We consider the system Applicative_first_order_05__#3.13. Alphabet: 0 : [] --> a cons : [d * e] --> e edge : [a * a * b] --> b empty : [] --> b eq : [a * a] --> c false : [] --> c filter : [d -> c * e] --> e filter2 : [c * d -> c * d * e] --> e if!fac6220reach!fac62201 : [c * a * a * b * b] --> c if!fac6220reach!fac62202 : [c * a * a * b * b] --> c map : [d -> d * e] --> e nil : [] --> e or : [c * c] --> c reach : [a * a * b * b] --> c s : [a] --> a true : [] --> c union : [b * b] --> b Rules: 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!fac6220reach!fac62201(eq(x, z), x, y, edge(z, u, v), w) if!fac6220reach!fac62201(true, x, y, edge(z, u, v), w) => if!fac6220reach!fac62202(eq(y, u), x, y, edge(z, u, v), w) if!fac6220reach!fac62201(false, x, y, edge(z, u, v), w) => reach(x, y, v, edge(z, u, w)) if!fac6220reach!fac62202(true, x, y, edge(z, u, v), w) => true if!fac6220reach!fac62202(false, x, y, edge(z, u, v), w) => or(reach(x, y, v, w), reach(u, y, union(v, w), empty)) 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] 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!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W) 3] reach#(X, Y, edge(Z, U, V), W) =#> eq#(X, Z) 4] if!fac6220reach!fac62201#(true, X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62202#(eq(Y, U), X, Y, edge(Z, U, V), W) 5] if!fac6220reach!fac62201#(true, X, Y, edge(Z, U, V), W) =#> eq#(Y, U) 6] if!fac6220reach!fac62201#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, edge(Z, U, W)) 7] if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> or#(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 8] if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, W) 9] if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(U, Y, union(V, W), empty) 10] if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> union#(V, W) 11] map#(F, cons(X, Y)) =#> map#(F, Y) 12] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y) 13] filter2#(true, F, X, Y) =#> filter#(F, Y) 14] filter2#(false, F, X, Y) =#> filter#(F, Y) 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!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) => reach(X, Y, V, edge(Z, U, W)) if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) => true if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) => or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 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 : 1 * 2 : 4, 5, 6 * 3 : 0 * 4 : 7, 8, 9, 10 * 5 : 0 * 6 : 2, 3 * 7 : * 8 : 2, 3 * 9 : 2, 3 * 10 : 1 * 11 : 11 * 12 : 13, 14 * 13 : 12 * 14 : 12 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!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201#(true, X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62202#(eq(Y, U), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, edge(Z, U, W)) if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(X, Y, V, W) if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) =#> reach#(U, Y, union(V, W), empty) 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 ::= 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!fac6220reach!fac62201(eq(X, Z), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201(true, X, Y, edge(Z, U, V), W) => if!fac6220reach!fac62202(eq(Y, U), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201(false, X, Y, edge(Z, U, V), W) => reach(X, Y, V, edge(Z, U, W)) if!fac6220reach!fac62202(true, X, Y, edge(Z, U, V), W) => true if!fac6220reach!fac62202(false, X, Y, edge(Z, U, V), W) => or(reach(X, Y, V, W), reach(U, Y, union(V, W), empty)) 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: 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!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201#(true, X, Y, edge(Z, U, V), W) >? if!fac6220reach!fac62202#(eq(Y, U), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201#(false, X, Y, edge(Z, U, V), W) >? reach#(X, Y, V, edge(Z, U, W)) if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) >? reach#(X, Y, V, W) if!fac6220reach!fac62202#(false, X, Y, edge(Z, U, V), W) >? reach#(U, Y, union(V, W), empty) 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. The following interpretation satisfies the requirements: 0 = 3 edge = \y0y1y2.1 + y2 empty = 0 eq = \y0y1.0 false = 0 if!fac6220reach!fac62201# = \y0y1y2y3y4.y3 + y4 if!fac6220reach!fac62202# = \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!fac6220reach!fac62201#(eq(_x0, _x2), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] [[if!fac6220reach!fac62201#(true, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 >= 1 + x4 + x5 = [[if!fac6220reach!fac62202#(eq(_x1, _x3), _x0, _x1, edge(_x2, _x3, _x4), _x5)]] [[if!fac6220reach!fac62201#(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 >= 1 + x4 + x5 = [[reach#(_x0, _x1, _x4, edge(_x2, _x3, _x5))]] [[if!fac6220reach!fac62202#(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 > x4 + x5 = [[reach#(_x0, _x1, _x4, _x5)]] [[if!fac6220reach!fac62202#(false, _x0, _x1, edge(_x2, _x3, _x4), _x5)]] = 1 + x4 + x5 > x4 + x5 = [[reach#(_x3, _x1, union(_x4, _x5), empty)]] [[eq(0, 0)]] = 0 >= 0 = [[true]] [[eq(0, s(_x0))]] = 0 >= 0 = [[false]] [[eq(s(_x0), 0)]] = 0 >= 0 = [[false]] [[eq(s(_x0), s(_x1))]] = 0 >= 0 = [[eq(_x0, _x1)]] [[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_1, computable, formative) by (P_7, R_1, computable, formative), where P_7 consists of: reach#(X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201#(true, X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62202#(eq(Y, U), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201#(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, 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 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_8: reach#(X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W) if!fac6220reach!fac62201#(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_7, R_1, m, f) by (P_8, R_1, m, f). 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 apply the subterm criterion with the following projection function: nu(if!fac6220reach!fac62201#) = 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!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W)) nu(if!fac6220reach!fac62201#(false, X, Y, edge(Z, U, V), W)) = edge(Z, U, V) |> V = nu(reach#(X, Y, V, edge(Z, U, W))) By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_8, R_1, computable, f) by (P_9, R_1, computable, f), where P_9 contains: reach#(X, Y, edge(Z, U, V), W) =#> if!fac6220reach!fac62201#(eq(X, Z), X, Y, edge(Z, U, V), W) Thus, the original system is terminating if each of (P_1, R_0, computable, formative), (P_2, R_0, computable, formative) and (P_9, R_1, computable, formative) is finite. We consider the dependency pair problem (P_9, 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(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 [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(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 [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.