We consider the system zipWith. Alphabet: 0 : [] --> nat cons : [nat * list] --> list false : [] --> bool gcd : [nat * nat] --> nat gcdlists : [list * list] --> list if : [bool * nat * nat] --> nat le : [nat * nat] --> bool minus : [nat * nat] --> nat nil : [] --> list s : [nat] --> nat true : [] --> bool zipWith : [nat -> nat -> nat * list * list] --> list Rules: le(0, x) => true le(s(x), 0) => false le(s(x), s(y)) => le(x, y) minus(x, 0) => x minus(s(x), s(y)) => minus(x, y) gcd(0, x) => 0 gcd(s(x), 0) => 0 gcd(s(x), s(y)) => if(le(y, x), s(x), s(y)) if(true, s(x), s(y)) => gcd(minus(x, y), s(y)) if(false, s(x), s(y)) => gcd(minus(y, x), s(x)) zipWith(f, x, nil) => nil zipWith(f, nil, x) => nil zipWith(f, cons(x, y), cons(z, u)) => cons(f x z, zipWith(f, y, u)) gcdlists(x, y) => zipWith(/\z./\u.gcd(z, u), x, 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#(s(X), s(Y)) =#> minus#(X, Y) 2] gcd#(s(X), s(Y)) =#> if#(le(Y, X), s(X), s(Y)) 3] gcd#(s(X), s(Y)) =#> le#(Y, X) 4] if#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) 5] if#(true, s(X), s(Y)) =#> minus#(X, Y) 6] if#(false, s(X), s(Y)) =#> gcd#(minus(Y, X), s(X)) 7] if#(false, s(X), s(Y)) =#> minus#(Y, X) 8] zipWith#(F, cons(X, Y), cons(Z, U)) =#> zipWith#(F, Y, U) 9] gcdlists#(X, Y) =#> zipWith#(/\x./\y.gcd(x, y), X, Y) 10] gcdlists#(X, Y) =#> gcd#(Z, U) Rules R_0: le(0, X) => true le(s(X), 0) => false le(s(X), s(Y)) => le(X, Y) minus(X, 0) => X minus(s(X), s(Y)) => minus(X, Y) gcd(0, X) => 0 gcd(s(X), 0) => 0 gcd(s(X), s(Y)) => if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) => gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) => gcd(minus(Y, X), s(X)) zipWith(F, X, nil) => nil zipWith(F, nil, X) => nil zipWith(F, cons(X, Y), cons(Z, U)) => cons(F X Z, zipWith(F, Y, U)) gcdlists(X, Y) => zipWith(/\x./\y.gcd(x, y), X, 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 : 1 * 2 : 4, 5, 6, 7 * 3 : 0 * 4 : 2, 3 * 5 : 1 * 6 : 2, 3 * 7 : 1 * 8 : 8 * 9 : 8 * 10 : 2, 3 This graph has the following strongly connected components: P_1: le#(s(X), s(Y)) =#> le#(X, Y) P_2: minus#(s(X), s(Y)) =#> minus#(X, Y) P_3: gcd#(s(X), s(Y)) =#> if#(le(Y, X), s(X), s(Y)) if#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) if#(false, s(X), s(Y)) =#> gcd#(minus(Y, X), s(X)) P_4: zipWith#(F, cons(X, Y), cons(Z, U)) =#> zipWith#(F, Y, U) 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) and (P_4, 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) 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(zipWith#) = 2 Thus, we can orient the dependency pairs as follows: nu(zipWith#(F, cons(X, Y), cons(Z, U))) = cons(X, Y) |> Y = nu(zipWith#(F, Y, U)) 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) minus(X, 0) => X minus(s(X), s(Y)) => minus(X, Y) gcd(0, X) => 0 gcd(s(X), 0) => 0 gcd(s(X), s(Y)) => if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) => gcd(minus(X, Y), s(Y)) if(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) minus(X, 0) => X minus(s(X), s(Y)) => 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#(le(Y, X), s(X), s(Y)) if#(true, s(X), s(Y)) >? gcd#(minus(X, Y), s(Y)) if#(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) minus(X, 0) >= X minus(s(X), s(Y)) >= 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# = \y0y1.3 + 2y1 + 3y0 if# = \y0y1y2.3 + 2y2 + 3y1 le = \y0y1.2y1 minus = \y0y1.y0 s = \y0.3 + 3y0 true = 0 Using this interpretation, the requirements translate to: [[gcd#(s(_x0), s(_x1))]] = 18 + 6x1 + 9x0 >= 18 + 6x1 + 9x0 = [[if#(le(_x1, _x0), s(_x0), s(_x1))]] [[if#(true, s(_x0), s(_x1))]] = 18 + 6x1 + 9x0 > 9 + 3x0 + 6x1 = [[gcd#(minus(_x0, _x1), s(_x1))]] [[if#(false, s(_x0), s(_x1))]] = 18 + 6x1 + 9x0 > 9 + 3x1 + 6x0 = [[gcd#(minus(_x1, _x0), s(_x0))]] [[le(0, _x0)]] = 2x0 >= 0 = [[true]] [[le(s(_x0), 0)]] = 6 >= 0 = [[false]] [[le(s(_x0), s(_x1))]] = 6 + 6x1 >= 2x1 = [[le(_x0, _x1)]] [[minus(_x0, 0)]] = x0 >= x0 = [[_x0]] [[minus(s(_x0), s(_x1))]] = 3 + 3x0 >= x0 = [[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_5, R_1, computable, formative), where P_5 consists of: gcd#(s(X), s(Y)) =#> if#(le(Y, X), s(X), 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_5, R_1, computable, formative) is finite. We consider the dependency pair problem (P_5, 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#) = 1 Thus, we can orient the dependency pairs as follows: nu(minus#(s(X), s(Y))) = s(X) |> X = 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.