4.17/1.88 YES 4.71/2.01 proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl 4.71/2.01 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.71/2.01 4.71/2.01 4.71/2.01 Left Termination of the query pattern 4.71/2.01 4.71/2.01 goal(g) 4.71/2.01 4.71/2.01 w.r.t. the given Prolog program could successfully be proven: 4.71/2.01 4.71/2.01 (0) Prolog 4.71/2.01 (1) PrologToPiTRSProof [SOUND, 13 ms] 4.71/2.01 (2) PiTRS 4.71/2.01 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 4.71/2.01 (4) PiDP 4.71/2.01 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 4.71/2.01 (6) AND 4.71/2.01 (7) PiDP 4.71/2.01 (8) UsableRulesProof [EQUIVALENT, 1 ms] 4.71/2.01 (9) PiDP 4.71/2.01 (10) PiDPToQDPProof [SOUND, 0 ms] 4.71/2.01 (11) QDP 4.71/2.01 (12) QDPSizeChangeProof [EQUIVALENT, 1 ms] 4.71/2.01 (13) YES 4.71/2.01 (14) PiDP 4.71/2.01 (15) UsableRulesProof [EQUIVALENT, 0 ms] 4.71/2.01 (16) PiDP 4.71/2.01 (17) PiDPToQDPProof [SOUND, 0 ms] 4.71/2.01 (18) QDP 4.71/2.01 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.71/2.01 (20) YES 4.71/2.01 4.71/2.01 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (0) 4.71/2.01 Obligation: 4.71/2.01 Clauses: 4.71/2.01 4.71/2.01 tree(nil). 4.71/2.01 tree(node(L, X, R)) :- ','(tree(L), tree(R)). 4.71/2.01 s2t(s(X), node(T, Y, T)) :- s2t(X, T). 4.71/2.01 s2t(s(X), node(nil, Y, T)) :- s2t(X, T). 4.71/2.01 s2t(s(X), node(T, Y, nil)) :- s2t(X, T). 4.71/2.01 s2t(s(X), node(nil, Y, nil)). 4.71/2.01 s2t(0, nil). 4.71/2.01 goal(X) :- ','(s2t(X, T), tree(T)). 4.71/2.01 4.71/2.01 4.71/2.01 Query: goal(g) 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (1) PrologToPiTRSProof (SOUND) 4.71/2.01 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 4.71/2.01 4.71/2.01 goal_in_1: (b) 4.71/2.01 4.71/2.01 s2t_in_2: (b,f) 4.71/2.01 4.71/2.01 tree_in_1: (b) 4.71/2.01 4.71/2.01 Transforming Prolog into the following Term Rewriting System: 4.71/2.01 4.71/2.01 Pi-finite rewrite system: 4.71/2.01 The TRS R consists of the following rules: 4.71/2.01 4.71/2.01 goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(T, Y, nil)) -> U5_ga(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) 4.71/2.01 s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) 4.71/2.01 U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) 4.71/2.01 U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) 4.71/2.01 U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) 4.71/2.01 U6_g(X, s2t_out_ga(X, T)) -> U7_g(X, tree_in_g(T)) 4.71/2.01 tree_in_g(nil) -> tree_out_g(nil) 4.71/2.01 tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) 4.71/2.01 U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) 4.71/2.01 U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) 4.71/2.01 U7_g(X, tree_out_g(T)) -> goal_out_g(X) 4.71/2.01 4.71/2.01 The argument filtering Pi contains the following mapping: 4.71/2.01 goal_in_g(x1) = goal_in_g(x1) 4.71/2.01 4.71/2.01 U6_g(x1, x2) = U6_g(x2) 4.71/2.01 4.71/2.01 s2t_in_ga(x1, x2) = s2t_in_ga(x1) 4.71/2.01 4.71/2.01 s(x1) = s(x1) 4.71/2.01 4.71/2.01 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 4.71/2.01 4.71/2.01 