24.73/12.06 NO 24.88/12.08 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 24.88/12.08 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.88/12.08 24.88/12.08 24.88/12.08 Termination w.r.t. Q of the given QTRS could be disproven: 24.88/12.08 24.88/12.08 (0) QTRS 24.88/12.08 (1) QTRSRRRProof [EQUIVALENT, 118 ms] 24.88/12.08 (2) QTRS 24.88/12.08 (3) QTRSRRRProof [EQUIVALENT, 46 ms] 24.88/12.08 (4) QTRS 24.88/12.08 (5) DependencyPairsProof [EQUIVALENT, 0 ms] 24.88/12.08 (6) QDP 24.88/12.08 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 24.88/12.08 (8) QDP 24.88/12.08 (9) MRRProof [EQUIVALENT, 75 ms] 24.88/12.08 (10) QDP 24.88/12.08 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 24.88/12.08 (12) AND 24.88/12.08 (13) QDP 24.88/12.08 (14) MRRProof [EQUIVALENT, 32 ms] 24.88/12.08 (15) QDP 24.88/12.08 (16) DependencyGraphProof [EQUIVALENT, 0 ms] 24.88/12.08 (17) QDP 24.88/12.08 (18) UsableRulesProof [EQUIVALENT, 0 ms] 24.88/12.08 (19) QDP 24.88/12.08 (20) QReductionProof [EQUIVALENT, 0 ms] 24.88/12.08 (21) QDP 24.88/12.08 (22) QDPSizeChangeProof [EQUIVALENT, 0 ms] 24.88/12.08 (23) YES 24.88/12.08 (24) QDP 24.88/12.08 (25) TransformationProof [EQUIVALENT, 0 ms] 24.88/12.08 (26) QDP 24.88/12.08 (27) DependencyGraphProof [EQUIVALENT, 0 ms] 24.88/12.08 (28) QDP 24.88/12.08 (29) TransformationProof [EQUIVALENT, 0 ms] 24.88/12.08 (30) QDP 24.88/12.08 (31) DependencyGraphProof [EQUIVALENT, 0 ms] 24.88/12.08 (32) QDP 24.88/12.08 (33) TransformationProof [EQUIVALENT, 0 ms] 24.88/12.08 (34) QDP 24.88/12.08 (35) TransformationProof [EQUIVALENT, 0 ms] 24.88/12.08 (36) QDP 24.88/12.08 (37) DependencyGraphProof [EQUIVALENT, 0 ms] 24.88/12.08 (38) AND 24.88/12.08 (39) QDP 24.88/12.08 (40) UsableRulesProof [EQUIVALENT, 0 ms] 24.88/12.08 (41) QDP 24.88/12.08 (42) QReductionProof [EQUIVALENT, 0 ms] 24.88/12.08 (43) QDP 24.88/12.08 (44) TransformationProof [EQUIVALENT, 0 ms] 24.88/12.08 (45) QDP 24.88/12.08 (46) NonTerminationLoopProof [COMPLETE, 0 ms] 24.88/12.08 (47) NO 24.88/12.08 (48) QDP 24.88/12.08 (49) QDPOrderProof [EQUIVALENT, 157 ms] 24.88/12.08 (50) QDP 24.88/12.08 (51) DependencyGraphProof [EQUIVALENT, 0 ms] 24.88/12.08 (52) QDP 24.88/12.08 24.88/12.08 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (0) 24.88/12.08 Obligation: 24.88/12.08 Q restricted rewrite system: 24.88/12.08 The TRS R consists of the following rules: 24.88/12.08 24.88/12.08 a__zeros -> cons(0, zeros) 24.88/12.08 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.08 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.08 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.08 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.08 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.08 a__length(nil) -> 0 24.88/12.08 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.08 a__take(0, IL) -> nil 24.88/12.08 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.08 mark(zeros) -> a__zeros 24.88/12.08 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.08 mark(length(X)) -> a__length(mark(X)) 24.88/12.08 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.08 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.08 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.08 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.08 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.08 mark(0) -> 0 24.88/12.08 mark(tt) -> tt 24.88/12.08 mark(s(X)) -> s(mark(X)) 24.88/12.08 mark(nil) -> nil 24.88/12.08 a__zeros -> zeros 24.88/12.08 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.08 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.08 a__length(X) -> length(X) 24.88/12.08 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.08 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.08 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.08 a__take(X1, X2) -> take(X1, X2) 24.88/12.08 24.88/12.08 The set Q consists of the following terms: 24.88/12.08 24.88/12.08 a__zeros 24.88/12.08 mark(zeros) 24.88/12.08 mark(U11(x0, x1)) 24.88/12.08 mark(U12(x0, x1)) 24.88/12.08 mark(length(x0)) 24.88/12.08 mark(U21(x0, x1, x2, x3)) 24.88/12.08 mark(U22(x0, x1, x2, x3)) 24.88/12.08 mark(U23(x0, x1, x2, x3)) 24.88/12.08 mark(take(x0, x1)) 24.88/12.08 mark(cons(x0, x1)) 24.88/12.08 mark(0) 24.88/12.08 mark(tt) 24.88/12.08 mark(s(x0)) 24.88/12.08 mark(nil) 24.88/12.08 a__U11(x0, x1) 24.88/12.08 a__U12(x0, x1) 24.88/12.08 a__length(x0) 24.88/12.08 a__U21(x0, x1, x2, x3) 24.88/12.08 a__U22(x0, x1, x2, x3) 24.88/12.08 a__U23(x0, x1, x2, x3) 24.88/12.08 a__take(x0, x1) 24.88/12.08 24.88/12.08 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (1) QTRSRRRProof (EQUIVALENT) 24.88/12.08 Used ordering: 24.88/12.08 Polynomial interpretation [POLO]: 24.88/12.08 24.88/12.08 POL(0) = 0 24.88/12.08 POL(U11(x_1, x_2)) = x_1 + x_2 24.88/12.08 POL(U12(x_1, x_2)) = 2*x_1 + x_2 24.88/12.08 POL(U21(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + x_2 + x_3 + x_4 24.88/12.08 POL(U22(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + x_2 + x_3 + x_4 24.88/12.08 POL(U23(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + x_2 + x_3 + x_4 24.88/12.08 POL(a__U11(x_1, x_2)) = x_1 + x_2 24.88/12.08 POL(a__U12(x_1, x_2)) = 2*x_1 + x_2 24.88/12.08 POL(a__U21(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + x_2 + x_3 + x_4 24.88/12.08 POL(a__U22(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + x_2 + x_3 + x_4 24.88/12.08 POL(a__U23(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + x_2 + x_3 + x_4 24.88/12.08 POL(a__length(x_1)) = x_1 24.88/12.08 POL(a__take(x_1, x_2)) = 1 + x_1 + x_2 24.88/12.08 POL(a__zeros) = 0 24.88/12.08 POL(cons(x_1, x_2)) = x_1 + x_2 24.88/12.08 POL(length(x_1)) = x_1 24.88/12.08 POL(mark(x_1)) = x_1 24.88/12.08 POL(nil) = 0 24.88/12.08 POL(s(x_1)) = x_1 24.88/12.08 POL(take(x_1, x_2)) = 1 + x_1 + x_2 24.88/12.08 POL(tt) = 0 24.88/12.08 POL(zeros) = 0 24.88/12.08 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 24.88/12.08 24.88/12.08 a__take(0, IL) -> nil 24.88/12.08 24.88/12.08 24.88/12.08 24.88/12.08 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (2) 24.88/12.08 Obligation: 24.88/12.08 Q restricted rewrite system: 24.88/12.08 The TRS R consists of the following rules: 24.88/12.08 24.88/12.08 a__zeros -> cons(0, zeros) 24.88/12.08 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.08 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.08 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.08 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.08 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.08 a__length(nil) -> 0 24.88/12.08 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.08 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.08 mark(zeros) -> a__zeros 24.88/12.08 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.08 mark(length(X)) -> a__length(mark(X)) 24.88/12.08 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.08 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.08 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.08 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.08 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.08 mark(0) -> 0 24.88/12.08 mark(tt) -> tt 24.88/12.08 mark(s(X)) -> s(mark(X)) 24.88/12.08 mark(nil) -> nil 24.88/12.08 a__zeros -> zeros 24.88/12.08 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.08 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.08 a__length(X) -> length(X) 24.88/12.08 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.08 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.08 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.08 a__take(X1, X2) -> take(X1, X2) 24.88/12.08 24.88/12.08 The set Q consists of the following terms: 24.88/12.08 24.88/12.08 a__zeros 24.88/12.08 mark(zeros) 24.88/12.08 mark(U11(x0, x1)) 24.88/12.08 mark(U12(x0, x1)) 24.88/12.08 mark(length(x0)) 24.88/12.08 mark(U21(x0, x1, x2, x3)) 24.88/12.08 mark(U22(x0, x1, x2, x3)) 24.88/12.08 mark(U23(x0, x1, x2, x3)) 24.88/12.08 mark(take(x0, x1)) 24.88/12.08 mark(cons(x0, x1)) 24.88/12.08 mark(0) 24.88/12.08 mark(tt) 24.88/12.08 mark(s(x0)) 24.88/12.08 mark(nil) 24.88/12.08 a__U11(x0, x1) 24.88/12.08 a__U12(x0, x1) 24.88/12.08 a__length(x0) 24.88/12.08 a__U21(x0, x1, x2, x3) 24.88/12.08 a__U22(x0, x1, x2, x3) 24.88/12.08 a__U23(x0, x1, x2, x3) 24.88/12.08 a__take(x0, x1) 24.88/12.08 24.88/12.08 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (3) QTRSRRRProof (EQUIVALENT) 24.88/12.08 Used ordering: 24.88/12.08 Polynomial interpretation [POLO]: 24.88/12.08 24.88/12.08 POL(0) = 0 24.88/12.08 POL(U11(x_1, x_2)) = x_1 + 2*x_2 24.88/12.08 POL(U12(x_1, x_2)) = 2*x_1 + 2*x_2 24.88/12.08 POL(U21(x_1, x_2, x_3, x_4)) = x_1 + 2*x_2 + 2*x_3 + 2*x_4 24.88/12.08 POL(U22(x_1, x_2, x_3, x_4)) = x_1 + 2*x_2 + 2*x_3 + 2*x_4 24.88/12.08 POL(U23(x_1, x_2, x_3, x_4)) = x_1 + 2*x_2 + 2*x_3 + 2*x_4 24.88/12.08 POL(a__U11(x_1, x_2)) = x_1 + 2*x_2 24.88/12.08 POL(a__U12(x_1, x_2)) = 2*x_1 + 2*x_2 24.88/12.08 POL(a__U21(x_1, x_2, x_3, x_4)) = x_1 + 2*x_2 + 2*x_3 + 2*x_4 24.88/12.08 POL(a__U22(x_1, x_2, x_3, x_4)) = x_1 + 2*x_2 + 2*x_3 + 2*x_4 24.88/12.08 POL(a__U23(x_1, x_2, x_3, x_4)) = x_1 + 2*x_2 + 2*x_3 + 2*x_4 24.88/12.08 POL(a__length(x_1)) = x_1 24.88/12.08 POL(a__take(x_1, x_2)) = x_1 + x_2 24.88/12.08 POL(a__zeros) = 0 24.88/12.08 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 24.88/12.08 POL(length(x_1)) = x_1 24.88/12.08 POL(mark(x_1)) = x_1 24.88/12.08 POL(nil) = 2 24.88/12.08 POL(s(x_1)) = 2*x_1 24.88/12.08 POL(take(x_1, x_2)) = x_1 + x_2 24.88/12.08 POL(tt) = 0 24.88/12.08 POL(zeros) = 0 24.88/12.08 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 24.88/12.08 24.88/12.08 a__length(nil) -> 0 24.88/12.08 24.88/12.08 24.88/12.08 24.88/12.08 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (4) 24.88/12.08 Obligation: 24.88/12.08 Q restricted rewrite system: 24.88/12.08 The TRS R consists of the following rules: 24.88/12.08 24.88/12.08 a__zeros -> cons(0, zeros) 24.88/12.08 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.08 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.08 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.08 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.08 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.08 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.08 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.08 mark(zeros) -> a__zeros 24.88/12.08 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.08 mark(length(X)) -> a__length(mark(X)) 24.88/12.08 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.08 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.08 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.08 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.08 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.08 mark(0) -> 0 24.88/12.08 mark(tt) -> tt 24.88/12.08 mark(s(X)) -> s(mark(X)) 24.88/12.08 mark(nil) -> nil 24.88/12.08 a__zeros -> zeros 24.88/12.08 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.08 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.08 a__length(X) -> length(X) 24.88/12.08 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.08 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.08 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.08 a__take(X1, X2) -> take(X1, X2) 24.88/12.08 24.88/12.08 The set Q consists of the following terms: 24.88/12.08 24.88/12.08 a__zeros 24.88/12.08 mark(zeros) 24.88/12.08 mark(U11(x0, x1)) 24.88/12.08 mark(U12(x0, x1)) 24.88/12.08 mark(length(x0)) 24.88/12.08 mark(U21(x0, x1, x2, x3)) 24.88/12.08 mark(U22(x0, x1, x2, x3)) 24.88/12.08 mark(U23(x0, x1, x2, x3)) 24.88/12.08 mark(take(x0, x1)) 24.88/12.08 mark(cons(x0, x1)) 24.88/12.08 mark(0) 24.88/12.08 mark(tt) 24.88/12.08 mark(s(x0)) 24.88/12.08 mark(nil) 24.88/12.08 a__U11(x0, x1) 24.88/12.08 a__U12(x0, x1) 24.88/12.08 a__length(x0) 24.88/12.08 a__U21(x0, x1, x2, x3) 24.88/12.08 a__U22(x0, x1, x2, x3) 24.88/12.08 a__U23(x0, x1, x2, x3) 24.88/12.08 a__take(x0, x1) 24.88/12.08 24.88/12.08 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (5) DependencyPairsProof (EQUIVALENT) 24.88/12.08 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (6) 24.88/12.08 Obligation: 24.88/12.08 Q DP problem: 24.88/12.08 The TRS P consists of the following rules: 24.88/12.08 24.88/12.08 A__U11(tt, L) -> A__U12(tt, L) 24.88/12.08 A__U12(tt, L) -> A__LENGTH(mark(L)) 24.88/12.08 A__U12(tt, L) -> MARK(L) 24.88/12.08 A__U21(tt, IL, M, N) -> A__U22(tt, IL, M, N) 24.88/12.08 A__U22(tt, IL, M, N) -> A__U23(tt, IL, M, N) 24.88/12.08 A__U23(tt, IL, M, N) -> MARK(N) 24.88/12.08 A__LENGTH(cons(N, L)) -> A__U11(tt, L) 24.88/12.08 A__TAKE(s(M), cons(N, IL)) -> A__U21(tt, IL, M, N) 24.88/12.08 MARK(zeros) -> A__ZEROS 24.88/12.08 MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) 24.88/12.08 MARK(U11(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(U12(X1, X2)) -> A__U12(mark(X1), X2) 24.88/12.08 MARK(U12(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(length(X)) -> A__LENGTH(mark(X)) 24.88/12.08 MARK(length(X)) -> MARK(X) 24.88/12.08 MARK(U21(X1, X2, X3, X4)) -> A__U21(mark(X1), X2, X3, X4) 24.88/12.08 MARK(U21(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(U22(X1, X2, X3, X4)) -> A__U22(mark(X1), X2, X3, X4) 24.88/12.08 MARK(U22(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(U23(X1, X2, X3, X4)) -> A__U23(mark(X1), X2, X3, X4) 24.88/12.08 MARK(U23(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) 24.88/12.08 MARK(take(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(take(X1, X2)) -> MARK(X2) 24.88/12.08 MARK(cons(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(s(X)) -> MARK(X) 24.88/12.08 24.88/12.08 The TRS R consists of the following rules: 24.88/12.08 24.88/12.08 a__zeros -> cons(0, zeros) 24.88/12.08 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.08 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.08 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.08 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.08 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.08 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.08 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.08 mark(zeros) -> a__zeros 24.88/12.08 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.08 mark(length(X)) -> a__length(mark(X)) 24.88/12.08 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.08 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.08 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.08 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.08 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.08 mark(0) -> 0 24.88/12.08 mark(tt) -> tt 24.88/12.08 mark(s(X)) -> s(mark(X)) 24.88/12.08 mark(nil) -> nil 24.88/12.08 a__zeros -> zeros 24.88/12.08 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.08 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.08 a__length(X) -> length(X) 24.88/12.08 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.08 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.08 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.08 a__take(X1, X2) -> take(X1, X2) 24.88/12.08 24.88/12.08 The set Q consists of the following terms: 24.88/12.08 24.88/12.08 a__zeros 24.88/12.08 mark(zeros) 24.88/12.08 mark(U11(x0, x1)) 24.88/12.08 mark(U12(x0, x1)) 24.88/12.08 mark(length(x0)) 24.88/12.08 mark(U21(x0, x1, x2, x3)) 24.88/12.08 mark(U22(x0, x1, x2, x3)) 24.88/12.08 mark(U23(x0, x1, x2, x3)) 24.88/12.08 mark(take(x0, x1)) 24.88/12.08 mark(cons(x0, x1)) 24.88/12.08 mark(0) 24.88/12.08 mark(tt) 24.88/12.08 mark(s(x0)) 24.88/12.08 mark(nil) 24.88/12.08 a__U11(x0, x1) 24.88/12.08 a__U12(x0, x1) 24.88/12.08 a__length(x0) 24.88/12.08 a__U21(x0, x1, x2, x3) 24.88/12.08 a__U22(x0, x1, x2, x3) 24.88/12.08 a__U23(x0, x1, x2, x3) 24.88/12.08 a__take(x0, x1) 24.88/12.08 24.88/12.08 We have to consider all minimal (P,Q,R)-chains. 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (7) DependencyGraphProof (EQUIVALENT) 24.88/12.08 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (8) 24.88/12.08 Obligation: 24.88/12.08 Q DP problem: 24.88/12.08 The TRS P consists of the following rules: 24.88/12.08 24.88/12.08 A__U12(tt, L) -> A__LENGTH(mark(L)) 24.88/12.08 A__LENGTH(cons(N, L)) -> A__U11(tt, L) 24.88/12.08 A__U11(tt, L) -> A__U12(tt, L) 24.88/12.08 A__U12(tt, L) -> MARK(L) 24.88/12.08 MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) 24.88/12.08 MARK(U11(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(U12(X1, X2)) -> A__U12(mark(X1), X2) 24.88/12.08 MARK(U12(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(length(X)) -> A__LENGTH(mark(X)) 24.88/12.08 MARK(length(X)) -> MARK(X) 24.88/12.08 MARK(U21(X1, X2, X3, X4)) -> A__U21(mark(X1), X2, X3, X4) 24.88/12.08 A__U21(tt, IL, M, N) -> A__U22(tt, IL, M, N) 24.88/12.08 A__U22(tt, IL, M, N) -> A__U23(tt, IL, M, N) 24.88/12.08 A__U23(tt, IL, M, N) -> MARK(N) 24.88/12.08 MARK(U21(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(U22(X1, X2, X3, X4)) -> A__U22(mark(X1), X2, X3, X4) 24.88/12.08 MARK(U22(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(U23(X1, X2, X3, X4)) -> A__U23(mark(X1), X2, X3, X4) 24.88/12.08 MARK(U23(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) 24.88/12.08 A__TAKE(s(M), cons(N, IL)) -> A__U21(tt, IL, M, N) 24.88/12.08 MARK(take(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(take(X1, X2)) -> MARK(X2) 24.88/12.08 MARK(cons(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(s(X)) -> MARK(X) 24.88/12.08 24.88/12.08 The TRS R consists of the following rules: 24.88/12.08 24.88/12.08 a__zeros -> cons(0, zeros) 24.88/12.08 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.08 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.08 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.08 