2.04/1.07 YES 2.04/1.08 We consider the system theBenchmark. 2.04/1.08 2.04/1.08 Alphabet: 2.04/1.08 2.04/1.08 0 : [] --> b 2.04/1.08 1 : [] --> b 2.04/1.08 c : [b] --> b 2.04/1.08 cons : [c * d] --> d 2.04/1.08 f : [b] --> a 2.04/1.08 false : [] --> a 2.04/1.08 filter : [c -> a * d] --> d 2.04/1.08 filter2 : [a * c -> a * c * d] --> d 2.04/1.08 g : [b * b] --> b 2.04/1.08 if : [a * b * b] --> b 2.04/1.08 map : [c -> c * d] --> d 2.04/1.08 nil : [] --> d 2.04/1.08 s : [b] --> b 2.04/1.08 true : [] --> a 2.04/1.08 2.04/1.08 Rules: 2.04/1.08 2.04/1.08 f(0) => true 2.04/1.08 f(1) => false 2.04/1.08 f(s(x)) => f(x) 2.04/1.08 if(true, s(x), s(y)) => s(x) 2.04/1.08 if(false, s(x), s(y)) => s(y) 2.04/1.08 g(x, c(y)) => c(g(x, y)) 2.04/1.08 g(x, c(y)) => g(x, if(f(x), c(g(s(x), y)), c(y))) 2.04/1.08 map(h, nil) => nil 2.04/1.08 map(h, cons(x, y)) => cons(h x, map(h, y)) 2.04/1.08 filter(h, nil) => nil 2.04/1.08 filter(h, cons(x, y)) => filter2(h x, h, x, y) 2.04/1.08 filter2(true, h, x, y) => cons(x, filter(h, y)) 2.04/1.08 filter2(false, h, x, y) => filter(h, y) 2.04/1.08 2.04/1.08 This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). 2.04/1.08 2.04/1.08 We observe that the rules contain a first-order subset: 2.04/1.08 2.04/1.08 f(0) => true 2.04/1.08 f(1) => false 2.04/1.08 f(s(X)) => f(X) 2.04/1.08 if(true, s(X), s(Y)) => s(X) 2.04/1.08 if(false, s(X), s(Y)) => s(Y) 2.04/1.08 g(X, c(Y)) => c(g(X, Y)) 2.04/1.08 g(X, c(Y)) => g(X, if(f(X), c(g(s(X), Y)), c(Y))) 2.04/1.08 2.04/1.08 Moreover, the system is finitely branching. Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is Ce-terminating when seen as a many-sorted first-order TRS. 2.04/1.08 2.04/1.08 According to the external first-order termination prover, this system is indeed Ce-terminating: 2.04/1.08 2.04/1.08 || proof of resources/system.trs 2.04/1.08 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616 2.04/1.08 || 2.04/1.08 || 2.04/1.08 || Termination w.r.t. Q of the given QTRS could be proven: 2.04/1.08 || 2.04/1.08 || (0) QTRS 2.04/1.08 || (1) DependencyPairsProof [EQUIVALENT] 2.04/1.08 || (2) QDP 2.04/1.08 || (3) DependencyGraphProof [EQUIVALENT] 2.04/1.08 || (4) AND 2.04/1.08 || (5) QDP 2.04/1.08 || (6) UsableRulesProof [EQUIVALENT] 2.04/1.08 || (7) QDP 2.04/1.08 || (8) QDPSizeChangeProof [EQUIVALENT] 2.04/1.08 || (9) YES 2.04/1.08 || (10) QDP 2.04/1.08 || (11) UsableRulesProof [EQUIVALENT] 2.04/1.08 || (12) QDP 2.04/1.08 || (13) QDPSizeChangeProof [EQUIVALENT] 2.04/1.08 || (14) YES 2.04/1.08 || 2.04/1.08 || 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (0) 2.04/1.08 || Obligation: 2.04/1.08 || Q restricted rewrite system: 2.04/1.08 || The TRS R consists of the following rules: 2.04/1.08 || 2.04/1.08 || f(0) -> true 2.04/1.08 || f(1) -> false 2.04/1.08 || f(s(%X)) -> f(%X) 2.04/1.08 || if(true, s(%X), s(%Y)) -> s(%X) 2.04/1.08 || if(false, s(%X), s(%Y)) -> s(%Y) 2.04/1.08 || g(%X, c(%Y)) -> c(g(%X, %Y)) 2.04/1.08 || g(%X, c(%Y)) -> g(%X, if(f(%X), c(g(s(%X), %Y)), c(%Y))) 2.04/1.08 || ~PAIR(%X, %Y) -> %X 2.04/1.08 || ~PAIR(%X, %Y) -> %Y 2.04/1.08 || 2.04/1.08 || Q is empty. 