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 4.71/2.01 4.71/2.01 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 4.71/2.01 4.71/2.01 s2t_out_ga(x1, x2) = s2t_out_ga(x2) 4.71/2.01 4.71/2.01 node(x1, x2, x3) = node(x1, x3) 4.71/2.01 4.71/2.01 0 = 0 4.71/2.01 4.71/2.01 U7_g(x1, x2) = U7_g(x2) 4.71/2.01 4.71/2.01 tree_in_g(x1) = tree_in_g(x1) 4.71/2.01 4.71/2.01 nil = nil 4.71/2.01 4.71/2.01 tree_out_g(x1) = tree_out_g 4.71/2.01 4.71/2.01 U1_g(x1, x2, x3, x4) = U1_g(x3, x4) 4.71/2.01 4.71/2.01 U2_g(x1, x2, x3, x4) = U2_g(x4) 4.71/2.01 4.71/2.01 goal_out_g(x1) = goal_out_g 4.71/2.01 4.71/2.01 4.71/2.01 4.71/2.01 4.71/2.01 4.71/2.01 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 4.71/2.01 4.71/2.01 4.71/2.01 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (2) 4.71/2.01 Obligation: 4.71/2.01 Pi-finite rewrite system: 4.71/2.01 The TRS R consists of the following rules: 4.71/2.01 4.71/2.01 goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(T, Y, nil)) -> U5_ga(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) 4.71/2.01 s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) 4.71/2.01 U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) 4.71/2.01 U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) 4.71/2.01 U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) 4.71/2.01 U6_g(X, s2t_out_ga(X, T)) -> U7_g(X, tree_in_g(T)) 4.71/2.01 tree_in_g(nil) -> tree_out_g(nil) 4.71/2.01 tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) 4.71/2.01 U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) 4.71/2.01 U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) 4.71/2.01 U7_g(X, tree_out_g(T)) -> goal_out_g(X) 4.71/2.01 4.71/2.01 The argument filtering Pi contains the following mapping: 4.71/2.01 goal_in_g(x1) = goal_in_g(x1) 4.71/2.01 4.71/2.01 U6_g(x1, x2) = U6_g(x2) 4.71/2.01 4.71/2.01 s2t_in_ga(x1, x2) = s2t_in_ga(x1) 4.71/2.01 4.71/2.01 s(x1) = s(x1) 4.71/2.01 4.71/2.01 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 4.71/2.01 4.71/2.01 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 4.71/2.01 4.71/2.01 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 4.71/2.01 4.71/2.01 s2t_out_ga(x1, x2) = s2t_out_ga(x2) 4.71/2.01 4.71/2.01 node(x1, x2, x3) = node(x1, x3) 4.71/2.01 4.71/2.01 0 = 0 4.71/2.01 4.71/2.01 U7_g(x1, x2) = U7_g(x2) 4.71/2.01 4.71/2.01 tree_in_g(x1) = tree_in_g(x1) 4.71/2.01 4.71/2.01 nil = nil 4.71/2.01 4.71/2.01 tree_out_g(x1) = tree_out_g 4.71/2.01 4.71/2.01 U1_g(x1, x2, x3, x4) = U1_g(x3, x4) 4.71/2.01 4.71/2.01 U2_g(x1, x2, x3, x4) = U2_g(x4) 4.71/2.01 4.71/2.01 goal_out_g(x1) = goal_out_g 4.71/2.01 4.71/2.01 4.71/2.01 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (3) DependencyPairsProof (EQUIVALENT) 4.71/2.01 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 4.71/2.01 Pi DP problem: 4.71/2.01 The TRS P consists of the following rules: 4.71/2.01 4.71/2.01 GOAL_IN_G(X) -> U6_G(X, s2t_in_ga(X, T)) 4.71/2.01 GOAL_IN_G(X) -> S2T_IN_GA(X, T) 4.71/2.01 S2T_IN_GA(s(X), node(T, Y, T)) -> U3_GA(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T) 4.71/2.01 S2T_IN_GA(s(X), node(nil, Y, T)) -> U4_GA(X, Y, T, s2t_in_ga(X, T)) 4.71/2.01 S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T) 4.71/2.01 S2T_IN_GA(s(X), node(T, Y, nil)) -> U5_GA(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) 4.71/2.01 U6_G(X, s2t_out_ga(X, T)) -> U7_G(X, tree_in_g(T)) 4.71/2.01 U6_G(X, s2t_out_ga(X, T)) -> TREE_IN_G(T) 4.71/2.01 TREE_IN_G(node(L, X, R)) -> U1_G(L, X, R, tree_in_g(L)) 4.71/2.01 TREE_IN_G(node(L, X, R)) -> TREE_IN_G(L) 4.71/2.01 U1_G(L, X, R, tree_out_g(L)) -> U2_G(L, X, R, tree_in_g(R)) 4.71/2.01 U1_G(L, X, R, tree_out_g(L)) -> TREE_IN_G(R) 4.71/2.01 4.71/2.01 The TRS R consists of the following rules: 4.71/2.01 4.71/2.01 goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(T, Y, nil)) -> U5_ga(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) 4.71/2.01 s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) 4.71/2.01 U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) 4.71/2.01 U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) 4.71/2.01 U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) 4.71/2.01 U6_g(X, s2t_out_ga(X, T)) -> U7_g(X, tree_in_g(T)) 4.71/2.01 tree_in_g(nil) -> tree_out_g(nil) 4.71/2.01 tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) 4.71/2.01 U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) 4.71/2.01 U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) 4.71/2.01 U7_g(X, tree_out_g(T)) -> goal_out_g(X) 4.71/2.01 4.71/2.01 The argument filtering Pi contains the following mapping: 4.71/2.01 goal_in_g(x1) = goal_in_g(x1) 4.71/2.01 4.71/2.01 U6_g(x1, x2) = U6_g(x2) 4.71/2.01 4.71/2.01 s2t_in_ga(x1, x2) = s2t_in_ga(x1) 4.71/2.01 4.71/2.01 s(x1) = s(x1) 4.71/2.01 4.71/2.01 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 4.71/2.01 4.71/2.01 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 4.71/2.01 4.71/2.01 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 4.71/2.01 4.71/2.01 s2t_out_ga(x1, x2) = s2t_out_ga(x2) 4.71/2.01 4.71/2.01 node(x1, x2, x3) = node(x1, x3) 4.71/2.01 4.71/2.01 0 = 0 4.71/2.01 4.71/2.01 U7_g(x1, x2) = U7_g(x2) 4.71/2.01 4.71/2.01 tree_in_g(x1) = tree_in_g(x1) 4.71/2.01 4.71/2.01 nil = nil 4.71/2.01 4.71/2.01 tree_out_g(x1) = tree_out_g 4.71/2.01 4.71/2.01 U1_g(x1, x2, x3, x4) = U1_g(x3, x4) 4.71/2.01 4.71/2.01 U2_g(x1, x2, x3, x4) = U2_g(x4) 4.71/2.01 4.71/2.01 goal_out_g(x1) = goal_out_g 4.71/2.01 4.71/2.01 GOAL_IN_G(x1) = GOAL_IN_G(x1) 4.71/2.01 4.71/2.01 U6_G(x1, x2) = U6_G(x2) 4.71/2.01 4.71/2.01 S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) 4.71/2.01 4.71/2.01 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 4.71/2.01 4.71/2.01 U4_GA(x1, x2, x3, x4) = U4_GA(x4) 4.71/2.01 4.71/2.01 U5_GA(x1, x2, x3, x4) = U5_GA(x4) 4.71/2.01 4.71/2.01 U7_G(x1, x2) = U7_G(x2) 4.71/2.01 4.71/2.01 TREE_IN_G(x1) = TREE_IN_G(x1) 4.71/2.01 4.71/2.01 U1_G(x1, x2, x3, x4) = U1_G(x3, x4) 4.71/2.01 4.71/2.01 U2_G(x1, x2, x3, x4) = U2_G(x4) 4.71/2.01 4.71/2.01 4.71/2.01 We have to consider all (P,R,Pi)-chains 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (4) 4.71/2.01 Obligation: 4.71/2.01 Pi DP problem: 4.71/2.01 The TRS P consists of the following rules: 4.71/2.01 4.71/2.01 GOAL_IN_G(X) -> U6_G(X, s2t_in_ga(X, T)) 