NO proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be disproven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 91 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 0 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 7 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 14 ms] (8) QTRS (9) QTRSRRRProof [EQUIVALENT, 0 ms] (10) QTRS (11) QTRSRRRProof [EQUIVALENT, 10 ms] (12) QTRS (13) QTRSRRRProof [EQUIVALENT, 0 ms] (14) QTRS (15) QTRSRRRProof [EQUIVALENT, 0 ms] (16) QTRS (17) QTRSRRRProof [EQUIVALENT, 8 ms] (18) QTRS (19) QTRSRRRProof [EQUIVALENT, 0 ms] (20) QTRS (21) DependencyPairsProof [EQUIVALENT, 0 ms] (22) QDP (23) DependencyGraphProof [EQUIVALENT, 0 ms] (24) AND (25) QDP (26) UsableRulesProof [EQUIVALENT, 0 ms] (27) QDP (28) QReductionProof [EQUIVALENT, 0 ms] (29) QDP (30) QDPSizeChangeProof [EQUIVALENT, 0 ms] (31) YES (32) QDP (33) UsableRulesProof [EQUIVALENT, 0 ms] (34) QDP (35) QReductionProof [EQUIVALENT, 0 ms] (36) QDP (37) MRRProof [EQUIVALENT, 7 ms] (38) QDP (39) TransformationProof [EQUIVALENT, 0 ms] (40) QDP (41) UsableRulesProof [EQUIVALENT, 0 ms] (42) QDP (43) QReductionProof [EQUIVALENT, 0 ms] (44) QDP (45) TransformationProof [EQUIVALENT, 0 ms] (46) QDP (47) UsableRulesProof [EQUIVALENT, 0 ms] (48) QDP (49) QReductionProof [EQUIVALENT, 0 ms] (50) QDP (51) TransformationProof [EQUIVALENT, 0 ms] (52) QDP (53) TransformationProof [EQUIVALENT, 0 ms] (54) QDP (55) NonTerminationLoopProof [COMPLETE, 0 ms] (56) NO ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U12(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__U11(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__U12(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__length(x_1)) = x_1 POL(a__zeros) = 0 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(length(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(nil) = 2 POL(s(x_1)) = 2*x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__length(nil) -> 0 ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1, x_2)) = x_1 + 2*x_2 POL(U12(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__U11(x_1, x_2)) = x_1 + 2*x_2 POL(a__U12(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__length(x_1)) = x_1 POL(a__zeros) = 0 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(length(x_1)) = x_1 POL(mark(x_1)) = 2*x_1 POL(nil) = 1 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(nil) -> nil ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1, x_2)) = x_1 + 2*x_2 POL(U12(x_1, x_2)) = x_1 + 2*x_2 POL(a__U11(x_1, x_2)) = x_1 + 2*x_2 POL(a__U12(x_1, x_2)) = x_1 + 2*x_2 POL(a__length(x_1)) = 1 + x_1 POL(a__zeros) = 0 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(length(x_1)) = 1 + x_1 POL(mark(x_1)) = 2*x_1 POL(s(x_1)) = x_1 POL(tt) = 1 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(length(X)) -> a__length(mark(X)) mark(tt) -> tt ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1, x_2)) = x_1 + 2*x_2 POL(U12(x_1, x_2)) = x_1 + 2*x_2 POL(a__U11(x_1, x_2)) = x_1 + 2*x_2 POL(a__U12(x_1, x_2)) = x_1 + 2*x_2 POL(a__length(x_1)) = 1 + x_1 POL(a__zeros) = 0 POL(cons(x_1, x_2)) = x_1 + 2*x_2 POL(length(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(s(x_1)) = x_1 POL(tt) = 1 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__length(X) -> length(X) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (9) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(U12(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(a__U11(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(a__U12(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(a__length(x_1)) = 2 + x_1 POL(a__zeros) = 0 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(mark(x_1)) = 2*x_1 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) ---------------------------------------- (10) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__zeros -> zeros The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (11) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U12(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(a__U11(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__U12(x_1, x_2)) = x_1 + 2*x_2 POL(a__length(x_1)) = x_1 POL(a__zeros) = 0 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(mark(x_1)) = x_1 POL(s(x_1)) = 2*x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(U12(X1, X2)) -> a__U12(mark(X1), X2) ---------------------------------------- (12) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__zeros -> zeros The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (13) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(a__U11(x_1, x_2)) = x_1 + 2*x_2 POL(a__U12(x_1, x_2)) = x_1 + 2*x_2 POL(a__length(x_1)) = x_1 POL(a__zeros) = 0 POL(cons(x_1, x_2)) = x_1 + 2*x_2 POL(mark(x_1)) = x_1 POL(s(x_1)) = 2*x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(U11(X1, X2)) -> a__U11(mark(X1), X2) ---------------------------------------- (14) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__zeros -> zeros The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (15) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(a__U11(x_1, x_2)) = x_1 + 2*x_2 POL(a__U12(x_1, x_2)) = x_1 + 2*x_2 POL(a__length(x_1)) = x_1 POL(a__zeros) = 2 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(mark(x_1)) = 2*x_1 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__zeros -> zeros ---------------------------------------- (16) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (17) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(a__U11(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__U12(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__length(x_1)) = 2*x_1 POL(a__zeros) = 2 POL(cons(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(mark(x_1)) = 2 + x_1 POL(s(x_1)) = x_1 POL(tt) = 2 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(0) -> 0 ---------------------------------------- (18) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (19) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(a__U11(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(a__U12(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(a__length(x_1)) = x_1 POL(a__zeros) = 1 POL(cons(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(mark(x_1)) = 