2.04/1.08 || 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (1) DependencyPairsProof (EQUIVALENT) 2.04/1.08 || Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (2) 2.04/1.08 || Obligation: 2.04/1.08 || Q DP problem: 2.04/1.08 || The TRS P consists of the following rules: 2.04/1.08 || 2.04/1.08 || F(s(%X)) -> F(%X) 2.04/1.08 || G(%X, c(%Y)) -> G(%X, %Y) 2.04/1.08 || G(%X, c(%Y)) -> G(%X, if(f(%X), c(g(s(%X), %Y)), c(%Y))) 2.04/1.08 || G(%X, c(%Y)) -> IF(f(%X), c(g(s(%X), %Y)), c(%Y)) 2.04/1.08 || G(%X, c(%Y)) -> F(%X) 2.04/1.08 || G(%X, c(%Y)) -> G(s(%X), %Y) 2.04/1.08 || 2.04/1.08 || The TRS R consists of the following rules: 2.04/1.08 || 2.04/1.08 || f(0) -> true 2.04/1.08 || f(1) -> false 2.04/1.08 || f(s(%X)) -> f(%X) 2.04/1.08 || if(true, s(%X), s(%Y)) -> s(%X) 2.04/1.08 || if(false, s(%X), s(%Y)) -> s(%Y) 2.04/1.08 || g(%X, c(%Y)) -> c(g(%X, %Y)) 2.04/1.08 || g(%X, c(%Y)) -> g(%X, if(f(%X), c(g(s(%X), %Y)), c(%Y))) 2.04/1.08 || ~PAIR(%X, %Y) -> %X 2.04/1.08 || ~PAIR(%X, %Y) -> %Y 2.04/1.08 || 2.04/1.08 || Q is empty. 2.04/1.08 || We have to consider all minimal (P,Q,R)-chains. 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (3) DependencyGraphProof (EQUIVALENT) 2.04/1.08 || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes. 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (4) 2.04/1.08 || Complex Obligation (AND) 2.04/1.08 || 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (5) 2.04/1.08 || Obligation: 2.04/1.08 || Q DP problem: 2.04/1.08 || The TRS P consists of the following rules: 2.04/1.08 || 2.04/1.08 || F(s(%X)) -> F(%X) 2.04/1.08 || 2.04/1.08 || The TRS R consists of the following rules: 2.04/1.08 || 2.04/1.08 || f(0) -> true 2.04/1.08 || f(1) -> false 2.04/1.08 || f(s(%X)) -> f(%X) 2.04/1.08 || if(true, s(%X), s(%Y)) -> s(%X) 2.04/1.08 || if(false, s(%X), s(%Y)) -> s(%Y) 2.04/1.08 || g(%X, c(%Y)) -> c(g(%X, %Y)) 2.04/1.08 || g(%X, c(%Y)) -> g(%X, if(f(%X), c(g(s(%X), %Y)), c(%Y))) 2.04/1.08 || ~PAIR(%X, %Y) -> %X 2.04/1.08 || ~PAIR(%X, %Y) -> %Y 2.04/1.08 || 2.04/1.08 || Q is empty. 2.04/1.08 || We have to consider all minimal (P,Q,R)-chains. 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (6) UsableRulesProof (EQUIVALENT) 2.04/1.08 || We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (7) 2.04/1.08 || Obligation: 2.04/1.08 || Q DP problem: 2.04/1.08 || The TRS P consists of the following rules: 2.04/1.08 || 2.04/1.08 || F(s(%X)) -> F(%X) 2.04/1.08 || 2.04/1.08 || R is empty. 2.04/1.08 || Q is empty. 2.04/1.08 || We have to consider all minimal (P,Q,R)-chains. 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (8) QDPSizeChangeProof (EQUIVALENT) 2.04/1.08 || By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 2.04/1.08 || 2.04/1.08 || From the DPs we obtained the following set of size-change graphs: 2.04/1.08 || *F(s(%X)) -> F(%X) 2.04/1.08 || The graph contains the following edges 1 > 1 2.04/1.08 || 2.04/1.08 || 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (9) 2.04/1.08 || YES 2.04/1.08 || 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (10) 2.04/1.08 || Obligation: 2.04/1.08 || Q DP problem: 2.04/1.08 || The TRS P consists of the following rules: 2.04/1.08 || 2.04/1.08 || G(%X, c(%Y)) -> G(s(%X), %Y) 2.04/1.08 || G(%X, c(%Y)) -> G(%X, %Y) 2.04/1.08 || 2.04/1.08 || The TRS R consists of the following rules: 2.04/1.08 || 2.04/1.08 || f(0) -> true 2.04/1.08 || f(1) -> false 2.04/1.08 || f(s(%X)) -> f(%X) 2.04/1.08 || if(true, s(%X), s(%Y)) -> s(%X) 2.04/1.08 || if(false, s(%X), s(%Y)) -> s(%Y) 2.04/1.08 || g(%X, c(%Y)) -> c(g(%X, %Y)) 2.04/1.08 || g(%X, c(%Y)) -> g(%X, if(f(%X), c(g(s(%X), %Y)), c(%Y))) 2.04/1.08 || ~PAIR(%X, %Y) -> %X 2.04/1.08 || ~PAIR(%X, %Y) -> %Y 2.04/1.08 || 2.04/1.08 || Q is empty. 2.04/1.08 || We have to consider all minimal (P,Q,R)-chains. 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (11) UsableRulesProof (EQUIVALENT) 2.04/1.08 || We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (12) 2.04/1.08 || Obligation: 2.04/1.08 || Q DP problem: 2.04/1.08 || The TRS P consists of the following rules: 2.04/1.08 || 2.04/1.08 || G(%X, c(%Y)) -> G(s(%X), %Y) 2.04/1.08 || G(%X, c(%Y)) -> G(%X, %Y) 2.04/1.08 || 2.04/1.08 || R is empty. 2.04/1.08 || Q is empty. 2.04/1.08 || We have to consider all minimal (P,Q,R)-chains. 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (13) QDPSizeChangeProof (EQUIVALENT) 2.04/1.08 || By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 2.04/1.08 || 2.04/1.08 || From the DPs we obtained the following set of size-change graphs: 2.04/1.08 || *G(%X, c(%Y)) -> G(s(%X), %Y) 2.04/1.08 || The graph contains the following edges 2 > 2 2.04/1.08 || 2.04/1.08 || 2.04/1.08 || *G(%X, c(%Y)) -> G(%X, %Y) 2.04/1.08 || The graph contains the following edges 1 >= 1, 2 > 2 2.04/1.08 || 2.04/1.08 || 2.04/1.08 || ---------------------------------------- 2.04/1.08 || 2.04/1.08 || (14) 2.04/1.08 || YES 2.04/1.08 || 2.04/1.08 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 [Kop13]). 2.04/1.08 2.04/1.08 We thus obtain the following dependency pair problem (P_0, R_0, static, formative): 2.04/1.08 2.04/1.08 Dependency Pairs P_0: 2.04/1.08 2.04/1.08 0] map#(F, cons(X, Y)) =#> map#(F, Y) 2.04/1.08 1] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y) 2.04/1.08 2] filter2#(true, F, X, Y) =#> filter#(F, Y) 2.04/1.08 3] filter2#(false, F, X, Y) =#> filter#(F, Y) 2.04/1.08 2.04/1.08 Rules R_0: 2.04/1.08 2.04/1.08 f(0) => true 2.04/1.08 f(1) => false 2.04/1.08 f(s(X)) => f(X) 2.04/1.08 if(true, s(X), s(Y)) => s(X) 2.04/1.08 if(false, s(X), s(Y)) => s(Y) 2.04/1.08 g(X, c(Y)) => c(g(X, Y)) 2.04/1.08 g(X, c(Y)) => g(X, if(f(X), c(g(s(X), Y)), c(Y))) 2.04/1.08 map(F, nil) => nil 2.04/1.08 map(F, cons(X, Y)) => cons(F X, map(F, Y)) 2.04/1.08 filter(F, nil) => nil 2.04/1.08 filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 2.04/1.08 filter2(true, F, X, Y) => cons(X, filter(F, Y)) 2.04/1.08 filter2(false, F, X, Y) => filter(F, Y) 2.04/1.08 2.04/1.08 Thus, the original system is terminating if (P_0, R_0, static, formative) is finite. 2.04/1.08 2.04/1.08 We consider the dependency pair problem (P_0, R_0, static, formative). 