4.71/2.01 GOAL_IN_G(X) -> S2T_IN_GA(X, T) 4.71/2.01 S2T_IN_GA(s(X), node(T, Y, T)) -> U3_GA(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T) 4.71/2.01 S2T_IN_GA(s(X), node(nil, Y, T)) -> U4_GA(X, Y, T, s2t_in_ga(X, T)) 4.71/2.01 S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T) 4.71/2.01 S2T_IN_GA(s(X), node(T, Y, nil)) -> U5_GA(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) 4.71/2.01 U6_G(X, s2t_out_ga(X, T)) -> U7_G(X, tree_in_g(T)) 4.71/2.01 U6_G(X, s2t_out_ga(X, T)) -> TREE_IN_G(T) 4.71/2.01 TREE_IN_G(node(L, X, R)) -> U1_G(L, X, R, tree_in_g(L)) 4.71/2.01 TREE_IN_G(node(L, X, R)) -> TREE_IN_G(L) 4.71/2.01 U1_G(L, X, R, tree_out_g(L)) -> U2_G(L, X, R, tree_in_g(R)) 4.71/2.01 U1_G(L, X, R, tree_out_g(L)) -> TREE_IN_G(R) 4.71/2.01 4.71/2.01 The TRS R consists of the following rules: 4.71/2.01 4.71/2.01 goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(T, Y, nil)) -> U5_ga(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) 4.71/2.01 s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) 4.71/2.01 U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) 4.71/2.01 U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) 4.71/2.01 U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) 4.71/2.01 U6_g(X, s2t_out_ga(X, T)) -> U7_g(X, tree_in_g(T)) 4.71/2.01 tree_in_g(nil) -> tree_out_g(nil) 4.71/2.01 tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) 4.71/2.01 U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) 4.71/2.01 U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) 4.71/2.01 U7_g(X, tree_out_g(T)) -> goal_out_g(X) 4.71/2.01 4.71/2.01 The argument filtering Pi contains the following mapping: 4.71/2.01 goal_in_g(x1) = goal_in_g(x1) 4.71/2.01 4.71/2.01 U6_g(x1, x2) = U6_g(x2) 4.71/2.01 4.71/2.01 s2t_in_ga(x1, x2) = s2t_in_ga(x1) 4.71/2.01 4.71/2.01 s(x1) = s(x1) 4.71/2.01 4.71/2.01 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 4.71/2.01 4.71/2.01 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 4.71/2.01 4.71/2.01 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 4.71/2.01 4.71/2.01 s2t_out_ga(x1, x2) = s2t_out_ga(x2) 4.71/2.01 4.71/2.01 node(x1, x2, x3) = node(x1, x3) 4.71/2.01 4.71/2.01 0 = 0 4.71/2.01 4.71/2.01 U7_g(x1, x2) = U7_g(x2) 4.71/2.01 4.71/2.01 tree_in_g(x1) = tree_in_g(x1) 4.71/2.01 4.71/2.01 nil = nil 4.71/2.01 4.71/2.01 tree_out_g(x1) = tree_out_g 4.71/2.01 4.71/2.01 U1_g(x1, x2, x3, x4) = U1_g(x3, x4) 4.71/2.01 4.71/2.01 U2_g(x1, x2, x3, x4) = U2_g(x4) 4.71/2.01 4.71/2.01 goal_out_g(x1) = goal_out_g 4.71/2.01 4.71/2.01 GOAL_IN_G(x1) = GOAL_IN_G(x1) 4.71/2.01 4.71/2.01 U6_G(x1, x2) = U6_G(x2) 4.71/2.01 4.71/2.01 S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) 4.71/2.01 4.71/2.01 U3_GA(x1, x2, x3, x4) = U3_GA(x4) 4.71/2.01 4.71/2.01 U4_GA(x1, x2, x3, x4) = U4_GA(x4) 4.71/2.01 4.71/2.01 U5_GA(x1, x2, x3, x4) = U5_GA(x4) 4.71/2.01 4.71/2.01 U7_G(x1, x2) = U7_G(x2) 4.71/2.01 4.71/2.01 TREE_IN_G(x1) = TREE_IN_G(x1) 4.71/2.01 4.71/2.01 U1_G(x1, x2, x3, x4) = U1_G(x3, x4) 4.71/2.01 4.71/2.01 U2_G(x1, x2, x3, x4) = U2_G(x4) 4.71/2.01 4.71/2.01 4.71/2.01 We have to consider all (P,R,Pi)-chains 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (5) DependencyGraphProof (EQUIVALENT) 4.71/2.01 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 8 less nodes. 