We consider the system Applicative_first_order_05__#3.48. Alphabet: 0 : [] --> b 1 : [] --> b c : [b] --> b cons : [c * d] --> d f : [b] --> a false : [] --> a filter : [c -> a * d] --> d filter2 : [a * c -> a * c * d] --> d g : [b * b] --> b if : [a * b * b] --> b map : [c -> c * d] --> d nil : [] --> d s : [b] --> b true : [] --> a Rules: f(0) => true f(1) => false f(s(x)) => f(x) if(true, s(x), s(y)) => s(x) if(false, s(x), s(y)) => s(y) g(x, c(y)) => c(g(x, y)) g(x, c(y)) => g(x, if(f(x), c(g(s(x), y)), c(y))) map(h, nil) => nil map(h, cons(x, y)) => cons(h x, map(h, y)) filter(h, nil) => nil filter(h, cons(x, y)) => filter2(h x, h, x, y) filter2(true, h, x, y) => cons(x, filter(h, y)) filter2(false, h, x, y) => filter(h, 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] f#(s(X)) =#> f#(X) 1] g#(X, c(Y)) =#> g#(X, Y) 2] g#(X, c(Y)) =#> g#(X, if(f(X), c(g(s(X), Y)), c(Y))) 3] g#(X, c(Y)) =#> if#(f(X), c(g(s(X), Y)), c(Y)) 4] g#(X, c(Y)) =#> f#(X) 5] g#(X, c(Y)) =#> g#(s(X), Y) 6] map#(F, cons(X, Y)) =#> map#(F, Y) 7] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y) 8] filter2#(true, F, X, Y) =#> filter#(F, Y) 9] filter2#(false, F, X, Y) =#> filter#(F, Y) Rules R_0: f(0) => true f(1) => false f(s(X)) => f(X) if(true, s(X), s(Y)) => s(X) if(false, s(X), s(Y)) => s(Y) g(X, c(Y)) => c(g(X, Y)) g(X, c(Y)) => g(X, if(f(X), c(g(s(X), Y)), c(Y))) 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, 3, 4, 5 * 2 : * 3 : * 4 : 0 * 5 : 1, 2, 3, 4, 5 * 6 : 6 * 7 : 8, 9 * 8 : 7 * 9 : 7 This graph has the following strongly connected components: P_1: f#(s(X)) =#> f#(X) P_2: g#(X, c(Y)) =#> g#(X, Y) g#(X, c(Y)) =#> g#(s(X), Y) P_3: map#(F, cons(X, Y)) =#> map#(F, Y) P_4: 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) 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(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_4, R_0, computable, f) by (P_5, R_0, computable, f), where P_5 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) and (P_5, R_0, computable, formative) is finite. We consider the dependency pair problem (P_5, 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) and (P_3, R_0, computable, formative) is finite. We consider the dependency pair problem (P_3, 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_3, 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) 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(g#) = 2 Thus, we can orient the dependency pairs as follows: nu(g#(X, c(Y))) = c(Y) |> Y = nu(g#(X, Y)) nu(g#(X, c(Y))) = c(Y) |> Y = nu(g#(s(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(f#) = 1 Thus, we can orient the dependency pairs as follows: nu(f#(s(X))) = s(X) |> X = nu(f#(X)) 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.