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.08 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.08 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.08 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.08 mark(zeros) -> a__zeros 24.88/12.08 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.08 mark(length(X)) -> a__length(mark(X)) 24.88/12.08 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.08 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.08 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.08 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.08 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.08 mark(0) -> 0 24.88/12.08 mark(tt) -> tt 24.88/12.08 mark(s(X)) -> s(mark(X)) 24.88/12.08 mark(nil) -> nil 24.88/12.08 a__zeros -> zeros 24.88/12.08 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.08 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.08 a__length(X) -> length(X) 24.88/12.08 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.08 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.08 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.08 a__take(X1, X2) -> take(X1, X2) 24.88/12.08 24.88/12.08 The set Q consists of the following terms: 24.88/12.08 24.88/12.08 a__zeros 24.88/12.08 mark(zeros) 24.88/12.08 mark(U11(x0, x1)) 24.88/12.08 mark(U12(x0, x1)) 24.88/12.08 mark(length(x0)) 24.88/12.08 mark(U21(x0, x1, x2, x3)) 24.88/12.08 mark(U22(x0, x1, x2, x3)) 24.88/12.08 mark(U23(x0, x1, x2, x3)) 24.88/12.08 mark(take(x0, x1)) 24.88/12.08 mark(cons(x0, x1)) 24.88/12.08 mark(0) 24.88/12.08 mark(tt) 24.88/12.08 mark(s(x0)) 24.88/12.08 mark(nil) 24.88/12.08 a__U11(x0, x1) 24.88/12.08 a__U12(x0, x1) 24.88/12.08 a__length(x0) 24.88/12.08 a__U21(x0, x1, x2, x3) 24.88/12.08 a__U22(x0, x1, x2, x3) 24.88/12.08 a__U23(x0, x1, x2, x3) 24.88/12.08 a__take(x0, x1) 24.88/12.08 24.88/12.08 We have to consider all minimal (P,Q,R)-chains. 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (9) MRRProof (EQUIVALENT) 24.88/12.08 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. 24.88/12.08 24.88/12.08 Strictly oriented dependency pairs: 24.88/12.08 24.88/12.08 MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) 24.88/12.08 MARK(U11(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(U12(X1, X2)) -> A__U12(mark(X1), X2) 24.88/12.08 MARK(U12(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(length(X)) -> A__LENGTH(mark(X)) 24.88/12.08 MARK(length(X)) -> MARK(X) 24.88/12.08 24.88/12.08 24.88/12.08 Used ordering: Polynomial interpretation [POLO]: 24.88/12.08 24.88/12.08 POL(0) = 0 24.88/12.08 POL(A__LENGTH(x_1)) = 2*x_1 24.88/12.08 POL(A__TAKE(x_1, x_2)) = 2*x_1 + 2*x_2 24.88/12.08 POL(A__U11(x_1, x_2)) = 2*x_1 + 2*x_2 24.88/12.08 POL(A__U12(x_1, x_2)) = x_1 + 2*x_2 24.88/12.08 POL(A__U21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + 2*x_4 24.88/12.08 POL(A__U22(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + 2*x_4 24.88/12.08 POL(A__U23(x_1, x_2, x_3, x_4)) = 2*x_1 + x_2 + x_3 + 2*x_4 24.88/12.08 POL(MARK(x_1)) = 2*x_1 24.88/12.08 POL(U11(x_1, x_2)) = 1 + x_1 + 2*x_2 24.88/12.08 POL(U12(x_1, x_2)) = 1 + x_1 + 2*x_2 24.88/12.08 POL(U21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + 2*x_4 24.88/12.08 POL(U22(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + 2*x_4 24.88/12.08 POL(U23(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + 2*x_4 24.88/12.08 POL(a__U11(x_1, x_2)) = 1 + x_1 + 2*x_2 24.88/12.08 POL(a__U12(x_1, x_2)) = 1 + x_1 + 2*x_2 24.88/12.08 POL(a__U21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + 2*x_4 24.88/12.08 POL(a__U22(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + 2*x_4 24.88/12.08 POL(a__U23(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + 2*x_4 24.88/12.08 POL(a__length(x_1)) = 1 + 2*x_1 24.88/12.08 POL(a__take(x_1, x_2)) = x_1 + x_2 24.88/12.08 POL(a__zeros) = 0 24.88/12.08 POL(cons(x_1, x_2)) = 2*x_1 + x_2 24.88/12.08 POL(length(x_1)) = 1 + 2*x_1 24.88/12.08 POL(mark(x_1)) = x_1 24.88/12.08 POL(nil) = 0 24.88/12.08 POL(s(x_1)) = x_1 24.88/12.08 POL(take(x_1, x_2)) = x_1 + x_2 24.88/12.08 POL(tt) = 0 24.88/12.08 POL(zeros) = 0 24.88/12.08 24.88/12.08 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (10) 24.88/12.08 Obligation: 24.88/12.08 Q DP problem: 24.88/12.08 The TRS P consists of the following rules: 24.88/12.08 24.88/12.08 A__U12(tt, L) -> A__LENGTH(mark(L)) 24.88/12.08 A__LENGTH(cons(N, L)) -> A__U11(tt, L) 24.88/12.08 A__U11(tt, L) -> A__U12(tt, L) 24.88/12.08 A__U12(tt, L) -> MARK(L) 24.88/12.08 MARK(U21(X1, X2, X3, X4)) -> A__U21(mark(X1), X2, X3, X4) 24.88/12.08 A__U21(tt, IL, M, N) -> A__U22(tt, IL, M, N) 24.88/12.08 A__U22(tt, IL, M, N) -> A__U23(tt, IL, M, N) 24.88/12.08 A__U23(tt, IL, M, N) -> MARK(N) 24.88/12.08 MARK(U21(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(U22(X1, X2, X3, X4)) -> A__U22(mark(X1), X2, X3, X4) 24.88/12.08 MARK(U22(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(U23(X1, X2, X3, X4)) -> A__U23(mark(X1), X2, X3, X4) 24.88/12.08 MARK(U23(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) 24.88/12.08 A__TAKE(s(M), cons(N, IL)) -> A__U21(tt, IL, M, N) 24.88/12.08 MARK(take(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(take(X1, X2)) -> MARK(X2) 24.88/12.08 MARK(cons(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(s(X)) -> MARK(X) 24.88/12.08 24.88/12.08 The TRS R consists of the following rules: 24.88/12.08 24.88/12.08 a__zeros -> cons(0, zeros) 24.88/12.08 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.08 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.08 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.08 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.08 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.08 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.08 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.08 mark(zeros) -> a__zeros 24.88/12.08 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.08 mark(length(X)) -> a__length(mark(X)) 24.88/12.08 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.08 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.08 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.08 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.08 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.08 mark(0) -> 0 24.88/12.08 mark(tt) -> tt 24.88/12.08 mark(s(X)) -> s(mark(X)) 24.88/12.08 mark(nil) -> nil 24.88/12.08 a__zeros -> zeros 24.88/12.08 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.08 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.08 a__length(X) -> length(X) 24.88/12.08 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.08 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.08 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.08 a__take(X1, X2) -> take(X1, X2) 24.88/12.08 24.88/12.08 The set Q consists of the following terms: 24.88/12.08 24.88/12.08 a__zeros 24.88/12.08 mark(zeros) 24.88/12.08 mark(U11(x0, x1)) 24.88/12.08 mark(U12(x0, x1)) 24.88/12.08 mark(length(x0)) 24.88/12.08 mark(U21(x0, x1, x2, x3)) 24.88/12.08 mark(U22(x0, x1, x2, x3)) 24.88/12.08 mark(U23(x0, x1, x2, x3)) 24.88/12.08 mark(take(x0, x1)) 24.88/12.08 mark(cons(x0, x1)) 24.88/12.08 mark(0) 24.88/12.08 mark(tt) 24.88/12.08 mark(s(x0)) 24.88/12.08 mark(nil) 24.88/12.08 a__U11(x0, x1) 24.88/12.08 a__U12(x0, x1) 24.88/12.08 a__length(x0) 24.88/12.08 a__U21(x0, x1, x2, x3) 24.88/12.08 a__U22(x0, x1, x2, x3) 24.88/12.08 a__U23(x0, x1, x2, x3) 24.88/12.08 a__take(x0, x1) 24.88/12.08 24.88/12.08 We have to consider all minimal (P,Q,R)-chains. 