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (6) 4.71/2.01 Complex Obligation (AND) 4.71/2.01 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (7) 4.71/2.01 Obligation: 4.71/2.01 Pi DP problem: 4.71/2.01 The TRS P consists of the following rules: 4.71/2.01 4.71/2.01 U1_G(L, X, R, tree_out_g(L)) -> TREE_IN_G(R) 4.71/2.01 TREE_IN_G(node(L, X, R)) -> U1_G(L, X, R, tree_in_g(L)) 4.71/2.01 TREE_IN_G(node(L, X, R)) -> TREE_IN_G(L) 4.71/2.01 4.71/2.01 The TRS R consists of the following rules: 4.71/2.01 4.71/2.01 goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(T, Y, nil)) -> U5_ga(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) 4.71/2.01 s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) 4.71/2.01 U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) 4.71/2.01 U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) 4.71/2.01 U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) 4.71/2.01 U6_g(X, s2t_out_ga(X, T)) -> U7_g(X, tree_in_g(T)) 4.71/2.01 tree_in_g(nil) -> tree_out_g(nil) 4.71/2.01 tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) 4.71/2.01 U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) 4.71/2.01 U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) 4.71/2.01 U7_g(X, tree_out_g(T)) -> goal_out_g(X) 4.71/2.01 4.71/2.01 The argument filtering Pi contains the following mapping: 4.71/2.01 goal_in_g(x1) = goal_in_g(x1) 4.71/2.01 4.71/2.01 U6_g(x1, x2) = U6_g(x2) 4.71/2.01 4.71/2.01 s2t_in_ga(x1, x2) = s2t_in_ga(x1) 4.71/2.01 4.71/2.01 s(x1) = s(x1) 4.71/2.01 4.71/2.01 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 4.71/2.01 4.71/2.01 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 4.71/2.01 4.71/2.01 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 4.71/2.01 4.71/2.01 s2t_out_ga(x1, x2) = s2t_out_ga(x2) 4.71/2.01 4.71/2.01 node(x1, x2, x3) = node(x1, x3) 4.71/2.01 4.71/2.01 0 = 0 4.71/2.01 4.71/2.01 U7_g(x1, x2) = U7_g(x2) 4.71/2.01 4.71/2.01 tree_in_g(x1) = tree_in_g(x1) 4.71/2.01 4.71/2.01 nil = nil 4.71/2.01 4.71/2.01 tree_out_g(x1) = tree_out_g 4.71/2.01 4.71/2.01 U1_g(x1, x2, x3, x4) = U1_g(x3, x4) 4.71/2.01 4.71/2.01 U2_g(x1, x2, x3, x4) = U2_g(x4) 4.71/2.01 4.71/2.01 goal_out_g(x1) = goal_out_g 4.71/2.01 4.71/2.01 TREE_IN_G(x1) = TREE_IN_G(x1) 4.71/2.01 4.71/2.01 U1_G(x1, x2, x3, x4) = U1_G(x3, x4) 4.71/2.01 4.71/2.01 4.71/2.01 We have to consider all (P,R,Pi)-chains 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (8) UsableRulesProof (EQUIVALENT) 4.71/2.01 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (9) 4.71/2.01 Obligation: 4.71/2.01 Pi DP problem: 4.71/2.01 The TRS P consists of the following rules: 4.71/2.01 4.71/2.01 U1_G(L, X, R, tree_out_g(L)) -> TREE_IN_G(R) 4.71/2.01 TREE_IN_G(node(L, X, R)) -> U1_G(L, X, R, tree_in_g(L)) 4.71/2.01 TREE_IN_G(node(L, X, R)) -> TREE_IN_G(L) 4.71/2.01 4.71/2.01 The TRS R consists of the following rules: 4.71/2.01 4.71/2.01 tree_in_g(nil) -> tree_out_g(nil) 