(X)) >= X minus(X, 0) >= X minus(X, s(Y)) >= pred(minus(X, Y)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 3 false = 0 gcd# = \y0y1.1 + y0 if!fac6220gcd# = \y0y1y2.y1 le = \y0y1.0 minus = \y0y1.y0 pred = \y0.y0 s = \y0.1 + 2y0 true = 0 Using this interpretation, the requirements translate to: [[gcd#(s(_x0), s(_x1))]] = 2 + 2x0 > 1 + 2x0 = [[if!fac6220gcd#(le(_x1, _x0), s(_x0), s(_x1))]] [[if!fac6220gcd#(true, s(_x0), s(_x1))]] = 1 + 2x0 >= 1 + x0 = [[gcd#(minus(_x0, _x1), s(_x1))]] [[le(0, _x0)]] = 0 >= 0 = [[true]] [[le(s(_x0), 0)]] = 0 >= 0 = [[false]] [[le(s(_x0), s(_x1))]] = 0 >= 0 = [[le(_x0, _x1)]] [[pred(s(_x0))]] = 1 + 2x0 >= x0 = [[_x0]] [[minus(_x0, 0)]] = x0 >= x0 = [[_x0]] [[minus(_x0, s(_x1))]] = x0 >= x0 = [[pred(minus(_x0, _x1))]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_7, R_1, computable, formative) by (P_8, R_1, computable, formative), where P_8 consists of: if!fac6220gcd#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) Thus, the original system is terminating if each of (P_1, R_0, computable, formative), (P_2, R_0, computable, formative) and (P_8, R_1, computable, formative) is finite. We consider the dependency pair problem (P_8, R_1, 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. Thus, the original system is terminating if each of (P_1, R_0, computable, formative) and (P_2, R_0, computable, formative) is finite. We consider the dependency pair problem (P_2, R_0, computable, formative). We apply the subterm criterion with the following projection function: nu(minus#) = 2 Thus, we can orient the dependency pairs as follows: nu(minus#(X, s(Y))) = s(Y) |> Y = nu(minus#(X, Y)) By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_2, R_0, computable, f) by ({}, R_0, computable, 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, 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(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 [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_1, R_0, computable, f) by ({}, R_0, computable, 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 +++ [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.