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (11) DependencyGraphProof (EQUIVALENT) 24.88/12.08 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (12) 24.88/12.08 Complex Obligation (AND) 24.88/12.08 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (13) 24.88/12.08 Obligation: 24.88/12.08 Q DP problem: 24.88/12.08 The TRS P consists of the following rules: 24.88/12.08 24.88/12.08 A__U21(tt, IL, M, N) -> A__U22(tt, IL, M, N) 24.88/12.08 A__U22(tt, IL, M, N) -> A__U23(tt, IL, M, N) 24.88/12.08 A__U23(tt, IL, M, N) -> MARK(N) 24.88/12.08 MARK(U21(X1, X2, X3, X4)) -> A__U21(mark(X1), X2, X3, X4) 24.88/12.08 MARK(U21(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(U22(X1, X2, X3, X4)) -> A__U22(mark(X1), X2, X3, X4) 24.88/12.08 MARK(U22(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(U23(X1, X2, X3, X4)) -> A__U23(mark(X1), X2, X3, X4) 24.88/12.08 MARK(U23(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) 24.88/12.08 A__TAKE(s(M), cons(N, IL)) -> A__U21(tt, IL, M, N) 24.88/12.08 MARK(take(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(take(X1, X2)) -> MARK(X2) 24.88/12.08 MARK(cons(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(s(X)) -> MARK(X) 24.88/12.08 24.88/12.08 The TRS R consists of the following rules: 24.88/12.08 24.88/12.08 a__zeros -> cons(0, zeros) 24.88/12.08 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.08 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.08 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.08 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.08 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.08 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.08 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.08 mark(zeros) -> a__zeros 24.88/12.08 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.08 mark(length(X)) -> a__length(mark(X)) 24.88/12.08 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.08 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.08 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.08 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.08 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.08 mark(0) -> 0 24.88/12.08 mark(tt) -> tt 24.88/12.08 mark(s(X)) -> s(mark(X)) 24.88/12.08 mark(nil) -> nil 24.88/12.08 a__zeros -> zeros 24.88/12.08 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.08 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.08 a__length(X) -> length(X) 24.88/12.08 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.08 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.08 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.08 a__take(X1, X2) -> take(X1, X2) 24.88/12.08 24.88/12.08 The set Q consists of the following terms: 24.88/12.08 24.88/12.08 a__zeros 24.88/12.08 mark(zeros) 24.88/12.08 mark(U11(x0, x1)) 24.88/12.08 mark(U12(x0, x1)) 24.88/12.08 mark(length(x0)) 24.88/12.08 mark(U21(x0, x1, x2, x3)) 24.88/12.08 mark(U22(x0, x1, x2, x3)) 24.88/12.08 mark(U23(x0, x1, x2, x3)) 24.88/12.08 mark(take(x0, x1)) 24.88/12.08 mark(cons(x0, x1)) 24.88/12.08 mark(0) 24.88/12.08 mark(tt) 24.88/12.08 mark(s(x0)) 24.88/12.08 mark(nil) 24.88/12.08 a__U11(x0, x1) 24.88/12.08 a__U12(x0, x1) 24.88/12.08 a__length(x0) 24.88/12.08 a__U21(x0, x1, x2, x3) 24.88/12.08 a__U22(x0, x1, x2, x3) 24.88/12.08 a__U23(x0, x1, x2, x3) 24.88/12.08 a__take(x0, x1) 24.88/12.08 24.88/12.08 We have to consider all minimal (P,Q,R)-chains. 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (14) MRRProof (EQUIVALENT) 24.88/12.08 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. 24.88/12.08 24.88/12.08 Strictly oriented dependency pairs: 24.88/12.08 24.88/12.08 A__U21(tt, IL, M, N) -> A__U22(tt, IL, M, N) 24.88/12.08 A__U22(tt, IL, M, N) -> A__U23(tt, IL, M, N) 24.88/12.08 MARK(U21(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(U22(X1, X2, X3, X4)) -> A__U22(mark(X1), X2, X3, X4) 24.88/12.08 MARK(U22(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(U23(X1, X2, X3, X4)) -> A__U23(mark(X1), X2, X3, X4) 24.88/12.08 MARK(U23(X1, X2, X3, X4)) -> MARK(X1) 24.88/12.08 MARK(take(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(take(X1, X2)) -> MARK(X2) 24.88/12.08 24.88/12.08 24.88/12.08 Used ordering: Polynomial interpretation [POLO]: 24.88/12.08 24.88/12.08 POL(0) = 0 24.88/12.08 POL(A__TAKE(x_1, x_2)) = 2 + 2*x_1 + x_2 24.88/12.08 POL(A__U21(x_1, x_2, x_3, x_4)) = 2 + 2*x_1 + x_2 + x_3 + 2*x_4 24.88/12.08 POL(A__U22(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + x_2 + x_3 + 2*x_4 24.88/12.08 POL(A__U23(x_1, x_2, x_3, x_4)) = 2*x_1 + x_2 + x_3 + 2*x_4 24.88/12.08 POL(MARK(x_1)) = 2*x_1 24.88/12.08 POL(U11(x_1, x_2)) = 2*x_1 + x_2 24.88/12.08 POL(U12(x_1, x_2)) = 2*x_1 + x_2 24.88/12.08 POL(U21(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + 2*x_3 + 2*x_4 24.88/12.08 POL(U22(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + 2*x_3 + 2*x_4 24.88/12.08 POL(U23(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + x_2 + 2*x_3 + 2*x_4 24.88/12.08 POL(a__U11(x_1, x_2)) = 2*x_1 + x_2 24.88/12.08 POL(a__U12(x_1, x_2)) = 2*x_1 + x_2 24.88/12.08 POL(a__U21(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + 2*x_3 + 2*x_4 24.88/12.08 POL(a__U22(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + 2*x_3 + 2*x_4 24.88/12.08 POL(a__U23(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + x_2 + 2*x_3 + 2*x_4 24.88/12.08 POL(a__length(x_1)) = x_1 24.88/12.08 POL(a__take(x_1, x_2)) = 1 + 2*x_1 + x_2 24.88/12.08 POL(a__zeros) = 0 24.88/12.08 POL(cons(x_1, x_2)) = 2*x_1 + x_2 24.88/12.08 POL(length(x_1)) = x_1 24.88/12.08 POL(mark(x_1)) = x_1 24.88/12.08 POL(nil) = 0 24.88/12.08 POL(s(x_1)) = x_1 24.88/12.08 POL(take(x_1, x_2)) = 1 + 2*x_1 + x_2 24.88/12.08 POL(tt) = 0 24.88/12.08 POL(zeros) = 0 24.88/12.08 24.88/12.08 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (15) 24.88/12.08 Obligation: 24.88/12.08 Q DP problem: 24.88/12.08 The TRS P consists of the following rules: 24.88/12.08 24.88/12.08 A__U23(tt, IL, M, N) -> MARK(N) 24.88/12.08 MARK(U21(X1, X2, X3, X4)) -> A__U21(mark(X1), X2, X3, X4) 24.88/12.08 MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) 24.88/12.08 A__TAKE(s(M), cons(N, IL)) -> A__U21(tt, IL, M, N) 24.88/12.08 MARK(cons(X1, X2)) -> MARK(X1) 24.88/12.08 MARK(s(X)) -> MARK(X) 24.88/12.08 24.88/12.08 The TRS R consists of the following rules: 24.88/12.08 24.88/12.08 a__zeros -> cons(0, zeros) 24.88/12.08 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.08 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.08 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.08 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.08 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.08 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.08 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.08 mark(zeros) -> a__zeros 24.88/12.08 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.08 mark(length(X)) -> a__length(mark(X)) 24.88/12.08 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.08 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.08 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.08 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.08 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.08 mark(0) -> 0 24.88/12.08 mark(tt) -> tt 24.88/12.08 mark(s(X)) -> s(mark(X)) 24.88/12.08 mark(nil) -> nil 24.88/12.08 a__zeros -> zeros 24.88/12.08 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.08 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.08 a__length(X) -> length(X) 24.88/12.08 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.08 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.08 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.08 a__take(X1, X2) -> take(X1, X2) 24.88/12.08 24.88/12.08 The set Q consists of the following terms: 24.88/12.08 24.88/12.08 a__zeros 24.88/12.08 mark(zeros) 24.88/12.08 mark(U11(x0, x1)) 24.88/12.08 mark(U12(x0, x1)) 24.88/12.08 mark(length(x0)) 24.88/12.08 mark(U21(x0, x1, x2, x3)) 24.88/12.08 mark(U22(x0, x1, x2, x3)) 24.88/12.08 mark(U23(x0, x1, x2, x3)) 24.88/12.08 mark(take(x0, x1)) 24.88/12.08 mark(cons(x0, x1)) 24.88/12.08 mark(0) 24.88/12.08 mark(tt) 24.88/12.08 mark(s(x0)) 24.88/12.08 mark(nil) 24.88/12.08 a__U11(x0, x1) 24.88/12.08 a__U12(x0, x1) 24.88/12.08 a__length(x0) 24.88/12.08 a__U21(x0, x1, x2, x3) 24.88/12.08 a__U22(x0, x1, x2, x3) 24.88/12.08 a__U23(x0, x1, x2, x3) 24.88/12.08 a__take(x0, x1) 24.88/12.08 24.88/12.08 We have to consider all minimal (P,Q,R)-chains. 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (16) DependencyGraphProof (EQUIVALENT) 24.88/12.08 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 24.88/12.08 ---------------------------------------- 24.88/12.08 24.88/12.08 (17) 24.88/12.08 Obligation: 24.88/12.08 Q DP problem: 24.88/12.08 The TRS P consists of the following rules: 24.88/12.08 24.88/12.08 MARK(s(X)) -> MARK(X) 24.88/12.08 MARK(cons(X1, X2)) -> MARK(X1) 24.88/12.08 24.88/12.08 The TRS R consists of the following rules: 24.88/12.08 24.88/12.08 a__zeros -> cons(0, zeros) 24.88/12.08 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.08 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.08 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.08 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.08 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.08 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.08 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.08 mark(zeros) -> a__zeros 24.88/12.08 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(0) -> 0 24.88/12.09 mark(tt) -> tt 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 mark(nil) -> nil 24.88/12.09 a__zeros -> zeros 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (18) UsableRulesProof (EQUIVALENT) 24.88/12.09 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. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (19) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 MARK(s(X)) -> MARK(X) 24.88/12.09 MARK(cons(X1, X2)) -> MARK(X1) 24.88/12.09 24.88/12.09 R is empty. 