4.71/2.01 tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) 4.71/2.01 U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) 4.71/2.01 U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) 4.71/2.01 4.71/2.01 The argument filtering Pi contains the following mapping: 4.71/2.01 node(x1, x2, x3) = node(x1, x3) 4.71/2.01 4.71/2.01 tree_in_g(x1) = tree_in_g(x1) 4.71/2.01 4.71/2.01 nil = nil 4.71/2.01 4.71/2.01 tree_out_g(x1) = tree_out_g 4.71/2.01 4.71/2.01 U1_g(x1, x2, x3, x4) = U1_g(x3, x4) 4.71/2.01 4.71/2.01 U2_g(x1, x2, x3, x4) = U2_g(x4) 4.71/2.01 4.71/2.01 TREE_IN_G(x1) = TREE_IN_G(x1) 4.71/2.01 4.71/2.01 U1_G(x1, x2, x3, x4) = U1_G(x3, x4) 4.71/2.01 4.71/2.01 4.71/2.01 We have to consider all (P,R,Pi)-chains 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (10) PiDPToQDPProof (SOUND) 4.71/2.01 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (11) 4.71/2.01 Obligation: 4.71/2.01 Q DP problem: 4.71/2.01 The TRS P consists of the following rules: 4.71/2.01 4.71/2.01 U1_G(R, tree_out_g) -> TREE_IN_G(R) 4.71/2.01 TREE_IN_G(node(L, R)) -> U1_G(R, tree_in_g(L)) 4.71/2.01 TREE_IN_G(node(L, R)) -> TREE_IN_G(L) 4.71/2.01 4.71/2.01 The TRS R consists of the following rules: 4.71/2.01 4.71/2.01 tree_in_g(nil) -> tree_out_g 4.71/2.01 tree_in_g(node(L, R)) -> U1_g(R, tree_in_g(L)) 4.71/2.01 U1_g(R, tree_out_g) -> U2_g(tree_in_g(R)) 4.71/2.01 U2_g(tree_out_g) -> tree_out_g 4.71/2.01 4.71/2.01 The set Q consists of the following terms: 4.71/2.01 4.71/2.01 tree_in_g(x0) 4.71/2.01 U1_g(x0, x1) 4.71/2.01 U2_g(x0) 4.71/2.01 4.71/2.01 We have to consider all (P,Q,R)-chains. 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (12) QDPSizeChangeProof (EQUIVALENT) 4.71/2.01 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. 4.71/2.01 4.71/2.01 From the DPs we obtained the following set of size-change graphs: 4.71/2.01 *TREE_IN_G(node(L, R)) -> U1_G(R, tree_in_g(L)) 4.71/2.01 The graph contains the following edges 1 > 1 4.71/2.01 4.71/2.01 4.71/2.01 *TREE_IN_G(node(L, R)) -> TREE_IN_G(L) 4.71/2.01 The graph contains the following edges 1 > 1 4.71/2.01 4.71/2.01 4.71/2.01 *U1_G(R, tree_out_g) -> TREE_IN_G(R) 4.71/2.01 The graph contains the following edges 1 >= 1 4.71/2.01 4.71/2.01 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (13) 4.71/2.01 YES 4.71/2.01 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (14) 4.71/2.01 Obligation: 4.71/2.01 Pi DP problem: 4.71/2.01 The TRS P consists of the following rules: 4.71/2.01 4.71/2.01 S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T) 4.71/2.01 S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T) 4.71/2.01 S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) 4.71/2.01 4.71/2.01 The TRS R consists of the following rules: 4.71/2.01 4.71/2.01 goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(T, Y, nil)) -> U5_ga(X, T, Y, s2t_in_ga(X, T)) 4.71/2.01 s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) 4.71/2.01 s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) 4.71/2.01 U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) 4.71/2.01 U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) 