2.04/1.08 2.04/1.08 We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: 2.04/1.08 2.04/1.08 * 0 : 0 2.04/1.08 * 1 : 2, 3 2.04/1.08 * 2 : 1 2.04/1.08 * 3 : 1 2.04/1.08 2.04/1.08 This graph has the following strongly connected components: 2.04/1.08 2.04/1.08 P_1: 2.04/1.08 2.04/1.08 map#(F, cons(X, Y)) =#> map#(F, Y) 2.04/1.08 2.04/1.08 P_2: 2.04/1.08 2.04/1.08 filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y) 2.04/1.08 filter2#(true, F, X, Y) =#> filter#(F, Y) 2.04/1.08 filter2#(false, F, X, Y) =#> filter#(F, Y) 2.04/1.08 2.04/1.08 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) and (P_2, R_0, m, f). 2.04/1.08 2.04/1.08 Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_2, R_0, static, formative) is finite. 2.04/1.08 2.04/1.08 We consider the dependency pair problem (P_2, R_0, static, formative). 2.04/1.08 2.04/1.08 We apply the subterm criterion with the following projection function: 2.04/1.08 2.04/1.08 nu(filter2#) = 4 2.04/1.08 nu(filter#) = 2 2.04/1.08 2.04/1.08 Thus, we can orient the dependency pairs as follows: 2.04/1.08 2.04/1.08 nu(filter#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter2#(F X, F, X, Y)) 2.04/1.08 nu(filter2#(true, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 2.04/1.08 nu(filter2#(false, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 2.04/1.08 2.04/1.08 By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_2, R_0, static, f) by (P_3, R_0, static, f), where P_3 contains: 2.04/1.08 2.04/1.08 filter2#(true, F, X, Y) =#> filter#(F, Y) 2.04/1.08 filter2#(false, F, X, Y) =#> filter#(F, Y) 2.04/1.08 2.04/1.08 Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_3, R_0, static, formative) is finite. 2.04/1.08 2.04/1.08 We consider the dependency pair problem (P_3, R_0, static, formative). 2.04/1.08 2.04/1.08 We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: 2.04/1.08 2.04/1.08 * 0 : 2.04/1.08 * 1 : 2.04/1.08 2.04/1.08 This graph has no strongly connected components. By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem. 2.04/1.08 2.04/1.08 Thus, the original system is terminating if (P_1, R_0, static, formative) is finite. 2.04/1.08 2.04/1.08 We consider the dependency pair problem (P_1, R_0, static, formative). 2.04/1.08 2.04/1.08 We apply the subterm criterion with the following projection function: 2.04/1.08 2.04/1.08 nu(map#) = 2 2.04/1.08 2.04/1.08 Thus, we can orient the dependency pairs as follows: 2.04/1.08 2.04/1.08 nu(map#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(map#(F, Y)) 2.04/1.08 2.04/1.08 By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_1, R_0, static, f) by ({}, R_0, static, f). By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed. 2.04/1.08 2.04/1.08 As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination. 2.04/1.08 2.04/1.08 2.04/1.08 +++ Citations +++ 2.04/1.08 2.04/1.08 [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. 2.04/1.08 [Kop13] C. Kop. Static Dependency Pairs with Accessibility. Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013. 2.04/1.08 [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. 2.04/1.08 EOF