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (20) QReductionProof (EQUIVALENT) 24.88/12.09 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (21) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 MARK(s(X)) -> MARK(X) 24.88/12.09 MARK(cons(X1, X2)) -> MARK(X1) 24.88/12.09 24.88/12.09 R is empty. 24.88/12.09 Q is empty. 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (22) QDPSizeChangeProof (EQUIVALENT) 24.88/12.09 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. 24.88/12.09 24.88/12.09 From the DPs we obtained the following set of size-change graphs: 24.88/12.09 *MARK(s(X)) -> MARK(X) 24.88/12.09 The graph contains the following edges 1 > 1 24.88/12.09 24.88/12.09 24.88/12.09 *MARK(cons(X1, X2)) -> MARK(X1) 24.88/12.09 The graph contains the following edges 1 > 1 24.88/12.09 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (23) 24.88/12.09 YES 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (24) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__LENGTH(cons(N, L)) -> A__U11(tt, L) 24.88/12.09 A__U11(tt, L) -> A__U12(tt, L) 24.88/12.09 A__U12(tt, L) -> A__LENGTH(mark(L)) 24.88/12.09 24.88/12.09 The TRS R consists of the following rules: 24.88/12.09 24.88/12.09 a__zeros -> cons(0, zeros) 24.88/12.09 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.09 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.09 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.09 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.09 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.09 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.09 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.09 mark(zeros) -> a__zeros 24.88/12.09 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.09 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(0) -> 0 24.88/12.09 mark(tt) -> tt 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 mark(nil) -> nil 24.88/12.09 a__zeros -> zeros 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (25) TransformationProof (EQUIVALENT) 24.88/12.09 By narrowing [LPAR04] the rule A__U12(tt, L) -> A__LENGTH(mark(L)) at position [0] we obtained the following new rules [LPAR04]: 24.88/12.09 24.88/12.09 (A__U12(tt, zeros) -> A__LENGTH(a__zeros),A__U12(tt, zeros) -> A__LENGTH(a__zeros)) 24.88/12.09 (A__U12(tt, U11(x0, x1)) -> A__LENGTH(a__U11(mark(x0), x1)),A__U12(tt, U11(x0, x1)) -> A__LENGTH(a__U11(mark(x0), x1))) 24.88/12.09 (A__U12(tt, U12(x0, x1)) -> A__LENGTH(a__U12(mark(x0), x1)),A__U12(tt, U12(x0, x1)) -> A__LENGTH(a__U12(mark(x0), x1))) 24.88/12.09 (A__U12(tt, length(x0)) -> A__LENGTH(a__length(mark(x0))),A__U12(tt, length(x0)) -> A__LENGTH(a__length(mark(x0)))) 24.88/12.09 (A__U12(tt, U21(x0, x1, x2, x3)) -> A__LENGTH(a__U21(mark(x0), x1, x2, x3)),A__U12(tt, U21(x0, x1, x2, x3)) -> A__LENGTH(a__U21(mark(x0), x1, x2, x3))) 24.88/12.09 (A__U12(tt, U22(x0, x1, x2, x3)) -> A__LENGTH(a__U22(mark(x0), x1, x2, x3)),A__U12(tt, U22(x0, x1, x2, x3)) -> A__LENGTH(a__U22(mark(x0), x1, x2, x3))) 24.88/12.09 (A__U12(tt, U23(x0, x1, x2, x3)) -> A__LENGTH(a__U23(mark(x0), x1, x2, x3)),A__U12(tt, U23(x0, x1, x2, x3)) -> A__LENGTH(a__U23(mark(x0), x1, x2, x3))) 24.88/12.09 (A__U12(tt, take(x0, x1)) -> A__LENGTH(a__take(mark(x0), mark(x1))),A__U12(tt, take(x0, x1)) -> A__LENGTH(a__take(mark(x0), mark(x1)))) 24.88/12.09 (A__U12(tt, cons(x0, x1)) -> A__LENGTH(cons(mark(x0), x1)),A__U12(tt, cons(x0, x1)) -> A__LENGTH(cons(mark(x0), x1))) 24.88/12.09 (A__U12(tt, 0) -> A__LENGTH(0),A__U12(tt, 0) -> A__LENGTH(0)) 24.88/12.09 (A__U12(tt, tt) -> A__LENGTH(tt),A__U12(tt, tt) -> A__LENGTH(tt)) 24.88/12.09 (A__U12(tt, s(x0)) -> A__LENGTH(s(mark(x0))),A__U12(tt, s(x0)) -> A__LENGTH(s(mark(x0)))) 24.88/12.09 (A__U12(tt, nil) -> A__LENGTH(nil),A__U12(tt, nil) -> A__LENGTH(nil)) 24.88/12.09 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (26) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__LENGTH(cons(N, L)) -> A__U11(tt, L) 24.88/12.09 A__U11(tt, L) -> A__U12(tt, L) 24.88/12.09 A__U12(tt, zeros) -> A__LENGTH(a__zeros) 24.88/12.09 A__U12(tt, U11(x0, x1)) -> A__LENGTH(a__U11(mark(x0), x1)) 24.88/12.09 A__U12(tt, U12(x0, x1)) -> A__LENGTH(a__U12(mark(x0), x1)) 24.88/12.09 A__U12(tt, length(x0)) -> A__LENGTH(a__length(mark(x0))) 24.88/12.09 A__U12(tt, U21(x0, x1, x2, x3)) -> A__LENGTH(a__U21(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, U22(x0, x1, x2, x3)) -> A__LENGTH(a__U22(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, U23(x0, x1, x2, x3)) -> A__LENGTH(a__U23(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, take(x0, x1)) -> A__LENGTH(a__take(mark(x0), mark(x1))) 24.88/12.09 A__U12(tt, cons(x0, x1)) -> A__LENGTH(cons(mark(x0), x1)) 24.88/12.09 A__U12(tt, 0) -> A__LENGTH(0) 24.88/12.09 A__U12(tt, tt) -> A__LENGTH(tt) 24.88/12.09 A__U12(tt, s(x0)) -> A__LENGTH(s(mark(x0))) 24.88/12.09 A__U12(tt, nil) -> A__LENGTH(nil) 24.88/12.09 24.88/12.09 The TRS R consists of the following rules: 24.88/12.09 24.88/12.09 a__zeros -> cons(0, zeros) 24.88/12.09 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.09 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.09 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.09 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.09 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.09 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.09 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.09 mark(zeros) -> a__zeros 24.88/12.09 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.09 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(0) -> 0 24.88/12.09 mark(tt) -> tt 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 mark(nil) -> nil 24.88/12.09 a__zeros -> zeros 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (27) DependencyGraphProof (EQUIVALENT) 24.88/12.09 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (28) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__U11(tt, L) -> A__U12(tt, L) 24.88/12.09 A__U12(tt, zeros) -> A__LENGTH(a__zeros) 24.88/12.09 A__LENGTH(cons(N, L)) -> A__U11(tt, L) 24.88/12.09 A__U12(tt, U11(x0, x1)) -> A__LENGTH(a__U11(mark(x0), x1)) 24.88/12.09 A__U12(tt, U12(x0, x1)) -> A__LENGTH(a__U12(mark(x0), x1)) 24.88/12.09 A__U12(tt, length(x0)) -> A__LENGTH(a__length(mark(x0))) 24.88/12.09 A__U12(tt, U21(x0, x1, x2, x3)) -> A__LENGTH(a__U21(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, U22(x0, x1, x2, x3)) -> A__LENGTH(a__U22(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, U23(x0, x1, x2, x3)) -> A__LENGTH(a__U23(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, take(x0, x1)) -> A__LENGTH(a__take(mark(x0), mark(x1))) 24.88/12.09 A__U12(tt, cons(x0, x1)) -> A__LENGTH(cons(mark(x0), x1)) 24.88/12.09 24.88/12.09 The TRS R consists of the following rules: 24.88/12.09 24.88/12.09 a__zeros -> cons(0, zeros) 24.88/12.09 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.09 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.09 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.09 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.09 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.09 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.09 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.09 mark(zeros) -> a__zeros 24.88/12.09 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.09 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(0) -> 0 24.88/12.09 mark(tt) -> tt 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 mark(nil) -> nil 24.88/12.09 a__zeros -> zeros 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (29) TransformationProof (EQUIVALENT) 24.88/12.09 By narrowing [LPAR04] the rule A__U12(tt, zeros) -> A__LENGTH(a__zeros) at position [0] we obtained the following new rules [LPAR04]: 24.88/12.09 24.88/12.09 (A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)),A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros))) 24.88/12.09 (A__U12(tt, zeros) -> A__LENGTH(zeros),A__U12(tt, zeros) -> A__LENGTH(zeros)) 24.88/12.09 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (30) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__U11(tt, L) -> A__U12(tt, L) 24.88/12.09 A__LENGTH(cons(N, L)) -> A__U11(tt, L) 24.88/12.09 A__U12(tt, U11(x0, x1)) -> A__LENGTH(a__U11(mark(x0), x1)) 24.88/12.09 A__U12(tt, U12(x0, x1)) -> A__LENGTH(a__U12(mark(x0), x1)) 24.88/12.09 A__U12(tt, length(x0)) -> A__LENGTH(a__length(mark(x0))) 24.88/12.09 A__U12(tt, U21(x0, x1, x2, x3)) -> A__LENGTH(a__U21(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, U22(x0, x1, x2, x3)) -> A__LENGTH(a__U22(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, U23(x0, x1, x2, x3)) -> A__LENGTH(a__U23(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, take(x0, x1)) -> A__LENGTH(a__take(mark(x0), mark(x1))) 24.88/12.09 A__U12(tt, cons(x0, x1)) -> A__LENGTH(cons(mark(x0), x1)) 24.88/12.09 A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) 24.88/12.09 A__U12(tt, zeros) -> A__LENGTH(zeros) 24.88/12.09 24.88/12.09 The TRS R consists of the following rules: 24.88/12.09 24.88/12.09 a__zeros -> cons(0, zeros) 24.88/12.09 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.09 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.09 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.09 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.09 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.09 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.09 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.09 mark(zeros) -> a__zeros 24.88/12.09 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.09 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(0) -> 0 24.88/12.09 mark(tt) -> tt 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 mark(nil) -> nil 24.88/12.09 a__zeros -> zeros 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (31) DependencyGraphProof (EQUIVALENT) 24.88/12.09 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (32) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__U12(tt, U11(x0, x1)) -> A__LENGTH(a__U11(mark(x0), x1)) 24.88/12.09 A__LENGTH(cons(N, L)) -> A__U11(tt, L) 24.88/12.09 A__U11(tt, L) -> A__U12(tt, L) 24.88/12.09 A__U12(tt, U12(x0, x1)) -> A__LENGTH(a__U12(mark(x0), x1)) 24.88/12.09 A__U12(tt, length(x0)) -> A__LENGTH(a__length(mark(x0))) 24.88/12.09 A__U12(tt, U21(x0, x1, x2, x3)) -> A__LENGTH(a__U21(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, U22(x0, x1, x2, x3)) -> A__LENGTH(a__U22(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, U23(x0, x1, x2, x3)) -> A__LENGTH(a__U23(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, take(x0, x1)) -> A__LENGTH(a__take(mark(x0), mark(x1))) 24.88/12.09 A__U12(tt, cons(x0, x1)) -> A__LENGTH(cons(mark(x0), x1)) 24.88/12.09 A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) 24.88/12.09 24.88/12.09 The TRS R consists of the following rules: 24.88/12.09 24.88/12.09 a__zeros -> cons(0, zeros) 24.88/12.09 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.09 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.09 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.09 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.09 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.09 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.09 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.09 mark(zeros) -> a__zeros 24.88/12.09 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.09 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(0) -> 0 24.88/12.09 mark(tt) -> tt 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 mark(nil) -> nil 24.88/12.09 a__zeros -> zeros 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (33) TransformationProof (EQUIVALENT) 24.88/12.09 By forward instantiating [JAR06] the rule A__U11(tt, L) -> A__U12(tt, L) we obtained the following new rules [LPAR04]: 24.88/12.09 24.88/12.09 (A__U11(tt, U11(y_0, y_1)) -> A__U12(tt, U11(y_0, y_1)),A__U11(tt, U11(y_0, y_1)) -> A__U12(tt, U11(y_0, y_1))) 24.88/12.09 (A__U11(tt, U12(y_0, y_1)) -> A__U12(tt, U12(y_0, y_1)),A__U11(tt, U12(y_0, y_1)) -> A__U12(tt, U12(y_0, y_1))) 24.88/12.09 (A__U11(tt, length(y_0)) -> A__U12(tt, length(y_0)),A__U11(tt, length(y_0)) -> A__U12(tt, length(y_0))) 24.88/12.09 (A__U11(tt, U21(y_0, y_1, y_2, y_3)) -> A__U12(tt, U21(y_0, y_1, y_2, y_3)),A__U11(tt, U21(y_0, y_1, y_2, y_3)) -> A__U12(tt, U21(y_0, y_1, y_2, y_3))) 24.88/12.09 (A__U11(tt, U22(y_0, y_1, y_2, y_3)) -> A__U12(tt, U22(y_0, y_1, y_2, y_3)),A__U11(tt, U22(y_0, y_1, y_2, y_3)) -> A__U12(tt, U22(y_0, y_1, y_2, y_3))) 24.88/12.09 (A__U11(tt, U23(y_0, y_1, y_2, y_3)) -> A__U12(tt, U23(y_0, y_1, y_2, y_3)),A__U11(tt, U23(y_0, y_1, y_2, y_3)) -> A__U12(tt, U23(y_0, y_1, y_2, y_3))) 24.88/12.09 (A__U11(tt, take(y_0, y_1)) -> A__U12(tt, take(y_0, y_1)),A__U11(tt, take(y_0, y_1)) -> A__U12(tt, take(y_0, y_1))) 24.88/12.09 (A__U11(tt, cons(y_0, y_1)) -> A__U12(tt, cons(y_0, y_1)),A__U11(tt, cons(y_0, y_1)) -> A__U12(tt, cons(y_0, y_1))) 24.88/12.09 (A__U11(tt, zeros) -> A__U12(tt, zeros),A__U11(tt, zeros) -> A__U12(tt, zeros)) 24.88/12.09 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (34) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__U12(tt, U11(x0, x1)) -> A__LENGTH(a__U11(mark(x0), x1)) 24.88/12.09 A__LENGTH(cons(N, L)) -> A__U11(tt, L) 24.88/12.09 A__U12(tt, U12(x0, x1)) -> A__LENGTH(a__U12(mark(x0), x1)) 24.88/12.09 A__U12(tt, length(x0)) -> A__LENGTH(a__length(mark(x0))) 24.88/12.09 A__U12(tt, U21(x0, x1, x2, x3)) -> A__LENGTH(a__U21(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, U22(x0, x1, x2, x3)) -> A__LENGTH(a__U22(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, U23(x0, x1, x2, x3)) -> A__LENGTH(a__U23(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, take(x0, x1)) -> A__LENGTH(a__take(mark(x0), mark(x1))) 24.88/12.09 A__U12(tt, cons(x0, x1)) -> A__LENGTH(cons(mark(x0), x1)) 24.88/12.09 A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) 24.88/12.09 A__U11(tt, U11(y_0, y_1)) -> A__U12(tt, U11(y_0, y_1)) 24.88/12.09 A__U11(tt, U12(y_0, y_1)) -> A__U12(tt, U12(y_0, y_1)) 24.88/12.09 A__U11(tt, length(y_0)) -> A__U12(tt, length(y_0)) 24.88/12.09 A__U11(tt, U21(y_0, y_1, y_2, y_3)) -> A__U12(tt, U21(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U22(y_0, y_1, y_2, y_3)) -> A__U12(tt, U22(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U23(y_0, y_1, y_2, y_3)) -> A__U12(tt, U23(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, take(y_0, y_1)) -> A__U12(tt, take(y_0, y_1)) 24.88/12.09 A__U11(tt, cons(y_0, y_1)) -> A__U12(tt, cons(y_0, y_1)) 24.88/12.09 A__U11(tt, zeros) -> A__U12(tt, zeros) 24.88/12.09 24.88/12.09 The TRS R consists of the following rules: 24.88/12.09 24.88/12.09 a__zeros -> cons(0, zeros) 24.88/12.09 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.09 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.09 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.09 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.09 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.09 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.09 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.09 mark(zeros) -> a__zeros 24.88/12.09 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.09 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(0) -> 0 24.88/12.09 mark(tt) -> tt 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 mark(nil) -> nil 24.88/12.09 a__zeros -> zeros 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (35) TransformationProof (EQUIVALENT) 24.88/12.09 By forward instantiating [JAR06] the rule A__LENGTH(cons(N, L)) -> A__U11(tt, L) we obtained the following new rules [LPAR04]: 24.88/12.09 24.88/12.09 (A__LENGTH(cons(x0, U11(y_0, y_1))) -> A__U11(tt, U11(y_0, y_1)),A__LENGTH(cons(x0, U11(y_0, y_1))) -> A__U11(tt, U11(y_0, y_1))) 24.88/12.09 (A__LENGTH(cons(x0, U12(y_0, y_1))) -> A__U11(tt, U12(y_0, y_1)),A__LENGTH(cons(x0, U12(y_0, y_1))) -> A__U11(tt, U12(y_0, y_1))) 24.88/12.09 (A__LENGTH(cons(x0, length(y_0))) -> A__U11(tt, length(y_0)),A__LENGTH(cons(x0, length(y_0))) -> A__U11(tt, length(y_0))) 24.88/12.09 (A__LENGTH(cons(x0, U21(y_0, y_1, y_2, y_3))) -> A__U11(tt, U21(y_0, y_1, y_2, y_3)),A__LENGTH(cons(x0, U21(y_0, y_1, y_2, y_3))) -> A__U11(tt, U21(y_0, y_1, y_2, y_3))) 24.88/12.09 (A__LENGTH(cons(x0, U22(y_0, y_1, y_2, y_3))) -> A__U11(tt, U22(y_0, y_1, y_2, y_3)),A__LENGTH(cons(x0, U22(y_0, y_1, y_2, y_3))) -> A__U11(tt, U22(y_0, y_1, y_2, y_3))) 24.88/12.09 (A__LENGTH(cons(x0, U23(y_0, y_1, y_2, y_3))) -> A__U11(tt, U23(y_0, y_1, y_2, y_3)),A__LENGTH(cons(x0, U23(y_0, y_1, y_2, y_3))) -> A__U11(tt, U23(y_0, y_1, y_2, y_3))) 24.88/12.09 (A__LENGTH(cons(x0, take(y_0, y_1))) -> A__U11(tt, take(y_0, y_1)),A__LENGTH(cons(x0, take(y_0, y_1))) -> A__U11(tt, take(y_0, y_1))) 24.88/12.09 (A__LENGTH(cons(x0, cons(y_0, y_1))) -> A__U11(tt, cons(y_0, y_1)),A__LENGTH(cons(x0, cons(y_0, y_1))) -> A__U11(tt, cons(y_0, y_1))) 24.88/12.09 (A__LENGTH(cons(x0, zeros)) -> A__U11(tt, zeros),A__LENGTH(cons(x0, zeros)) -> A__U11(tt, zeros)) 24.88/12.09 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (36) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__U12(tt, U11(x0, x1)) -> A__LENGTH(a__U11(mark(x0), x1)) 24.88/12.09 A__U12(tt, U12(x0, x1)) -> A__LENGTH(a__U12(mark(x0), x1)) 24.88/12.09 A__U12(tt, length(x0)) -> A__LENGTH(a__length(mark(x0))) 24.88/12.09 A__U12(tt, U21(x0, x1, x2, x3)) -> A__LENGTH(a__U21(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, U22(x0, x1, x2, x3)) -> A__LENGTH(a__U22(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, U23(x0, x1, x2, x3)) -> A__LENGTH(a__U23(mark(x0), x1, x2, x3)) 24.88/12.09 A__U12(tt, take(x0, x1)) -> A__LENGTH(a__take(mark(x0), mark(x1))) 24.88/12.09 A__U12(tt, cons(x0, x1)) -> A__LENGTH(cons(mark(x0), x1)) 24.88/12.09 A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) 24.88/12.09 A__U11(tt, U11(y_0, y_1)) -> A__U12(tt, U11(y_0, y_1)) 24.88/12.09 A__U11(tt, U12(y_0, y_1)) -> A__U12(tt, U12(y_0, y_1)) 24.88/12.09 A__U11(tt, length(y_0)) -> A__U12(tt, length(y_0)) 24.88/12.09 A__U11(tt, U21(y_0, y_1, y_2, y_3)) -> A__U12(tt, U21(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U22(y_0, y_1, y_2, y_3)) -> A__U12(tt, U22(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U23(y_0, y_1, y_2, y_3)) -> A__U12(tt, U23(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, take(y_0, y_1)) -> A__U12(tt, take(y_0, y_1)) 24.88/12.09 A__U11(tt, cons(y_0, y_1)) -> A__U12(tt, cons(y_0, y_1)) 24.88/12.09 A__U11(tt, zeros) -> A__U12(tt, zeros) 24.88/12.09 A__LENGTH(cons(x0, U11(y_0, y_1))) -> A__U11(tt, U11(y_0, y_1)) 24.88/12.09 A__LENGTH(cons(x0, U12(y_0, y_1))) -> A__U11(tt, U12(y_0, y_1)) 24.88/12.09 A__LENGTH(cons(x0, length(y_0))) -> A__U11(tt, length(y_0)) 24.88/12.09 A__LENGTH(cons(x0, U21(y_0, y_1, y_2, y_3))) -> A__U11(tt, U21(y_0, y_1, y_2, y_3)) 24.88/12.09 A__LENGTH(cons(x0, U22(y_0, y_1, y_2, y_3))) -> A__U11(tt, U22(y_0, y_1, y_2, y_3)) 24.88/12.09 A__LENGTH(cons(x0, U23(y_0, y_1, y_2, y_3))) -> A__U11(tt, U23(y_0, y_1, y_2, y_3)) 24.88/12.09 A__LENGTH(cons(x0, take(y_0, y_1))) -> A__U11(tt, take(y_0, y_1)) 24.88/12.09 A__LENGTH(cons(x0, cons(y_0, y_1))) -> A__U11(tt, cons(y_0, y_1)) 24.88/12.09 A__LENGTH(cons(x0, zeros)) -> A__U11(tt, zeros) 24.88/12.09 24.88/12.09 The TRS R consists of the following rules: 24.88/12.09 24.88/12.09 a__zeros -> cons(0, zeros) 24.88/12.09 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.09 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.09 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.09 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.09 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.09 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.09 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.09 mark(zeros) -> a__zeros 24.88/12.09 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.09 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(0) -> 0 24.88/12.09 mark(tt) -> tt 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 mark(nil) -> nil 24.88/12.09 a__zeros -> zeros 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (37) DependencyGraphProof (EQUIVALENT) 24.88/12.09 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (38) 24.88/12.09 Complex Obligation (AND) 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (39) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__LENGTH(cons(x0, zeros)) -> A__U11(tt, zeros) 24.88/12.09 A__U11(tt, zeros) -> A__U12(tt, zeros) 24.88/12.09 A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) 24.88/12.09 24.88/12.09 The TRS R consists of the following rules: 24.88/12.09 24.88/12.09 a__zeros -> cons(0, zeros) 24.88/12.09 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.09 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.09 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.09 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.09 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.09 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.09 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.09 mark(zeros) -> a__zeros 24.88/12.09 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.09 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(0) -> 0 24.88/12.09 mark(tt) -> tt 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 mark(nil) -> nil 24.88/12.09 a__zeros -> zeros 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (40) UsableRulesProof (EQUIVALENT) 24.88/12.09 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. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (41) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__LENGTH(cons(x0, zeros)) -> A__U11(tt, zeros) 24.88/12.09 A__U11(tt, zeros) -> A__U12(tt, zeros) 24.88/12.09 A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) 24.88/12.09 24.88/12.09 R is empty. 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (42) QReductionProof (EQUIVALENT) 24.88/12.09 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (43) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__LENGTH(cons(x0, zeros)) -> A__U11(tt, zeros) 24.88/12.09 A__U11(tt, zeros) -> A__U12(tt, zeros) 24.88/12.09 A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) 24.88/12.09 24.88/12.09 R is empty. 24.88/12.09 Q is empty. 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (44) TransformationProof (EQUIVALENT) 24.88/12.09 By instantiating [LPAR04] the rule A__LENGTH(cons(x0, zeros)) -> A__U11(tt, zeros) we obtained the following new rules [LPAR04]: 24.88/12.09 24.88/12.09 (A__LENGTH(cons(0, zeros)) -> A__U11(tt, zeros),A__LENGTH(cons(0, zeros)) -> A__U11(tt, zeros)) 24.88/12.09 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (45) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__U11(tt, zeros) -> A__U12(tt, zeros) 24.88/12.09 A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) 24.88/12.09 A__LENGTH(cons(0, zeros)) -> A__U11(tt, zeros) 24.88/12.09 24.88/12.09 R is empty. 24.88/12.09 Q is empty. 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (46) NonTerminationLoopProof (COMPLETE) 24.88/12.09 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 24.88/12.09 Found a loop by narrowing to the left: 24.88/12.09 24.88/12.09 s = A__U12(tt, zeros) evaluates to t =A__U12(tt, zeros) 24.88/12.09 24.88/12.09 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 24.88/12.09 * Matcher: [ ] 24.88/12.09 * Semiunifier: [ ] 24.88/12.09 24.88/12.09 -------------------------------------------------------------------------------- 24.88/12.09 Rewriting sequence 24.88/12.09 24.88/12.09 A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) 24.88/12.09 with rule A__U12(tt, zeros) -> A__LENGTH(cons(0, zeros)) at position [] and matcher [ ] 24.88/12.09 24.88/12.09 A__LENGTH(cons(0, zeros)) -> A__U11(tt, zeros) 24.88/12.09 with rule A__LENGTH(cons(0, zeros)) -> A__U11(tt, zeros) at position [] and matcher [ ] 24.88/12.09 24.88/12.09 A__U11(tt, zeros) -> A__U12(tt, zeros) 24.88/12.09 with rule A__U11(tt, zeros) -> A__U12(tt, zeros) 24.88/12.09 24.88/12.09 Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence 24.88/12.09 24.88/12.09 24.88/12.09 All these steps are and every following step will be a correct step w.r.t to Q. 24.88/12.09 24.88/12.09 24.88/12.09 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (47) 24.88/12.09 NO 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (48) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__LENGTH(cons(x0, U11(y_0, y_1))) -> A__U11(tt, U11(y_0, y_1)) 24.88/12.09 A__U11(tt, U11(y_0, y_1)) -> A__U12(tt, U11(y_0, y_1)) 24.88/12.09 A__U12(tt, U11(x0, x1)) -> A__LENGTH(a__U11(mark(x0), x1)) 24.88/12.09 A__LENGTH(cons(x0, U12(y_0, y_1))) -> A__U11(tt, U12(y_0, y_1)) 24.88/12.09 A__U11(tt, U12(y_0, y_1)) -> A__U12(tt, U12(y_0, y_1)) 24.88/12.09 A__U12(tt, U12(x0, x1)) -> A__LENGTH(a__U12(mark(x0), x1)) 24.88/12.09 A__LENGTH(cons(x0, length(y_0))) -> A__U11(tt, length(y_0)) 24.88/12.09 A__U11(tt, length(y_0)) -> A__U12(tt, length(y_0)) 24.88/12.09 A__U12(tt, length(x0)) -> A__LENGTH(a__length(mark(x0))) 24.88/12.09 A__LENGTH(cons(x0, U21(y_0, y_1, y_2, y_3))) -> A__U11(tt, U21(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U21(y_0, y_1, y_2, y_3)) -> A__U12(tt, U21(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U12(tt, U21(x0, x1, x2, x3)) -> A__LENGTH(a__U21(mark(x0), x1, x2, x3)) 24.88/12.09 A__LENGTH(cons(x0, U22(y_0, y_1, y_2, y_3))) -> A__U11(tt, U22(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U22(y_0, y_1, y_2, y_3)) -> A__U12(tt, U22(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U12(tt, U22(x0, x1, x2, x3)) -> A__LENGTH(a__U22(mark(x0), x1, x2, x3)) 24.88/12.09 A__LENGTH(cons(x0, U23(y_0, y_1, y_2, y_3))) -> A__U11(tt, U23(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U23(y_0, y_1, y_2, y_3)) -> A__U12(tt, U23(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U12(tt, U23(x0, x1, x2, x3)) -> A__LENGTH(a__U23(mark(x0), x1, x2, x3)) 24.88/12.09 A__LENGTH(cons(x0, take(y_0, y_1))) -> A__U11(tt, take(y_0, y_1)) 24.88/12.09 A__U11(tt, take(y_0, y_1)) -> A__U12(tt, take(y_0, y_1)) 24.88/12.09 A__U12(tt, take(x0, x1)) -> A__LENGTH(a__take(mark(x0), mark(x1))) 24.88/12.09 A__LENGTH(cons(x0, cons(y_0, y_1))) -> A__U11(tt, cons(y_0, y_1)) 24.88/12.09 A__U11(tt, cons(y_0, y_1)) -> A__U12(tt, cons(y_0, y_1)) 24.88/12.09 A__U12(tt, cons(x0, x1)) -> A__LENGTH(cons(mark(x0), x1)) 24.88/12.09 24.88/12.09 The TRS R consists of the following rules: 24.88/12.09 24.88/12.09 a__zeros -> cons(0, zeros) 24.88/12.09 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.09 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.09 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.09 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.09 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.09 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.09 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.09 mark(zeros) -> a__zeros 24.88/12.09 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.09 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(0) -> 0 24.88/12.09 mark(tt) -> tt 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 mark(nil) -> nil 24.88/12.09 a__zeros -> zeros 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (49) QDPOrderProof (EQUIVALENT) 24.88/12.09 We use the reduction pair processor [LPAR04,JAR06]. 24.88/12.09 24.88/12.09 24.88/12.09 The following pairs can be oriented strictly and are deleted. 24.88/12.09 24.88/12.09 A__U12(tt, U11(x0, x1)) -> A__LENGTH(a__U11(mark(x0), x1)) 24.88/12.09 A__U12(tt, U12(x0, x1)) -> A__LENGTH(a__U12(mark(x0), x1)) 24.88/12.09 A__U12(tt, length(x0)) -> A__LENGTH(a__length(mark(x0))) 24.88/12.09 The remaining pairs can at least be oriented weakly. 24.88/12.09 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 24.88/12.09 24.88/12.09 POL( A__LENGTH_1(x_1) ) = x_1 24.88/12.09 POL( a__U11_2(x_1, x_2) ) = max{0, -2} 24.88/12.09 POL( a__U12_2(x_1, x_2) ) = max{0, -2} 24.88/12.09 POL( a__U21_4(x_1, ..., x_4) ) = 2 24.88/12.09 POL( a__U22_4(x_1, ..., x_4) ) = 2 24.88/12.09 POL( a__U23_4(x_1, ..., x_4) ) = 2 24.88/12.09 POL( a__length_1(x_1) ) = max{0, -2} 24.88/12.09 POL( a__take_2(x_1, x_2) ) = 2 24.88/12.09 POL( cons_2(x_1, x_2) ) = 2 24.88/12.09 POL( s_1(x_1) ) = max{0, -2} 24.88/12.09 POL( mark_1(x_1) ) = 2 24.88/12.09 POL( zeros ) = 0 24.88/12.09 POL( a__zeros ) = 0 24.88/12.09 POL( U11_2(x_1, x_2) ) = 0 24.88/12.09 POL( U12_2(x_1, x_2) ) = 0 24.88/12.09 POL( length_1(x_1) ) = 0 24.88/12.09 POL( U21_4(x_1, ..., x_4) ) = 0 24.88/12.09 POL( U22_4(x_1, ..., x_4) ) = 2 24.88/12.09 POL( U23_4(x_1, ..., x_4) ) = 0 24.88/12.09 POL( take_2(x_1, x_2) ) = 0 24.88/12.09 POL( 0 ) = 0 24.88/12.09 POL( tt ) = 0 24.88/12.09 POL( nil ) = 1 24.88/12.09 POL( A__U11_2(x_1, x_2) ) = 2x_1 + 2 24.88/12.09 POL( A__U12_2(x_1, x_2) ) = 2x_1 + 2 24.88/12.09 24.88/12.09 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 24.88/12.09 24.88/12.09 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.09 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (50) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__LENGTH(cons(x0, U11(y_0, y_1))) -> A__U11(tt, U11(y_0, y_1)) 24.88/12.09 A__U11(tt, U11(y_0, y_1)) -> A__U12(tt, U11(y_0, y_1)) 24.88/12.09 A__LENGTH(cons(x0, U12(y_0, y_1))) -> A__U11(tt, U12(y_0, y_1)) 24.88/12.09 A__U11(tt, U12(y_0, y_1)) -> A__U12(tt, U12(y_0, y_1)) 24.88/12.09 A__LENGTH(cons(x0, length(y_0))) -> A__U11(tt, length(y_0)) 24.88/12.09 A__U11(tt, length(y_0)) -> A__U12(tt, length(y_0)) 24.88/12.09 A__LENGTH(cons(x0, U21(y_0, y_1, y_2, y_3))) -> A__U11(tt, U21(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U21(y_0, y_1, y_2, y_3)) -> A__U12(tt, U21(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U12(tt, U21(x0, x1, x2, x3)) -> A__LENGTH(a__U21(mark(x0), x1, x2, x3)) 24.88/12.09 A__LENGTH(cons(x0, U22(y_0, y_1, y_2, y_3))) -> A__U11(tt, U22(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U22(y_0, y_1, y_2, y_3)) -> A__U12(tt, U22(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U12(tt, U22(x0, x1, x2, x3)) -> A__LENGTH(a__U22(mark(x0), x1, x2, x3)) 24.88/12.09 A__LENGTH(cons(x0, U23(y_0, y_1, y_2, y_3))) -> A__U11(tt, U23(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U23(y_0, y_1, y_2, y_3)) -> A__U12(tt, U23(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U12(tt, U23(x0, x1, x2, x3)) -> A__LENGTH(a__U23(mark(x0), x1, x2, x3)) 24.88/12.09 A__LENGTH(cons(x0, take(y_0, y_1))) -> A__U11(tt, take(y_0, y_1)) 24.88/12.09 A__U11(tt, take(y_0, y_1)) -> A__U12(tt, take(y_0, y_1)) 24.88/12.09 A__U12(tt, take(x0, x1)) -> A__LENGTH(a__take(mark(x0), mark(x1))) 24.88/12.09 A__LENGTH(cons(x0, cons(y_0, y_1))) -> A__U11(tt, cons(y_0, y_1)) 24.88/12.09 A__U11(tt, cons(y_0, y_1)) -> A__U12(tt, cons(y_0, y_1)) 24.88/12.09 A__U12(tt, cons(x0, x1)) -> A__LENGTH(cons(mark(x0), x1)) 24.88/12.09 24.88/12.09 The TRS R consists of the following rules: 24.88/12.09 24.88/12.09 a__zeros -> cons(0, zeros) 24.88/12.09 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.09 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.09 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.09 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.09 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.09 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.09 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.09 mark(zeros) -> a__zeros 24.88/12.09 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.09 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(0) -> 0 24.88/12.09 mark(tt) -> tt 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 mark(nil) -> nil 24.88/12.09 a__zeros -> zeros 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (51) DependencyGraphProof (EQUIVALENT) 24.88/12.09 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 24.88/12.09 ---------------------------------------- 24.88/12.09 24.88/12.09 (52) 24.88/12.09 Obligation: 24.88/12.09 Q DP problem: 24.88/12.09 The TRS P consists of the following rules: 24.88/12.09 24.88/12.09 A__LENGTH(cons(x0, U21(y_0, y_1, y_2, y_3))) -> A__U11(tt, U21(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U21(y_0, y_1, y_2, y_3)) -> A__U12(tt, U21(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U12(tt, U21(x0, x1, x2, x3)) -> A__LENGTH(a__U21(mark(x0), x1, x2, x3)) 24.88/12.09 A__LENGTH(cons(x0, U22(y_0, y_1, y_2, y_3))) -> A__U11(tt, U22(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U22(y_0, y_1, y_2, y_3)) -> A__U12(tt, U22(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U12(tt, U22(x0, x1, x2, x3)) -> A__LENGTH(a__U22(mark(x0), x1, x2, x3)) 24.88/12.09 A__LENGTH(cons(x0, U23(y_0, y_1, y_2, y_3))) -> A__U11(tt, U23(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U11(tt, U23(y_0, y_1, y_2, y_3)) -> A__U12(tt, U23(y_0, y_1, y_2, y_3)) 24.88/12.09 A__U12(tt, U23(x0, x1, x2, x3)) -> A__LENGTH(a__U23(mark(x0), x1, x2, x3)) 24.88/12.09 A__LENGTH(cons(x0, take(y_0, y_1))) -> A__U11(tt, take(y_0, y_1)) 24.88/12.09 A__U11(tt, take(y_0, y_1)) -> A__U12(tt, take(y_0, y_1)) 24.88/12.09 A__U12(tt, take(x0, x1)) -> A__LENGTH(a__take(mark(x0), mark(x1))) 24.88/12.09 A__LENGTH(cons(x0, cons(y_0, y_1))) -> A__U11(tt, cons(y_0, y_1)) 24.88/12.09 A__U11(tt, cons(y_0, y_1)) -> A__U12(tt, cons(y_0, y_1)) 24.88/12.09 A__U12(tt, cons(x0, x1)) -> A__LENGTH(cons(mark(x0), x1)) 24.88/12.09 24.88/12.09 The TRS R consists of the following rules: 24.88/12.09 24.88/12.09 a__zeros -> cons(0, zeros) 24.88/12.09 a__U11(tt, L) -> a__U12(tt, L) 24.88/12.09 a__U12(tt, L) -> s(a__length(mark(L))) 24.88/12.09 a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) 24.88/12.09 a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) 24.88/12.09 a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 24.88/12.09 a__length(cons(N, L)) -> a__U11(tt, L) 24.88/12.09 a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) 24.88/12.09 mark(zeros) -> a__zeros 24.88/12.09 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 24.88/12.09 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 24.88/12.09 mark(length(X)) -> a__length(mark(X)) 24.88/12.09 mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) 24.88/12.09 mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) 24.88/12.09 mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) 24.88/12.09 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 24.88/12.09 mark(cons(X1, X2)) -> cons(mark(X1), X2) 24.88/12.09 mark(0) -> 0 24.88/12.09 mark(tt) -> tt 24.88/12.09 mark(s(X)) -> s(mark(X)) 24.88/12.09 mark(nil) -> nil 24.88/12.09 a__zeros -> zeros 24.88/12.09 a__U11(X1, X2) -> U11(X1, X2) 24.88/12.09 a__U12(X1, X2) -> U12(X1, X2) 24.88/12.09 a__length(X) -> length(X) 24.88/12.09 a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) 24.88/12.09 a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) 24.88/12.09 a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) 24.88/12.09 a__take(X1, X2) -> take(X1, X2) 24.88/12.09 24.88/12.09 The set Q consists of the following terms: 24.88/12.09 24.88/12.09 a__zeros 24.88/12.09 mark(zeros) 24.88/12.09 mark(U11(x0, x1)) 24.88/12.09 mark(U12(x0, x1)) 24.88/12.09 mark(length(x0)) 24.88/12.09 mark(U21(x0, x1, x2, x3)) 24.88/12.09 mark(U22(x0, x1, x2, x3)) 24.88/12.09 mark(U23(x0, x1, x2, x3)) 24.88/12.09 mark(take(x0, x1)) 24.88/12.09 mark(cons(x0, x1)) 24.88/12.09 mark(0) 24.88/12.09 mark(tt) 24.88/12.09 mark(s(x0)) 24.88/12.09 mark(nil) 24.88/12.09 a__U11(x0, x1) 24.88/12.09 a__U12(x0, x1) 24.88/12.09 a__length(x0) 24.88/12.09 a__U21(x0, x1, x2, x3) 24.88/12.09 a__U22(x0, x1, x2, x3) 24.88/12.09 a__U23(x0, x1, x2, x3) 24.88/12.09 a__take(x0, x1) 24.88/12.09 24.88/12.09 We have to consider all minimal (P,Q,R)-chains. 25.10/12.27 EOF