4.71/2.01 U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) 4.71/2.01 U6_g(X, s2t_out_ga(X, T)) -> U7_g(X, tree_in_g(T)) 4.71/2.01 tree_in_g(nil) -> tree_out_g(nil) 4.71/2.01 tree_in_g(node(L, X, R)) -> U1_g(L, X, R, tree_in_g(L)) 4.71/2.01 U1_g(L, X, R, tree_out_g(L)) -> U2_g(L, X, R, tree_in_g(R)) 4.71/2.01 U2_g(L, X, R, tree_out_g(R)) -> tree_out_g(node(L, X, R)) 4.71/2.01 U7_g(X, tree_out_g(T)) -> goal_out_g(X) 4.71/2.01 4.71/2.01 The argument filtering Pi contains the following mapping: 4.71/2.01 goal_in_g(x1) = goal_in_g(x1) 4.71/2.01 4.71/2.01 U6_g(x1, x2) = U6_g(x2) 4.71/2.01 4.71/2.01 s2t_in_ga(x1, x2) = s2t_in_ga(x1) 4.71/2.01 4.71/2.01 s(x1) = s(x1) 4.71/2.01 4.71/2.01 U3_ga(x1, x2, x3, x4) = U3_ga(x4) 4.71/2.01 4.71/2.01 U4_ga(x1, x2, x3, x4) = U4_ga(x4) 4.71/2.01 4.71/2.01 U5_ga(x1, x2, x3, x4) = U5_ga(x4) 4.71/2.01 4.71/2.01 s2t_out_ga(x1, x2) = s2t_out_ga(x2) 4.71/2.01 4.71/2.01 node(x1, x2, x3) = node(x1, x3) 4.71/2.01 4.71/2.01 0 = 0 4.71/2.01 4.71/2.01 U7_g(x1, x2) = U7_g(x2) 4.71/2.01 4.71/2.01 tree_in_g(x1) = tree_in_g(x1) 4.71/2.01 4.71/2.01 nil = nil 4.71/2.01 4.71/2.01 tree_out_g(x1) = tree_out_g 4.71/2.01 4.71/2.01 U1_g(x1, x2, x3, x4) = U1_g(x3, x4) 4.71/2.01 4.71/2.01 U2_g(x1, x2, x3, x4) = U2_g(x4) 4.71/2.01 4.71/2.01 goal_out_g(x1) = goal_out_g 4.71/2.01 4.71/2.01 S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) 4.71/2.01 4.71/2.01 4.71/2.01 We have to consider all (P,R,Pi)-chains 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (15) UsableRulesProof (EQUIVALENT) 4.71/2.01 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (16) 4.71/2.01 Obligation: 4.71/2.01 Pi DP problem: 4.71/2.01 The TRS P consists of the following rules: 4.71/2.01 4.71/2.01 S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T) 4.71/2.01 S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T) 4.71/2.01 S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) 4.71/2.01 4.71/2.01 R is empty. 4.71/2.01 The argument filtering Pi contains the following mapping: 4.71/2.01 s(x1) = s(x1) 4.71/2.01 4.71/2.01 node(x1, x2, x3) = node(x1, x3) 4.71/2.01 4.71/2.01 nil = nil 4.71/2.01 4.71/2.01 S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) 4.71/2.01 4.71/2.01 4.71/2.01 We have to consider all (P,R,Pi)-chains 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (17) PiDPToQDPProof (SOUND) 4.71/2.01 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (18) 4.71/2.01 Obligation: 4.71/2.01 Q DP problem: 4.71/2.01 The TRS P consists of the following rules: 4.71/2.01 4.71/2.01 S2T_IN_GA(s(X)) -> S2T_IN_GA(X) 4.71/2.01 4.71/2.01 R is empty. 4.71/2.01 Q is empty. 4.71/2.01 We have to consider all (P,Q,R)-chains. 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (19) QDPSizeChangeProof (EQUIVALENT) 4.71/2.01 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. 4.71/2.01 4.71/2.01 From the DPs we obtained the following set of size-change graphs: 4.71/2.01 *S2T_IN_GA(s(X)) -> S2T_IN_GA(X) 4.71/2.01 The graph contains the following edges 1 > 1 4.71/2.01 4.71/2.01 4.71/2.01 ---------------------------------------- 4.71/2.01 4.71/2.01 (20) 4.71/2.01 YES 4.79/2.03 EOF