1 + 2*x_1 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(cons(X1, X2)) -> cons(mark(X1), X2) ---------------------------------------- (20) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(s(X)) -> s(mark(X)) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (21) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: A__U11(tt, L) -> A__U12(tt, L) A__U12(tt, L) -> A__LENGTH(mark(L)) A__U12(tt, L) -> MARK(L) A__LENGTH(cons(N, L)) -> A__U11(tt, L) MARK(zeros) -> A__ZEROS MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(s(X)) -> s(mark(X)) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. ---------------------------------------- (24) Complex Obligation (AND) ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(s(X)) -> s(mark(X)) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(s(X)) -> MARK(X) R is empty. The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(s(X)) -> MARK(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) 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: *MARK(s(X)) -> MARK(X) The graph contains the following edges 1 > 1 ---------------------------------------- (31) YES ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: A__U12(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U11(tt, L) A__U11(tt, L) -> A__U12(tt, L) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(s(X)) -> s(mark(X)) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: A__U12(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U11(tt, L) A__U11(tt, L) -> A__U12(tt, L) The TRS R consists of the following rules: mark(zeros) -> a__zeros mark(s(X)) -> s(mark(X)) a__zeros -> cons(0, zeros) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) a__U11(x0, x1) a__U12(x0, x1) a__length(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. a__U11(x0, x1) a__U12(x0, x1) a__length(x0) ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: A__U12(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U11(tt, L) A__U11(tt, L) -> A__U12(tt, L) The TRS R consists of the following rules: mark(zeros) -> a__zeros mark(s(X)) -> s(mark(X)) a__zeros -> cons(0, zeros) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: mark(s(X)) -> s(mark(X)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(A__LENGTH(x_1)) = x_1 POL(A__U11(x_1, x_2)) = x_1 + 2*x_2 POL(A__U12(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__zeros) = 0 POL(cons(x_1, x_2)) = x_1 + 2*x_2 POL(mark(x_1)) = 2*x_1 POL(s(x_1)) = 2 + 2*x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: A__U12(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U11(tt, L) A__U11(tt, L) -> A__U12(tt, L) The TRS R consists of the following rules: mark(zeros) -> a__zeros a__zeros -> cons(0, zeros) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A__U12(tt, L) -> A__LENGTH(mark(L)) at position [0] we obtained the following new rules [LPAR04]: (A__U12(tt, zeros) -> A__LENGTH(a__zeros),A__U12(tt, zeros) -> A__LENGTH(a__zeros)) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: A__LENGTH(cons(N, L)) -> A__U11(tt, L) A__U11(tt, L) -> A__U12(tt, L) A__U12(tt, zeros) -> A__LENGTH(a__zeros) The TRS R consists of the following rules: mark(zeros) -> a__zeros a__zeros -> cons(0, zeros) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: A__LENGTH(cons(N, L)) -> A__U11(tt, L) A__U11(tt, L) -> A__U12(tt, L) A__U12(tt, zeros) -> A__LENGTH(a__zeros) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) The set Q consists of the following terms: a__zeros mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. mark(zeros) mark(U11(x0, x1)) mark(U12(x0, x1)) mark(length(x0)) mark(cons(x0, x1)) mark(0) mark(tt) mark(s(x0)) mark(nil) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: A__LENGTH(cons(N, L)) -> A__U11(tt, L) A__U11(tt, L) -> A__U12(tt, L) A__U12(tt, zeros) -> A__LENGTH(a__zeros) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) The set Q consists of the following terms: a__zeros We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule A__U12(tt, zeros) -> A__LENGTH(a__zeros) at position [0] we obtained the following new rules [LPAR04]: (A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)),A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros))) ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: A__LENGTH(cons(N, L)) -> A__U11(tt, L) A__U11(tt, L) -> A__U12(tt, L) A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) The set Q consists of the following terms: a__zeros We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: A__LENGTH(cons(N, L)) -> A__U11(tt, L) A__U11(tt, L) -> A__U12(tt, L) A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) R is empty. The set Q consists of the following terms: a__zeros We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. a__zeros ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: A__LENGTH(cons(N, L)) -> A__U11(tt, L) A__U11(tt, L) -> A__U12(tt, L) A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule A__LENGTH(cons(N, L)) -> A__U11(tt, L) we obtained the following new rules [LPAR04]: (A__LENGTH(cons(0, zeros)) -> A__U11(tt, zeros),A__LENGTH(cons(0, zeros)) -> A__U11(tt, zeros)) ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: A__U11(tt, L) -> A__U12(tt, L) A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) A__LENGTH(cons(0, zeros)) -> A__U11(tt, zeros) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule A__U11(tt, L) -> A__U12(tt, L) we obtained the following new rules [LPAR04]: (A__U11(tt, zeros) -> A__U12(tt, zeros),A__U11(tt, zeros) -> A__U12(tt, zeros)) ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) A__LENGTH(cons(0, zeros)) -> A__U11(tt, zeros) A__U11(tt, zeros) -> A__U12(tt, zeros) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = A__LENGTH(cons(0, zeros)) evaluates to t =A__LENGTH(cons(0, zeros)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence A__LENGTH(cons(0, zeros)) -> A__U11(tt, zeros) with rule A__LENGTH(cons(0, zeros)) -> A__U11(tt, zeros) at position [] and matcher [ ] A__U11(tt, zeros) -> A__U12(tt, zeros) with rule A__U11(tt, zeros) -> A__U12(tt, zeros) at position [] and matcher [ ] A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) with rule A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (56) NO