/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern goal(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 5 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: Clauses: tree(nil). tree(node(L, X, R)) :- ','(tree(L), tree(R)). s2t(s(X), node(T, Y, T)) :- s2t(X, T). s2t(s(X), node(nil, Y, T)) :- s2t(X, T). s2t(s(X), node(T, Y, nil)) :- s2t(X, T). s2t(s(X), node(nil, Y, nil)). s2t(0, nil). goal(X) :- ','(s2t(X, T), tree(T)). Query: goal(g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: goal_in_1: (b) s2t_in_2: (b,f) tree_in_1: (b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U5_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U6_g(X, s2t_out_ga(X, T)) -> U7_g(X, tree_in_g(T)) tree_in_g(nil) -> tree_out_g(nil) tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) U7_g(X, tree_out_g(T)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U6_g(x1, x2) = U6_g(x1, x2) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U7_g(x1, x2) = U7_g(x1, x2) tree_in_g(x1) = tree_in_g(x1) nil = nil tree_out_g(x1) = tree_out_g(x1) U1_g(x1, x2, x3, x4) = U1_g(x1, x3, x4) U2_g(x1, x2, x3, x4) = U2_g(x1, x3, x4) goal_out_g(x1) = goal_out_g(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U5_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U6_g(X, s2t_out_ga(X, T)) -> U7_g(X, tree_in_g(T)) tree_in_g(nil) -> tree_out_g(nil) tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) U7_g(X, tree_out_g(T)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U6_g(x1, x2) = U6_g(x1, x2) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U7_g(x1, x2) = U7_g(x1, x2) tree_in_g(x1) = tree_in_g(x1) nil = nil tree_out_g(x1) = tree_out_g(x1) U1_g(x1, x2, x3, x4) = U1_g(x1, x3, x4) U2_g(x1, x2, x3, x4) = U2_g(x1, x3, x4) goal_out_g(x1) = goal_out_g(x1) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: GOAL_IN_G(X) -> U6_G(X, s2t_in_ga(X, T)) GOAL_IN_G(X) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, T)) -> U3_GA(X, T, Y, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(nil, Y, T)) -> U4_GA(X, Y, T, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, nil)) -> U5_GA(X, T, Y, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) U6_G(X, s2t_out_ga(X, T)) -> U7_G(X, tree_in_g(T)) U6_G(X, s2t_out_ga(X, T)) -> TREE_IN_G(T) TREE_IN_G(node(L, X, R)) -> U1_G(L, X, R, tree_in_g(L)) TREE_IN_G(node(L, X, R)) -> TREE_IN_G(L) U1_G(L, X, R, tree_out_g(L)) -> U2_G(L, X, R, tree_in_g(R)) U1_G(L, X, R, tree_out_g(L)) -> TREE_IN_G(R) The TRS R consists of the following rules: goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U5_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U6_g(X, s2t_out_ga(X, T)) -> U7_g(X, tree_in_g(T)) tree_in_g(nil) -> tree_out_g(nil) tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) U7_g(X, tree_out_g(T)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U6_g(x1, x2) = U6_g(x1, x2) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U7_g(x1, x2) = U7_g(x1, x2) tree_in_g(x1) = tree_in_g(x1) nil = nil tree_out_g(x1) = tree_out_g(x1) U1_g(x1, x2, x3, x4) = U1_g(x1, x3, x4) U2_g(x1, x2, x3, x4) = U2_g(x1, x3, x4) goal_out_g(x1) = goal_out_g(x1) GOAL_IN_G(x1) = GOAL_IN_G(x1) U6_G(x1, x2) = U6_G(x1, x2) S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x1, x4) U4_GA(x1, x2, x3, x4) = U4_GA(x1, x4) U5_GA(x1, x2, x3, x4) = U5_GA(x1, x4) U7_G(x1, x2) = U7_G(x1, x2) TREE_IN_G(x1) = TREE_IN_G(x1) U1_G(x1, x2, x3, x4) = U1_G(x1, x3, x4) U2_G(x1, x2, x3, x4) = U2_G(x1, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: GOAL_IN_G(X) -> U6_G(X, s2t_in_ga(X, T)) GOAL_IN_G(X) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, T)) -> U3_GA(X, T, Y, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(nil, Y, T)) -> U4_GA(X, Y, T, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, nil)) -> U5_GA(X, T, Y, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) U6_G(X, s2t_out_ga(X, T)) -> U7_G(X, tree_in_g(T)) U6_G(X, s2t_out_ga(X, T)) -> TREE_IN_G(T) TREE_IN_G(node(L, X, R)) -> U1_G(L, X, R, tree_in_g(L)) TREE_IN_G(node(L, X, R)) -> TREE_IN_G(L) U1_G(L, X, R, tree_out_g(L)) -> U2_G(L, X, R, tree_in_g(R)) U1_G(L, X, R, tree_out_g(L)) -> TREE_IN_G(R) The TRS R consists of the following rules: goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U5_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U6_g(X, s2t_out_ga(X, T)) -> U7_g(X, tree_in_g(T)) tree_in_g(nil) -> tree_out_g(nil) tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) U7_g(X, tree_out_g(T)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U6_g(x1, x2) = U6_g(x1, x2) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U7_g(x1, x2) = U7_g(x1, x2) tree_in_g(x1) = tree_in_g(x1) nil = nil tree_out_g(x1) = tree_out_g(x1) U1_g(x1, x2, x3, x4) = U1_g(x1, x3, x4) U2_g(x1, x2, x3, x4) = U2_g(x1, x3, x4) goal_out_g(x1) = goal_out_g(x1) GOAL_IN_G(x1) = GOAL_IN_G(x1) U6_G(x1, x2) = U6_G(x1, x2) S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x1, x4) U4_GA(x1, x2, x3, x4) = U4_GA(x1, x4) U5_GA(x1, x2, x3, x4) = U5_GA(x1, x4) U7_G(x1, x2) = U7_G(x1, x2) TREE_IN_G(x1) = TREE_IN_G(x1) U1_G(x1, x2, x3, x4) = U1_G(x1, x3, x4) U2_G(x1, x2, x3, x4) = U2_G(x1, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 8 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_G(L, X, R, tree_out_g(L)) -> TREE_IN_G(R) TREE_IN_G(node(L, X, R)) -> U1_G(L, X, R, tree_in_g(L)) TREE_IN_G(node(L, X, R)) -> TREE_IN_G(L) The TRS R consists of the following rules: goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U5_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U6_g(X, s2t_out_ga(X, T)) -> U7_g(X, tree_in_g(T)) tree_in_g(nil) -> tree_out_g(nil) tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) U7_g(X, tree_out_g(T)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U6_g(x1, x2) = U6_g(x1, x2) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U7_g(x1, x2) = U7_g(x1, x2) tree_in_g(x1) = tree_in_g(x1) nil = nil tree_out_g(x1) = tree_out_g(x1) U1_g(x1, x2, x3, x4) = U1_g(x1, x3, x4) U2_g(x1, x2, x3, x4) = U2_g(x1, x3, x4) goal_out_g(x1) = goal_out_g(x1) TREE_IN_G(x1) = TREE_IN_G(x1) U1_G(x1, x2, x3, x4) = U1_G(x1, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_G(L, X, R, tree_out_g(L)) -> TREE_IN_G(R) TREE_IN_G(node(L, X, R)) -> U1_G(L, X, R, tree_in_g(L)) TREE_IN_G(node(L, X, R)) -> TREE_IN_G(L) The TRS R consists of the following rules: tree_in_g(nil) -> tree_out_g(nil) tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) The argument filtering Pi contains the following mapping: node(x1, x2, x3) = node(x1, x3) tree_in_g(x1) = tree_in_g(x1) nil = nil tree_out_g(x1) = tree_out_g(x1) U1_g(x1, x2, x3, x4) = U1_g(x1, x3, x4) U2_g(x1, x2, x3, x4) = U2_g(x1, x3, x4) TREE_IN_G(x1) = TREE_IN_G(x1) U1_G(x1, x2, x3, x4) = U1_G(x1, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: U1_G(L, R, tree_out_g(L)) -> TREE_IN_G(R) TREE_IN_G(node(L, R)) -> U1_G(L, R, tree_in_g(L)) TREE_IN_G(node(L, R)) -> TREE_IN_G(L) The TRS R consists of the following rules: tree_in_g(nil) -> tree_out_g(nil) tree_in_g(node(L, R)) -> U1_g(L, R, tree_in_g(L)) U1_g(L, R, tree_out_g(L)) -> U2_g(L, R, tree_in_g(R)) U2_g(L, R, tree_out_g(R)) -> tree_out_g(node(L, R)) The set Q consists of the following terms: tree_in_g(x0) U1_g(x0, x1, x2) U2_g(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *TREE_IN_G(node(L, R)) -> U1_G(L, R, tree_in_g(L)) The graph contains the following edges 1 > 1, 1 > 2 *TREE_IN_G(node(L, R)) -> TREE_IN_G(L) The graph contains the following edges 1 > 1 *U1_G(L, R, tree_out_g(L)) -> TREE_IN_G(R) The graph contains the following edges 2 >= 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) The TRS R consists of the following rules: goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U5_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U6_g(X, s2t_out_ga(X, T)) -> U7_g(X, tree_in_g(T)) tree_in_g(nil) -> tree_out_g(nil) tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) U7_g(X, tree_out_g(T)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U6_g(x1, x2) = U6_g(x1, x2) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U7_g(x1, x2) = U7_g(x1, x2) tree_in_g(x1) = tree_in_g(x1) nil = nil tree_out_g(x1) = tree_out_g(x1) U1_g(x1, x2, x3, x4) = U1_g(x1, x3, x4) U2_g(x1, x2, x3, x4) = U2_g(x1, x3, x4) goal_out_g(x1) = goal_out_g(x1) S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) node(x1, x2, x3) = node(x1, x3) nil = nil S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: S2T_IN_GA(s(X)) -> S2T_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *S2T_IN_GA(s(X)) -> S2T_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES